29/11/2018
TASK 01
| Explain the construction and application of pressure measuring devices. (Manometers and mechanical gauges). Write the limitations of manometers.
Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure pressure are called pressure gauges or vacuum gauges.
A manometer is an instrument that uses a column of liquid to measure pressure, although the term is currently often used to mean any pressure measuring instrument.
A vacuum gauge is used to measure the pressure in a vacuum—which is further divided into two subcategories: high and low vacuum (and sometimes ultra-high vacuum). The applicable pressure ranges of many of the techniques used to measure vacuums have an overlap. Hence, by combining several different types of gauge, it is possible to measure system pressure continuously from 10 mbar down to 10−11 mbar.
There are basically two types of manometers.
U-Tube Manometer
Well Type Manometer
There are also variations of the above said basic types called Enlarged-Leg Type Manometer, and Inclined Tube Manometer. Another manometer used commercially is the Ring-Balance Type Manometer.
U-Tube Manometer
A simple u-tube manometer is shown below. If ‘dm‘is the manometric fluid density, ‘d1’ is the density of the fluid over the manometer, ‘P2’ is the atmospheric pressure (for general measurement of gas pressure) and ‘P1’ is the gas pressure, and also if d1<<dm, then the differential pressure can be obtained by the relation
p1-p2 = h (dm-d1)
U-Tube Manometer
An enhanced version of a manometer is shown below with a seal liquid over the manometer liquid to separate the process fluid from the manometer fluid for any probable source of trouble like absorption, mixing or explosion and so on. Seal pots with large diameters are also placed for increasing the range. The equation for differential pressure is the same as mentioned above.
Manometer With Large Seal Pots
Well-Type Manometer
The main difference between a U-tube manometer and a well type manometer is that the U-tube is substituted by a large well such that the variation in the level in the well will be negligible and instead of measuring a differential height, a single height in the remaining column is measured. If a1 and a2 are the areas of the well and the capillary, and if (h1-h2) is the difference in height in the well due to the pressure difference (p1-p2) as shown, at balance, then
p1-p2 = dm.h (1+a2/a1)
The figure of a well-type manometer is shown below.
Well-Type Manometer
Enlarged-Leg Manometer
In the enlarged-leg manometer, a2 is not negligible compared to a1. It has a float in the enlarged-leg which is utilized for indication or recording. The two legs can be changed for changing the measurement span. Thus, the equation becomes,
p1-p2 = dm.h
The figure of an enlarged-leg manometer is shown below.
Enlarged Leg Manometer
Inclined Tube Manometer
The inclined tube manometer is an enlarged leg manometer with its measuring leg inclined to the vertical axis by an angle b. This is done to expand the scale and thereby to increase the sensitivity. The differential pressure can be written by the equation,
p1-p2 = dm.h.Cosb (1+a2/a1)
The factor cosb expands the scale of the instrument. When b is quite large, h can be increased such that (h.cosb) remains constant. The figure of an inclined tube manometer is shown below.
Inclined Tube Manometer
Micromanometer
The micromanometer is another variation of liquid column manometers that is based on the principle of inclined tube manometer and is used for the measurement of extremely small differences of pressure. The meniscus of the inclined tube is at a reference level as shown in the figure below, viewing through a magnifier provided with cross hair line. This is done for the condition, p1=p2. The adjustment is done by moving the well up and down a micrometer. For the condition p1 not equal to p2, the shift in the meniscus position is restored to zero by raising or lowering the well as before and the difference between these two readings gives the pressure difference in terms of height.
Micromanometer
Manometer is shown above as a static measuring device. Its dynamics can rarely be ignored. Considering manometric fluid as a free body, the forces acting on it are
The weight distributed over the entire fluid.
The drag force due to its motion and the corresponding tube wall shearing stress.
The force due to differential pressure.
Surface tension force at the two ends.
Ring-Balance Manometer
This device cannot be actually called a manometer, but it is often considered so. The tube is made of polythene or other light and transparent material. This tube is bent into in to the form of a ring and is supported at the centre by a suitable pivot. The tubular chamber is divided in to two parts by spilling, sealing, and filling with a suitable light liquid like kerosene or paraffin oil for isolating the two pressures. Pressure taps are made with two flexible tubings. Pressures p1 and p2 act against the sealed walls as shown in the figure below, and rotate the ring which is balanced by the counter weight w.
Of the various manometric fluids used, mercury has many advantages like low vapour pressure, non-sticky nature, and wide temperature range from -20 degree Celsius to 350 degree Celsius. Its high density is disadvantageous for low differential pressure measurements. The device installation and maintenance is known to be quite expensive.
Application of Manometers
The manometer has many advantages in this age of technology. Containing no mechanical moving parts, needing nothing but the simplest of measurements, the primary standard manometer is readily available at modest cost. The principle of the manometer has not changed since its inception, however great strides have been made in its arrangement and the application of the instrument to various industrial measurement requirements. Whereas formerly the manometer was considered a laboratory instrument, today we find the manometer commonly used to measure pressures ranging from as high as 600 inches of mercury to space vacuums.
The manometer utilizes the hydrostatic (standing liquid) balance principle wherein a pressure is measured by the height of the liquid it will support. For example, the weight of a column of mercury at 0 degree C that is one inch high and one inch in cross sectional area is .4892 pounds. Thus we can say that a column of mercury one inch high imposes a force of .4892 pounds per square inch or .4892 PSI.
Throughout history, mercury-filled manometer has been a very important device needed in the construction of aqueducts, bridges, installing of swimming pools, and other engineering applications. The said measuring instrument has also been implemented for a number of industrial applications like for visual monitoring of air and gas pressure in compressors as well as in vacuum equipment and specialty tank applications. The manometer is also necessary in avionics and climate forecasting. It principally study the stress of fluid at the same time measure the speed at which a stream of air is flowing.
Mechanical gauges
Mechanical gauges are best suitable for measuring very high fluid pressures. In case of steam boilers, where manometers cannot be used, a mechanical gauges can be conveniently used.
Working of Mechanical gauges
Mechanical gauges utilize an internal bourdon tube. One end of the bourdon tube is connected to a gear and shaft assembly that moves a pointer. When the pressure inside the bourdon tube increases, the bourdon tube uncoils slightly. The amount of uncoiling that occurs is proportional to the pressure inside the bourdon tube. As the tube uncoils, its motion activates the gear and shaft system that turns the pointer on the gauge. While all that you see when you look at the gauge is the pointer moving, you should understand that there is a small, bent tube (the bourdon tube) that's coiling and uncoiling with each change in the pressure inside that tube.
Mechanical pressure gauges are connected directly to the process fluid being measured (i.e. oil). As the process fluid pressure changes the pressure on the bourdon tube also changes which in turn moves the pointer on the gauge.
Mechanical temperature gauges also utilize a bourdon tube. They have a sealed capillary tube and bulb assembly that is filled with temperature sensitive liquid that produces a proportional vapour pressure on the bourdon tube. As the temperature changes, the pressure inside the bourdon tube changes, which in turn moves the pointer on the gauge.
Types of mechanical gauges
Ruler and scales: they are used to measure lengths and other geometrical parameters. They can be single steel plate or flexible tape type tool.
Callipers: they are normally of two types- inside and outside calliper. They are used to measure internal and external size (for e.g. Diameter) of an object. It requires external scale to compare the measured value. Some callipers are provided with measuring scale. Other types are odd leg and divider calliper.
Venire calliper: It is a precision tool used to measure a small distance with high accuracy. It has got two different jaws to measure outside and inside dimension of an object. It can be a scale, dial or digital type venire calliper.
Micrometre: It is a fine precision tool which is used to measure small distances and is more accurate than the venire calliper. Another type is a large micrometre calliper which is used to measure large outside diameter or distance.
Feeler gauge: Feelers gauges are a bunch of fine thickened steel strips with marked thickness which are used to measure gap width or clearance between surface and bearings.
Bridge gauge: Bridge gauges are used to measure the amount of wear of Main engine bearing. Normally the upper bearing keep is removed and clearance is measured with respect to journal. Feeler gauge can be used to complete the process.
B) Explain the two different system of pressure measurement and their relation.
Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure pressure are called pressure gauges or vacuum gauges. A manometer is an instrument that uses a column of liquid to measure pressure, although the term is currently often used to mean any pressure measuring instrument.
A vacuum gauge is used to measure the pressure in a vacuum—which is further divided into two subcategories: high and low vacuum (and sometimes ultra-high vacuum). The applicable pressure ranges of many of the techniques used to measure vacuums have an overlap. Hence, by combining several different types of gauge, it is possible to measure system pressure continuously from 10 mbar down to 10−11 mbar.
Everyday pressure measurements, such as for tire pressure, are usually made relative to ambient air pressure. In other cases measurements are made relative to a vacuum or to some other specific reference. When distinguishing between these zero references, the following terms are used:
Absolute pressure is zero-referenced against a perfect vacuum, using an absolute scale, so it is equal to gauge pressure plus atmospheric pressure.
Gauge pressure is zero-referenced against ambient air pressure, so it is equal to absolute pressure minus atmospheric pressure. Negative signs are usually omitted. To distinguish a negative pressure, the value may be appended with the word "vacuum" or the gauge may be labeled a "vacuum gauge."
| A U - tube differential mercury manometer is connected between two pipes Xand Y. Pipe X contains carbon tetra chloride (Sp.gr. 1.59) under a pressure of 103KN/m2 and pipe Y contains oil (Sp.gr. 0.8) under a pressure of 172 kN/m2. Pipe X is 2.5 m above pipe Y. Mercury level in the limb connected to pipe X is 1.5 m below the center line of pipe Y. Find the manometer reading as shown by a centimeter Scale attached to it. (P.1.2)
Given Data:
Pipe x and left limb contains carbon tetra chloride of Specific gravity = Sp.gr. = 1.59
Under pressure of = Px = 103 KNm2 = 103 × 1000 = 103,000 N/m2
Pipe Y and right limb contains oil of Specific gravity = Sp.gr. = 0.8
Under pressure of = Py = 172 KNm2 = 172 × 1000 = 172,000 N/m2
Height = h + x = 2.5 + 1.5 = 4 m
Solution:
ρx = 1.59 × 1000 = 1590 kg/m³
ρy = 0.8 × 1000 = 800 kg/m³
Px = 103 KNm2 = 103 × 1000 = 103,000 N/m2
Py = 172 KNm2 = 172 × 1000 = 172,000 N/m2
Taking X-X as datum line
Pressure above X-X in the Left Limb
Left limb = (13.6 × 1000 × 9.81 × a) + (1590 × 9.81 × 4) + 103,000
Left limb = 13.6 × 1000 × 9.81 × a + 6360 × 9.81 + 103,000
Pressure above X-X in the Right Limb
Right Limb = 800 × 9.81 × (1.5 + a) + 172, 000
Equating the two pressures we get
13.6 × 1000 × 9.81 × a + 6360 × 9.81 + 103,000 = 800 × 9.81 × (1.5 + a) + 172, 000
Dividing both sides by 1000 × 9.81
13.6 × 1000 × 9.81 × a + 6360 × 9.81 + 103,000 = 800 × 9.81 × (1.5 + a) + 172, 000
1000 × 9.81 1000 × 9.81 1000 × 9.81 1000 × 9.81 1000 × 9.81
13.6 a + 6.36 + 10.49 = 0.8 × (1.5 + a) + 17.53
13.6 a + 16.85 = 1.2 + 0.8a + 17.53
13.6 a + 16.85 = 0.8a + 18.73
13.6 a - 0.8a = 18.73 - 16.85
a (13.6 - 0.8) = 18.73 - 16.85
12.8a = 1.88
a = 1.88 / 12.8
a = 0.146 m = 0.146 × 100 = 14.6 cm
a = 14.6 cm
Manometer Reading = 14.6 cm of Hg.
| A pipe connected with a tank (diameter 3 m) has an inclination of q with the horizontal and the diameter of the pipe is 20 cm. Determine the angle? Which will give a deflection of 5 m in the pipe for a gauge pressure of 1 m water in the tank. Liquid in the tank has a specific gravity of 0.88. (M 1.1).
Given Data:
Diameter of the tank= D = 3m.
Diameter of the pipe connected with tank = d = 20cm.
Deflection in the pipe for a gauge pressure of 1 m water in the tank = L = 5m
Specific gravity of the liquid enclosed in the tank = S.g = 0.88
Gauge = 1m of water.
To Find:
Determine the angle = θ=?
Solution:
Volume displaced in the tank = volume raise in the tube
Hence:
(3)2 × X = (0.2)2 × L
9 X = 0.04 L
X = 0.04/9 L
Difference in head = X + h
= 0.04/9 L + L Sin θ
= 0.02 + 5 Sin θ
Whereas:
Gauge pressure = 1 meter of water
ρ g h = 1
ρ g ( h + X) = 1
0.88 × 1000 × 9.81 (0.02 + 5 Sin θ) = 1
8632.8 (0.02 + 5 Sin θ) = 1
172.656 + 43164 Sin θ = 1
43164 Sin θ = -172.656
Sin θ = (-172.656)/43164
Sin θ = 0.004
θ = 0.22o
| A manometer connected to a pipe indicates a negative gauge pressure of 70mm of mercury. What is the pressure in the pipe in N/m2? (M.1.2)
Given data:
Negative gauge pressure = 70mm of mercury = 0.07
Standard atmospheric pressure= 101.325 KN.
To Find:
Pressure inside the Pipe N/m² =?
Solution:
We know,
Pabsolute = Patmospheric + P gauge
= Patmospheric + ρ g h
ρ g h = 13.6 × 1000 × 9.81 × 0.07
ρ g h = 9339.12 N/m2
ρ g h = - 9.33 KN/m2
Pabsolute = Patmospheric + ρ g h
Pabsolute = 101.213 + (-9.33)
Pabsolute = 91.8 KN/m2
| As shown in figure water flows through pipe A and B. The pressure difference of these two points is to be measured by multiple tube manometers. Oil with specific gravity 0.88 is in the upper portion of inverted U-tube and mercury in the bottom of both bends. Determine the pressure difference. (D.1.1)
Given Data:
Oil with specific gravity = Sp.gr = 0.88
Mercury specific gravity = Sp.gr = 13.6
To Find:
Pressure difference = PA - PB
Solution:
Pressure at X is given as
= PX = PA +ρ g h
= PX = PA + (1000 × 9.81 × 0.1)
= PX = PA + 981
Pressure at Υ is given as
= PY = PX – ρ g h
= PY = (PA + 981) – (13.6 × 1000 × 9.81× 0.3)
= PY = (PA + 981) – 40024.8
Pressure at Z is given as
= Pz = Py + ρ g h
= Pz = (PA + 981 – 40024.8) + 0.88 × 1000 × 9.81× 0.4
= Pz = (PA + 981 – 40024.8) + 3453.12
Pressure at U is given as = Pu = Pz - ρ g h
= Pu = (PA + 981 – 40024.8 + 3453.12) – 13.6× 1000 × 9.81× 0.5
= Pu = (PA + 981 – 40024.8 + 3453.12) – 66,708
Pressure at B is given as = PB = Pu - ρ g h
= PB = (PA + 981 – 40024.8 + 3453.12 – 66,708) – 1000 × 9.81 × 0.8
= PB = (PA + 981 – 40024.8 + 3453.12 – 66,708) – 7848
Summing all the equations above
The pressure difference is given by = PA - PB
= PA - PB = 103.28 Pwg
The pressure difference is given by = PA - PB = 10.131 KPa
| Estimate the vapor pressure of the oil at 27ₒ, if the water height h was 3 cm when the gas of absolute pressure 101047.7 Pascal was trapped. (M 1.2)
Given Data:
Absolute Pressure = 101047.7 Pascal
Water height = 3cm
Atmospheric Pressure = 101325 Pa
To Find:
Vapor pressure of the oil at 27ₒ =?
Solution:
Vapor Pressure = P atmospheric- P absolute
Vapor Pressure = 101325 - 101047.7
Vapor Pressure = 277.3 Pa
Or
Vapor Pressure = 0.2773 KPa
| A vertical gate of 5 m height and 3 m wide closes a tunnel running full with water. The pressure at the bottom of the gate is 195 kN/m 2. Determine the total pressure on the gate and position of the center of the pressure
Given Data:
Height of gate = 5 m
Width of gate = 3 m
Area of the gate = A = height x width = 5 x 3 = 15 m²
The pressure at the bottom of the gate = 195 kN/m²
And we have to find:
The total pressure on the gate = P =?
Position of the center of the pressure = C.P=?
The force is given by the expression = F = ρ g A x
x = distance of center of gravity from fee surface
x = h – d/2
Suppose Height = h = 10 m
x = 10 + 5/2
x = 12.5
The total force on the gate = F = ρ g A x
Putting the values
= F = 1000 × 9.81× 15× 12.5
The total pressure on the gate = F = 1,839,375 N
The total pressure on the gate = P = F / A
The total pressure on the gate = P =1,839,375 / 15
The total pressure on the gate = P = 122625 Pa Or 122.625 KN / m²
Where IG is the 2nd moment of area about a line through the centroid of the rectangle and
IG for rectangular shapes BD³ / 12
Now we are putting the values
Position of the center of the pressure = C.P= 0.172 m
Task 02
2.1| A horizontal pipe line 50m long is connected to the water tank at one end and discharges freely into the atmosphere at the other end. For the first 30m length from the tank the pipe is 200mm diameter and it diameter suddenly enlarged to 400mm afterwards. The height of water level in the tank is 10m above the center of the pipe.
A. Determine the loss of head in the pipe.
Solution:
From Bernoulli’s equation
P1/(ρ g ) + V1²/2g + Z = P2/(ρ g ) + V2²/2g + head loss
At both ends pressure is equal due to expose to atmospheric pressure ad water is static so velocity = 0
So above equation can be written as
10 = V2²/2g + head loss
Head loss = HL (entrance) + Hf1 + HL (enlargement) + Hf2 + HL (exit)
HL (entrance) = head loss at entrance (Sudden Contraction)
Hf1= head loss due to friction in pipe length L1, 30-meter length.
HL (enlargement) = Head loss due to enlargement.
Hf2= head loss due to friction in pipe L2, 20 meter length.
HL (exit) = Head loss at exit of the pipe.
10 = V2²/2g + head loss
10 = V2²/2g + ( 0.5 V1²)/2g + (4.f.L1.V1²)/(2g.d1) + (V1²-V2²)/2g +(4.f.L2.V2²)/(2g.d2) + ( V2²)/2g
From continuity equation
A1 V1 = A2 V2
V1 = (A2 . V2)/A1
V1 = (π/4 ×0.4 ²)/(π/4 ×0.2²)V2
V1= 4V2
Friction factor = f = 0.01
Putting the value of V1 in head loss equation above
10 = ( v2²)/(2 ×9.81) + ( 0.5(4 . v2)²)/2g + (4 ×0.01 × 30 × (4.v2)²)/(2 ×9.81 ×0.2) + ((4.v2)²-V2²)/(2 ×9.81) + (4 ×0.01 ×20 × v2²)/(2 ×9.81 ×0.4) + ( V2²)/( 2 ×9.81)
Calculating value for V2
Q = A2 × V2
Q = 164.13 L/s = 0.16413 m3 /s
A2 = π/4 ×0.4 ² = 0.1256
V2 = Q /A = 0.16413 / 0.1256 = 1.3
V2 = 1.3 ms-1
Putting values
10 = ( (1.3)²)/(2 × 9.81) + ( 0.5 (4 ×1.3)²)/(2 ×9.81) + (4 ×0.01 ×30 × (4 ×1.3)²)/(2 ×9.81 ×0.2) + ((4×1.3)²-1.3²)/(2 ×9.81)
+ (4 ×0.01 ×20 × 1.3²)/(2 ×9.81 ×0.4) + (1.3 ²)/(2 ×9.81)
10 = 0.086 + 0.689 + 8.269 + 1.29 + 0.1724 + 0.086
10 = 10.59
Loss of head in pipe = 10 – 10.59 = 0.59
B. Determine the rate of flow and the power required to maintain the flow.
Power required to maintain the flow = P = (ρgQ . h)/1000 k watts
P = (ρgQ . h)/1000 k watts
ρ = 1000
g = 9.81
h = 10m
Q = 164.13 L/s = 0.16413 m3 /s
Putting values
P = (1000 ×9.81 ×0.16413 ×10)/1000 kwatts
P = 16.10 k watts power is required to maintain the flow.
2.2 | An irrigation channel (trapezoidal in cross section) excavated in smooth earth (n = 0.030) is to carry a flow rate of 12 m3/s at a uniform flow depth of 2 m. The bed width of the channel is 2.7m and the side slope of 2 horizontal to 1 vertical. Calculate the bed slope of the channel
Given Data:
Flow rate is = Q = 12 m3/s
Depth of flow = h = 2 m
The bed width of the channel is = 2.7m
Side slop = 2 horizontal to 1 vertical
n = 0.030
To Find:
Bed slope of the channel = i =?
Solution
To calculate top widthdepth of flow is 2 m and side slop is 2 horizontal to 1 vertical.
Therefore for top width =( 2)/1 × 2
= 4m
Top width = 4 + 4 + 2.7 = 10.7 m
Area of flow = 1/2 (a + b) h
Area of flow = 1/2 (2 +10.7) ×2
Area of flow = 12.7 m²
Hydraulic mean depth = m = Area/Perimeter
Perimeter =?
Area of wetted perimeter is given as
Perimeter = 4.47 + 2.7 + 4.47 = 11.87
Area = 12.7 m²
Perimeter = 11.87 m
m = Area/Perimeter
m = 12.7/11.87
m = 1.06
Applying manning’s equation to find Chassis’s constant C
Manning’s equation = C = 1/n . m 1/6
= C = 1/0.030× 1.06 1/6
= C = 33.65
To find velocity V use flow rate equation
Flow rate = Q = A × V
Or V = Q / A
V = 12 / 12.7
V = 0.94 ms-1
We have calculated
V = 0.94 ms-1
C = 33.65
m = 1.06
Now apply chassis’s equation to find bed slope of the channel = i
Chassis’s equation = V = C √(m . i)
Putting values
0.94 = 33.65√(1.06. i)
Taking square on both sides
0.94² = (33.65√(1.06. i) )2
0.8836 = (33.65)2 × 1.06 × i
0.8836 = 1200.26 × i
Or
Bed slope of the channel = i = 0.8836/1200.26
Bed slope of the channel = i = 7.3 × 〖10〗^(-04)
Or
i = 0.00073
2.3 |
a. Show that in a rectangular channel, critical depth is two third of Specific energy. [M1.2]
Specific energy & critical depth relationship for rectangular channels
Solution of the specific energy equation for rectangular channels Consider a specific energy equation for the case of a rectangular channel.
The concept of specific energy as it applies to open channels with small slopes is given below.
If bed slope is taking datum line then Z = 0
Energy = yc + V²/2g equation (i)
Whereas yc is critical depth
This indicates that the specific energy is the sum of the depth of water and the velocity head.
Solution of the specific energy equation for rectangular channels. Consider a specific energy equation for the case of a rectangular channel
Energy = yc + V²/2g
Discharge = Q = A × V
V = Q / A
V² = (Q / A) ²
V² = Q² / b² × y² equation (ii)
Where b is the width of the channel and y is the depth of flow
Put the equation (ii) in equation (i)
Substituting this in the specific energy equation it can be written as
Energy = yc+ Q²/(b^2.y^2 )2g
Defining q= Q / b
Then Energy = yc+ q²/2gy² equation (iii)
= q²/g = yc³ =>yc = (q² / g) 1/3
Put the value of q²/g = yc³in equation iii
E min = yc + yc³/2yc²
E min = yc3/2
yc = 2/3 E min
So, Yc minimum, we can write as E minimum is equal to 3 by 2 into Y c or this can be written in the form that Y c is equal to 2 by 3 E minimum. So, once we know the energy E minimum, then we can calculate the Y c directly from this or if we know the Y c, we can calculate E minimum directly from this. So, this relation shows that, there is a definite relation for critical depth computation or for computation of minimum specific energy in critical condition.
b. Calculate the minimum Specific energy and critical velocity of flow for a rectangular channel of width 6m and discharge 18 m3/sec when the depth of flow of water is 1.8 m. Also classify the type of flow in the channel.[D1.1]
Given Data:
Rectangular channel of width = b = 6m
Discharge = Q = 18 m3/sec
Depth of flow of water is = y = 1.8 m
To Find:
Minimum Specific energy = E Min =?
Critical velocity = Vc=?
Solution:
For Flow velocity we have
Discharge = Q = A × VC
VC = Q / A
Area of Rectangular channel is given as A = width of channel × Depth of flow
A = b × y
Flow velocity V = Q / b × y
By substituting the values
Flow velocity = V = 18/(6 ×1.8)= 1.67 m s-1
To calculate critical velocity we have to consider
Critical depth = hc = (q² / g) 1/3
q = 18 / 6 = 3 m2 /s
= hc = (q² / g) 1/3
= hc = (3² / 9.81) 1/3 = 0.3058 m
Critical depth = hc = 0.3058m
Now critical velocity is given as
Critical velocity = Vc = √(g ×hc )
Critical velocity = Vc = √(9.81 ×0.3058 )
Critical velocity = Vc = 1.732 m s-1
Minimum Specific energy = E min = hc 3/2
Critical depth hc is calculated above the value is 0.3058m
Substituting the value for Minimum Specific energy
E min = 0.3058 ×3/2
E min = 0.4587
The type of flow in the channel
The type of flow in the channel is based on Froude number. The Froude number, Fr, is a dimensionless value that describes different flow regimes of open channel flow. The Froude number is a ratio of inertial and gravitational forces.
If Froude number is equal to 1 the flow is critical = Fr = 1 = Critical Flow
If Froude number is less than 1 the flow is super critical = Fr> 1 = Super Critical Flow
Type of flow in the channel for given task
To calculate the type of flow for given task we have
Froude number = Fr = V/√gD
Where
V = Flow velocity = 1.67 m s-1
g = Gravitational acceleration = 9.81 m s-2
D = Area/(Top width of channel) = (10.8 )/6 = 1.8
Substituting the values
Froude number = Fr = 1.666/√(9.81 ×1.8)
Froude number = Fr = 0.396
From the above value Froude number is less than 1 = Fr> 1
Hence proved flow is Super Critical.
TASK 03
3.1 |Explain the importance of Duty point (point of maximum efficiency) in matching a pump to a pipeline system. Use suitable pump curves to explain your answer.
Water is to be pumped through a 200mm diameter pipe from a pile hole (bore hole) of 50 m deep to a river one kilometre away the point of dewatering. The pump available has the discharge – head performance characteristic as shown in table 1.
Table 1 : Performance Characteristic
Head / (m)
Discharge / (lit/min)
30
2000
50
1750
65
1410
80
800
Neglecting the local losses, calculate the duty point of the pump when the friction factor of the pipe f = 0.04.
Duty Point: Characterized as that Duty Point on the H-Q bend where a radial pump works at greatest effectiveness. It is dictated by the Flow Rate and the Total Head at the individual pump speed. The point on the pump execution bend where the stream and head coordinate the application's prerequisite is known as the obligation point. An outward pump dependably works at the point on its execution bend where its head coordinates the resistance in the pipeline. The obligation point is at the crossing point between the Q-H bend and the framework attributes.
To pick a pump for a framework, it is important to know both the framework qualities and the pump bend. The pump is dimensioned so it can give a sufficient execution at the most extreme framework load. The pump execution bend is a chart demonstrating the association between stream (Q) and produced head (H) or weight. The framework's qualities demonstrate the weight misfortune in the framework as a component of the stream. The obligation point is the convergence between the pump execution bend and the framework's qualities.
The significance of Duty point
Pumps are constantly characterized by the essential Pump attributes beneath. They demonstrate the relationship between head, force and effectiveness against stream. It is vital to see exactly how "peaky" the productivity may be; demonstrating that running at an obligation (head and/or stream) beneath evaluated obligation is liable to prompt a huge lessening in pump effectiveness. The Best Efficiency Point (BEP) of a pump is in a perfect world at the appraised obligation point. Pump productivity can diminish altogether when the pump is working far from the planned Best Efficiency Point. Over-determining the obligation while indicating a pump will hence mean tremendously expanded vitality costs. Pumps are not made to standard obligations, which make contrasting efficiencies less basic than and items that are made to standard obligations, (for example, engines). A producer of pumps with a high outline (BEP) effectiveness might regularly miss out to another makers of a less proficient (BEP) pump, contingent upon where the real obligation point asked for exists in the execution bends of the pumps. The same pump will more often than not be offered with distinctive impellers to give great execution at lower obligations. The same pump will frequently be offered with diverse velocity engines to permit it to cover a much more extensive scope of obligations.
Calculating the duty point of the pump
Given Data:
Diameter of the pipe = 200 mm = 0.2m
Static Head = Hs = 50 m
Length = 1 km = 1000 m
Friction factor of the pipe = f = 0.04
Neglecting the local losses
To Find:
The duty point of the pump
Solution:
Converting given discharge rates liter/min to m3/s to plot graph.
2000 liter/min = 0.033 m3/s
1750 liter/min = 0.029 m3/s
1410 liter/min = 0.0235 m3/s
800 liter/min = 0.0133 m3/s
ΔH = Static Head + Delivery Head + minor losses
As per given condition neglecting the local losses so,
ΔH = Hs + Hf + 0
ΔH = 50m + 4flv²/2gd + v²/2g
To calculate velocity using the equation
Q = A × V
Q = (π d²)/4× V
V = (4 Q)/(π d²)
Substituting the values
V = (4 Q)/(3.14 ×0.2²)
V = 31.84 Q
Putting the value of V in above equation ΔH = 50m + 4flv²/2gd +v²/2g
ΔH = 50m + (4×0.04 ×1000 ×(31.84 Q)²)/(2×9.81 ×0.2) + (31.84 Q)²)/(2 ×9.81)
ΔH = 50m + 41,336.8 + 51.69 Q2
ΔH = 50m + 41,388.49 Q2
By putting values of Q
If
Q = 0, ΔH = 50
Q = 0.0133 , ΔH = 57
Q= 0.0235 , ΔH = 72
Q= 0.029 , ΔH = 84
Q= 0.033 , ΔH = 95
At duty point the value of ΔH = 67m and value of ΔQ is 0.023 m3/s.
This is the Best Efficiency Point (BEP). The point at which highest efficiency occurs, at this point we can get low cost per meter cube of pumping
Task 04
Task 4.1
4.1.1 List different methods of measuring flow rates of open channels. [P 4.1]
Open-channel flow, a branch of hydraulics and fluid mechanics, is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow.
Open channels are used to conduct liquids in most sewer systems, sewage treatment plants, industrial waste applications, and irrigation systems.
There are many methods of determining the rate of flow in open channels. Some of the more common include
The timed gravimetric.
Dilution.
Velocity-area.
Hydraulic structures.
Slope-hydraulic radius-area methods.
Timed Gravimetric Method
The stream rate is ascertained by measuring the whole substance of the stream that was gathered in a compartment for an altered period of time. This is commonsense for little surges of under 25 to 30 gallons for each minutes (gpm) and is not appropriate for constant estimation.
Dilution Method: The stream rate is measured by deciding how much the streaming water weakens an included tracer arrangement.
Velocity-Area Method: Measuring the mean stream speed over a cross segment and increasing it by the zone by then to ascertain the stream rate. The region speed technique computes stream rate by increasing the region of the stream by its normal speed. This is regularly alluded to as the progression comparison
Q = A ×V
The flowmeter converts this level into the area of the flow based on the size and shape of the channel. The main advantage of the area velocity method is that it can be used to measure flow under a wide range of conditions.
• Open Channel
• Surcharged
• Full Pipe
• Submerged
• Reverse Flow
In addition, the area velocity method does not require the installation of a weir or flume.
Hydraulic Structure Method
This method uses a hydraulic structure placed in the flow stream of the channel to produce flow properties that are characterized by known relationships between the water level measurement at some location and the flow rate of the stream. Therefore, the flow rate is determined by taking a single measurement of the water surface level in or near the restriction of the hydraulic structure
Slope-hydraulic Radius-Area Method
Measurement of water surface slope, cross-sectional area, and wetted perimeter over a length of uniform section channel are used to calculate the flow rate, by using a resistant equation such as the Manning formula.
The Gravitational, Dilution, and the Velocity Area methods are more commonly used for calibration purposes. The Depth-Related methods (Hydraulic Structures) are the most common. The depth-related technique measures flow rate from a measurement of the water depth, or head. Weir and flumes are the oldest and most common devices used for measuring open channel flows.
Various resistance equations are used to estimate flow rate based on measurements of the water surface slope, cross-sectional area, and wetted perimeter over a length of uniform channel. The most popular of these equations is the Manning formula.
Q = K A R2/3 S1/2 n
Where:
Q = flow rate
A = cross sectional area of flow
R = hydraulic radius (cross sectional area divided by wetted perimeter)
S = slope of the hydraulic gradient
n = roughness coefficient based on channel material and condition
K = constant dependent upon units
4.1.2 Explain the method used in this experiment to measure the flow rates. [P 4.1]
In this experiment hydraulic structure technique is utilized it is the most widely recognized strategy for measuring open channel stream is the pressure driven structures system. An aligned confinement embedded into the channel controls the shape and speed of the stream. The stream rate is then controlled by measuring the fluid level in or close to the limitation.
The confining structures are called essential measuring gadgets. They might be partitioned into two general classes weirs and flumes.
A weir is a hindrance or dam constructed over an open channel over which the fluid streams, frequently through an exceptionally molded opening. Weirs are arranged by state of this opening. The most well-known sorts of weirs are the triangular (or V-indent) weir, the rectangular weir, and the trapezoidal (or Cipolletti) weir.
The stream rate over a weir is dictated by measuring the fluid profundity in the pool upstream from the weir. Weirs are straightforward and reasonable to manufacture and introduce. Regular materials of development incorporate metal, fiberglass and wood. In any case, they speak to a huge loss of head, and are not suitable for measuring streams with solids that might stick to the weir or collect upstream from it.
A flume is an exceptionally formed open channel stream segment giving a confinement in channel territory and/or an adjustment in channel incline. The stream rate in the channel is controlled by measuring the fluid profundity at a predetermined point in the flume.
The most well-known flume is the Parshall flume. The stream rate through a Parshall flume is dictated by measuring the fluid level 33% of the route into the uniting segment. Parshall flumes are assigned by the width of the throat, which goes from one inch to 50 feet. The throat width and every other measurement must be entirely taken after with the goal that standard release tables can be utilized. Likewise, take note of the drop in the floor of the flume, which makes it hard to introduce a Parshall flume in a current channel.
4.1.3 Calculate the flow rate coefficient of discharge [D 1.1]
The release coefficient is the proportion of the real stream to the hypothetical stream, expecting unit coefficients of compression and speed, equivalent to the result of these coefficients
Cd = (Q actual)/( Q theoretical)
To calculate the flow rate coefficient of discharge, suppose the values from the task 2.2 and considering the channel is rectangular.
Flow rate is = Q = 12 m3/s
Depth of flow = h = 2 m
The bed width of the channel is = 2.7m
Area of the channel = A = 5.4 m2
The above equation can be written as
Cd = Q/(A . √(2 . g .Δh ))
Where
A: Area
Cd: Discharge coefficient
g: Gravity 9.8 m/s²
Q: Flow rate
Δh : Head Drop (high of fluid)
By substituting the values, we can get coefficient of discharge.
Cd = 12/(5.4 ×√(2 × 9.81 × 2 ))
Cd = 0.354
4.1.4 Plot the flow profiles and identify the types of flow. [M 1.1]
There are mainly two types of flow.
Uniform Flow: Steady uniform flow. Conditions do not change with position in the stream or with time. An example is the flow of water in a pipe of constant diameter at constant velocity.
Non Uniform Flow: Non steady Flow. Conditions do changes with position in the stream or with time.
Flow profiles for Steady and Unsteady flow
Steady flow
A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ from point to point but do not change with time.
Unsteady flow
If at any point in the fluid, the conditions change with time, the flow is described as unsteady. (In practice there is always slight variations in velocity and pressure, but if the average values are constant, the flow is considered steady
Combining the above we can classify any flow in to one of four type:
Steady uniform flow. Conditions do not change with position in the stream or with time. An example is the flow of water in a pipe of constant diameter at constant velocity.
Steady non-uniform flow. Conditions change from point to point in the stream but do not change with time. An example is flow in a tapering pipe with constant velocity at the inlet - velocity will change as you move along the length of the pipe toward the exit.
Unsteady uniform flow. At a given instant in time the conditions at every point are the same, but will change with time. An example is a pipe of constant diameter connected to a pump pumping at a constant rate which is then switched off.
Unsteady non-uniform flow. Every condition of the flow may change from point to point and with time at every point. For example waves in a channel.
Flow Profiles of uniform flow, Rapidly Varied flow and gradually varied flow in open channels
Rapidly Varied flow (RVF) occurs over a short distance near the obstacle.
Gradually varied flow (GVF) occurs over larger distances and usually connects uniform flow and Rapidly Varied flow
Flow Profiles of critical, sub critical and super critical flow
The behavior of a flow over a rise and then a fall of channel bottom, when approaching sub critical flow is forced to the critical condition by raising the step by a significantly large increment.
Task 4.2 Clearly explain the procedure for conducting characteristic test on single stage Centrifugal pump to find out the optimum discharge and head. Plot all the required graphs. Explain how you will increase the operating head. [M1.1, D1.1]
Centrifugal pumps are the most commonly used kinetic-energy pump. Centrifugal force pushes the liquid outward from the eye of the impeller where it enters the casing. Differential head can be increased by turning the impeller faster, using a larger impeller, or by increasing the number of impellers. The impeller and the fluid being pumped are isolated from the outside by packing or mechanical seals. Shaft radial and thrust bearings restrict the movement of the shaft and reduce the friction of rotation. Single stage centrifugal pumps are the most common pump for fluid transfer in high flow rate, low pressure installations. If a lower flow rate or a higher pressure is needed over what a single stage centrifugal pump can provide. In this type, one impeller keyed to the shaft. It is usually low lift pump.
The aim for conducting characteristic test on single stage Centrifugal pump to find out the optimum discharge, head, efficiency and performance characteristics.
Procedure for conducting characteristic test on single stage Centrifugal pump
The setting of test data is done semi-automatically and requires appropriate test experience and accuracy. For conducting characteristic test on single stage Centrifugal pump only pure cold water with a density of ρ= 1 kg/dm3 is used as the test medium. Pump block units are normally tested with installed motor.
Priming of the radiating pump is completed utilizing a twofold acting responding pump which fills water in diffusive pump taking water from repository.
After the preparing is finished, the force is supplied to the radiating pump and the data force is measured as far as KWh at general interims of time.
Also suction and the conveyance head are measured with the assistance of gages mounted on the suction and conveyance channels itself. The previous is gotten as far as mm of mercury while the last is acquired as far as meter of water segment.
The release is measured as far as the stature of water segment over the V-score. The stature is measured in a square segment in which snare gage is embedded to watch the level ascent. As the sump and square segment are joined the level demonstrated are same as in sump. Perceptions of every single above estimation are noted in perception table.
The release, information power, yield power and effectiveness are figured by equations and taking into account this outcome table is made.
Graphs are plotted for - release v/s yield power, release v/s effectiveness and release v/s all out head and release v/s Compact disc.To obtain the test data for main characteristics, the pump is run at a constant speed and the discharge is varied over the desired range. Measurements are taken for suction head, delivery head, shaft power and height above V-notch for each discharge and calculations are made for the pump efficiency. Curves are then plotted for discharge V/s head, power and efficiency.
Measurements of optimum Discharge (Q)
The optimum Discharge is measured using magnetic inductive flow measuring devices having the specified stabilization distances in front and after the measuring unit. The setting of the capacity is effected by control valves in the discharge pipe.
Q=volume of liquid flowing per second = Area x velocity of flow Q = πD2B2Vf2
Where
D2=Diameter of the impeller at outlet
B2is the width of the impeller at the outlet
Vf2 are the velocities of flow at outlet
Heads on a centrifugal pump:
Suction head (hs): it is the vertical distance between the liquid level in the sump and the centre line of the pump. It is expressed as meters.
Delivery head (hd): It is the vertical distance between the centre line of the pump and the liquid level in the overhead tank or the supply point. It is expressed in meters.
Static head (Hs): It is the vertical difference between the liquid levels. In the overhead tank and the sump, when the pump is not working. It is expressed as meters.
Therefore, HS= (hs+ hd)
Friction head (hf): It is the sum of the head loss due to the friction in the suction and delivery pipes. The friction loss in both the pipes is calculated using the Darcy’s equation,
hf=(fLV2/2gD).
Total head (H): It is the sum of the static head Hs, friction head (hf) and the velocity head in the delivery pipe (Vd 2/2g). Where, Vd =velocity in the delivery pipe.
Hm = hs+ hd + hf+ Vd²/2g
Manometric head (Hm): It is the total head developed by the pump. This head is slightly less than the head generated by the impeller due to some losses in the pump
Hm = H +Vs²/2g -Vd²/2g
Figure 1: Characteristic Curves for a Single Stage Centrifugal Pump
Figure 2: Head versus capacity characteristics curve for a single-stage pump
Figure 2 shows a typical head, H (ft), versus capacity (discharge), Q (gpm), curve for a single-stage pump. This curve relates head produced by a pump to the volume of water pumped per unit time. Generally, the head produced decreases as the amount of water pumped increases. The shape of the curve varies with pump’s specific speed and impeller design. Usually, the highest head is produced at zero discharge and it is called the shut-off head.
How to Increase Operating Head
To increase the operating head connect an additional pump in series with a main pump may be used in time of larger flow demands.
Figure 3: Two Pumps Connected in Series
To associate two pumps in arrangement implies that the release from the first pump is funneled into the gulf side of the second pump. In this kind of game plan all the stream progressively goes starting with one pump then onto the next with every pump adding more vitality to the water.
This is a common game plan in multi-stage turbine or submersible pump where the same release goes through all stages and every forms extra head. Frequently, arrangement setups are utilized when head prerequisites of the framework surpass what can be supplied by individual pumps. They are additionally utilized as a part of frameworks with variable head necessities. A little radial pump utilized as a sponsor pump for corner watering system on an inside turn framework or, so far as that is concerned, any supporter pump, in any water framework, which works notwithstanding the primary water pump. Figure 4 demonstrates head-release bends for two pumps working in arrangement.
Figure4: Head versus Discharge characteristics curves for pumps operating in Series.
Task 4.3
4.3.1 Explain the method used to measure the flow rates in pipes [P 4.1]
Introduction
There are a wide variety of methods for measuring discharge and velocity in pipes, or closed conduits. Many of these methods can provide very accurate measurements others give only rough estimates. But, in general, it is easier to obtain a given measurement accuracy in pipes when compared to measurement in open channels. Some of the devices used are very expensive and are more suited to industrial and municipal systems than for agricultural irrigation systems.
Pitot Tubes
The Pitot tube can be used not only for measuring flow velocity in open channels (such as canals and rivers), but in closed conduits as well. There are several variations of pitot tubes for measuring flow velocity, and many of these are commercially available. Pitot tubes can be very simple devices with no moving parts. More sophisticated versions can provide greater accuracy (e.g. differential head meters that separate the static pressure head from the velocity head). The pitot static tube in one variation of the device which allows the static head (P/γ) and dynamic (total) head (P/γ + V2/2g) to be separately measured.
The static head equals the depth if open-channel flow. Calibrations are required because the velocity profile can change with the flow rate, and because measurement(s) are only a sampling of the velocities in the pipe. The measurement from a pitot tube can be accurate to ±1% of the true velocity, even if the submerged end of the tube is up to ±15% out of alignment from the flow direction. The velocity reading from a pitot tube must be multiplied by cross-sectional area to obtain the flow rate (it is a velocity-area method). Pitot tubes tend to become clogged unless the water in the pipe is very clean. Also, pitot tubes may be impractical if there is a large head, unless a manometer is used with a dense liquid like mercury.
Pitot tube
Venturi Meters
Venturi meters have only a small head loss, no moving parts, and do not clog easily. The principle under which these devices operate is that some pressure head is converted to velocity head when the cross-sectional area of flow decreases (Bernoulli equation). Thus, the head differential can be measured between the upstream section and the throat section to give an estimation of flow velocity, and this can be multiplied by flow area to arrive at a discharge value. The converging section is usually about 21º, and the diverging section is usually from 5 to 7º.
A form of the calibration equation is:
where C is a dimensionless coefficient from approximately 0.935 (small throat velocity and diameter) to 0.988 (large throat velocity and diameter); β is the ratio of D2/D1; D1 and D2 are the inside diameters at the upstream and throat sections, respectively; A2 is the area of the throat section; ∆h is the head differential; and “sg” is the specific gravity of the manometer liquid.
The discharge coefficient, C, is a constant value for given venturi dimensions. Note that if D2 = D1, then β = 1, and Q is undefined; if D0 > D1, you get the square root of a negative number (but neither condition applies to a venturi). The coefficient, C, must be adjusted to accommodate variations in water temperature.
Flow Nozzles
Flow nozzles operate on the same principle as venturi meters, but the head loss tends to be much greater due to the absence of a downstream diverging section. There is an upstream converging section, like a venturi, but there is no downstream diverging section to reduce energy loss.
Flow nozzles can be less expensive than venturi meters, and can provide comparable accuracy. The same equation as for venturi meters is used for flow nozzles. The head differential across the nozzle can be measured using a manometer or some kind of differential pressure gauge. The upstream tap should be within ½D1 to D1 upstream of the entrance to the nozzle. The downstream tap should be approximately at the outlet of the nozzle.
Orifice Meters
These devices use a thin plate with an orifice, smaller than the pipe ID, to create a pressure differential. The pressure differential can be measured, as in venturi and nozzle meters, and the same equation as for venturi meters can be used however, the discharge coefficient is different for orifice meters. It is easy to make and install an orifice meter in a pipeline easier than a nozzle.
Orifice meters can give accurate measurements of Q, and they are simple and inexpensive to build. But, orifice meters cause a higher head loss than either the venturi or flow nozzle meters. As with venturi meters and flow nozzles, orifice meters can provide values within ±1% (or better) of the true discharge. As with venturi meters, there should be a straight section of pipe no less than 10 diameters upstream.
Some engineers have used eccentric orifices to allow passage of sediments – the orifice is located at the bottom of a horizontal pipe, not in the centre of the pipe cross section. The orifice opening can be “sharp” (bevelled) for better accuracy. But don’t use a bevelled orifice opening if you are going to use it to measure flow in both directions.
An Orifice Meter in a Pipe
Elbow Meters
An elbow in a pipe can be used as a flow measuring device much in the same way as a venturi or orifice plate. The head differential across the elbow (from inside to outside) is measured, and according to a calibration the discharge can be estimated. The taps are usually located in the center of the elbow (e.g. at a 45° angle for a 90° elbow), but can be at other locations toward the upstream side of the elbow. Some companies manufacture elbow meters for flow measurement, but almost any pipe elbow can be calibrated.
Elbow meters are not as potentially accurate as venturi, nozzle, and orifice meters. Typical accuracy is about ±4% of Q. One advantage of elbow meters is that there need not be any additional head loss in the piping system as a result of flow measurement
Variable Area Meters
These are vertical cylinders with a uniformly expanding cross-section in the upward direction. A float inside the cylinder stabilizes at a certain elevation depending on the flow rate through the cylinder. Note that the outside walls are usually transparent to allow direct readings by eye.
Horizontal Trajectory Method
From physics, an accelerating object will travel a distance x in time t according to the following equation (based on Newton’s 2nd law):
x = vo t + at2
2
Where x is the distance; vo is the initial velocity at time 0; t is the elapsed time; and a is the acceleration
Flow emanating from a horizontal pipe will fall a height y over a distance x. The horizontal component (x-direction) has almost no acceleration, and the vertical component (y-direction) has an initial velocity of zero. The vertical acceleration is equal to the ratio of weight to mass, or g = 9.81 m/s2 (32.2 ft/s2). Therefore,
x = vo t and, y = gt2
2
Then by getting rid of t, knowing that Q = VA, and the equation for the area of a circle, the flow rate is calculated as follows:
Q = πD2x
4 2y
g
Where D is the inside diameter of the circular pipe
California Pipe Method
This is the horizontal pipe method for partially-full pipes. It is somewhat analogous to the calibration for a weir or free over fall.
The following equation is in English units:
Where a and D are defined in the figure below (ft); and Q is discharge in cfs
Vertical Trajectory Method
As with pipes discharging horizontally into the air, there is a method to measure the flow rate from vertical pipes. This is accomplished by assuming a translation of velocity head into the measurable height of a column of water above the top of the pipe. Thus, to estimate the flow rate from pipes discharging vertically into the air it is only necessary to measure the:
Inside diameter of the pipe, D; and,
the height of the jet, H, above the pipe
This is a nice idea on “paper,” but in practice, it can be difficult to measure the height of the column of water because of sloshing, surging, and splashing. Also, the act of measuring the height of the column can significantly alter the measured value.
Vortex Shedding Meters
The vortex shedding meter can be accurate to within ±½% to ±1% of the true discharge. The basic principal is that an object placed in the flow will cause turbulence and vortices in the downstream direction, and the rate of fluctuation of the vortices can be measured by detecting pressure variations just downstream.
This rate increases with increasing velocity, and it can be used to give an estimate of the discharge. This requires calibration for a particular pipe material, pipe size, element shape and size, fluid type, and temperature. It is essentially a velocity-area flow measurement method, but it is calibrated to give discharge directly
Ultrasonic Meters
1. Doppler
An emitted pressure wave reflects off a deflector plate. Difference between transmitted and reflected frequencies correlates to flow velocity. Liquid does not have to be clean – in fact, it may not work well if the liquid is “too clean” because it needs particles to reflect the signal
2. Transit-time
Also called “time-of-flight”. The liquid should be fairly clean with this method. Devices generates high-frequency (≈1 MHz) pressure wave(s). Time to reach an opposing wall (inside the pipe) depends on:
Flow velocity
Beam orientation (angle)
Speed of sound through the liquid medium
Upstream straightening vanes may be needed to avoid swirling flow.
May have a single or multiple transmitted sound beams
Some Other Measurement Devices
Collins meters.
Commercial propeller flow meters.
Electromagnetic flow meters.
Volumetric tank
4.3.2 Explain the principle used to measure the pressure across the valves and pipes.
When a fluid is moving in a closed channel such as a pipe and valves operating principle to measure the pressure are based on equations developed by Daniel Bernoulli, a late 18th century Swiss Scientist. His experiments related to the pressure and velocity of flowing water. He determined that at any point in a closed pipe there were three types of head pressure present:
1. Static Head Pressure due to elevation.
2. Static Head Pressure due to applied pressure.
3. Velocity Head Pressure.
The Bernoulli Principle states that the sum of the kinetic, potential, and flow energies (all per unit mass) of a fluid particle is constant along a streamline during steady flow.
According to Bernoulli’s Principle along a stream line
The value of the constant in above equation can be evaluated at any point on the streamline where the pressure, density, velocity, and elevation are known. The Bernoulli equation can also be written between any two points on the same streamline as
Bernoulli equation for unsteady, compressible flow is
Static, Dynamic, and Stagnation Pressures
The Bernoulli equation states that the sum of the flow, kinetic, and potential energies of a fluid particle along a streamline is constant. Therefore, the kinetic and potential energies of the fluid can be converted to flow energy (and vice versa) during flow, causing the pressure to change. This phenomenon can be made more visible by multiplying the Bernoulli equation by the density ρ,
Each term in above equation has pressure units, and thus each term represents some kind of pressure:
• P is the static pressure (it does not incorporate any dynamic effects); it represents the actual thermodynamic pressure of the fluid. This is the same as the pressure used in thermodynamics and property tables.
• ρV2 /2 is the dynamic pressure; it represents the pressure rise when the m fluid in motion is brought to a stop isentropic ally.
ρgz is the hydrostatic pressure term, which is not pressure in a real sense since its value depends on the reference level selected; it accounts for the elevation effects, i.e., fluid weight on pressure. (Be careful of the sign unlike hydrostatic pressure ρgh which increases with fluid depth h, the hydrostatic pressure term ρgz decreases with fluid depth.)
Total Pressure
The sum of the static, dynamic, and hydrostatic pressures is called the total pressure. Therefore, the Bernoulli equation states that the total pressure along a streamline is constant.
Total Pressure = Static Pressure + Dynamic Pressure + hydrostatic pressures
Stagnation pressure
The sum of the static and dynamic pressures is called the stagnation pressure, and it is expressed as
The stagnation pressure represents the pressure at a point where the fluid is brought to a complete stop isentropically.
The static, dynamic, and stagnation pressures measured using piezometer tubes.
Differential pressure flowmeters use Bernoulli’s principle to measure the flow of fluid across the pipes and valves. Differential pressure flowmeters introduce a constriction in the pipe that creates a pressure drop across the flowmeter. When the flow increases, more pressure drop is created. Impulse piping routes the upstream and downstream pressures of the flowmeter to the transmitter that measures the differential pressure to determine the fluid flow.
4.3.3 Plot the required graphs (Head Vs Velocity or Discharge [P 4.2]
Head and Velocity graph for pipelines connected in series
4.3.4 Find local friction coefficient of the pipe and minor friction loss of the valves. [P4.2]
Friction coefficient of the pipe
Local friction coefficient of the pipe friction factor f, in general, f depends on the Reynolds Number R of the pipe flow, and the relative roughness e/D of the pipe wall,
The roughness measure e is the average size of the bumps on the pipe wall. The relative roughness e/D is therefore the size of the bumps compared to the diameter of the pipe
For laminar flow if Reynolds Number R is less than 2000, (R < 2000 in pipes), f can be deduced analytically by the equation
f= 64/Re
For the smooth pipe Turbulent flow if Reynolds Number R is greater than 4000, (R > 4000 in pipes), to calculate friction coefficient equation will be
f= 0.316/(Re¼)
Minor friction loss of the valves
Minor losses in pipes are caused by fittings, bends, valves, sudden enlargement and contraction, change in velocity at entrance and exit of pipe and due to obstruction in the flow.
Minor losses in comparison to friction losses which are considered major losses.
Minor losses for Valves & Fittings = h L = kv²/2g
Where the value of K depends upon the type of valve and degrees of opening
K is computed as = K = (Le / D) ft
Le = equivalent length (length of pipe with same resistance as the fitting/valve)
ft= friction factor
The loss coefficient K for the various pipe components is given in the table below.
The loss coefficient K for the various pipe components
4.3.5 Reason the possibilities of the variation among the trails in the experiment. [M 1.1]
There are many reasons which cause variations among trails in the experiment tends to change the experiment readings taken at different trails and alters the results at the end. It is impossible to make an exact measurement. Therefore, all experimental results are not 100 % accurate. Just how wrong they are depends on the kinds of errors that were made in the experiment.
All experimental data collected among the trails is imperfect. We cannot eliminate this but however, we can struggle to minimize errors.
Some Reason for the variation among the trails in the experiment
Random Errors during the experiment trails are unpredictable. They are chance variations in the measurements over which experimenter have little or no control. There is just as great a chance that the measurement is too big as that it is too small. Since the errors are equally likely to be high as low, averaging a sufficiently large number of results will, in principle, reduce their effect.
Systematic Errors are caused by the way in which the experiment was conducted. In other words, they are caused by the design of the system. Systematic errors cannot be eliminated by averaging in principle, they can always be eliminated by changing the way in which the experiment was done. In actual fact though, you may not even know that the error exists.
Human error is also something that is not an error at all, and that is human error. This kind of error that is caused by your eye's inability to read the exact level of liquid in a graduated cylinder, then that is a random error. If you mean the kind of error that is caused by a poor design of the experiment after all a human designed it then that is a systematic error. These two kinds of errors are the only errors you should ever have in your experimental results.
Human errors are really mistakes during the experiment trials. Spilling part of a solution, dropping part of a solid from the weighing paper, or doing a calculation wrong are blunders, not errors.
There are some other reasons for the variation
Experimental errors can occur due to poor operation and handling of apparatus and the apparatus used for experiment is not accurate.
Slight variations in the level of your eye while reading the meniscus in the graduated cylinder.
Vibration in the floor or air currents that cause fluctuation in the balance.
Errors that may occur in the execution of a statistical experiment.
Reading errors while taking readings and measurement errors.
A miss calibrated balance will cause all the measured masses to be wrong.
Density depends on temperature. The temperature was not specified or controlled.
4.3.6 How do you make sure the flows in the pipe are turbulent? [D 1.1]
Fluid flow in pipes
Fluid flow in circular and noncircular pipes is commonly encountered in practice. Fluid flow is classified as external and internal, depending on whether the fluid is forced to flow over a surface or in a conduit. When calculating heat transfer or pressure and head loss it is important to know if the fluid flow is laminar, transitional or turbulent
Laminar and turbulent flows
An attentive examination of stream in a channel reveals that the fluid stream is streamlined at low speeds however turns confounded as the velocity is extended over a fundamental worth, as showed up in The stream organization in the first case is said to be laminar, depicted by smooth streamlines and exceedingly asked for development, and turbulent in the second case, where it is depicted by rate changes and especially scattered development. The move from laminar to turbulent stream does not happen out of the blue; rather, it happens over some locale in which the stream changes amidst laminar and turbulent streams before it ends up being totally turbulent. Most streams experienced eventually are turbulent. Laminar stream is experienced when exceedingly thick fluids, for instance, oils stream in little pipes or restrict sections. To make sure the flows in the pipe is turbulent by injecting dye streaks into the flow.
It can be verified the existence of these laminar, transitional, and turbulent flow regimes by injecting some dye streaks into the flow in a glass pipe.
It can be observed that the dye streak forms a straight and smooth line at low velocities when the flow is laminar figure (a) (we may see some blurring because of molecular diffusion), has bursts of fluctuations in the transitional regime, and zigzags rapidly and randomly when the flow becomes fully turbulent figure (b). These zigzags and the dispersion of the dye are indicative of the fluctuations in the main flow and the rapid mixing of fluid particles from adjacent layers.
The intense mixing of the fluid in turbulent flow as a result of rapid fluctuations enhances momentum transfer between fluid particles, which increases the friction force on the surface and thus the required pumping power. The friction factor reaches a maximum when the flow becomes fully turbulent.
The Figure below is illustrating the behavior of colored fluid injected into the flow in laminar and turbulent flows in a pipe.
In turbulent flow vortices, eddies and wakes make the flow unpredictable. Turbulent flow happens in general at high flow rates and with larger pipes. Shear stress for turbulent flow is a function of the density.
Turbulent flow is characterized by random and rapid fluctuations of swirling regions of fluid, called eddies, throughout the flow. These fluctuations provide an additional mechanism for momentum and energy transfer. In laminar flow, fluid particles flow in an orderly manner along path lines, and momentum and energy are transferred across streamlines by molecular diffusion. In turbulent flow, the swirling eddies transport mass, momentum, and energy to other regions of flow much more rapidly than molecular diffusion, greatly enhancing mass, momentum, and heat transfer. As a result, turbulent flow is associated with much higher values of friction, heat transfer, and mass transfer coefficients.
Notwithstanding when the normal stream is relentless, the swirl movement in turbulent stream causes critical vacillations in the estimations of speed, temperature, weight, and even thickness (in compressible stream). Figure beneath demonstrates the variety of the momentary speed part u with time at a predefined area, as can be measured with a hot-wire anemometer test or other touchy gadget. We watch that the prompt estimations of the speed change around a normal quality, which recommends that the speed can be communicated as the entirety of a normal worth u– and a fluctuating part
Fluctuations of the velocity component u with time at a specified location in turbulent flow.
Reynolds Number
The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow velocity, surface temperature, and type of fluid, among other things.
After exhaustive experiments in the 1880s, Osborne Reynolds discovered that the flow regime depends mainly on the ratio of inertial forces to viscous forces in the fluid. This ratio is called the Reynolds number and is expressed for internal flow in a circular pipe as
The Reynolds number is important in analyzing any type of flow when there is substantial velocity gradient (i.e. shear.) It indicates the relative significance of the viscous effect compared to the inertia effect. The Reynolds number is proportional to inertial force divided by viscous force.
It certainly is desirable to have precise values of Reynolds numbers for laminar, transitional, and turbulent flows, but this is not the case in practice. It turns out that the transition from laminar to turbulent flow also depends on the degree of disturbance of the flow by surface roughness, pipe vibrations, and fluctuations in the flow. Under most practical conditions, the flow in a circular pipe is laminar for Re & 2300, turbulent for Re * 4000, and transitional in between. That is,
Re ≥ 4000 turbulent flow
2300 ≤ Re ≤ 4000 transitional flow and Re ≤ 2300 laminar flow
In transitional flow, the flow switches between laminar and turbulent randomly. It should be kept in mind that laminar flow can be maintained at much higher Reynolds numbers in very smooth pipes by avoiding flow disturbances and pipe vibrations. In such carefully controlled experiments, laminar flow has been maintained at Reynolds numbers of up to 100,000.
In the transitional flow region of 2300 ≤ Re ≤ 4000, the flow switches between laminar and turbulent randomly.
4.3.7 Discuss on the application of k values in finding the flow rates when pipes are connected and run in parallel.
When two pipes are connected and run in parallel to find the flow rates. In the figure below A and Bare described as nodes or junctions of pipes connected in parallel. In the steady state the known incoming flow at node A must balance with the outgoing flow in pipe 1 and 2. Similarly the incoming flow at node B in the incoming pipes 1 and 2 must equal the know withdrawal at node B.
QA = QB = Q1 + Q2 + Q3 equation (i)
The head loss in the pipe 1 and 2 must be same. Since all begin at a single node (A) and all end at single node (B) and the difference in head between those two nodes is unique, regardless of the pipe characteristics the head loss in the is the same or
HA - HB = hL1 = hL2 = hL3 = hL equation (ii)
Where HA and HB are the total heads at the nodes A and B respectively, hL1 is the head loss in pipe 1 and hL2 is the loss in pipe 2 and hL is the single value of head loss between nodes A and B.
Identify all the terms that make up hL,
Such as pipe major losses, hL= f (L/D) v2/2g
And for minor losses, hL = Kv2/2g for each branch.
If k values are given for each pipe connected in parallel then
For pipe 1 = hL1 = K1v12/2g and
For pipe 2 = hL2 = K2v22/2g)
For pipe 3 = hL3 = K3v22/2g)
And hL1 = hL2 = hL3
Now using above we can determine the head loss and flow for each pipe and an equivalent pipe coefficient, Kpeq
For steady flow conditions in the network we have a total two pipe flowsQ1, Q2 and Q3, and two head losses hL1, hL2 and hL3
Eq. (ii) provides independent equation relating the head losses (hL,1=hL,2 = hL3). The third equation is that the head loss in any pipe equals the difference in head between nodes A and B. Conservation of mass at node A (Eq. (i) the final equations are the head loss versus discharge equations will be
Equation (iii)
We can substitute above equation in the mass balance equations. (i) With hL equal to each pipe’s head loss or
Equation (IV)
In this equation, all terms except for hL are known. After solving for hL, the unknown pipe flows can be computed by equation (iii)
Like pipes in series, an equivalent pipe coefficient can be computed for parallel pipes. In Eq. (IV) hL can be pulled from each term on the left hand side or for a general discharge and two parallel pipes:
The equivalent coefficient k is then:
The head loss between the two end nodes in terms of flow rate and K is:
Reference:
http://www.instrumentationtoday.com/manometer/2011/09/
https://en.wikipedia.org/wiki/Pressure_measurement
http://www.calright.com/manometers
http://www.transcat.com/calibration-resources/application-notes/manometer-meriam-principles/
The Hydraulics of Open Channel Flow: An Introduction Basic principles, sediment motion, hydraulic modelling, design of hydraulic structures Second Edition by Hubert Chanson.
A textbook of Fluid Mechanics and Hydraulic Machines -Dr. R. K. Bansal.
Fluid Mechanics for civil engineers by Bruce Hunt.
HYDRAULICS IN CIVIL AND ENVIRONMENTAL ENGINEERING By Andrew Chadwick and John Morfett.
Hydraulics and Hydraulic Machines.
Student Solutions Manual to accompany Engineering Fluid Mechanics, 7th Edition by Clayton T. Crowe and Donald F. Elger October 1, 2001.
Flow Meters in Pipes by Gray P. Merkely.
BOOKLET 7 FLOW MEASUREMENT METHODS IN OPEN CHANNELS English Version of The Original Publishing.
Open Channel Flowmeters By Brian Dawson, Isco, Inc.
BERNOULLI AND ENERGY EQUATIONS/
CENTRIFUGAL PUMPS e-notes: Dr.N.Balasubramanya.
Energy and Momentum Principle Lecture No. # 02 Computation of Critical Depth by Prof. Dr. Arup Kumar Sarma
TASK 01
| Explain the construction and application of pressure measuring devices. (Manometers and mechanical gauges). Write the limitations of manometers.
Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure pressure are called pressure gauges or vacuum gauges.
A manometer is an instrument that uses a column of liquid to measure pressure, although the term is currently often used to mean any pressure measuring instrument.
A vacuum gauge is used to measure the pressure in a vacuum—which is further divided into two subcategories: high and low vacuum (and sometimes ultra-high vacuum). The applicable pressure ranges of many of the techniques used to measure vacuums have an overlap. Hence, by combining several different types of gauge, it is possible to measure system pressure continuously from 10 mbar down to 10−11 mbar.
There are basically two types of manometers.
U-Tube Manometer
Well Type Manometer
There are also variations of the above said basic types called Enlarged-Leg Type Manometer, and Inclined Tube Manometer. Another manometer used commercially is the Ring-Balance Type Manometer.
U-Tube Manometer
A simple u-tube manometer is shown below. If ‘dm‘is the manometric fluid density, ‘d1’ is the density of the fluid over the manometer, ‘P2’ is the atmospheric pressure (for general measurement of gas pressure) and ‘P1’ is the gas pressure, and also if d1<<dm, then the differential pressure can be obtained by the relation
p1-p2 = h (dm-d1)
U-Tube Manometer
An enhanced version of a manometer is shown below with a seal liquid over the manometer liquid to separate the process fluid from the manometer fluid for any probable source of trouble like absorption, mixing or explosion and so on. Seal pots with large diameters are also placed for increasing the range. The equation for differential pressure is the same as mentioned above.
Manometer With Large Seal Pots
Well-Type Manometer
The main difference between a U-tube manometer and a well type manometer is that the U-tube is substituted by a large well such that the variation in the level in the well will be negligible and instead of measuring a differential height, a single height in the remaining column is measured. If a1 and a2 are the areas of the well and the capillary, and if (h1-h2) is the difference in height in the well due to the pressure difference (p1-p2) as shown, at balance, then
p1-p2 = dm.h (1+a2/a1)
The figure of a well-type manometer is shown below.
Well-Type Manometer
Enlarged-Leg Manometer
In the enlarged-leg manometer, a2 is not negligible compared to a1. It has a float in the enlarged-leg which is utilized for indication or recording. The two legs can be changed for changing the measurement span. Thus, the equation becomes,
p1-p2 = dm.h
The figure of an enlarged-leg manometer is shown below.
Enlarged Leg Manometer
Inclined Tube Manometer
The inclined tube manometer is an enlarged leg manometer with its measuring leg inclined to the vertical axis by an angle b. This is done to expand the scale and thereby to increase the sensitivity. The differential pressure can be written by the equation,
p1-p2 = dm.h.Cosb (1+a2/a1)
The factor cosb expands the scale of the instrument. When b is quite large, h can be increased such that (h.cosb) remains constant. The figure of an inclined tube manometer is shown below.
Inclined Tube Manometer
Micromanometer
The micromanometer is another variation of liquid column manometers that is based on the principle of inclined tube manometer and is used for the measurement of extremely small differences of pressure. The meniscus of the inclined tube is at a reference level as shown in the figure below, viewing through a magnifier provided with cross hair line. This is done for the condition, p1=p2. The adjustment is done by moving the well up and down a micrometer. For the condition p1 not equal to p2, the shift in the meniscus position is restored to zero by raising or lowering the well as before and the difference between these two readings gives the pressure difference in terms of height.
Micromanometer
Manometer is shown above as a static measuring device. Its dynamics can rarely be ignored. Considering manometric fluid as a free body, the forces acting on it are
The weight distributed over the entire fluid.
The drag force due to its motion and the corresponding tube wall shearing stress.
The force due to differential pressure.
Surface tension force at the two ends.
Ring-Balance Manometer
This device cannot be actually called a manometer, but it is often considered so. The tube is made of polythene or other light and transparent material. This tube is bent into in to the form of a ring and is supported at the centre by a suitable pivot. The tubular chamber is divided in to two parts by spilling, sealing, and filling with a suitable light liquid like kerosene or paraffin oil for isolating the two pressures. Pressure taps are made with two flexible tubings. Pressures p1 and p2 act against the sealed walls as shown in the figure below, and rotate the ring which is balanced by the counter weight w.
Of the various manometric fluids used, mercury has many advantages like low vapour pressure, non-sticky nature, and wide temperature range from -20 degree Celsius to 350 degree Celsius. Its high density is disadvantageous for low differential pressure measurements. The device installation and maintenance is known to be quite expensive.
Application of Manometers
The manometer has many advantages in this age of technology. Containing no mechanical moving parts, needing nothing but the simplest of measurements, the primary standard manometer is readily available at modest cost. The principle of the manometer has not changed since its inception, however great strides have been made in its arrangement and the application of the instrument to various industrial measurement requirements. Whereas formerly the manometer was considered a laboratory instrument, today we find the manometer commonly used to measure pressures ranging from as high as 600 inches of mercury to space vacuums.
The manometer utilizes the hydrostatic (standing liquid) balance principle wherein a pressure is measured by the height of the liquid it will support. For example, the weight of a column of mercury at 0 degree C that is one inch high and one inch in cross sectional area is .4892 pounds. Thus we can say that a column of mercury one inch high imposes a force of .4892 pounds per square inch or .4892 PSI.
Throughout history, mercury-filled manometer has been a very important device needed in the construction of aqueducts, bridges, installing of swimming pools, and other engineering applications. The said measuring instrument has also been implemented for a number of industrial applications like for visual monitoring of air and gas pressure in compressors as well as in vacuum equipment and specialty tank applications. The manometer is also necessary in avionics and climate forecasting. It principally study the stress of fluid at the same time measure the speed at which a stream of air is flowing.
Mechanical gauges
Mechanical gauges are best suitable for measuring very high fluid pressures. In case of steam boilers, where manometers cannot be used, a mechanical gauges can be conveniently used.
Working of Mechanical gauges
Mechanical gauges utilize an internal bourdon tube. One end of the bourdon tube is connected to a gear and shaft assembly that moves a pointer. When the pressure inside the bourdon tube increases, the bourdon tube uncoils slightly. The amount of uncoiling that occurs is proportional to the pressure inside the bourdon tube. As the tube uncoils, its motion activates the gear and shaft system that turns the pointer on the gauge. While all that you see when you look at the gauge is the pointer moving, you should understand that there is a small, bent tube (the bourdon tube) that's coiling and uncoiling with each change in the pressure inside that tube.
Mechanical pressure gauges are connected directly to the process fluid being measured (i.e. oil). As the process fluid pressure changes the pressure on the bourdon tube also changes which in turn moves the pointer on the gauge.
Mechanical temperature gauges also utilize a bourdon tube. They have a sealed capillary tube and bulb assembly that is filled with temperature sensitive liquid that produces a proportional vapour pressure on the bourdon tube. As the temperature changes, the pressure inside the bourdon tube changes, which in turn moves the pointer on the gauge.
Types of mechanical gauges
Ruler and scales: they are used to measure lengths and other geometrical parameters. They can be single steel plate or flexible tape type tool.
Callipers: they are normally of two types- inside and outside calliper. They are used to measure internal and external size (for e.g. Diameter) of an object. It requires external scale to compare the measured value. Some callipers are provided with measuring scale. Other types are odd leg and divider calliper.
Venire calliper: It is a precision tool used to measure a small distance with high accuracy. It has got two different jaws to measure outside and inside dimension of an object. It can be a scale, dial or digital type venire calliper.
Micrometre: It is a fine precision tool which is used to measure small distances and is more accurate than the venire calliper. Another type is a large micrometre calliper which is used to measure large outside diameter or distance.
Feeler gauge: Feelers gauges are a bunch of fine thickened steel strips with marked thickness which are used to measure gap width or clearance between surface and bearings.
Bridge gauge: Bridge gauges are used to measure the amount of wear of Main engine bearing. Normally the upper bearing keep is removed and clearance is measured with respect to journal. Feeler gauge can be used to complete the process.
B) Explain the two different system of pressure measurement and their relation.
Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure pressure are called pressure gauges or vacuum gauges. A manometer is an instrument that uses a column of liquid to measure pressure, although the term is currently often used to mean any pressure measuring instrument.
A vacuum gauge is used to measure the pressure in a vacuum—which is further divided into two subcategories: high and low vacuum (and sometimes ultra-high vacuum). The applicable pressure ranges of many of the techniques used to measure vacuums have an overlap. Hence, by combining several different types of gauge, it is possible to measure system pressure continuously from 10 mbar down to 10−11 mbar.
Everyday pressure measurements, such as for tire pressure, are usually made relative to ambient air pressure. In other cases measurements are made relative to a vacuum or to some other specific reference. When distinguishing between these zero references, the following terms are used:
Absolute pressure is zero-referenced against a perfect vacuum, using an absolute scale, so it is equal to gauge pressure plus atmospheric pressure.
Gauge pressure is zero-referenced against ambient air pressure, so it is equal to absolute pressure minus atmospheric pressure. Negative signs are usually omitted. To distinguish a negative pressure, the value may be appended with the word "vacuum" or the gauge may be labeled a "vacuum gauge."
| A U - tube differential mercury manometer is connected between two pipes Xand Y. Pipe X contains carbon tetra chloride (Sp.gr. 1.59) under a pressure of 103KN/m2 and pipe Y contains oil (Sp.gr. 0.8) under a pressure of 172 kN/m2. Pipe X is 2.5 m above pipe Y. Mercury level in the limb connected to pipe X is 1.5 m below the center line of pipe Y. Find the manometer reading as shown by a centimeter Scale attached to it. (P.1.2)
Given Data:
Pipe x and left limb contains carbon tetra chloride of Specific gravity = Sp.gr. = 1.59
Under pressure of = Px = 103 KNm2 = 103 × 1000 = 103,000 N/m2
Pipe Y and right limb contains oil of Specific gravity = Sp.gr. = 0.8
Under pressure of = Py = 172 KNm2 = 172 × 1000 = 172,000 N/m2
Height = h + x = 2.5 + 1.5 = 4 m
Solution:
ρx = 1.59 × 1000 = 1590 kg/m³
ρy = 0.8 × 1000 = 800 kg/m³
Px = 103 KNm2 = 103 × 1000 = 103,000 N/m2
Py = 172 KNm2 = 172 × 1000 = 172,000 N/m2
Taking X-X as datum line
Pressure above X-X in the Left Limb
Left limb = (13.6 × 1000 × 9.81 × a) + (1590 × 9.81 × 4) + 103,000
Left limb = 13.6 × 1000 × 9.81 × a + 6360 × 9.81 + 103,000
Pressure above X-X in the Right Limb
Right Limb = 800 × 9.81 × (1.5 + a) + 172, 000
Equating the two pressures we get
13.6 × 1000 × 9.81 × a + 6360 × 9.81 + 103,000 = 800 × 9.81 × (1.5 + a) + 172, 000
Dividing both sides by 1000 × 9.81
13.6 × 1000 × 9.81 × a + 6360 × 9.81 + 103,000 = 800 × 9.81 × (1.5 + a) + 172, 000
1000 × 9.81 1000 × 9.81 1000 × 9.81 1000 × 9.81 1000 × 9.81
13.6 a + 6.36 + 10.49 = 0.8 × (1.5 + a) + 17.53
13.6 a + 16.85 = 1.2 + 0.8a + 17.53
13.6 a + 16.85 = 0.8a + 18.73
13.6 a - 0.8a = 18.73 - 16.85
a (13.6 - 0.8) = 18.73 - 16.85
12.8a = 1.88
a = 1.88 / 12.8
a = 0.146 m = 0.146 × 100 = 14.6 cm
a = 14.6 cm
Manometer Reading = 14.6 cm of Hg.
| A pipe connected with a tank (diameter 3 m) has an inclination of q with the horizontal and the diameter of the pipe is 20 cm. Determine the angle? Which will give a deflection of 5 m in the pipe for a gauge pressure of 1 m water in the tank. Liquid in the tank has a specific gravity of 0.88. (M 1.1).
Given Data:
Diameter of the tank= D = 3m.
Diameter of the pipe connected with tank = d = 20cm.
Deflection in the pipe for a gauge pressure of 1 m water in the tank = L = 5m
Specific gravity of the liquid enclosed in the tank = S.g = 0.88
Gauge = 1m of water.
To Find:
Determine the angle = θ=?
Solution:
Volume displaced in the tank = volume raise in the tube
Hence:
(3)2 × X = (0.2)2 × L
9 X = 0.04 L
X = 0.04/9 L
Difference in head = X + h
= 0.04/9 L + L Sin θ
= 0.02 + 5 Sin θ
Whereas:
Gauge pressure = 1 meter of water
ρ g h = 1
ρ g ( h + X) = 1
0.88 × 1000 × 9.81 (0.02 + 5 Sin θ) = 1
8632.8 (0.02 + 5 Sin θ) = 1
172.656 + 43164 Sin θ = 1
43164 Sin θ = -172.656
Sin θ = (-172.656)/43164
Sin θ = 0.004
θ = 0.22o
| A manometer connected to a pipe indicates a negative gauge pressure of 70mm of mercury. What is the pressure in the pipe in N/m2? (M.1.2)
Given data:
Negative gauge pressure = 70mm of mercury = 0.07
Standard atmospheric pressure= 101.325 KN.
To Find:
Pressure inside the Pipe N/m² =?
Solution:
We know,
Pabsolute = Patmospheric + P gauge
= Patmospheric + ρ g h
ρ g h = 13.6 × 1000 × 9.81 × 0.07
ρ g h = 9339.12 N/m2
ρ g h = - 9.33 KN/m2
Pabsolute = Patmospheric + ρ g h
Pabsolute = 101.213 + (-9.33)
Pabsolute = 91.8 KN/m2
| As shown in figure water flows through pipe A and B. The pressure difference of these two points is to be measured by multiple tube manometers. Oil with specific gravity 0.88 is in the upper portion of inverted U-tube and mercury in the bottom of both bends. Determine the pressure difference. (D.1.1)
Given Data:
Oil with specific gravity = Sp.gr = 0.88
Mercury specific gravity = Sp.gr = 13.6
To Find:
Pressure difference = PA - PB
Solution:
Pressure at X is given as
= PX = PA +ρ g h
= PX = PA + (1000 × 9.81 × 0.1)
= PX = PA + 981
Pressure at Υ is given as
= PY = PX – ρ g h
= PY = (PA + 981) – (13.6 × 1000 × 9.81× 0.3)
= PY = (PA + 981) – 40024.8
Pressure at Z is given as
= Pz = Py + ρ g h
= Pz = (PA + 981 – 40024.8) + 0.88 × 1000 × 9.81× 0.4
= Pz = (PA + 981 – 40024.8) + 3453.12
Pressure at U is given as = Pu = Pz - ρ g h
= Pu = (PA + 981 – 40024.8 + 3453.12) – 13.6× 1000 × 9.81× 0.5
= Pu = (PA + 981 – 40024.8 + 3453.12) – 66,708
Pressure at B is given as = PB = Pu - ρ g h
= PB = (PA + 981 – 40024.8 + 3453.12 – 66,708) – 1000 × 9.81 × 0.8
= PB = (PA + 981 – 40024.8 + 3453.12 – 66,708) – 7848
Summing all the equations above
The pressure difference is given by = PA - PB
= PA - PB = 103.28 Pwg
The pressure difference is given by = PA - PB = 10.131 KPa
| Estimate the vapor pressure of the oil at 27ₒ, if the water height h was 3 cm when the gas of absolute pressure 101047.7 Pascal was trapped. (M 1.2)
Given Data:
Absolute Pressure = 101047.7 Pascal
Water height = 3cm
Atmospheric Pressure = 101325 Pa
To Find:
Vapor pressure of the oil at 27ₒ =?
Solution:
Vapor Pressure = P atmospheric- P absolute
Vapor Pressure = 101325 - 101047.7
Vapor Pressure = 277.3 Pa
Or
Vapor Pressure = 0.2773 KPa
| A vertical gate of 5 m height and 3 m wide closes a tunnel running full with water. The pressure at the bottom of the gate is 195 kN/m 2. Determine the total pressure on the gate and position of the center of the pressure
Given Data:
Height of gate = 5 m
Width of gate = 3 m
Area of the gate = A = height x width = 5 x 3 = 15 m²
The pressure at the bottom of the gate = 195 kN/m²
And we have to find:
The total pressure on the gate = P =?
Position of the center of the pressure = C.P=?
The force is given by the expression = F = ρ g A x
x = distance of center of gravity from fee surface
x = h – d/2
Suppose Height = h = 10 m
x = 10 + 5/2
x = 12.5
The total force on the gate = F = ρ g A x
Putting the values
= F = 1000 × 9.81× 15× 12.5
The total pressure on the gate = F = 1,839,375 N
The total pressure on the gate = P = F / A
The total pressure on the gate = P =1,839,375 / 15
The total pressure on the gate = P = 122625 Pa Or 122.625 KN / m²
Where IG is the 2nd moment of area about a line through the centroid of the rectangle and
IG for rectangular shapes BD³ / 12
Now we are putting the values
Position of the center of the pressure = C.P= 0.172 m
Task 02
2.1| A horizontal pipe line 50m long is connected to the water tank at one end and discharges freely into the atmosphere at the other end. For the first 30m length from the tank the pipe is 200mm diameter and it diameter suddenly enlarged to 400mm afterwards. The height of water level in the tank is 10m above the center of the pipe.
A. Determine the loss of head in the pipe.
Solution:
From Bernoulli’s equation
P1/(ρ g ) + V1²/2g + Z = P2/(ρ g ) + V2²/2g + head loss
At both ends pressure is equal due to expose to atmospheric pressure ad water is static so velocity = 0
So above equation can be written as
10 = V2²/2g + head loss
Head loss = HL (entrance) + Hf1 + HL (enlargement) + Hf2 + HL (exit)
HL (entrance) = head loss at entrance (Sudden Contraction)
Hf1= head loss due to friction in pipe length L1, 30-meter length.
HL (enlargement) = Head loss due to enlargement.
Hf2= head loss due to friction in pipe L2, 20 meter length.
HL (exit) = Head loss at exit of the pipe.
10 = V2²/2g + head loss
10 = V2²/2g + ( 0.5 V1²)/2g + (4.f.L1.V1²)/(2g.d1) + (V1²-V2²)/2g +(4.f.L2.V2²)/(2g.d2) + ( V2²)/2g
From continuity equation
A1 V1 = A2 V2
V1 = (A2 . V2)/A1
V1 = (π/4 ×0.4 ²)/(π/4 ×0.2²)V2
V1= 4V2
Friction factor = f = 0.01
Putting the value of V1 in head loss equation above
10 = ( v2²)/(2 ×9.81) + ( 0.5(4 . v2)²)/2g + (4 ×0.01 × 30 × (4.v2)²)/(2 ×9.81 ×0.2) + ((4.v2)²-V2²)/(2 ×9.81) + (4 ×0.01 ×20 × v2²)/(2 ×9.81 ×0.4) + ( V2²)/( 2 ×9.81)
Calculating value for V2
Q = A2 × V2
Q = 164.13 L/s = 0.16413 m3 /s
A2 = π/4 ×0.4 ² = 0.1256
V2 = Q /A = 0.16413 / 0.1256 = 1.3
V2 = 1.3 ms-1
Putting values
10 = ( (1.3)²)/(2 × 9.81) + ( 0.5 (4 ×1.3)²)/(2 ×9.81) + (4 ×0.01 ×30 × (4 ×1.3)²)/(2 ×9.81 ×0.2) + ((4×1.3)²-1.3²)/(2 ×9.81)
+ (4 ×0.01 ×20 × 1.3²)/(2 ×9.81 ×0.4) + (1.3 ²)/(2 ×9.81)
10 = 0.086 + 0.689 + 8.269 + 1.29 + 0.1724 + 0.086
10 = 10.59
Loss of head in pipe = 10 – 10.59 = 0.59
B. Determine the rate of flow and the power required to maintain the flow.
Power required to maintain the flow = P = (ρgQ . h)/1000 k watts
P = (ρgQ . h)/1000 k watts
ρ = 1000
g = 9.81
h = 10m
Q = 164.13 L/s = 0.16413 m3 /s
Putting values
P = (1000 ×9.81 ×0.16413 ×10)/1000 kwatts
P = 16.10 k watts power is required to maintain the flow.
2.2 | An irrigation channel (trapezoidal in cross section) excavated in smooth earth (n = 0.030) is to carry a flow rate of 12 m3/s at a uniform flow depth of 2 m. The bed width of the channel is 2.7m and the side slope of 2 horizontal to 1 vertical. Calculate the bed slope of the channel
Given Data:
Flow rate is = Q = 12 m3/s
Depth of flow = h = 2 m
The bed width of the channel is = 2.7m
Side slop = 2 horizontal to 1 vertical
n = 0.030
To Find:
Bed slope of the channel = i =?
Solution
To calculate top widthdepth of flow is 2 m and side slop is 2 horizontal to 1 vertical.
Therefore for top width =( 2)/1 × 2
= 4m
Top width = 4 + 4 + 2.7 = 10.7 m
Area of flow = 1/2 (a + b) h
Area of flow = 1/2 (2 +10.7) ×2
Area of flow = 12.7 m²
Hydraulic mean depth = m = Area/Perimeter
Perimeter =?
Area of wetted perimeter is given as
Perimeter = 4.47 + 2.7 + 4.47 = 11.87
Area = 12.7 m²
Perimeter = 11.87 m
m = Area/Perimeter
m = 12.7/11.87
m = 1.06
Applying manning’s equation to find Chassis’s constant C
Manning’s equation = C = 1/n . m 1/6
= C = 1/0.030× 1.06 1/6
= C = 33.65
To find velocity V use flow rate equation
Flow rate = Q = A × V
Or V = Q / A
V = 12 / 12.7
V = 0.94 ms-1
We have calculated
V = 0.94 ms-1
C = 33.65
m = 1.06
Now apply chassis’s equation to find bed slope of the channel = i
Chassis’s equation = V = C √(m . i)
Putting values
0.94 = 33.65√(1.06. i)
Taking square on both sides
0.94² = (33.65√(1.06. i) )2
0.8836 = (33.65)2 × 1.06 × i
0.8836 = 1200.26 × i
Or
Bed slope of the channel = i = 0.8836/1200.26
Bed slope of the channel = i = 7.3 × 〖10〗^(-04)
Or
i = 0.00073
2.3 |
a. Show that in a rectangular channel, critical depth is two third of Specific energy. [M1.2]
Specific energy & critical depth relationship for rectangular channels
Solution of the specific energy equation for rectangular channels Consider a specific energy equation for the case of a rectangular channel.
The concept of specific energy as it applies to open channels with small slopes is given below.
If bed slope is taking datum line then Z = 0
Energy = yc + V²/2g equation (i)
Whereas yc is critical depth
This indicates that the specific energy is the sum of the depth of water and the velocity head.
Solution of the specific energy equation for rectangular channels. Consider a specific energy equation for the case of a rectangular channel
Energy = yc + V²/2g
Discharge = Q = A × V
V = Q / A
V² = (Q / A) ²
V² = Q² / b² × y² equation (ii)
Where b is the width of the channel and y is the depth of flow
Put the equation (ii) in equation (i)
Substituting this in the specific energy equation it can be written as
Energy = yc+ Q²/(b^2.y^2 )2g
Defining q= Q / b
Then Energy = yc+ q²/2gy² equation (iii)
= q²/g = yc³ =>yc = (q² / g) 1/3
Put the value of q²/g = yc³in equation iii
E min = yc + yc³/2yc²
E min = yc3/2
yc = 2/3 E min
So, Yc minimum, we can write as E minimum is equal to 3 by 2 into Y c or this can be written in the form that Y c is equal to 2 by 3 E minimum. So, once we know the energy E minimum, then we can calculate the Y c directly from this or if we know the Y c, we can calculate E minimum directly from this. So, this relation shows that, there is a definite relation for critical depth computation or for computation of minimum specific energy in critical condition.
b. Calculate the minimum Specific energy and critical velocity of flow for a rectangular channel of width 6m and discharge 18 m3/sec when the depth of flow of water is 1.8 m. Also classify the type of flow in the channel.[D1.1]
Given Data:
Rectangular channel of width = b = 6m
Discharge = Q = 18 m3/sec
Depth of flow of water is = y = 1.8 m
To Find:
Minimum Specific energy = E Min =?
Critical velocity = Vc=?
Solution:
For Flow velocity we have
Discharge = Q = A × VC
VC = Q / A
Area of Rectangular channel is given as A = width of channel × Depth of flow
A = b × y
Flow velocity V = Q / b × y
By substituting the values
Flow velocity = V = 18/(6 ×1.8)= 1.67 m s-1
To calculate critical velocity we have to consider
Critical depth = hc = (q² / g) 1/3
q = 18 / 6 = 3 m2 /s
= hc = (q² / g) 1/3
= hc = (3² / 9.81) 1/3 = 0.3058 m
Critical depth = hc = 0.3058m
Now critical velocity is given as
Critical velocity = Vc = √(g ×hc )
Critical velocity = Vc = √(9.81 ×0.3058 )
Critical velocity = Vc = 1.732 m s-1
Minimum Specific energy = E min = hc 3/2
Critical depth hc is calculated above the value is 0.3058m
Substituting the value for Minimum Specific energy
E min = 0.3058 ×3/2
E min = 0.4587
The type of flow in the channel
The type of flow in the channel is based on Froude number. The Froude number, Fr, is a dimensionless value that describes different flow regimes of open channel flow. The Froude number is a ratio of inertial and gravitational forces.
If Froude number is equal to 1 the flow is critical = Fr = 1 = Critical Flow
If Froude number is less than 1 the flow is super critical = Fr> 1 = Super Critical Flow
Type of flow in the channel for given task
To calculate the type of flow for given task we have
Froude number = Fr = V/√gD
Where
V = Flow velocity = 1.67 m s-1
g = Gravitational acceleration = 9.81 m s-2
D = Area/(Top width of channel) = (10.8 )/6 = 1.8
Substituting the values
Froude number = Fr = 1.666/√(9.81 ×1.8)
Froude number = Fr = 0.396
From the above value Froude number is less than 1 = Fr> 1
Hence proved flow is Super Critical.
TASK 03
3.1 |Explain the importance of Duty point (point of maximum efficiency) in matching a pump to a pipeline system. Use suitable pump curves to explain your answer.
Water is to be pumped through a 200mm diameter pipe from a pile hole (bore hole) of 50 m deep to a river one kilometre away the point of dewatering. The pump available has the discharge – head performance characteristic as shown in table 1.
Table 1 : Performance Characteristic
Head / (m)
Discharge / (lit/min)
30
2000
50
1750
65
1410
80
800
Neglecting the local losses, calculate the duty point of the pump when the friction factor of the pipe f = 0.04.
Duty Point: Characterized as that Duty Point on the H-Q bend where a radial pump works at greatest effectiveness. It is dictated by the Flow Rate and the Total Head at the individual pump speed. The point on the pump execution bend where the stream and head coordinate the application's prerequisite is known as the obligation point. An outward pump dependably works at the point on its execution bend where its head coordinates the resistance in the pipeline. The obligation point is at the crossing point between the Q-H bend and the framework attributes.
To pick a pump for a framework, it is important to know both the framework qualities and the pump bend. The pump is dimensioned so it can give a sufficient execution at the most extreme framework load. The pump execution bend is a chart demonstrating the association between stream (Q) and produced head (H) or weight. The framework's qualities demonstrate the weight misfortune in the framework as a component of the stream. The obligation point is the convergence between the pump execution bend and the framework's qualities.
The significance of Duty point
Pumps are constantly characterized by the essential Pump attributes beneath. They demonstrate the relationship between head, force and effectiveness against stream. It is vital to see exactly how "peaky" the productivity may be; demonstrating that running at an obligation (head and/or stream) beneath evaluated obligation is liable to prompt a huge lessening in pump effectiveness. The Best Efficiency Point (BEP) of a pump is in a perfect world at the appraised obligation point. Pump productivity can diminish altogether when the pump is working far from the planned Best Efficiency Point. Over-determining the obligation while indicating a pump will hence mean tremendously expanded vitality costs. Pumps are not made to standard obligations, which make contrasting efficiencies less basic than and items that are made to standard obligations, (for example, engines). A producer of pumps with a high outline (BEP) effectiveness might regularly miss out to another makers of a less proficient (BEP) pump, contingent upon where the real obligation point asked for exists in the execution bends of the pumps. The same pump will more often than not be offered with distinctive impellers to give great execution at lower obligations. The same pump will frequently be offered with diverse velocity engines to permit it to cover a much more extensive scope of obligations.
Calculating the duty point of the pump
Given Data:
Diameter of the pipe = 200 mm = 0.2m
Static Head = Hs = 50 m
Length = 1 km = 1000 m
Friction factor of the pipe = f = 0.04
Neglecting the local losses
To Find:
The duty point of the pump
Solution:
Converting given discharge rates liter/min to m3/s to plot graph.
2000 liter/min = 0.033 m3/s
1750 liter/min = 0.029 m3/s
1410 liter/min = 0.0235 m3/s
800 liter/min = 0.0133 m3/s
ΔH = Static Head + Delivery Head + minor losses
As per given condition neglecting the local losses so,
ΔH = Hs + Hf + 0
ΔH = 50m + 4flv²/2gd + v²/2g
To calculate velocity using the equation
Q = A × V
Q = (π d²)/4× V
V = (4 Q)/(π d²)
Substituting the values
V = (4 Q)/(3.14 ×0.2²)
V = 31.84 Q
Putting the value of V in above equation ΔH = 50m + 4flv²/2gd +v²/2g
ΔH = 50m + (4×0.04 ×1000 ×(31.84 Q)²)/(2×9.81 ×0.2) + (31.84 Q)²)/(2 ×9.81)
ΔH = 50m + 41,336.8 + 51.69 Q2
ΔH = 50m + 41,388.49 Q2
By putting values of Q
If
Q = 0, ΔH = 50
Q = 0.0133 , ΔH = 57
Q= 0.0235 , ΔH = 72
Q= 0.029 , ΔH = 84
Q= 0.033 , ΔH = 95
At duty point the value of ΔH = 67m and value of ΔQ is 0.023 m3/s.
This is the Best Efficiency Point (BEP). The point at which highest efficiency occurs, at this point we can get low cost per meter cube of pumping
Task 04
Task 4.1
4.1.1 List different methods of measuring flow rates of open channels. [P 4.1]
Open-channel flow, a branch of hydraulics and fluid mechanics, is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow.
Open channels are used to conduct liquids in most sewer systems, sewage treatment plants, industrial waste applications, and irrigation systems.
There are many methods of determining the rate of flow in open channels. Some of the more common include
The timed gravimetric.
Dilution.
Velocity-area.
Hydraulic structures.
Slope-hydraulic radius-area methods.
Timed Gravimetric Method
The stream rate is ascertained by measuring the whole substance of the stream that was gathered in a compartment for an altered period of time. This is commonsense for little surges of under 25 to 30 gallons for each minutes (gpm) and is not appropriate for constant estimation.
Dilution Method: The stream rate is measured by deciding how much the streaming water weakens an included tracer arrangement.
Velocity-Area Method: Measuring the mean stream speed over a cross segment and increasing it by the zone by then to ascertain the stream rate. The region speed technique computes stream rate by increasing the region of the stream by its normal speed. This is regularly alluded to as the progression comparison
Q = A ×V
The flowmeter converts this level into the area of the flow based on the size and shape of the channel. The main advantage of the area velocity method is that it can be used to measure flow under a wide range of conditions.
• Open Channel
• Surcharged
• Full Pipe
• Submerged
• Reverse Flow
In addition, the area velocity method does not require the installation of a weir or flume.
Hydraulic Structure Method
This method uses a hydraulic structure placed in the flow stream of the channel to produce flow properties that are characterized by known relationships between the water level measurement at some location and the flow rate of the stream. Therefore, the flow rate is determined by taking a single measurement of the water surface level in or near the restriction of the hydraulic structure
Slope-hydraulic Radius-Area Method
Measurement of water surface slope, cross-sectional area, and wetted perimeter over a length of uniform section channel are used to calculate the flow rate, by using a resistant equation such as the Manning formula.
The Gravitational, Dilution, and the Velocity Area methods are more commonly used for calibration purposes. The Depth-Related methods (Hydraulic Structures) are the most common. The depth-related technique measures flow rate from a measurement of the water depth, or head. Weir and flumes are the oldest and most common devices used for measuring open channel flows.
Various resistance equations are used to estimate flow rate based on measurements of the water surface slope, cross-sectional area, and wetted perimeter over a length of uniform channel. The most popular of these equations is the Manning formula.
Q = K A R2/3 S1/2 n
Where:
Q = flow rate
A = cross sectional area of flow
R = hydraulic radius (cross sectional area divided by wetted perimeter)
S = slope of the hydraulic gradient
n = roughness coefficient based on channel material and condition
K = constant dependent upon units
4.1.2 Explain the method used in this experiment to measure the flow rates. [P 4.1]
In this experiment hydraulic structure technique is utilized it is the most widely recognized strategy for measuring open channel stream is the pressure driven structures system. An aligned confinement embedded into the channel controls the shape and speed of the stream. The stream rate is then controlled by measuring the fluid level in or close to the limitation.
The confining structures are called essential measuring gadgets. They might be partitioned into two general classes weirs and flumes.
A weir is a hindrance or dam constructed over an open channel over which the fluid streams, frequently through an exceptionally molded opening. Weirs are arranged by state of this opening. The most well-known sorts of weirs are the triangular (or V-indent) weir, the rectangular weir, and the trapezoidal (or Cipolletti) weir.
The stream rate over a weir is dictated by measuring the fluid profundity in the pool upstream from the weir. Weirs are straightforward and reasonable to manufacture and introduce. Regular materials of development incorporate metal, fiberglass and wood. In any case, they speak to a huge loss of head, and are not suitable for measuring streams with solids that might stick to the weir or collect upstream from it.
A flume is an exceptionally formed open channel stream segment giving a confinement in channel territory and/or an adjustment in channel incline. The stream rate in the channel is controlled by measuring the fluid profundity at a predetermined point in the flume.
The most well-known flume is the Parshall flume. The stream rate through a Parshall flume is dictated by measuring the fluid level 33% of the route into the uniting segment. Parshall flumes are assigned by the width of the throat, which goes from one inch to 50 feet. The throat width and every other measurement must be entirely taken after with the goal that standard release tables can be utilized. Likewise, take note of the drop in the floor of the flume, which makes it hard to introduce a Parshall flume in a current channel.
4.1.3 Calculate the flow rate coefficient of discharge [D 1.1]
The release coefficient is the proportion of the real stream to the hypothetical stream, expecting unit coefficients of compression and speed, equivalent to the result of these coefficients
Cd = (Q actual)/( Q theoretical)
To calculate the flow rate coefficient of discharge, suppose the values from the task 2.2 and considering the channel is rectangular.
Flow rate is = Q = 12 m3/s
Depth of flow = h = 2 m
The bed width of the channel is = 2.7m
Area of the channel = A = 5.4 m2
The above equation can be written as
Cd = Q/(A . √(2 . g .Δh ))
Where
A: Area
Cd: Discharge coefficient
g: Gravity 9.8 m/s²
Q: Flow rate
Δh : Head Drop (high of fluid)
By substituting the values, we can get coefficient of discharge.
Cd = 12/(5.4 ×√(2 × 9.81 × 2 ))
Cd = 0.354
4.1.4 Plot the flow profiles and identify the types of flow. [M 1.1]
There are mainly two types of flow.
Uniform Flow: Steady uniform flow. Conditions do not change with position in the stream or with time. An example is the flow of water in a pipe of constant diameter at constant velocity.
Non Uniform Flow: Non steady Flow. Conditions do changes with position in the stream or with time.
Flow profiles for Steady and Unsteady flow
Steady flow
A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ from point to point but do not change with time.
Unsteady flow
If at any point in the fluid, the conditions change with time, the flow is described as unsteady. (In practice there is always slight variations in velocity and pressure, but if the average values are constant, the flow is considered steady
Combining the above we can classify any flow in to one of four type:
Steady uniform flow. Conditions do not change with position in the stream or with time. An example is the flow of water in a pipe of constant diameter at constant velocity.
Steady non-uniform flow. Conditions change from point to point in the stream but do not change with time. An example is flow in a tapering pipe with constant velocity at the inlet - velocity will change as you move along the length of the pipe toward the exit.
Unsteady uniform flow. At a given instant in time the conditions at every point are the same, but will change with time. An example is a pipe of constant diameter connected to a pump pumping at a constant rate which is then switched off.
Unsteady non-uniform flow. Every condition of the flow may change from point to point and with time at every point. For example waves in a channel.
Flow Profiles of uniform flow, Rapidly Varied flow and gradually varied flow in open channels
Rapidly Varied flow (RVF) occurs over a short distance near the obstacle.
Gradually varied flow (GVF) occurs over larger distances and usually connects uniform flow and Rapidly Varied flow
Flow Profiles of critical, sub critical and super critical flow
The behavior of a flow over a rise and then a fall of channel bottom, when approaching sub critical flow is forced to the critical condition by raising the step by a significantly large increment.
Task 4.2 Clearly explain the procedure for conducting characteristic test on single stage Centrifugal pump to find out the optimum discharge and head. Plot all the required graphs. Explain how you will increase the operating head. [M1.1, D1.1]
Centrifugal pumps are the most commonly used kinetic-energy pump. Centrifugal force pushes the liquid outward from the eye of the impeller where it enters the casing. Differential head can be increased by turning the impeller faster, using a larger impeller, or by increasing the number of impellers. The impeller and the fluid being pumped are isolated from the outside by packing or mechanical seals. Shaft radial and thrust bearings restrict the movement of the shaft and reduce the friction of rotation. Single stage centrifugal pumps are the most common pump for fluid transfer in high flow rate, low pressure installations. If a lower flow rate or a higher pressure is needed over what a single stage centrifugal pump can provide. In this type, one impeller keyed to the shaft. It is usually low lift pump.
The aim for conducting characteristic test on single stage Centrifugal pump to find out the optimum discharge, head, efficiency and performance characteristics.
Procedure for conducting characteristic test on single stage Centrifugal pump
The setting of test data is done semi-automatically and requires appropriate test experience and accuracy. For conducting characteristic test on single stage Centrifugal pump only pure cold water with a density of ρ= 1 kg/dm3 is used as the test medium. Pump block units are normally tested with installed motor.
Priming of the radiating pump is completed utilizing a twofold acting responding pump which fills water in diffusive pump taking water from repository.
After the preparing is finished, the force is supplied to the radiating pump and the data force is measured as far as KWh at general interims of time.
Also suction and the conveyance head are measured with the assistance of gages mounted on the suction and conveyance channels itself. The previous is gotten as far as mm of mercury while the last is acquired as far as meter of water segment.
The release is measured as far as the stature of water segment over the V-score. The stature is measured in a square segment in which snare gage is embedded to watch the level ascent. As the sump and square segment are joined the level demonstrated are same as in sump. Perceptions of every single above estimation are noted in perception table.
The release, information power, yield power and effectiveness are figured by equations and taking into account this outcome table is made.
Graphs are plotted for - release v/s yield power, release v/s effectiveness and release v/s all out head and release v/s Compact disc.To obtain the test data for main characteristics, the pump is run at a constant speed and the discharge is varied over the desired range. Measurements are taken for suction head, delivery head, shaft power and height above V-notch for each discharge and calculations are made for the pump efficiency. Curves are then plotted for discharge V/s head, power and efficiency.
Measurements of optimum Discharge (Q)
The optimum Discharge is measured using magnetic inductive flow measuring devices having the specified stabilization distances in front and after the measuring unit. The setting of the capacity is effected by control valves in the discharge pipe.
Q=volume of liquid flowing per second = Area x velocity of flow Q = πD2B2Vf2
Where
D2=Diameter of the impeller at outlet
B2is the width of the impeller at the outlet
Vf2 are the velocities of flow at outlet
Heads on a centrifugal pump:
Suction head (hs): it is the vertical distance between the liquid level in the sump and the centre line of the pump. It is expressed as meters.
Delivery head (hd): It is the vertical distance between the centre line of the pump and the liquid level in the overhead tank or the supply point. It is expressed in meters.
Static head (Hs): It is the vertical difference between the liquid levels. In the overhead tank and the sump, when the pump is not working. It is expressed as meters.
Therefore, HS= (hs+ hd)
Friction head (hf): It is the sum of the head loss due to the friction in the suction and delivery pipes. The friction loss in both the pipes is calculated using the Darcy’s equation,
hf=(fLV2/2gD).
Total head (H): It is the sum of the static head Hs, friction head (hf) and the velocity head in the delivery pipe (Vd 2/2g). Where, Vd =velocity in the delivery pipe.
Hm = hs+ hd + hf+ Vd²/2g
Manometric head (Hm): It is the total head developed by the pump. This head is slightly less than the head generated by the impeller due to some losses in the pump
Hm = H +Vs²/2g -Vd²/2g
Figure 1: Characteristic Curves for a Single Stage Centrifugal Pump
Figure 2: Head versus capacity characteristics curve for a single-stage pump
Figure 2 shows a typical head, H (ft), versus capacity (discharge), Q (gpm), curve for a single-stage pump. This curve relates head produced by a pump to the volume of water pumped per unit time. Generally, the head produced decreases as the amount of water pumped increases. The shape of the curve varies with pump’s specific speed and impeller design. Usually, the highest head is produced at zero discharge and it is called the shut-off head.
How to Increase Operating Head
To increase the operating head connect an additional pump in series with a main pump may be used in time of larger flow demands.
Figure 3: Two Pumps Connected in Series
To associate two pumps in arrangement implies that the release from the first pump is funneled into the gulf side of the second pump. In this kind of game plan all the stream progressively goes starting with one pump then onto the next with every pump adding more vitality to the water.
This is a common game plan in multi-stage turbine or submersible pump where the same release goes through all stages and every forms extra head. Frequently, arrangement setups are utilized when head prerequisites of the framework surpass what can be supplied by individual pumps. They are additionally utilized as a part of frameworks with variable head necessities. A little radial pump utilized as a sponsor pump for corner watering system on an inside turn framework or, so far as that is concerned, any supporter pump, in any water framework, which works notwithstanding the primary water pump. Figure 4 demonstrates head-release bends for two pumps working in arrangement.
Figure4: Head versus Discharge characteristics curves for pumps operating in Series.
Task 4.3
4.3.1 Explain the method used to measure the flow rates in pipes [P 4.1]
Introduction
There are a wide variety of methods for measuring discharge and velocity in pipes, or closed conduits. Many of these methods can provide very accurate measurements others give only rough estimates. But, in general, it is easier to obtain a given measurement accuracy in pipes when compared to measurement in open channels. Some of the devices used are very expensive and are more suited to industrial and municipal systems than for agricultural irrigation systems.
Pitot Tubes
The Pitot tube can be used not only for measuring flow velocity in open channels (such as canals and rivers), but in closed conduits as well. There are several variations of pitot tubes for measuring flow velocity, and many of these are commercially available. Pitot tubes can be very simple devices with no moving parts. More sophisticated versions can provide greater accuracy (e.g. differential head meters that separate the static pressure head from the velocity head). The pitot static tube in one variation of the device which allows the static head (P/γ) and dynamic (total) head (P/γ + V2/2g) to be separately measured.
The static head equals the depth if open-channel flow. Calibrations are required because the velocity profile can change with the flow rate, and because measurement(s) are only a sampling of the velocities in the pipe. The measurement from a pitot tube can be accurate to ±1% of the true velocity, even if the submerged end of the tube is up to ±15% out of alignment from the flow direction. The velocity reading from a pitot tube must be multiplied by cross-sectional area to obtain the flow rate (it is a velocity-area method). Pitot tubes tend to become clogged unless the water in the pipe is very clean. Also, pitot tubes may be impractical if there is a large head, unless a manometer is used with a dense liquid like mercury.
Pitot tube
Venturi Meters
Venturi meters have only a small head loss, no moving parts, and do not clog easily. The principle under which these devices operate is that some pressure head is converted to velocity head when the cross-sectional area of flow decreases (Bernoulli equation). Thus, the head differential can be measured between the upstream section and the throat section to give an estimation of flow velocity, and this can be multiplied by flow area to arrive at a discharge value. The converging section is usually about 21º, and the diverging section is usually from 5 to 7º.
A form of the calibration equation is:
where C is a dimensionless coefficient from approximately 0.935 (small throat velocity and diameter) to 0.988 (large throat velocity and diameter); β is the ratio of D2/D1; D1 and D2 are the inside diameters at the upstream and throat sections, respectively; A2 is the area of the throat section; ∆h is the head differential; and “sg” is the specific gravity of the manometer liquid.
The discharge coefficient, C, is a constant value for given venturi dimensions. Note that if D2 = D1, then β = 1, and Q is undefined; if D0 > D1, you get the square root of a negative number (but neither condition applies to a venturi). The coefficient, C, must be adjusted to accommodate variations in water temperature.
Flow Nozzles
Flow nozzles operate on the same principle as venturi meters, but the head loss tends to be much greater due to the absence of a downstream diverging section. There is an upstream converging section, like a venturi, but there is no downstream diverging section to reduce energy loss.
Flow nozzles can be less expensive than venturi meters, and can provide comparable accuracy. The same equation as for venturi meters is used for flow nozzles. The head differential across the nozzle can be measured using a manometer or some kind of differential pressure gauge. The upstream tap should be within ½D1 to D1 upstream of the entrance to the nozzle. The downstream tap should be approximately at the outlet of the nozzle.
Orifice Meters
These devices use a thin plate with an orifice, smaller than the pipe ID, to create a pressure differential. The pressure differential can be measured, as in venturi and nozzle meters, and the same equation as for venturi meters can be used however, the discharge coefficient is different for orifice meters. It is easy to make and install an orifice meter in a pipeline easier than a nozzle.
Orifice meters can give accurate measurements of Q, and they are simple and inexpensive to build. But, orifice meters cause a higher head loss than either the venturi or flow nozzle meters. As with venturi meters and flow nozzles, orifice meters can provide values within ±1% (or better) of the true discharge. As with venturi meters, there should be a straight section of pipe no less than 10 diameters upstream.
Some engineers have used eccentric orifices to allow passage of sediments – the orifice is located at the bottom of a horizontal pipe, not in the centre of the pipe cross section. The orifice opening can be “sharp” (bevelled) for better accuracy. But don’t use a bevelled orifice opening if you are going to use it to measure flow in both directions.
An Orifice Meter in a Pipe
Elbow Meters
An elbow in a pipe can be used as a flow measuring device much in the same way as a venturi or orifice plate. The head differential across the elbow (from inside to outside) is measured, and according to a calibration the discharge can be estimated. The taps are usually located in the center of the elbow (e.g. at a 45° angle for a 90° elbow), but can be at other locations toward the upstream side of the elbow. Some companies manufacture elbow meters for flow measurement, but almost any pipe elbow can be calibrated.
Elbow meters are not as potentially accurate as venturi, nozzle, and orifice meters. Typical accuracy is about ±4% of Q. One advantage of elbow meters is that there need not be any additional head loss in the piping system as a result of flow measurement
Variable Area Meters
These are vertical cylinders with a uniformly expanding cross-section in the upward direction. A float inside the cylinder stabilizes at a certain elevation depending on the flow rate through the cylinder. Note that the outside walls are usually transparent to allow direct readings by eye.
Horizontal Trajectory Method
From physics, an accelerating object will travel a distance x in time t according to the following equation (based on Newton’s 2nd law):
x = vo t + at2
2
Where x is the distance; vo is the initial velocity at time 0; t is the elapsed time; and a is the acceleration
Flow emanating from a horizontal pipe will fall a height y over a distance x. The horizontal component (x-direction) has almost no acceleration, and the vertical component (y-direction) has an initial velocity of zero. The vertical acceleration is equal to the ratio of weight to mass, or g = 9.81 m/s2 (32.2 ft/s2). Therefore,
x = vo t and, y = gt2
2
Then by getting rid of t, knowing that Q = VA, and the equation for the area of a circle, the flow rate is calculated as follows:
Q = πD2x
4 2y
g
Where D is the inside diameter of the circular pipe
California Pipe Method
This is the horizontal pipe method for partially-full pipes. It is somewhat analogous to the calibration for a weir or free over fall.
The following equation is in English units:
Where a and D are defined in the figure below (ft); and Q is discharge in cfs
Vertical Trajectory Method
As with pipes discharging horizontally into the air, there is a method to measure the flow rate from vertical pipes. This is accomplished by assuming a translation of velocity head into the measurable height of a column of water above the top of the pipe. Thus, to estimate the flow rate from pipes discharging vertically into the air it is only necessary to measure the:
Inside diameter of the pipe, D; and,
the height of the jet, H, above the pipe
This is a nice idea on “paper,” but in practice, it can be difficult to measure the height of the column of water because of sloshing, surging, and splashing. Also, the act of measuring the height of the column can significantly alter the measured value.
Vortex Shedding Meters
The vortex shedding meter can be accurate to within ±½% to ±1% of the true discharge. The basic principal is that an object placed in the flow will cause turbulence and vortices in the downstream direction, and the rate of fluctuation of the vortices can be measured by detecting pressure variations just downstream.
This rate increases with increasing velocity, and it can be used to give an estimate of the discharge. This requires calibration for a particular pipe material, pipe size, element shape and size, fluid type, and temperature. It is essentially a velocity-area flow measurement method, but it is calibrated to give discharge directly
Ultrasonic Meters
1. Doppler
An emitted pressure wave reflects off a deflector plate. Difference between transmitted and reflected frequencies correlates to flow velocity. Liquid does not have to be clean – in fact, it may not work well if the liquid is “too clean” because it needs particles to reflect the signal
2. Transit-time
Also called “time-of-flight”. The liquid should be fairly clean with this method. Devices generates high-frequency (≈1 MHz) pressure wave(s). Time to reach an opposing wall (inside the pipe) depends on:
Flow velocity
Beam orientation (angle)
Speed of sound through the liquid medium
Upstream straightening vanes may be needed to avoid swirling flow.
May have a single or multiple transmitted sound beams
Some Other Measurement Devices
Collins meters.
Commercial propeller flow meters.
Electromagnetic flow meters.
Volumetric tank
4.3.2 Explain the principle used to measure the pressure across the valves and pipes.
When a fluid is moving in a closed channel such as a pipe and valves operating principle to measure the pressure are based on equations developed by Daniel Bernoulli, a late 18th century Swiss Scientist. His experiments related to the pressure and velocity of flowing water. He determined that at any point in a closed pipe there were three types of head pressure present:
1. Static Head Pressure due to elevation.
2. Static Head Pressure due to applied pressure.
3. Velocity Head Pressure.
The Bernoulli Principle states that the sum of the kinetic, potential, and flow energies (all per unit mass) of a fluid particle is constant along a streamline during steady flow.
According to Bernoulli’s Principle along a stream line
The value of the constant in above equation can be evaluated at any point on the streamline where the pressure, density, velocity, and elevation are known. The Bernoulli equation can also be written between any two points on the same streamline as
Bernoulli equation for unsteady, compressible flow is
Static, Dynamic, and Stagnation Pressures
The Bernoulli equation states that the sum of the flow, kinetic, and potential energies of a fluid particle along a streamline is constant. Therefore, the kinetic and potential energies of the fluid can be converted to flow energy (and vice versa) during flow, causing the pressure to change. This phenomenon can be made more visible by multiplying the Bernoulli equation by the density ρ,
Each term in above equation has pressure units, and thus each term represents some kind of pressure:
• P is the static pressure (it does not incorporate any dynamic effects); it represents the actual thermodynamic pressure of the fluid. This is the same as the pressure used in thermodynamics and property tables.
• ρV2 /2 is the dynamic pressure; it represents the pressure rise when the m fluid in motion is brought to a stop isentropic ally.
ρgz is the hydrostatic pressure term, which is not pressure in a real sense since its value depends on the reference level selected; it accounts for the elevation effects, i.e., fluid weight on pressure. (Be careful of the sign unlike hydrostatic pressure ρgh which increases with fluid depth h, the hydrostatic pressure term ρgz decreases with fluid depth.)
Total Pressure
The sum of the static, dynamic, and hydrostatic pressures is called the total pressure. Therefore, the Bernoulli equation states that the total pressure along a streamline is constant.
Total Pressure = Static Pressure + Dynamic Pressure + hydrostatic pressures
Stagnation pressure
The sum of the static and dynamic pressures is called the stagnation pressure, and it is expressed as
The stagnation pressure represents the pressure at a point where the fluid is brought to a complete stop isentropically.
The static, dynamic, and stagnation pressures measured using piezometer tubes.
Differential pressure flowmeters use Bernoulli’s principle to measure the flow of fluid across the pipes and valves. Differential pressure flowmeters introduce a constriction in the pipe that creates a pressure drop across the flowmeter. When the flow increases, more pressure drop is created. Impulse piping routes the upstream and downstream pressures of the flowmeter to the transmitter that measures the differential pressure to determine the fluid flow.
4.3.3 Plot the required graphs (Head Vs Velocity or Discharge [P 4.2]
Head and Velocity graph for pipelines connected in series
4.3.4 Find local friction coefficient of the pipe and minor friction loss of the valves. [P4.2]
Friction coefficient of the pipe
Local friction coefficient of the pipe friction factor f, in general, f depends on the Reynolds Number R of the pipe flow, and the relative roughness e/D of the pipe wall,
The roughness measure e is the average size of the bumps on the pipe wall. The relative roughness e/D is therefore the size of the bumps compared to the diameter of the pipe
For laminar flow if Reynolds Number R is less than 2000, (R < 2000 in pipes), f can be deduced analytically by the equation
f= 64/Re
For the smooth pipe Turbulent flow if Reynolds Number R is greater than 4000, (R > 4000 in pipes), to calculate friction coefficient equation will be
f= 0.316/(Re¼)
Minor friction loss of the valves
Minor losses in pipes are caused by fittings, bends, valves, sudden enlargement and contraction, change in velocity at entrance and exit of pipe and due to obstruction in the flow.
Minor losses in comparison to friction losses which are considered major losses.
Minor losses for Valves & Fittings = h L = kv²/2g
Where the value of K depends upon the type of valve and degrees of opening
K is computed as = K = (Le / D) ft
Le = equivalent length (length of pipe with same resistance as the fitting/valve)
ft= friction factor
The loss coefficient K for the various pipe components is given in the table below.
The loss coefficient K for the various pipe components
4.3.5 Reason the possibilities of the variation among the trails in the experiment. [M 1.1]
There are many reasons which cause variations among trails in the experiment tends to change the experiment readings taken at different trails and alters the results at the end. It is impossible to make an exact measurement. Therefore, all experimental results are not 100 % accurate. Just how wrong they are depends on the kinds of errors that were made in the experiment.
All experimental data collected among the trails is imperfect. We cannot eliminate this but however, we can struggle to minimize errors.
Some Reason for the variation among the trails in the experiment
Random Errors during the experiment trails are unpredictable. They are chance variations in the measurements over which experimenter have little or no control. There is just as great a chance that the measurement is too big as that it is too small. Since the errors are equally likely to be high as low, averaging a sufficiently large number of results will, in principle, reduce their effect.
Systematic Errors are caused by the way in which the experiment was conducted. In other words, they are caused by the design of the system. Systematic errors cannot be eliminated by averaging in principle, they can always be eliminated by changing the way in which the experiment was done. In actual fact though, you may not even know that the error exists.
Human error is also something that is not an error at all, and that is human error. This kind of error that is caused by your eye's inability to read the exact level of liquid in a graduated cylinder, then that is a random error. If you mean the kind of error that is caused by a poor design of the experiment after all a human designed it then that is a systematic error. These two kinds of errors are the only errors you should ever have in your experimental results.
Human errors are really mistakes during the experiment trials. Spilling part of a solution, dropping part of a solid from the weighing paper, or doing a calculation wrong are blunders, not errors.
There are some other reasons for the variation
Experimental errors can occur due to poor operation and handling of apparatus and the apparatus used for experiment is not accurate.
Slight variations in the level of your eye while reading the meniscus in the graduated cylinder.
Vibration in the floor or air currents that cause fluctuation in the balance.
Errors that may occur in the execution of a statistical experiment.
Reading errors while taking readings and measurement errors.
A miss calibrated balance will cause all the measured masses to be wrong.
Density depends on temperature. The temperature was not specified or controlled.
4.3.6 How do you make sure the flows in the pipe are turbulent? [D 1.1]
Fluid flow in pipes
Fluid flow in circular and noncircular pipes is commonly encountered in practice. Fluid flow is classified as external and internal, depending on whether the fluid is forced to flow over a surface or in a conduit. When calculating heat transfer or pressure and head loss it is important to know if the fluid flow is laminar, transitional or turbulent
Laminar and turbulent flows
An attentive examination of stream in a channel reveals that the fluid stream is streamlined at low speeds however turns confounded as the velocity is extended over a fundamental worth, as showed up in The stream organization in the first case is said to be laminar, depicted by smooth streamlines and exceedingly asked for development, and turbulent in the second case, where it is depicted by rate changes and especially scattered development. The move from laminar to turbulent stream does not happen out of the blue; rather, it happens over some locale in which the stream changes amidst laminar and turbulent streams before it ends up being totally turbulent. Most streams experienced eventually are turbulent. Laminar stream is experienced when exceedingly thick fluids, for instance, oils stream in little pipes or restrict sections. To make sure the flows in the pipe is turbulent by injecting dye streaks into the flow.
It can be verified the existence of these laminar, transitional, and turbulent flow regimes by injecting some dye streaks into the flow in a glass pipe.
It can be observed that the dye streak forms a straight and smooth line at low velocities when the flow is laminar figure (a) (we may see some blurring because of molecular diffusion), has bursts of fluctuations in the transitional regime, and zigzags rapidly and randomly when the flow becomes fully turbulent figure (b). These zigzags and the dispersion of the dye are indicative of the fluctuations in the main flow and the rapid mixing of fluid particles from adjacent layers.
The intense mixing of the fluid in turbulent flow as a result of rapid fluctuations enhances momentum transfer between fluid particles, which increases the friction force on the surface and thus the required pumping power. The friction factor reaches a maximum when the flow becomes fully turbulent.
The Figure below is illustrating the behavior of colored fluid injected into the flow in laminar and turbulent flows in a pipe.
In turbulent flow vortices, eddies and wakes make the flow unpredictable. Turbulent flow happens in general at high flow rates and with larger pipes. Shear stress for turbulent flow is a function of the density.
Turbulent flow is characterized by random and rapid fluctuations of swirling regions of fluid, called eddies, throughout the flow. These fluctuations provide an additional mechanism for momentum and energy transfer. In laminar flow, fluid particles flow in an orderly manner along path lines, and momentum and energy are transferred across streamlines by molecular diffusion. In turbulent flow, the swirling eddies transport mass, momentum, and energy to other regions of flow much more rapidly than molecular diffusion, greatly enhancing mass, momentum, and heat transfer. As a result, turbulent flow is associated with much higher values of friction, heat transfer, and mass transfer coefficients.
Notwithstanding when the normal stream is relentless, the swirl movement in turbulent stream causes critical vacillations in the estimations of speed, temperature, weight, and even thickness (in compressible stream). Figure beneath demonstrates the variety of the momentary speed part u with time at a predefined area, as can be measured with a hot-wire anemometer test or other touchy gadget. We watch that the prompt estimations of the speed change around a normal quality, which recommends that the speed can be communicated as the entirety of a normal worth u– and a fluctuating part
Fluctuations of the velocity component u with time at a specified location in turbulent flow.
Reynolds Number
The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow velocity, surface temperature, and type of fluid, among other things.
After exhaustive experiments in the 1880s, Osborne Reynolds discovered that the flow regime depends mainly on the ratio of inertial forces to viscous forces in the fluid. This ratio is called the Reynolds number and is expressed for internal flow in a circular pipe as
The Reynolds number is important in analyzing any type of flow when there is substantial velocity gradient (i.e. shear.) It indicates the relative significance of the viscous effect compared to the inertia effect. The Reynolds number is proportional to inertial force divided by viscous force.
It certainly is desirable to have precise values of Reynolds numbers for laminar, transitional, and turbulent flows, but this is not the case in practice. It turns out that the transition from laminar to turbulent flow also depends on the degree of disturbance of the flow by surface roughness, pipe vibrations, and fluctuations in the flow. Under most practical conditions, the flow in a circular pipe is laminar for Re & 2300, turbulent for Re * 4000, and transitional in between. That is,
Re ≥ 4000 turbulent flow
2300 ≤ Re ≤ 4000 transitional flow and Re ≤ 2300 laminar flow
In transitional flow, the flow switches between laminar and turbulent randomly. It should be kept in mind that laminar flow can be maintained at much higher Reynolds numbers in very smooth pipes by avoiding flow disturbances and pipe vibrations. In such carefully controlled experiments, laminar flow has been maintained at Reynolds numbers of up to 100,000.
In the transitional flow region of 2300 ≤ Re ≤ 4000, the flow switches between laminar and turbulent randomly.
4.3.7 Discuss on the application of k values in finding the flow rates when pipes are connected and run in parallel.
When two pipes are connected and run in parallel to find the flow rates. In the figure below A and Bare described as nodes or junctions of pipes connected in parallel. In the steady state the known incoming flow at node A must balance with the outgoing flow in pipe 1 and 2. Similarly the incoming flow at node B in the incoming pipes 1 and 2 must equal the know withdrawal at node B.
QA = QB = Q1 + Q2 + Q3 equation (i)
The head loss in the pipe 1 and 2 must be same. Since all begin at a single node (A) and all end at single node (B) and the difference in head between those two nodes is unique, regardless of the pipe characteristics the head loss in the is the same or
HA - HB = hL1 = hL2 = hL3 = hL equation (ii)
Where HA and HB are the total heads at the nodes A and B respectively, hL1 is the head loss in pipe 1 and hL2 is the loss in pipe 2 and hL is the single value of head loss between nodes A and B.
Identify all the terms that make up hL,
Such as pipe major losses, hL= f (L/D) v2/2g
And for minor losses, hL = Kv2/2g for each branch.
If k values are given for each pipe connected in parallel then
For pipe 1 = hL1 = K1v12/2g and
For pipe 2 = hL2 = K2v22/2g)
For pipe 3 = hL3 = K3v22/2g)
And hL1 = hL2 = hL3
Now using above we can determine the head loss and flow for each pipe and an equivalent pipe coefficient, Kpeq
For steady flow conditions in the network we have a total two pipe flowsQ1, Q2 and Q3, and two head losses hL1, hL2 and hL3
Eq. (ii) provides independent equation relating the head losses (hL,1=hL,2 = hL3). The third equation is that the head loss in any pipe equals the difference in head between nodes A and B. Conservation of mass at node A (Eq. (i) the final equations are the head loss versus discharge equations will be
Equation (iii)
We can substitute above equation in the mass balance equations. (i) With hL equal to each pipe’s head loss or
Equation (IV)
In this equation, all terms except for hL are known. After solving for hL, the unknown pipe flows can be computed by equation (iii)
Like pipes in series, an equivalent pipe coefficient can be computed for parallel pipes. In Eq. (IV) hL can be pulled from each term on the left hand side or for a general discharge and two parallel pipes:
The equivalent coefficient k is then:
The head loss between the two end nodes in terms of flow rate and K is:
Reference:
http://www.instrumentationtoday.com/manometer/2011/09/
https://en.wikipedia.org/wiki/Pressure_measurement
http://www.calright.com/manometers
http://www.transcat.com/calibration-resources/application-notes/manometer-meriam-principles/
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A textbook of Fluid Mechanics and Hydraulic Machines -Dr. R. K. Bansal.
Fluid Mechanics for civil engineers by Bruce Hunt.
HYDRAULICS IN CIVIL AND ENVIRONMENTAL ENGINEERING By Andrew Chadwick and John Morfett.
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Flow Meters in Pipes by Gray P. Merkely.
BOOKLET 7 FLOW MEASUREMENT METHODS IN OPEN CHANNELS English Version of The Original Publishing.
Open Channel Flowmeters By Brian Dawson, Isco, Inc.
BERNOULLI AND ENERGY EQUATIONS/
CENTRIFUGAL PUMPS e-notes: Dr.N.Balasubramanya.
Energy and Momentum Principle Lecture No. # 02 Computation of Critical Depth by Prof. Dr. Arup Kumar Sarma
