(a) A convergent-divergent nozzle is designed to expand air from a chamber in which the pressure is 800 kPa and temperature is 40°C to give Mach 2·5. The throat area of the nozzle is 0·0025 m². Find the following:

(i) The flow rate through the nozzl

Question

(a) A convergent-divergent nozzle is designed to expand air from a chamber in which the pressure is 800 kPa and temperature is 40°C to give Mach 2·5. The throat area of the nozzle is 0·0025 m². Find the following:

(i) The flow rate through the nozzle under design conditions.

(ii) The exit area of the nozzle.

(iii) The design back-pressure and the temperature of the air leaving the nozzle with this back-pressure.

(iv) The lowest back-pressure for which there is only subsonic flow in the nozzle.

(v) The back-pressure at which there is a normal shock wave on the exit plane of the nozzle.

(vi) The back-pressure below which there are no shock waves in the nozzle.

(vii) The back-pressure over which there are oblique shock waves in the exhaust from the nozzle.

(viii) The back-pressure over which there are expansion waves in the exhaust from the nozzle.

Use Isentropic and Shock tables attached at the end.

(20 marks)

(b) A stainless steel plate of 1 m length, 1 m width and 10 mm thickness is kept horizontal. The thermal conductivity of the plate is 10 W/mK. Bottom surface of the plate is exposed to hot gases at 700°C with heat transfer coefficient at the bottom surface of 50 W/m²K. Top surface of the plate is cooled by air at 69°C and flowing parallel to the top surface. Any part of the plate should not exceed a maximum permissible temperature of 400°C to avoid failure. Find the minimum permissible velocity of the air required to ensure the plate does not get over-heated beyond the permissible limit. Neglect the heat loss from the side surfaces of the plate and assume one-dimensional heat transfer. Use the following correlation to solve the problem:

$$\overline{	ext{Nu}}_{	ext{L}} = 	ext{Pr}^{0.333} \left[ 0.037 	ext{ Re}_{	ext{L}}^{0.8} - 871 ight]$$

Take the appropriate properties of the air from the table attached at the end.

(20 marks)

(c) (i) A vertical flat plate is maintained at a temperature of "T_w °C" and exposed to a stagnant atmospheric air "T_a °C". If T_w < T_a, show the shape of thermal and velocity boundary layers developed on the surface of the plate. Also show the variation "h_x" along the vertical surface of the plate. Assume the flow is within the laminar region and consider only free convection.

(5 marks)

(ii) A fluid flows through a tube exposed to constant heat flux condition. The flow is in the turbulent flow regime for which the Dittus-Boelter correlation is applicable for determining the Nu number as given below:

Nu_d = (0·023) Re^0.8 Pr^0.4

In order to reduce the surface temperature of the tube, it is suggested to double the velocity of the flow. Find the percentage increase in the heat transfer coefficient due to increased velocity.

(5 marks)

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) given its 8 sub-parts and complexity in isentropic/shock table usage; 35% to part (b) involving iterative heat transfer coefficient determination; and 25% to part (c) requiring sketches and percentage calculation. Begin each part with stated assumptions, proceed with systematic calculations using provided tables/correlations, and conclude with clearly labelled final answers for each sub-part.
- Part (a): Correct use of isentropic relations/tables for M=2.5 to find exit pressure ratio (0.0585), temperature ratio (0.444), and area ratio (2.637); mass flow rate at throat using choked flow condition with gamma=1.4
- Part (a)(iii)-(viii): Design back-pressure (46.8 kPa), subsonic-only limit (528 kPa), normal shock at exit (136 kPa), no-shock limit (46.8 kPa), oblique shock regime (46.8-136 kPa), expansion wave regime (<46.8 kPa)
- Part (b): Energy balance q_bottom = q_top + q_conduction; iterative solution for h_top using given Nu correlation; verify T_max ≤ 400°C at bottom surface; minimum velocity ≈ 8-10 m/s
- Part (c)(i): Free convection on cooled vertical plate (T_w < T_a): velocity and thermal boundary layer profiles showing flow reversal, h_x ∝ x^(-1/4) decreasing along height
- Part (c)(ii): Nu ∝ Re^0.8 ∝ V^0.8, therefore doubling velocity gives 2^0.8 = 1.741 → 74.1% increase in h
- Appropriate air properties at film temperature: Pr ≈ 0.7, k ≈ 0.026 W/mK, ν ≈ 1.6×10^-5 m²/s for part (b)

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies choked flow at throat for part (a); recognizes part (a)(iv) requires isentropic subsonic solution at same area ratio; applies correct thermal resistance network for part (b) with conduction and two convection resistances; correctly sketches inverted boundary layer profiles for cooled plate in (c)(i) and applies Dittus-Boelter exponent correctly in (c)(ii).	Uses correct basic formulas but confuses design vs. operating conditions in nozzle; treats part (b) as simple convection without conduction resistance; draws standard heated plate profiles for (c)(i); uses linear proportionality for (c)(ii).	Applies incompressible flow equations to nozzle; ignores conduction completely in (b); draws no sketches or completely wrong profiles; fundamental errors in applying correlations.
Numerical accuracy	25%	12.5	Mass flow rate ≈ 7.25 kg/s; exit area ≈ 0.00659 m²; all back-pressure values within 5% of correct values (46.8, 136, 528 kPa); part (b) h_top ≈ 35-40 W/m²K yielding V_min ≈ 8.5 m/s; part (c)(ii) exactly 74.1% or 2^0.8-1 expressed properly.	Correct methodology but errors in table interpolation; part (b) within 20% of correct h and V; part (c)(ii) shows 80% or uses wrong exponent.	Order-of-magnitude errors in flow rate or areas; completely wrong back-pressure sequence; part (b) yields unrealistic velocities (<1 m/s or >100 m/s); part (c)(ii) gives 100% or 0%.
Diagram quality	15%	7.5	Clear T-s diagram for nozzle showing all operating points (a)(iii)-(viii); part (c)(i) shows velocity profile with u=0 at wall and y=δ, negative velocity maximum, thermal profile with δ_t > δ or < δ indicated, and h_x vs x curve with proper decay.	Basic nozzle sketch without state points; boundary layer sketches show correct qualitative shapes but missing key features (no flow direction, no velocity reversal); h_x curve not labelled.	No diagrams; or diagrams with wrong shapes (e.g., linear profiles, no boundary layer concept); missing entirely for part (c)(i).
Step-by-step derivation	25%	12.5	Shows A/A* = 2.637 lookup and interpolation; explicit energy balance for part (b) with T_bottom expressed in terms of h_top; iterative solution structure shown; part (c)(ii) derives h2/h1 = (V2/V1)^0.8 explicitly.	Jumps to final formulas with minimal working; part (b) shows final equation but not iteration setup; part (c)(ii) states answer without showing exponent manipulation.	No derivations; answers stated without formulae; no indication of table usage; part (b) has no equation setup.
Practical interpretation	15%	7.5	Comments on nozzle operating regimes (over-expanded, under-expanded, design) with applications to rocket nozzles, steam turbines; part (b) relates to heat exchanger design, thermal protection systems; notes laminar-turbulent transition in free convection for (c)(i); discusses practical limits of Dittus-Boelter in (c)(ii).	Brief mention of applications without elaboration; standard concluding statements without specific engineering context.	No interpretation; purely mathematical treatment; no physical insight into flow phenomena or engineering relevance.

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