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Fluid Mechanics

John F. Douglas, Janusz M. Gasiorek, John A. Swaffield

Chapter 14

Steady Incompressible Flow in Pipe and Duct Systems - all with Video Answers

Educators

Chapter Questions

17:03

Problem 1

Two vessels in which the difference of surface levels is maintained constant at $2.4 \mathrm{m}$ are connected by a $75 \mathrm{mm}$ diameter pipeline $15 \mathrm{m}$ long. If the frictional coefficient $f$ may be taken as 0.008 , determine the volume rate of flow through the pipe.

Ronald Prasad
Ronald Prasad
Numerade Educator
17:03

Problem 2

The difference in surface levels in two reservoirs connected by a siphon is $7.5 \mathrm{m}$. The diameter of the siphon is $300 \mathrm{mm}$ and its length $750 \mathrm{m}$. The friction coefficient $f$ is $0.0064 .$ If air is liberated from solution when the absolute pressure is less than $1.2 \mathrm{m}$ of water, what will be the maximum length of the inlet leg of the siphon to run full, if the highest point is $5.4 \mathrm{m}$ above the surface level in the upper reservoir? What will be the discharge?

Ronald Prasad
Ronald Prasad
Numerade Educator
08:59

Problem 3

Two reservoirs whose difference of level is $15 \mathrm{m}$ are connected by a pipe ABC whose highest point $\mathrm{B}$ is $2 \mathrm{m}$ below the level in the upper reservoir A. The portion AB has a diameter of $200 \mathrm{mm}$ and the portion $\mathrm{BC}$ a diameter of $150 \mathrm{mm}$, the friction coefficient being the same for both portions. The total length of the pipe is $3 \mathrm{km}$

Find the maximum allowable length of the portion $\mathrm{AB}$ if the pressure head at $\mathrm{B}$ is not to be more than $2 \mathrm{m}$ below atmospheric pressure. Neglect the secondary losses

Ronald Prasad
Ronald Prasad
Numerade Educator
12:36

Problem 4

A pipeline $30 \mathrm{m}$ long connects two tanks which have a difference of water level of $12 \mathrm{m}$. The first $10 \mathrm{m}$ of pipeline from the upper tank is of $40 \mathrm{mm}$ diameter and the next $20 \mathrm{m}$ is of $60 \mathrm{mm}$ diameter. At the change in section a valve is fitted. Calculate the rate of flow when the valve is fully opened assuming that its resistance is negligible and that $f$ for both pipes is $0.0054 .$ In order to restrict the flow the valve is then partially closed. If $k$ for the valve is now 5.6 find the percentage reduction in flow.

Ronald Prasad
Ronald Prasad
Numerade Educator
00:37

Problem 5

A smooth walled tube is used in a $3000 \mathrm{m}$ long pipeline carrying water at $15^{\circ} \mathrm{C}$ between two reservoirs whose surface elevations are $6 \mathrm{m}$ apart. Entry is sharp edged and the outlet is also abrupt to the downstream reservoir. The pipeline contains six $45^{\circ}$ bends and two globe valves. Determine the necessary pipe diameter so that the discharge should be 28 litres $^{-1}$ to the lower reservoir.
Take the equivalent length of each bend as 26.5 diameters the valves as 75 diameters and the entry as 30 diameters.

Ameer Said
Ameer Said
Numerade Educator
01:17

Problem 6

A horizontal duct system draws atmospheric air into a circular duct of $0.3 \mathrm{m}$ diameter, $20 \mathrm{m}$ long, then through a centrifugal fan and discharges it to atmosphere through a rectangular duct $0.25 \mathrm{m}$ by $0.20 \mathrm{m}, 50 \mathrm{m}$ long. Assuming that the friction factor for each duct is 0.01 and accounting for an inlet loss of one-half of the velocity head and also for the kinetic energy at outlet, find the total pressure rise across the fan to produce a flow of $0.5 \mathrm{m}^{3} \mathrm{s}^{-1}$
Sketch also the total energy and hydraulic gradient lines putting in the most important values. Assume the density of air to be $1.2 \mathrm{kg} \mathrm{m}^{-3}$. $\quad\left[695 \mathrm{Nm}^{-2}\right]$

Dominador Tan
Dominador Tan
Numerade Educator
01:40

Problem 7

For flow through pipes at high Reynolds number, the coefficient of friction is given by the following relation,
\[
\frac{1}{\sqrt{f}}-4 \log _{10}\left(\frac{r}{\varepsilon}\right)=3.48
\]
where $r=$ pipe radius and $\varepsilon=$ mean height of roughness projections. A pipe of internal diameter $0.15 \mathrm{m}$ is formed of a material for which $\varepsilon$ is 0.00038 m. The pipe is 1524 m long and it connects two water reservoirs whose surface levels are maintained at the same height. Water may be pumped along the pipe and the maximum pumping power available is $82 \mathrm{kW}$. Calculate the maximum rate of flow in the pipe.
\[
\left[0.06 \mathrm{m}^{3} \mathrm{s}^{-1}\right]
\]

Narayan Hari
Narayan Hari
Numerade Educator
01:37

Problem 8

A pipeline conveying water between reservoirs $\mathrm{A}$ and $\mathrm{B}$ is of $30.5 \mathrm{cm}$ diameter and $366 \mathrm{m}$ long. The difference of head between the two surfaces is $4.12 \mathrm{m}$. Determine the flow rate if $f=0.005$factor are unchanged and minor losses are ignored, find the length of the second pipe which is of the same diameter as the first.
\[
\left[0.134 \mathrm{m}^{3} \mathrm{s}^{-1} ; 270 \mathrm{m}\right]
\]
It is required to increase the flow by 50 per cent by duplicating a portion of the pipe. If the head and friction

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
08:18

Problem 9

There is a pressure loss of $300 \mathrm{kN} \mathrm{m}^{-2}$ when water is pumped through pipeline $A$ at a rate of $2 m^{3} s^{-1}$ and there is a pressure loss of $250 \mathrm{kN} \mathrm{m}^{-2}$ when water is pumped at a rate of $1.4 \mathrm{m}^{3} \mathrm{s}^{-1}$ through pipeline $\mathrm{B}$. Calculate the pressure loss which will occur when $1.5 \mathrm{m}^{3} \mathrm{s}^{-1}$ of water are pumped through pipes $A$ and $B$ jointly if they are connected $(a)$ in series
$(b)$ in parallel, assuming that junction losses may be neglected. In the latter case calculate the volume rate of flow through each pipe.
$\left[(a) 456 \mathrm{kN} \mathrm{m}^{-2},(b) 54.1 \mathrm{kN} \mathrm{m}^{-2}\right.$
\[
\left.0.849 \mathrm{m}^{3} \mathrm{s}^{-1}, 0.651 \mathrm{m}^{3} \mathrm{s}^{-1}\right]
\]

Ronald Prasad
Ronald Prasad
Numerade Educator
02:09

Problem 10

A complex ventilation system for a coal mine may be reduced to the system shown in Fig. $14.22,$ where $R_{1}, R_{2}$ and $R_{3}$ represent the equivalent resistances of the three main sections of the mine. Assuming an air density of $1.17 \mathrm{kg} \mathrm{m}^{-3}$ these resistances are:
Discharge $Q\left(\mathrm{m}^{3} \mathrm{s}^{-1}\right) \quad 0 \quad 100 \quad 150 \quad 200$
Fan total pressure
\[
\begin{array}{cllll}
\text { (mm of water) } & 175 & 180 & 175 & 160 \\
\text { Discharge } Q\left(\mathrm{m}^{3} \mathrm{s}^{-1}\right) & 250 & 300 & 350 &
\end{array}
\]
Fan total pressure
\[
\text { (mm of water) } 135 \quad 100 \quad 60
\]
(a) Determine the volume rate of flow handled by the fan and the fan total pressure.
(b) If, owing to the increased length of workings, the resistance of the whole system changes and is found to be $150 \mathrm{mm}$ of water total at $200 \mathrm{m}^{3} \mathrm{s}^{-1}$ and density $1.17 \mathrm{kg} \mathrm{m}^{-3},$ determine the percentage increase of fan speed required to maintain the same flow through the fan.

Penny Riley
Penny Riley
Numerade Educator
03:00

Problem 11

Water flows in the parallel pipe system shown in Fig. 14.23 for which the following data are available:
The supply pipe to point $A$ is of 0.30 m diameter and the mean velocity of water in it is $3 \mathrm{m} \mathrm{s}^{-1}$. If the elevation of point $A$ is $100 \mathrm{m}$ and the elevation of point $B$ is $30 \mathrm{m}$ above datum, calculate the pressure at point $B$ if that at $A$ is $200 \mathrm{kN} \mathrm{m}^{-2} .$ What is the discharge in each pipe? Neglect all minor losses.
\[
\left[559.7 \mathrm{kN} \mathrm{m}^{-2}, 0.024 \mathrm{m}^{3} \mathrm{s}^{-1}, 0.075 \mathrm{m}^{3} \mathrm{s}^{-1}, 0.114 \mathrm{m}^{3} \mathrm{s}^{-1}\right]
\]

Chai Santi
Chai Santi
Numerade Educator
03:20

Problem 12

Water is handled by a system of pipes as shown in Fig. $14.24,$ the details being as follows:
The elevation of outlets $D, E$ and $F$ is $100 \mathrm{m}$ above the elevation of inlets $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$. All outlets and inlets are open to atmosphere. If the mean velocity in the pipes $A_{1} B$ and $\mathrm{A}_{2} \mathrm{B}$ is $2.5 \mathrm{m} \mathrm{s}^{-1},$ calculate the flow rate through the pump $\mathrm{P}$ the pressure difference across the pump and the power consumed. Take the pump efficiency as 76 per cent.
\[
\left[0.98 \mathrm{m}^{3} \mathrm{s}^{-1}, 1538 \mathrm{kNm}^{-2}, 1983 \mathrm{kW}\right]
\]

Chai Santi
Chai Santi
Numerade Educator
05:57

Problem 13

A horizontal water main comprises $1500 \mathrm{m}$ of $150 \mathrm{mm}$ diameter pipe followed by $900 \mathrm{m}$ of $100 \mathrm{mm}$ diameter pipe the friction factor $f$ for each pipe being 0.007 . All the water is drawn off at a uniform rate per unit length along the pipe. If the total input to the system is $25 \mathrm{dm}^{3} \mathrm{s}^{-1}$, find the total pressure drop along the main, neglecting all losses other than pipe friction. Also draw the hydraulic gradient taking the pressure head at inlet as $54 \mathrm{m}$
\[
[20.50 \mathrm{m}]
\]

Ronald Prasad
Ronald Prasad
Numerade Educator
05:57

Problem 14

A 675 mm water main runs horizontally for $1500 \mathrm{m}$ and then branches into two $450 \mathrm{mm}$ mains each $3000 \mathrm{m}$ long. In one of these branches the whole of the water entering is drawn off at a uniform rate along the length of the pipe. In the other branch one-half of the quantity entering is drawn off at a uniform rate along the length of the pipe. If $f=0.006$ throughout, calculate the total difference of head between inlet and outlet when the inflow to the system is $0.28 \mathrm{m}^{3} \mathrm{s}^{-1}$. Consider only frictional losses and assume atmospheric pressure at the end of each branch.

Ronald Prasad
Ronald Prasad
Numerade Educator
01:37

Problem 15

The head loss for flow in a duct can be written as $h$ $=r Q^{n},$ where $r$ is the pipe resistance and $Q$ is the volume rate of flow. The fuel gallery for a small gas turbine is shown in Fig. $14.25 .$ Each injection nozzle passes 5 litres of kerosene per minute.
The relationships between the pipe resistances are as follows:

Narayan Hari
Narayan Hari
Numerade Educator
01:43

Problem 16

The head loss for flow in a duct can be written as $h_{f}$ $=r Q^{n},$ where $r$ is the pipe resistance and $Q$ is the volume rate of flow. The fuel gallery for a small gas turbine is shown in Fig. $14.25 .$ Each injection nozzle passes 5 litres of kerosene per minute.
The relationships between the pipe resistances are as follows:
If the pipe $\mathrm{OC}$ is $2 \mathrm{m}$ long, $0.01 \mathrm{m}$ in diameter and the friction factor $f$ is 0.010 in the formula
\[
h_{f}=\frac{4 f L}{d} \frac{v^{2}}{2 g}
\]
find the pressure drop between $\mathrm{O}$ and $\mathrm{C}$. $\quad\left[3360 \mathrm{N} \mathrm{m}^{-2}\right]$

Manik Pulyani
Manik Pulyani
Numerade Educator
02:45

Problem 17

A vertical cylindrical tank, $0.4 \mathrm{m}$ in diameter and $3 \mathrm{m}$ high, is used as part of a flow calibration diverter unit. If the water collected in the tank, up to a depth of $2.5 \mathrm{m}$, is discharged through an orifice and valve in the tank base, which may be represented by a $50 \mathrm{mm}$ diameter orifice of discharge coefficient 0.6, calculate the time to empty half the collected volume and express as a percentage of the time to empty fully.
\[
[22.3 \mathrm{s}, 29.3 \mathrm{per} \mathrm{cent}]
\]

James Kiss
James Kiss
Numerade Educator
01:20

Problem 18

A vertical axis tank is conical in shape, the diameter increasing uniformly from $1 \mathrm{m}$ at the base to $1.75 \mathrm{m}$ diameter at a height of $3 \mathrm{m}$. The tank is to be emptied by means of a $50 \mathrm{mm}$ orifice in the base having a discharge coefficient of $0.6 .$ Calculate the time to reduce the water level from $2 \mathrm{m}$ to $1 \mathrm{m}$ above the base.
\[
[233.8 \mathrm{s}]
\]

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
05:09

Problem 19

For the case set out in Problem 14.18 above, calculate the inflow necessary to hold the liquid level at $1.5 \mathrm{m}$ above the base.
\[
\left[383 \text { litres } \min ^{-1}\right]
\]

Chai Santi
Chai Santi
Numerade Educator
11:27

Problem 20

For the tank in Problem 14.18 , calculate the time to discharge the full contents of the tank if the discharge is carried away by a $25 \mathrm{mm}$ diameter pipe, length $4 \mathrm{m}$, friction factor $0.005 .$ Assume that the effect of the orifice to pipe connection can be represented by a separation loss having a $k$ value of $2,$ and that final discharge is at tank base level.

Ronald Prasad
Ronald Prasad
Numerade Educator
03:46

Problem 21

A rectangular cross-section tank, $2 \mathrm{m} \times 3 \mathrm{m}$, is filled with water up to a depth of $2 \mathrm{m}$. Calculate the time to reduce the volume in the tank by 50 per cent if the discharge is via a $40 \mathrm{mm}$ diameter pipe, $6 \mathrm{m}$ long, for which a friction factor of 0.005 may be assumed and the separation losses may be represented by a $k$ value of $0.9 .$ Assume final discharge $2 \mathrm{m}$ below tank base level.
\[
[1276 \mathrm{s}]
\]

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:56

Problem 22

A $1.2 \mathrm{m}$ deep rectangular tank is $2 \mathrm{m} \times 1 \mathrm{m}$ in area and has a vee notch in one side. The lowest point of the vee notch is $770 \mathrm{mm}$ above the base of the tank. A water supply to the tank of 1036 litres $\min ^{-1}$ establishes a steady depth of $1 \mathrm{m}$ above the base of the tank. If the water inflow ceases, calculate the time needed for the level to fall to $150 \mathrm{mm}$

Vipender Yadav
Vipender Yadav
Numerade Educator
01:20

Problem 23

A cylindrical tank is $1.8 \mathrm{m}$ in diameter and $3 \mathrm{m}$ long and is mounted horizontally. Oil of specific gravity 0.87 stored in the tank is drawn off through an orifice, $20 \mathrm{mm}$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
06:34

Problem 24

$\mathrm{A} 2 \mathrm{m}$ deep tank is $2 \mathrm{m} \times 3 \mathrm{m}$ in area and is divided into two equal halves by a vertical separation plate. Flow from one tank to the other takes place through a square orifice, $1 \mathrm{cm}$ side, having a discharge coefficient of $0.6 .$ If water is initially at $1.5 \mathrm{m}$ depth on one side of the plate and $0.5 \mathrm{m}$ depth on the other, calculate the time taken for the depths in both tanks to be equal. $[11290 \mathrm{s}]$

Ronald Prasad
Ronald Prasad
Numerade Educator
05:17

Problem 25

A rectangular cross-section tank of $12 \mathrm{m}^{2}$ surface area is filled to a depth of $2.5 \mathrm{m}$ with water. Calculate the time to reduce the depth in the tank by $2.0 \mathrm{m}$ if it is drained through a $3 \mathrm{m}$ long, $0.04 \mathrm{m}$ diameter pipe discharging $2 \mathrm{m}$ below the base of the tank and having a friction factor of 0.01 and separation losses due to bends etc. of $0.9 .$ Compare the integral for discharge time with a range of $\Delta h$ incremental values to demonstrate the accuracy of a finite difference approach. $[5144.59 \text { seconds }]$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:12

Problem 26

For the tank and discharge pipe described in Problem $14.25,$ with an initial tank depth of $4.5 \mathrm{m}$, determine the time taken for the difference in depth in the two tanks to fall from $6.0 \mathrm{m}$ to $5.0 \mathrm{m}$ depth if the discharge pipe is connected, at its previous discharge level of $2 \mathrm{m}$ below the base of the first tank, into a second tank, surface area $4.5 \mathrm{m}^{2}$. The initial depth in the second tank is $0.5 \mathrm{m}$ above the discharge pipe entry and the entry loss coefficient may be assumed to be 1.0

Ronald Prasad
Ronald Prasad
Numerade Educator