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# Thermal Engineering MCQ’s – Relationship Between Area, Velocity and Pressure in Nozzle Flow

Which of the following expressions correctly represents the relationship between cross-sectional area of a nozzle, fluid velocity and specific volume?

a) dAA+dCC=dvv

b) dAA−dCC=dvv

c) dAA+dCC+dvv = 0

d) dAA+dvv=

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Answer: aExplanation: The correct relationship between cross-sectional area, fluid velocity and specific volume is represented by the following equation – dAA+dCC=dvv The nature of change (positive or negative) in any two parameters in the equation above predicts the nature of change in the third parameter.

Which of the following expressions is correct? (P – fluid pressure, M- Mach number, A – cross-sectional area of nozzle/diffuser)

a) dAA=1γdPP{M21−M2}

b) dAA=1γdPP{M2−1M2}

c) dAA=1γdPP{1−M2M2}

d) 1γdAA=dPP{1−M2M2}

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Answer: c

Explanation: The correct expression relating the pressure, cross-sectional area of nozzle/diffuser and Mach number is –

dAA=1γdPP{1−M2M2}

or

dAA=1γdPPCs2C2−1

where, Cs – sonic velocity

C – fluid velocity

What is Mach number?

a) It is the ratio of sonic velocity of a fluid at N.T.P. to the local sonic velocity of the same fluid

b) It is the ratio of fluid velocity to sonic velocity of the same fluid at N.T.P.

c) It is the ratio of local sonic velocity to fluid velocity

d) It is the ratio of fluid velocity to local sonic velocity

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Answer: dExplanation: Mach number is the ratio of fluid velocity to the local sonic velocity. It is the ratio of the same quantity and hence is dimensionless. The fluid velocity is called subsonic is Mach number is less than one and supersonic if the Mach number is greater than one.

In case of accelerated flow, when the pressure decreases along the flow direction and Mach number is less than one, it corresponds to _____

a) Convergent part of a nozzle

b) Divergent part of a nozzle

c) Throat of a nozzle

d) Convergent part of a diffuser

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Answer: aExplanation: The flow is accelerated, hence it’s a nozzle. Since dPP is negative and Mach number is less than one, for the following equation to hold – dAA=1γdPP{1−M2M2} L.H.S should also be negative. This implies that dAA should be negative, which corresponds to convergent part of the nozzle.

Which of the following statements regarding the Mach number is TRUE, when the fluid reaches the throat of a nozzle?

a) It becomes unity

b) It is less than one

c) It is greater than one

d) Mach number is not defined at throat of a nozzle

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Answer: aExplanation: At the throat of the nozzle, there is no change in cross-sectional are of the nozzle i.e. dAA=0. Therefore, from the following equation – dAA=1γdPP{1−M2M2} It is evident that the R.H.S. should also be zero. This implies that the Mach number must be one. The fluid velocity attains the value of local sonic velocity at the throat.

Which of the following conditions corresponds to divergent part of a nozzle?

a) M 0

b) M < 1 and dPP < 0

c) M > 1 and dPP < 0

d) M > 1 and dPP > 0

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Answer: c

Explanation: Consider the following equation –

dAA=1γdPP{1−M2M2}

For a nozzle dPP<0, and for divergent part, dAA>0. For the above equation to hold M > 1.

Therefore, M > 1 and dPP < 0 is the correct answer.

A decelerated flow, having fluid velocity greater than the local sonic velocity corresponds to _____

a) Convergent part of a nozzle

b) Divergent part of a nozzle

c) Convergent part of a diffuser

d) Divergent part of a diffuser

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Answer: cExplanation: For a diffuser, dPP>0. It is given that fluid velocity is greater than the local sonic velocity i.e. M > 1. Therefore, according to the following equation – dAA=1γdPP{1−M2M2}

Which of the following conditions correspond to divergent type diffuser?

a) M 0

b) M 0

c) M < 1 and dAA<0

d) M > 1 and dAA<0

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Answer: a

Explanation: Diffuser promotes decelerated flow, \frac{dP}{P}>0. Consider the following equation –

dAA=1γdPP{1−M2M2}

For the divergent part, \frac{dA}{A}>0. For the above equation to hold under the listed conditions, the Mach number must be less than one. Hence, M < 1 and dAA>0 is the correct answer.

The purpose of a steam injector is to force water into the boiler under pressure.

a) True

b) False

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Answer: aExplanation: A steam injector is used to force water into the boiler under pressure. It employees the principle of steam nozzles. It makes use of the kinetic energy of a steam jet for increasing the pressure and velocity of a corresponding amount of water.

Air at 18 bar and 100°C enters a convergent nozzle. Assume the flow to be isentropic and calculate the sonic velocity. Take adiabatic index equal to 1.4.

a) 353.40 m/s

b) 321.56 m/s

c) 360.87 m/s

d) 400.32 m/s

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Answer: a

Explanation: P = 18 bar, T = 100°C or 373 K, γ = 1.4

Critical pressure, P* = P(2γ+1)γγ−1 = 18(21.4+1)1.41.4−1 = 9.51 bar

T1 = T(P∗P)γ−1γ = (373)(9.5118)1.4−11.4 = 310.83 K

Sonic velocity, Cs = γR(T1)−−−−−−√=1.4∗287∗310.83−−−−−−−−−−−−−−√ = 353.40 m/s.

Air enters a frictionless adiabatic horizontal nozzle at 12 bar and 167°C with inlet velocity 50 m/s and leaves at 3 bar. Take adiabatic index equal to 1.4 and cp = 1.005 kJ/kg-K.

a) 654.78 m/s

b) 321.75 m/s

c) 552.45 m/s

d) 456.87 m/s

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Answer: c

Explanation: P1 = 13 bar, T1 = 167°C or 440 K, C1 = 50 m/s, P2 = 3 bar, cp = 1.005 kJ/kg-K, γ = 1.4

We know that, T2 = T1 (P2P1)γ−1γ = 440 (313)1.4−11.4 = 289.40 K

Applying the energy equation at inlet and outlet of the nozzle, we get

m[h1+c212+Z1*g]+Q=m[h2+c222+Z2*g]+W

Q = 0, W = 0, Z1 = Z2

h1+c212=h2+c222

c22=2(h1-h2)+c12

C2 = 2(h1−h2)+c21−−−−−−−−−−−−√

C2 = 2cp(T1−T2)+c21−−−−−−−−−−−−−−√

C2 = 2∗1.005∗103(440−289.40)+502−−−−−−−−−−−−−−−−−−−−−−−−−−−√

C2 = 552.45 m/s.