Impact of Jets and Turbines

Question 1
Consider the reactor shown in the figure. The flow rate through the reactor is Q \; m^3/h. The concentrations (in mg/L) of a compound in the influent and effluent are C_0 and C, respectively. The compound is degraded in the reactor following the first order reaction. The mixing condition of the reactor can be varied such that the reactor becomes either a completely mixed flow reactor (CMFR) or a plug-flow reactor (PFR). The length of the reactor can be adjusted in these two mixing conditions to L_{CMFR} and L_{PFR} while keeping the cross-section of the reactor constant. Assuming steady stateand for C/C_0 =0.8, the value of L_{CMFR}/L_{PFR} (round off to 2 decimal places) is _______
A
1.12
B
1.85
C
2.44
D
2.97
GATE CE 2019 SET-2      Impact of Jets and Turbines
Question 1 Explanation: 
Hydrocarbons and nitrogen oxides are considered primary air pollutants.
c=\frac{c_0}{1+kt}
For (PFR) plug flow reactor
\begin{aligned} c&=c_0e^{-kt} \\ \text{As}\;\; c/c_0 &=0.8 \\ \text{For CMFR} \;\; 0.8&=\frac{1}{1+kt_{CMFR}} \\ \Rightarrow \;\;t_{CMFR} &=\frac{0.25}{k} \\ \text{For PFR} \;\; 0.8&=e^{-kt_{PFR}} \\ \Rightarrow \;\; t_{PFR}&=\frac{0.22314}{k} \end{aligned}
Now for steady state
\begin{aligned} v&=\text{constant}\\ L&=vt\\ So, \;\;\frac{L_{CMFR}}{L_{PFR}}&=\frac{vt_{CMFR}}{vt_{PFR}}\\ &=\frac{0.25}{0.22314}=1.12 \end{aligned}
Question 2
A penstock of 1 m diameter and 5 km length is used to supply water from a reservoir to an impulse turbine. A nozzle of 15 cm diameter is fixed at the end of the penstock. The elevation difference between the turbine and water level in the reservoir is 500 m. Consider the head loss due to friction as 5% of the velocity head available at the jet. Assume unit weight of water = 10\: kN/m^{3} and acceleration due to gravity (g)=10\: m/s^{2}. If the overall efficiency 80%, power generated (expressed in kW and rounded to nearest integer) is ___________
A
7570
B
8212.5
C
6570
D
8874.8
GATE CE 2016 SET-2      Impact of Jets and Turbines
Question 2 Explanation: 
Apply energy equaiton at the free surlace of
reservior and exit of nozzle
\begin{aligned} 500&=\text{Head loss } \frac{v_{1}^{2}}{2 g}\\ 500&=0.05 \frac{v_{1}^{2}}{2 g}+\frac{v_{1}^{2}}{2 g}\\ \sqrt{\frac{2 \times 10 \times 500}{1.05}}&=v_{1} \\ V_{1}&=97.59 \mathrm{m} / \mathrm{sec}\\ \text{Water Power WP }\\ &=\frac{1}{2} m v_{1}^{2} \\ &=\frac{1}{2}\left(10^{2}\right) \times \frac{\pi}{4}(0.15)^{2}(9759) \\ &=8212178 \mathrm{kW} \\ \text{Now,}\quad \eta_{0}&=\frac{\text { Shaft power }(\$ P)}{W P} \\ 08&=\frac{S P}{8212178} \\ S P&=6569.74 \mathrm{kW} \\ &\simeq 6570 \mathrm{kW} \end{aligned}
Question 3
A square plate is suspended vertically from one of its edges using a hinge support as shown in figure. A water jet of 20 mm diameter having a velocity of 10 m/s strikes the plate at its mid-point, at an angle of 30^{\circ} with the vertical. Consider g as 9.81\: m/s^{2} and neglect the self-weight of the plate. The force F (expressed in N) required to keep the plate in its vertical position is _______________
A
7.85
B
8.8
C
6.63
D
0.763
GATE CE 2016 SET-2      Impact of Jets and Turbines
Question 3 Explanation: 


Force exerted by jet in x direction
\begin{array}{l} F_{x}=m[V \sin \theta-0 \mid \\ =\rho Q \times V \sin \theta \\ =\rho A V \times V \sin \theta \\ =1000 \times \frac{\pi}{4}(0.02)^{2} \times(10)^{2} \sin 30^{\circ} \\ =157079 \mathrm{N} \end{array}
Taking mornent about hinge
\begin{array}{r} F_{x} \times \frac{0.2}{2}=F \times 02 \\ F=7.85 \mathrm{N} \end{array}
Question 4
A horizontal nozzle of 30 mm diameter discharges a steady jet of water into the atmosphere at a rate of 15 liters per second. The diameter of inlet to the nozzle is 100 mm. The jet Impinges normal to a flat stationary plate held close to the nozzle end. Neglecting air friction and considering the density of water as 1000 kg/m^3, force exerted by the jet (in N) on the plate is _______
A
318.3
B
328.3
C
246.8
D
338.3
GATE CE 2014 SET-2      Impact of jets and Turbines
Question 4 Explanation: 
\text{Force }=\rho a V^{2}
\mathrm{Q}=\mathrm{a} \mathrm{V}=15 \; litre/sec (given)
\text{Force }=\rho \frac{\mathrm{Q}^{2}}{\mathrm{a}}
=1000 \times \frac{15^{2}}{\pi} \times 30^{2} \mathrm{m}^{3} \times 10^{-6} \frac{\mathrm{m}^{6}}{\mathrm{s}^{2}} \times \frac{1}{10^{-6}} \mathrm{m}^{2}
=1000 \times \frac{1}{\pi} \mathrm{N}=318.3 \mathrm{N}
Question 5
A horizontal jet of water with its cross-sectional area of 0.0028 m^2 hits a fixed vertical plate with a velocity of 5 m/s. After impact the jet splits symmetrically in a plane parallel to the plane of the plate. The force of impact (in N) of the jet on the plate is
A
90
B
80
C
70
D
60
GATE CE 2014 SET-1      Impact of jets and Turbines
Question 5 Explanation: 


\begin{aligned} \text { Force on plate } &=\left(\rho_{w} a V\right) \mathrm{V} \\ &=\rho_{w} a V^{2} \\ &=1000 \times 0.0028 \times(5)^{2} \\ &=70 \mathrm{N} \end{aligned}
Question 6
A horizontal water jet with a velocity of 10 m/s and cross sectional area of 10 mm^{2} strikes a flat plate held normal to the flow direction. The density of water is 1000 kg/m^{3}. The total force on the plate due to the jet is
A
100N
B
10N
C
1N
D
0.1N
GATE CE 2007      Impact of Jets and Turbines
Question 6 Explanation: 
The force is given by
\begin{aligned} F &=\rho a V^{2} \\ \therefore F &=1000 \times 10 \times 10^{-6} \times(10)^{2} \\ &=1 \mathrm{N} \end{aligned}
Question 7
A tank and a deflector are placed on a frictionless trolley. The tank issues water jet (mass density of water = 1000 kg/m^{3}), which strikes the deflector and turns by 45^{\circ}. If the velocity of jet leaving the deflector is 4 m/s and discharge is 0.1 m^{3}/s, the force recorded by the spring will be
A
100N
B
100\sqrt{2}N
C
200N
D
200\sqrt{2}N
GATE CE 2005      Impact of Jets and Turbines
Question 7 Explanation: 
Force in spring will be the force in horizonta direction.
\begin{aligned} \therefore \quad F_{H} &=\rho Q V \cos \theta \\ &=1000 \times 0.1 \times 4 \times \cos 45^{\circ} \\ &=\frac{400}{\sqrt{2}}=200 \sqrt{2} \mathrm{N} \end{aligned}
Question 8
A horizontal jet strikes a frictionless vertical plate (the plan view is shown in the figure). It is then divided into two parts, as shown in the figure. If the impact loss is neglected, what is the value of \theta?
A
15^{\circ}
B
30^{\circ}
C
45^{\circ}
D
60^{\circ}
GATE CE 2003      Impact of Jets and Turbines
Question 8 Explanation: 
From continuity equation, we get,
\begin{aligned} & & Q_{0}=Q_{1}+Q_{2} \\ \Rightarrow & & Q_{0}=0.25 Q_{0}+Q_{2} \\ \Rightarrow & & Q_{2}=0.75 Q_{0} \end{aligned}
since, the impact losses are neglected, the velocity will remain unchanged in the direction of Q_{1} and Q_{2}
\text{i.e.},\quad V_{0}=V_{1}=V_{2}
Applying impulse momentum equation, we get,
\begin{aligned} \rho Q_{0} V_{0} \sin \theta&=\rho Q_{2} V_{2}-\rho Q_{1} V_{1} \\ \Rightarrow \quad Q_{0} \sin \theta&=0.75 Q_{0}-0.25 Q_{0} \\ \Rightarrow \quad \sin \theta&=1 / 2 \\ \Rightarrow \quad \theta&=30^{\circ} \end{aligned}
There are 8 questions to complete.

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