(a)(i) Show that the effective conductance, $(A_1\bar{F}_{12})$ for two black, parallel plates of equal area connected by re-radiating walls at constant temperature is $A_1\bar{F}_{1-2} = A_1\left(\dfrac{1+F_{1-2}}{2}\right)$. (ii) Determine the stea

Question

(a)(i) Show that the effective conductance, $(A_1\bar{F}_{12})$ for two black, parallel plates of equal area connected by re-radiating walls at constant temperature is $A_1\bar{F}_{1-2} = A_1\left(\dfrac{1+F_{1-2}}{2}ight)$. (ii) Determine the steady-state temperatures of two radiation shields placed in the evacuated space between two infinite planes at temperatures of 555 K and 278 K. The emissivity of all surfaces is 0.8. [$\sigma$ = Stefan-Boltzmann constant = $5.670 	imes 10^{-8}$ W/m²K⁴] (20 marks)

(b) Assume that the velocity distribution in the turbulent core for tube flow may be represented by $\dfrac{u}{u_c} = \left(1-\dfrac{r}{r_o}ight)^{\frac{1}{7}}$ where $u_c$ is the velocity at the centre of the tube and $r_o$ is the tube radius. Using the Blasius relation for friction factor, derive an equation for the thickness of the laminar sublayer. For this problem the average flow velocity may be calculated using the turbulent velocity distribution. Assume linear profile in sublayer. (20 marks)

(c) Explain how the process of reheating in a gas turbine affects its operational performance. (10 marks)

UPSC Answer Check · Accepted Answer

Derive the effective conductance expression for part (a)(i) using radiation network analysis with re-radiating surfaces; for (a)(ii) set up heat balance equations for two shields and solve the resulting system for steady-state temperatures. For part (b), derive the laminar sublayer thickness by equating shear stresses at the interface using the Blasius friction factor and the 1/7th power law velocity profile. For part (c), explain reheating effects on thermal efficiency, specific work output, and turbine blade cooling requirements, citing operational trade-offs in Indian power plants like NTPC gas turbine stations.
- Part (a)(i): Derivation using radiation network with two black surfaces and re-radiating walls; application of reciprocity and summation rules to obtain A₁F̄₁₂ = A₁(1+F₁₂)/2
- Part (a)(ii): Heat balance equations for two shields: q = σ(T₁⁴-Tₛ₁⁴)/(1/ε₁+1/εₛ₁-1) = σ(Tₛ₁⁴-Tₛ₂⁴)/(1/εₛ₁+1/εₛ₂-1) = σ(Tₛ₂⁴-T₂⁴)/(1/εₛ₂+1/ε₂-1); solution yields Tₛ₁ ≈ 467 K, Tₛ₂ ≈ 389 K
- Part (b): Blasius relation f = 0.0791/Re^0.25; wall shear stress τ_w = fρū²/8; velocity gradient in sublayer du/dy = τ_w/μ; equate with 1/7th law at edge of sublayer to obtain δ_s = 11.6ν/u* where u* = √(τ_w/ρ)
- Part (b): Integration of 1/7th power law to find average velocity ū = 49u_c/60; substitution to express δ_s in terms of Re and pipe diameter
- Part (c): Reheating increases specific work output (w_net ↑) but decreases thermal efficiency (η_th ↓) due to additional heat addition at lower average temperature; reduces compressor work ratio; mention blade cooling challenges in Indian tropical conditions

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly applies radiation network theory for re-radiating surfaces in (a)(i); recognizes that shields reduce heat transfer geometrically in (a)(ii); properly identifies that laminar sublayer thickness depends on friction velocity and Reynolds number in (b); accurately explains reheating trade-offs between work output and efficiency in (c), citing Brayton cycle modifications.	Uses correct general formulas but misapplies view factor algebra in (a)(i) or makes sign errors in heat balance; understands sublayer concept in (b) but confuses integration limits; describes reheating effects in (c) without quantitative efficiency discussion.	Treats re-radiating walls as black surfaces at fixed temperature in (a); ignores Blasius relation in (b) or uses incorrect velocity profile integration; describes reheating only as 'good for power' without thermodynamic reasoning in (c).
Numerical accuracy	20%	10	Shield temperatures Tₛ₁ = 467 K and Tₛ₂ = 389 K obtained correctly with σ = 5.67×10⁻⁸ W/m²K⁴; heat flux q ≈ 267 W/m²; final numerical expression for δ_s in (b) correctly simplified to δ_s/D = 62.2/Re^(7/8) or equivalent; all exponents and coefficients in Blasius integration handled precisely.	Shield temperatures within 5% due to algebraic slip in solving simultaneous equations; correct approach in (b) but arithmetic error in 49/60 integration constant; order of magnitude correct for sublayer thickness.	Shield temperatures wrong by >10% due to incorrect resistance network; uses ε = 0.8 in Stefan-Boltzmann law directly instead of in resistance terms; order-of-magnitude error in δ_s (e.g., mm instead of μm); σ value misquoted.
Diagram quality	15%	7.5	Clear radiation network diagram for (a) showing surface resistances (zero for black), space resistances, and re-radiating node; velocity profile sketch for (b) showing laminar sublayer, buffer layer, and turbulent core with 1/7th curve; T-s diagram for (c) showing ideal Brayton cycle with reheat stage, pressure and temperature levels labelled.	Diagrams present but missing key labels (e.g., no 1/7th power annotation, or missing re-radiating node in network); T-s diagram drawn but without clear reheat pressure level indication.	No diagrams provided despite question requiring visual representation; or diagrams drawn without any labels or incorrect shapes (e.g., linear velocity profile shown for entire turbulent core).
Step-by-step derivation	25%	12.5	Complete derivation of effective conductance starting from J₁, J₂, J₃ definitions, applying J₃ = σT₃⁴ for re-radiating surface, eliminating T₃ algebraically to reach final expression; explicit integration of (1-r/r₀)^(1/7)r dr for ū; clear differentiation of u/u_c to find du/dr at r = r₀-δ_s; systematic solution of three-equation system for shield temperatures.	Key steps shown but skips algebraic manipulation (e.g., states 'solving gives' without showing T₃ elimination); integration shown but limits not explicitly stated; heat balance equations written but solution method unclear.	Final formulas stated without derivation; jumps from Blasius to sublayer thickness without velocity profile analysis; shield temperatures stated without showing simultaneous equation setup; no derivation of effective conductance formula.
Practical interpretation	20%	10	Discusses that radiation shields find application in cryogenic storage (ISRO LH₂ tanks) and high-temperature furnaces; notes that laminar sublayer thickness determines critical for heat transfer enhancement (roughness, additives); explains why Indian gas turbines (e.g., NTPC Anta, Auraiya) use reheat for peak load despite efficiency penalty, and mentions combined cycle integration to recover efficiency; comments on material limits for reheat temperatures.	Mentions practical applications in general terms without Indian context; states that reheat increases power but decreases efficiency without explaining why utilities accept this trade-off.	No practical interpretation provided; or irrelevant discussion (e.g., discusses conduction in solids when question is on radiation); treats all parts as purely academic exercises.

Q3

Directive word: Derive

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Mechanical Engineering 2021 Paper II