Turbines and Pumps

Question 1
A vertical shaft Francis turbine rotates at 300 rpm. The available head at the inlet to the turbine is 200 m. The tip speed of the rotor is 40 m/s. Water leaves the runner of the turbine without whirl. Velocity at the exit of the draft tube is 3.5 m/s. The head losses in different components of the turbine are: (i) stator and guide vanes: 5.0 m, (ii) rotor: 10 m, and (iii) draft tube: 2 m. Flow rate through the turbine is 20 m^3/s. Take g=9.8 m/s^2. The hydraulic efficiency of the turbine is ________% (round off to one decimal place).
A
91.2
B
88.4
C
122.4
D
68.2
GATE ME 2021 SET-2   Fluid Mechanics
Question 1 Explanation: 
\begin{aligned} H_{\text {inlet }} &=H_{\text {turbine }}+H_{L, \text { stator }+\text { Guide }}+H_{\text {rotor }}+H_{\text {draftuen }}+\frac{V_{3}^{2}}{2 g} \\ 200 &=H_{\text {turbine }}+5+10+2+\frac{3.5^{2}}{2 \times 9.8} \\ H_{\text {turbine }} &=182.375 \mathrm{~m} \\ \eta_{H} &=\frac{H_{\text {turbine }}}{H_{\text {inlet }}}=\frac{182.375}{200}=0.9118=91.187 \% \end{aligned}
Question 2
An object is moving with a Mach number of 0.6 in an ideal gas environment, which is at a temperature of 350 K. The gas constant is 320 J/kg.K and ratio of specific heats is 1.3. The speed of object is _______m/s (round off to the nearest integer).
A
147
B
187
C
229
D
312
GATE ME 2021 SET-2   Fluid Mechanics
Question 2 Explanation: 
\begin{aligned} M &=\frac{V}{C}=\frac{V}{\sqrt{\gamma R T}} \\ 0.6 &=\frac{V}{\sqrt{1.3 \times 350 \times 320}} \\ V &=228.94 \mathrm{~m} / \mathrm{s} \simeq 229 \mathrm{~m} / \mathrm{s} \end{aligned}
Question 3
A single jet Pelton wheel operates at 300 rpm. The mean diameter of the wheel is 2 m. Operating head and dimensions of jet are such that water comes out of the jet with a velocity of 40 m/s and flow rate of 5 m^3/s. The jet is deflected by the bucket at an angle of 165^{\circ}. Neglecting all losses, the power developed by the Pelton wheel is _________MW (round off to two decimal places).
A
1.54
B
6.25
C
2.65
D
4.22
GATE ME 2021 SET-1   Fluid Mechanics
Question 3 Explanation: 
\begin{aligned} P &=? &P=\dot{m}\left[V_{w_{1}}-V_{w_{2}}\right] u \\ N &=300 \mathrm{rpm} \\ D &=2 \mathrm{~m} \\ V_{1} &=40 \mathrm{~m} / \mathrm{s} \\ Q &=5 \mathrm{~m}^{3} / \mathrm{s} \\ \beta &=180-165=15^{\circ} \end{aligned}




\begin{aligned} u &=\frac{\pi D N}{60}=\frac{\pi \times 2 \times 300}{60} \\ u_{1} &=u_{2}=u=31.42 \mathrm{~m} / \mathrm{s} \\ V_{r 1} &=v_{1}-u \\ V_{r 1} &=40-31.42 \\ V_{r 1} &=8.58\\ V_{r 1}&=V_{r 2}=8.58 \mathrm{~m} / \mathrm{s} \text { [neglecting blade friction] } \\ V_{r 2} \cos \beta_{2} &=u_{2}-V_{w 2} \\ P &=\rho Q\left[V_{w-1}-V_{w 2}\right] \\ &=2.65 \\ P &=2.65 \mathrm{MW} \end{aligned}

Question 4
For a Kaplan (axial flow) turbine, the outlet blade velocity diagram at a section is shown in figure.

The diameter at this section is 3m. The hub and tip diameters of the blade are 2m and 4m, respectively. The water volume flow rate is 100 m^3/s. The rotational speed of the turbine is 300 rpm. The blade outlet angle \beta is _________ degrees (round off to one decimal place).
A
8.6
B
12.7
C
18.2
D
14.6
GATE ME 2020 SET-1   Fluid Mechanics
Question 4 Explanation: 


\begin{array}{rl} D_{b}=2 \mathrm{m} & Q=100 \mathrm{m}^{3} / \mathrm{sec} \\ D_{o}=4 \mathrm{m} & N=300 \mathrm{rpm} \\ \tan \beta & =\frac{C_{F}}{C_{b}}=\frac{V_{F 2}}{u_{2}} \\ u_{2} & =C_{b}=\frac{\pi D N}{60}=\frac{\pi \times 3 \times 300}{60} \\ u_{2} & =C_{b}=47.12 \mathrm{m} / \mathrm{s} \\ Q & =\frac{\pi}{4}\left(4^{2}-2^{2}\right) \times V_{F 2} \\ V_{F 2} & =C_{F}=10.61 \mathrm{m} / \mathrm{s} \\ \tan \beta & =\frac{10.61}{47.12} \\ \therefore \quad \beta & =12.69^{\circ} \simeq 12.7^{\circ} \end{array}
Question 5
Air discharges steadily through a horizontal nozzle and impinges on a stationay vertical plate as shown in figure.

The inlet and outlet areas of the nozzle are 0.1 m^2 and 0.02 m^2, respectively. Take air density as constant and equal to 1.2 kg/m^3. If the inlet gauge pressure of air is 0.36 kPa, the gauge pressure at point O on the plate is ________ kPa (round off to two decimal places).
A
375
B
0.375
C
3.75
D
37.5
GATE ME 2020 SET-1   Fluid Mechanics
Question 5 Explanation: 
On applying continuity equation,
\begin{aligned} \dot{m} &=\rho_{1} \cdot A_{1} \cdot v_{1}=\rho_{2} \cdot A_{2} \cdot v_{2} \\ \Rightarrow 0.1 V_{1} &=0.02 V_{2} \\ \Rightarrow V_{2} &=\frac{10}{2} V_{1}=5 V_{1} \end{aligned}
Now on applying Bernoulli between 1 and 2 section


\begin{aligned} \frac{P_{1}}{\rho g}+\frac{V_{1}^{2}}{2 g}+z_{1} &=\frac{P_{2} /^{0}}{\rho g}+\frac{V_{2}^{2}}{2 g}+z_{2} \\ \Rightarrow \quad \frac{0.36 \times 10^{3}}{1.21 \times 9.81}+\frac{V_{1}^{2}}{2 g} &=\frac{25 V_{1}^{2}}{2 g} \\ \Rightarrow \quad V_{1} &=4.98 \mathrm{m} / \mathrm{s}\\ \Rightarrow \quad V_{2}&=24.89 \mathrm{m} / \mathrm{s} \end{aligned}
On applying Bernoulli between 2 and 3 sections
\begin{aligned} \frac{P_{2} }{\rho g}+\frac{V_{2}^{2}}{2 g}+z_{2} &=\frac{P_{3}}{\rho g}+\frac{V_{3}^{2}}{z g}+z_{3} \\ P_{3} &=\frac{\rho g \cdot V_{2}^{2}}{2 g}=\frac{1.21 \times(24.89)^{2}}{2}=375 \mathrm{Pa} \\ &=0.375 \mathrm{kPa} \text { (gauge) } \end{aligned}
Question 6
An idealized centrifugal pump (blade outer radius of 50 mm) consumes 2 kW power while running at 3000 rpm. The entry of the liquid into the pump is axial and exit from the pump is radial with respect to impeller. If the losses are neglected, then the mass flow rate of the liquid through the pump is ______kg/s (round off to two decimal places).
A
4.52
B
8.11
C
6.25
D
2.65
GATE ME 2019 SET-2   Fluid Mechanics
Question 6 Explanation: 
The theoretical head (Euler's head) is given by
\begin{aligned} \mathrm{H}_{\mathrm{e}} &=\frac{\mathrm{u}_{2} \mathrm{V}_{\mathrm{w} 2}}{\mathrm{g}} \\ &=\frac{\mathrm{u}_{2}^{2}}{\mathrm{g}} \text { as }\left(\mathrm{V}_{\mathrm{w} 2}=\mathrm{u}_{2} \text { for radial exit }\right) \end{aligned}
Thus,
P_{t h}=\rho g Q H_{c}
=\rho g Q \times \frac{u_{2}^{2}}{g}
=\rho \mathrm{Q} \mathrm{u}_{2}^{2}
P_{t h}=\dot{m} u_{2}^{2}
or,
\dot{\mathrm{m}}=\frac{\mathrm{P}_{\mathrm{th}}}{\mathrm{u}_{2}^{2}}
where, u_{2}=\frac{\pi \mathrm{d}_{2} \mathrm{N}}{60}=\pi \times 0.1 \times \frac{3000}{60}=5 \pi \mathrm{m} / \mathrm{s}
Hence, \dot{\mathrm{m}}=\frac{2 \times 10^{3}}{(5 \pi)^{2}}=8.11 \mathrm{kg} / \mathrm{s}
Question 7
As per common design practice, the three types of hydraulic turbines, in descending order of flow rate, are
A
Kaplan, Francis, Pelton
B
Pelton, Francis, Kaplan
C
Francis, Kaplan, Pelton
D
Pelton, Kaplan, Francis
GATE ME 2019 SET-1   Fluid Mechanics
Question 7 Explanation: 
Kaplan turbine has highest flow area hence it can handle highest discharge. On the other hand, Pelton turbine has lowest flow area hence it works on low discharge.
\therefore \mathrm{Q}_{\mathrm{Kaplan}} \gt \mathrm{Q}_{\mathrm{Francis}}>\mathrm{Q}_{\mathrm{Pclton}}
Question 8
A test is conducted on a one-fifth scale model of a Francis turbine under a head of 2 m and volumetric flow rate of 1 m^{3}/s at 450 rpm. Take the water density and the acceleration due to gravity as 10^{3} kg/m^{3} and 10 m/s^{2}, respectively. Assume no losses both in model and prototype turbines. The power (in MW) of a full sized turbine while working under a head of 30 m is _______ (correct to two decimal places).
A
29.05
B
25.88
C
41.52
D
45.36
GATE ME 2018 SET-2   Fluid Mechanics
Question 8 Explanation: 
\begin{aligned} \frac{D_{m}}{D_{p}}&=\frac{1}{5} \\ &\text { Model } &\text{Prototype}\\ H&=2 \mathrm{m} &H=30 \mathrm{m} \\ Q&=1 \mathrm{m}^{3} / \mathrm{s} &P=? \\ N&= 450 \mathrm{rpm} \\ \rho&= 1000 \mathrm{kg} / \mathrm{m}^{3} \\ g&=10 \mathrm{m} / \mathrm{s}^{2}\\ \text{For model}\quad \rho_{m}&=\rho_{g} Q H\\ &=1000 \times 10 \times 1 \times 2=20 \mathrm{kW} \\ \left.\frac{H}{D^{2} N^{2}}\right|_{P}&=\left.\frac{H}{D^{2} N^{2}}\right|_{M}\\ N_{P} &=\sqrt{\frac{H_{P}}{H_{M}}} \times \frac{D_{M}}{D_{P}} \times N_{M} \\ &=\sqrt{\frac{30}{2}} \times \frac{1}{5} \times 450=348.56 \mathrm{rpm} \\ \left.\frac{P}{D^{5} N^{3}}\right|_{P} &=\left.\frac{P}{D^{5} N^{3}}\right|_{M}\\ P_{P}&=P_{M} \times \frac{D_{P}^{5}}{D_{M}^{5}} \times \frac{N_{P}^{3}}{N_{M}^{3}} \\ P_{P}&=20 \times 5^{5} \times \frac{348.56^{3}}{450^{3}} \\ P_{P}&=29.05 \mathrm{MW} \end{aligned}
Question 9
Select the correct statement for 50% reaction stage in a steam turbine.
A
The rotor blade is symmetric.
B
The stator blade is symmetric.
C
The absolute inlet flow angle is equal to absolute exit flow angle.
D
The absolute exit flow angle is equal to inlet angle of rotor blade.
GATE ME 2018 SET-2   Fluid Mechanics
Question 9 Explanation: 
For 50% reaction stage in a steam turbine, the absolute exit flow angle is equal to inlet angle of the rotor blade.
Question 10
Steam flows through a nozzle at a mass flow rate of \dot{m}=0.1 kg/s with a heat loss of 5 kW. The enthalpies at inlet and exit are2500kJ/kg and 2350 kJ/kg, respectively. Assuming negligible velocity at inlet (C_{1}\approx 0 ), the velocity ( C_{2} ) of steam (in m/s) at the nozzle exit is __________ (correct to two decimal places).
A
260.68
B
320.87
C
410.56
D
447.21
GATE ME 2018 SET-1   Fluid Mechanics
Question 10 Explanation: 


\dot{m}=0.1 \mathrm{kg} / \mathrm{s}, \dot{Q}=5 \mathrm{kW} \text { (heat loss) }
Applying SFEE
\begin{aligned} \dot{m}\left(h_{1}+\frac{1}{2} c_{1}^{2}+g z_{1}\right)+\dot{Q}&=\dot{m}\left(h_{2}+\frac{1}{2} c_{2}^{2}+g z_{2}\right)+\dot{w}_{c v}\\ c_{1}=0 \text{ and }\dot{w}_{c v}&=0 \\ z_{1} &=z_{2}(\text { assume }) \\ \Rightarrow \quad \dot{m} h_{1}+\dot{Q} &=\dot{m} h_{2}+\dot{m} \frac{1}{2} c_{2}^{2} \\ \Rightarrow \quad 0.1 \times \frac{1}{2} c_{2}^{2} \times 10^{-3} &=0.1(2500-2350)-5 \\ c_{2} &=447.213 \mathrm{m} / \mathrm{s} \end{aligned}
There are 10 questions to complete.

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