Internal Flow
Reynolds Number
Learning Objectives:

Calculate Reynold’s Number and understand its significance.

Define a flow as laminar, turbulent or transition based on Reynolds Number.
It is the ratio of inertia force to the viscous force. i.e.
\[Re =\dfrac{vD\rho }{\mu } = \dfrac{vD}{\nu }\]
Re = Reynold’s Number,
D= Diameter of pipe,
v= Mean velocity
\(\mu\) = Dynamic viscosity,
\(\nu\) =Kinematics viscosity of flow,
\(\rho\) = Fluid density
It has been found that for flow in circular pipes the critical Reynold’s number is about 2000. At this point the laminar flow changes to turbulent flow. The transition from laminar to turbulent does not exactly happen at Re=2000 but varies from one experiments to another experiments due to experimental condition.
Solved Example: 63101
For pipes, laminar flow occurs when Reynold's number is: (HPCL Asst Maintenance Mech 2019)
A. Less than 2000
B. Between 2000 and 4000
C. More than 4000
D. Less than 4000
Laminar flow generally occurs when the Reynolds number is less than 2000. Turbulent flow occurs at an Re above 4000 and transitional flow occurs between 2000 and 4000.
Correct Answer: A
Solved Example: 63102
A large Reynold's number is indication of: (ISRO IPRC Tech Asst Mech Aug 2016)
A. Smooth and streamline flow
B. Laminar flow
C. Steady flow
D. Highly turbulent flow
Reynold's number indicate whether the flow is laminar or turbulent. It is the ratio of inertial forces to viscous forces. At higher Reynold's number, inertial forces are dominant as compared to viscous forces, which indicate the flow is highly turbulent.
Correct Answer: D
Solved Example: 63103
For pipes, turbulent flow occurs when Reynolds number is:
A. Less than 2000
B. Between 2000 and 4000
C. More than 4000
D. Less than 4000
Correct Answer: C
Solved Example: 63104
Reynold's number is significant in:
A. Supersonics, as with projectile and jet propulsion
B. Full immersion or completely enclosed flow, as with pipes, aircraft wings, nozzles etc.
C. Simultaneous motion through two fluids where there is a surface of discontinuity, gravity forces, and wave making effect, as with ship's hulls
D. All of the above
Correct Answer: B
Solved Example: 63105
The velocity at which the laminar flow stops is known as: (ISRO Scientist ME 2015)
A. Velocity of approach
B. Lower critical velocity
C. Higher critical velocity
D. None of the above
Correct Answer: B
Solved Example: 63106
Water enters a circular pipe of length L = 5.0 m and diameter D = 0.20 m with Reynolds number $Re_D$ = 500. The velocity profile at the inlet of the pipe is uniform while it is parabolic at the exit. The Reynolds number at the exit of the pipe is: (GATE ME 2019)
A. 400
B. 500
C. 600
D. 2000
Correct Answer: B
Losses in Pipes
Learning Objectives:

Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow

Understand various velocity and flow rate and flow rate measurement techniques and learn their advantages and disadvantages

Explain the Darcy–Weisbach equation.

Explain the various types of joints used in fluid power.

Evaluate the head losses for laminar and turbulent flow.

Calculate the major and minor losses associated with pipe flow in piping networks

Determine flow in piping networks and determine the pumping power requirements

Explain the various types of losses in fittings and valves.
Types of Losses:

Major Loss:
Loss of head due to friction:\[h_{f} = \frac{flv^{2}}{2gD}\]

Minor Loss: The loss of energy due to change of velocity of the flowing fluid in magnitude or direction is called minor loss of energy. The minor loss of energy includes the following cases:

Loss of head due to Contraction: The head loss due to sudden contraction equation is \[h_c =k \left(\dfrac{{V_2}^2}{2g}\right)\] Where k = \((\dfrac{1}{C_c}1)^2\) , \(V_2\) is the velocity at section 2

Loss of head due to enlargement The head loss due to sudden expansion equation is \[h_e = \dfrac{(V_1 V_2)^2}{2g}\] Where \(V_1\) is the velocity at section 1, and \(V_2\) is the velocity at section 2

Loss of heads at the entrance of pipe: \[h_L = K_e \left(\dfrac{V^2}{2g}\right)\] Where:
\(h_L\) = Head Loss
\(K_e\) = Head Loss Coefficient
V = Velocity in the barrel
g = Acceleration due to gravity 
Loss of head at the exit of a pipe
In the equation of loss due to sudden enlargement, \(A_2 \rightarrow \infty\), then the head loss at an abrupt enlargement tends to \(\dfrac{{V_1}^2}{2g}\). The physical resemblance of this situation is the submerged outlet of a pipe discharging into a large reservoir.
Since the fluid velocities are arrested in the large reservoir, the entire kinetic energy at the outlet of the pipe is dissipated into intermolecular energy of the reservoir through the creation of turbulent eddies.In such circumstances, the loss is usually termed as the exit loss for the pipe and equals to the velocity head at the discharge end of the pipe.

Loss of head due to bend in pipe The head loss due to bending equation is: \[h_b= k \left(\dfrac{V^2}{2g}\right)\]
Where V is the velocity of the flow. k is the coefficient of the bend ,which depends on the angle of the bend, radius of curvature of bend and diameter of the pipe.

Solved Example: 63201
Oil flows through a 200 mm diameter horizontal cast iron pipe (friction factor, f = 0.0225) of length 500 m. The volumetric flow rate is 0.2 $m^3/s$. The head loss (in m) due to friction is (assume g = 9.81 $m/s^2$) (GATE ME 2012)
A. 18.22
B. 232.36
C. 0.116
D. 116.18
First calculate the velocity. \[v = \dfrac{Q}{A} = \dfrac{0.2}{\dfrac{\pi}{4}(200 \times 10^{3})^2} = 6.36 \ m/s\] Now calculate the friction loss head, \[h_{f} = \dfrac{flv^{2}}{2gD} = \dfrac{0.0225 \times 500 \times 6.36^{2}}{2\times 9.81 \times 200 \times 10^{3}} = 116.18 \ m\]
Correct Answer: D
Solved Example: 63202
The head loss in a sudden expansion from 8 cm diameter pipe to 16 cm diameter pipe in terms of velocity V$_1$, in the smaller pipe is: (ESE Mechanical 2015)
A. $\dfrac{1}{4}\left( {\dfrac{{V_1^2}}{{2g}}} \right)$
B. $\dfrac{3}{{16}}\left( {\dfrac{{V_1^2}}{{2g}}} \right)$
C. $\dfrac{1}{{64}}\left( {\dfrac{{V_1^2}}{{2g}}} \right)$
D. $\dfrac{9}{{16}}\left( {\dfrac{{V_1^2}}{{2g}}} \right)$
Correct Answer: D
Solved Example: 63203
Loss of head at Exit of pipe is calculated by: (NPCIL ST ME Nov 2019, ShiftII)
A. $0.375 \dfrac{v^2}{2g}$
B. $0.5\dfrac{v^2}{2g}$
C. $0.25\dfrac{v^2}{2g}$
D. $\dfrac{v^2}{2g}$
Correct Answer: D
Solved Example: 63204
Head loss due to friction in a circular pipe of diameter D, under laminar flow, is inversely proportional to: (SSC JE CE Oct 2020 Morning)
A. $D^3$
B. $D^2$
C. $D^5$
D. $D^4$
Correct Answer: B
Solved Example: 63205
A head loss in a sudden expansion from 6 cm diameter pipe to 12 cm diameter pipe in terms of velocity v_{1} in the smaller diameter pipe is: (BPSC AE Paper 5 (Mechanical) 2019)
A. $\dfrac{3}{{16}}\dfrac{{v_1^2}}{{2g}}$
B. $\dfrac{5}{{16}}\dfrac{{v_1^2}}{{2g}}$
C. $\dfrac{7}{{16}}\dfrac{{v_1^2}}{{2g}}$
D. $\dfrac{9}{{16}}\dfrac{{v_1^2}}{{2g}}$
Correct Answer: D
Solved Example: 63206
Which of the following is not a minor energy loss? (TNTRB 2017 ME)
A. Loss due to sudden enlargement
B. Loss due to friction
C. Loss due to entrance of pipe
D. Loss due to bend in pipe
Correct Answer: B
Solved Example: 63207
Minor losses in a piping system are: (VIZAG MT Mechanical 2015)
A. Less than the friction losses $f\dfrac{l}{d^2}\dfrac{v^2}{g}$
B. Due to the viscous stresses
C. Assumed to vary linearly with the velocity
D. Found by using loss coefficients
Correct Answer: D
Solved Example: 63208
The loss of energy of the flowing fluid is due to: (NPCIL ST ME Nov 2019 ShiftII)
A. Sudden contraction
B. Sudden enlargement
C. Bends
D. All of the options
Correct Answer: D
Solved Example: 63209
A smooth pipe of diameter 200 mm carries water. The pressure in the pipe at section S1 (elevation: 10 m) is 50kPa. At Section S2 (elevation: 12 m) the pressure is 20 kPa and velocity is 2 ms^{1}. Density of water is 1000 kgm^{3} and acceleration due to gravity is 9.8 ms^{2}. Which of the following is TRUE? (ISRO Scientist ME 2013)
A. flow from S1 to S2 and head loss is 0.53 m
B. flow from S2 to S1 and head loss is 0.53 m
C. flow from S1 to S2 and head loss is 1.06 m
D. flow from S2 to S1 and head loss is 1.06 m
Correct Answer: C
Solved Example: 63210
Water is flowing through a horizontal pipe of diameter 200 mm at a velocity of 3 m/s. A circular solid plate of diameter 150 mm is placed in the pipe to obstruct the flow. What is the loss of head due to obstruction in the pipe if C_{c}= 0.62?
(Take g = 9.81 m/s^{2}) (ESE Mechanical 2021)
A. 3.27 m
B. 4.27 m
C. 5.27 m
D. 6.27 m
Correct Answer: A
Solved Example: 63211
Major energy losses occur due to: (JKSSB JE CE Shift II Oct 2021)
A. Bend in pipe
B. Pipe fittings
C. Expansion of pipes
D. Friction
Correct Answer: D
Solved Example: 63212
Water flows through a vertical contraction form a pipe of a diameter d to another of diameter of d/2. Exit velocity from bigger pipe to contraction is 2 m/s and pressure 200 kN/m$^2$. If height of contraction measures 2 m, the pressure at exit of contraction will be nearly to: (ESE Mechanical 2013)
A. 192 kN/m$^2$
B. 150 kN/m$^2$
C. 165 kN/m$^2$
D. 175 kN/m$^2$
Correct Answer: B
Pipes Connected in Series
Learning Objectives:

Calculate the flow and head characteristics for pipes connected in series.
Flow remains same in all pipes. \[Q = A_1v_1 = A_2v_2\] Total head loss of head= Loss of head in pipe 1 + Loss of head in pipe 2 + Loss of head in pipe 3. \[h_f = h_{f_1} + h_{f_2} + h_{f_3} + ....\]
Solved Example: 63301
Simplified equation of continuity is represented as: (Based on UPPCL JE CE Paper I2014)
A. $A_1v_1 = A_2v_2$
B. $A_1v_2 = A_2v_2$
C. $A_1v_1 = A_1v_2$
D. $A_2v_1 = A_1v_1$
Continuity equation in its simplified format says: The sum of all influx and outflux into and out of the system must sum up to zero.
Since mass = $\rho \times$ V = $\rho \times$ Area $\times$ velocity
If the $\rho$ remains constant, then it can be cancelled.
Hence, $A_1v_1 = A_2v_2$
Correct Answer: A
Pipes Connected in Parallel
Learning Objectives:

Calculate the flow and head characteristics for pipes connected in parallel.
Pressure across each pipe remains same.
Total discharge \[Q= Q_1+Q_2+Q_3\] Also, head loss across each pipe branch will be same.
\[h_{f_1} = h_{f_2} = h_{f_3} = ...\]
Solved Example: 63401
A main pipe divides into two parallel pipes which again forms one pipe. The length and diameter of the first parallel pipe are 2000 m and 1.0 m respectively. While the length and diameter of the second parallel pipe are 2000 m and 0.8 m.Find the rate of flow in each parallel pipe, if total flow in main is 3000 lit/sec. Take f= 0.005.
A. $Q_1$=1206 lit/sec, $Q_2$= 1764 Lit/sec
B. $Q_1$=1676 lit/sec, $Q_2$= 1012 Lit/sec
C. $Q_1$=1824 lit/sec, $Q_2$= 1554 Lit/sec
D. $Q_1$=1906 lit/sec, $Q_2$= 1094 Lit/sec
Correct Answer: D
Solved Example: 63402
Two pipe lines of equal length and diameters of 10 cm and 40 cm are connected in parallel between two reservoirs. If friction factor f is same for both the pipes, the ratio of the discharges in the larger to the smaller pipe is: (ISRO RAC 2019)
A. 4
B. 16
C. 32
D. 64
Correct Answer: C
Friction Factor
Learning Objectives:

Explain the concept of friction factor for fluid friction in pipe flow considering pipe roughness.

For laminar flow, \(f= \dfrac{64}{Re}\), which is independent of the relative roughness.

For very large Reynolds numbers, \(f=\phi (\dfrac{\epsilon}{D})\), which is independent of the Reynolds numbers.

For flows with very large value of Re, commonly termed completely turbulent flow (or wholly turbulent flow), the laminar sublayer is so thin (its thickness decrease with increasing Re) that the surface roughness completely dominates the character of the flow near the wall.

For flows with moderate value of Re, the friction factor \(f=\phi (Re, \dfrac{\epsilon}{D})\), and can be referred from Moody (Stanton) Diagram.

For many practical applications the Reynolds number is large enough so that the flow through the component is dominated by inertial effects, with viscous effects being of secondary importance.

In a flow that is dominated by inertia effects rather than viscous effects, it is usually found that pressure drops and head losses correlate directly with the dynamic pressure.

This is the reason why the friction factor for very large Reynolds number, fully developed pipe flow is independent of the Reynolds number.
Solved Example: 63501
For a fully developed flow of water in a pipe of dia. = 10 cm, V = 0.1 m/sec. Kinematic viscosity = $10^{–5} m^2/sec$. Find Darcy friction factor.
A. 0.064
B. 0.032
C. 0.164
D. 0.134
Correct Answer: A
Solved Example: 63502
Calculate the discharge through a pipe of diameter 200 mm when the difference of pressure head between the two ends of pipe 500 m apart is 4.0 m of water. Take f= 0.009.
A. 19.3 lit/sec
B. 32.1 lit/sec
C. 12.9 lit/sec
D. 29.3 lit/sec
First calculate the velocity from the friction loss head, \[h_{f} = \frac{flv^{2}}{2gD}\] \[4 = \frac{0.009 \times 500 \times v^{2}}{2\times 9.81 \times 200 \times 10^{3}}\] \[v = .... \ m/s\] Now, use the discharge formula, \[Q = A \times v = (\dfrac{\pi}{4}d^2) \times v = 29.3 \mathrm{lit/sec}\]
Correct Answer: D
Solved Example: 63503
Select the correct statement.
A. The absolute roughness of a pipe decreases with time.
B. A pipe becomes smooth after using for long time.
C. The friction factor decreases with time.
D. The absolute roughness increases with time.
Correct Answer: D
Solved Example: 63504
For fully developed laminar flow through a circular pipe with Reynolds number Re the friction factor is: (ESE Mechanical 2014)
A. Inversely proportional to Re
B. Proportional to Re
C. Proportional to square of Re
D. Independent of Re
Correct Answer: A
Solved Example: 63505
The head loss due to friction in a pipe of 1 m diameter and 1.5 km long when water is flowing with a velocity of 1 m/s is: (Darcy's friction factor f= 0.02 and acceleration due to gravity g = 10 m/s$^2$) (ISRO VSSC Tech Asst Mechanical Jun 2019)
A. 1.5 m
B. 0.5 m
C. 1.0 m
D. 2.0 m
Correct Answer: A
Solved Example: 63506
The Moody's chart is a logarithmic chart plotted against Darcy Weisbach friction factor and which one of the following parameters? (JKSSB JE CE 2015)
A. Density of fluids
B. Reynolds number
C. Viscosity of the fluid
D. Slope of the inclination of the fluid
Correct Answer: B
Solved Example: 63507
Friction factor for fluid flow in pipe does not depend upon the: (EKT Computer Science Paper II Feb 2015)
A. pipe length
B. pipe roughness
C. fluid density and viscosity
D. mass flow rate of fluid
Correct Answer: A
Noncircular Ducts
Learning Objectives:

Hydraulic diameter (DH) for noncircular ducts.

Study flow through noncircular ducts.
The empirical correlations for pipe flow may be used for computations involving noncircular ducts, provided their cross sections are not too exaggerated. The correlation for turbulent pipe flow are extended for use with noncircular geometries by introducing the hydraulic diameter, defined as: For a circular duct, \[D_h = \dfrac{4A}{P} = D\] For a rectangular duct of width b and height h, \[D_h = \dfrac{4A}{P} = \dfrac{4bh}{2(b+h)}\]
Solved Example: 63601
The hydraulic diameter, $D_H$ of a rectangular duct with sides a and b is: (ISRO Scientist ME 2016)
A. $D_H = \dfrac{4ab}{2a+b}$
B. $D_H = \dfrac{2ab}{a+b}$
C. $D_H = \dfrac{2ab}{2a+b}$
D. $D_H = \dfrac{4ab}{a+b}$
\begin{align*} D_H &= \dfrac{4A}{P}\\ &= \dfrac{4 \times ab}{2(a+b)}\\ &= \dfrac{2ab}{(a+b)} \end{align*}
Correct Answer: B