NivaarExam PrepOfficial exam papers ↗

22-Mec-A1 Applied Thermodynamics and Heat Transfer · December 2018

Question 7 of 8: Liquid-sodium heat exchanger duct in a molten-lead pool

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Examination 16-Mec-A1, 3 hours, open book. Eight questions of equal value: Part A — Thermodynamics (Q1–Q4) and Part B — Heat Transfer (Q5–Q8). A complete paper is any five questions (three from one part and two from the other).

Reference texts: Çengel & Boles, Thermodynamics: An Engineering Approach (9th ed., McGraw-Hill) — ideal-gas mixtures, the air-standard Otto cycle, wet-region steam properties, the throttling calorimeter, the steady-flow energy equation, the regenerative gas-turbine (Brayton) cycle and vapour-compression refrigeration; Çengel & Ghajar, Heat and Mass Transfer (6th ed.) and Incropera, DeWitt, Bergman & Lavine, Fundamentals of Heat and Mass Transfer (8th ed., Wiley) — radial conduction through composite cylinders, conduction with internal heat generation, internal-flow convection with a constant surrounding-fluid temperature, and the effectiveness–NTU method for s​hell-and-tube exchangers. Steam properties are IAPWS-consistent (equivalent to the steam tables); ammonia properties are read from the saturated- and superheated-ammonia tables appended to the examination; air and combustion gases are treated as ideal gases with constant specific heats ($\gamma=1.4$, $R=0.287\ \text{kJ/kg}\cdot\text{K}$, $c_p=1.005\ \text{kJ/kg}\cdot\text{K}$).

Question 7 — Liquid-sodium heat exchanger duct in a molten-lead pool (Part B, equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Equilateral-triangular duct, side $a=3$ cm, length $L=4$ m, immersed in a large molten-lead pool held at $T_\text{ml}=600$ K. Liquid sodium enters at $T_i=478$ K, $\dot m=3.6$ kg/s, $c_p=1340\ \text{J/kg°C}$; $\bar h_i=89{,}140$, $\bar h_o=8687\ \text{W/m}^2\text{°C}$ (thin duct wall neglected).

Find. (i) the sodium outlet temperature, (ii) the total heat transfer.

molten lead pool, $T_\text{ml}=600$ K Na in 478 K 532 K $L=4$ m, $\dot m=3.6$ kg/s $a=3$ cm section
Figure 7 — The equilateral-triangular sodium duct runs 4 m through the molten-lead pool. With a nearly constant surrounding temperature, the sodium bulk temperature approaches 600 K exponentially along the duct, reaching 532 K at exit.

Approach. The pool is effectively an infinite reservoir at constant $T_\text{ml}$, so the duct is a single-stream exchanger with a fixed external temperature. Build $U$ from the two films, form $UA$ on the triangular perimeter, then use the exponential approach relation for the outlet temperature and an energy balance for the duty.

  1. Geometry and overall coefficient. Perimeter $P=3a=0.09$ m, so the heat-transfer area is $A=PL=0.09(4)=0.36\ \text{m}^2$. Neglecting the thin wall, $$U=\left(\frac{1}{\bar h_i}+\frac{1}{\bar h_o}\right)^{-1}=\left(\frac{1}{89140}+\frac{1}{8687}\right)^{-1}=7916\ \text{W/m}^2\text{°C}$$ $$UA=7916(0.36)=2850\ \text{W/°C},\qquad \dot m c_p=3.6(1340)=4824\ \text{W/°C}$$
  2. Outlet temperature (constant-$T_\text{ml}$ approach). $$\frac{T_\text{ml}-T_o}{T_\text{ml}-T_i}=\exp\!\left(-\frac{UA}{\dot m c_p}\right)=\exp\!\left(-\frac{2850}{4824}\right)=e^{-0.591}=0.554$$ $$T_o=600-0.554(600-478)=600-67.6=532.4\ \text{K}$$ $T_o\approx 532\ \text{K}\ (259\ ^\circ\text{C})$
  3. Heat transfer. From the sodium energy balance, $$\dot Q=\dot m c_p(T_o-T_i)=4824(532.4-478)=2.62\times10^{5}\ \text{W}$$ $\dot Q\approx 262\ \text{kW}$
QuantityResult
Overall coefficient $U$7916 W/m²°C
Sodium outlet temperature532 K (259 °C)
Heat transfer $\dot Q$262 kW