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17-Phys-B6 Applied Thermodynamics and Heat Transfer · December 2018

Question 7 of 8: Liquid-Sodium Duct Immersed in Molten Lead — Outlet Temperature and Heat Transfer

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

Notes on this paper

Paper format. 17-Phys-B6 Applied Thermodynamics and Heat Transfer, National Examination December 2018 — a three-hour open-book examination; candidates are expected to bring both a thermodynamics text and a heat-transfer text to make use of the property tables and graphs. A complete examination is five questions — either three from Part A (Thermodynamics, Q1–Q4) and two from Part B (Heat Transfer, Q5–Q8), or two from Part A and three from Part B — every question carrying equal value; all eight are solved below as a complete study set. Question 2 is solved as one connected narrative: the wet steam whose quality is measured by the throttling calorimeter in part (a) is the same steam entering the turbine in part (b), which is what makes part (c)'s "isentropic despite heat loss" observation checkable. Question 3 gives every cycle temperature directly from the printed diagram but no pressures, so it is solved purely from energy balances (constant specific heat, cold-air-standard) rather than isentropic pressure ratios — the intended reading, since no compressor/turbine pressure ratio is given anywhere on the page.

Reference texts. Y. A. Çengel and M. A. Boles, Thermodynamics: An Engineering Approach, 8th ed. (ideal-gas mixtures, air-standard Otto and Brayton cycles, throttling calorimeters, steam turbines, vapour-compression refrigeration); F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed. (composite cylindrical conduction with convection at both surfaces, heat generation in a solid cylinder, internal/external convection combined via an overall coefficient, effectiveness–NTU heat-exchanger analysis). Ammonia and steam saturation/superheat property values were computed (Bell et al., IAPWS-95 / REFPROP-quality equations of state) and cross-checked against the printed appendix tables on pages 5–6 of the source exam and standard steam tables, which they matched to 3–4 significant figures throughout.

Question 7: Liquid-Sodium Duct Immersed in Molten Lead — Outlet Temperature and Heat Transfer

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. An equilateral-triangular duct (side $a=3\text{ cm}$, length $L=4\text{ m}$) immersed in an effectively infinite, constant-temperature pool of molten lead at $T_{ml}=600\text{ K}$; liquid sodium enters at $T_i=478\text{ K}$, $\dot m=3.6\text{ kg/s}$, $c_p=1340\text{ J/kg}\cdot{}^{\circ}\text{C}$, with $\bar h_i=89{,}140\text{ W/m}^2\cdot{}^{\circ}\text{C}$ inside and $\bar h_o=8687\text{ W/m}^2\cdot{}^{\circ}\text{C}$ outside.

Given data
QuantitySymbolValue
Duct side / length$a,L$3 cm / 4 m
Lead pool temperature$T_{ml}$600 K
Sodium inlet temperature$T_i$478 K
Sodium flow rate, specific heat$\dot m,c_p$3.6 kg/s, 1340 J/kg·°C
Inside / outside coefficients$\bar h_i,\bar h_o$89,140 / 8687 W/m²·°C

Find. (i) The sodium outlet temperature $T_o$; (ii) the heat transfer rate between the lead and the sodium.

molten lead pool, T_ml = 600 Kliquid Nah̄_i = 89,140 W/m²°Ch̄_o = 8687 W/m²°C (lead side)side = 3 cm, L = 4 mṁ=3.6 kg/s, T_i=478 KT_o = ? (find)
Sodium flows the length of a triangular duct immersed in a large, effectively isothermal molten-lead bath — a "heating in a constant-temperature reservoir" configuration.
Check: the duct is given only ONE side length (not separate inner/outer dimensions), so the wall is treated as thin — the inside and outside convection surfaces share the same perimeter/area, and wall conduction resistance is neglected.

Approach. Combine $\bar h_i$ and $\bar h_o$ into an overall coefficient $U$ (thin wall, same area both sides); because the lead pool is a large reservoir its temperature stays essentially constant, so the sodium's approach to $T_{ml}$ follows the standard exponential "heating-in-a-constant-temperature-bath" relation, from which $T_o$ and then $\dot Q$ follow directly.

  1. Surface area and overall coefficient. $$A=(3a)L=(3\times0.03)\times4=0.36\text{ m}^2$$ $$\frac{1}{U}=\frac{1}{\bar h_i}+\frac{1}{\bar h_o}=\frac{1}{89{,}140}+\frac{1}{8687}$$ $$\boxed{U=7916\text{ W/m}^2\cdot{}^{\circ}\text{C}}$$ $$UA=7916\times0.36=2850\text{ W/}{}^{\circ}\text{C}$$
  2. Sodium outlet temperature (constant-reservoir heating relation). $$\frac{T_{ml}-T_o}{T_{ml}-T_i}=\exp\left(-\frac{UA}{\dot mc_p}\right)$$ $$\dot mc_p=3.6\times1340=4824\text{ W/}{}^{\circ}\text{C},\qquad\frac{UA}{\dot mc_p}=\frac{2850}{4824}=0.5907$$ $$T_{ml}-T_o=(T_{ml}-T_i)\,e^{-0.5907}=122\times0.5539$$ $$\boxed{T_o=T_{ml}-67.6=532.4\text{ K}=259.3\,{}^{\circ}\text{C}}$$
  3. Heat transfer rate, part (ii). $$\dot Q=\dot mc_p(T_o-T_i)=4824\times(532.4-478)$$ $$\boxed{\dot Q=262.5\text{ kW}}$$ Cross-check via LMTD: $\Delta T_1=T_{ml}-T_i=122$, $\Delta T_2=T_{ml}-T_o=67.6$, $LMTD=(122-67.6)/\ln(122/67.6)=92.2\,{}^{\circ}\text{C}$, and $\dot Q=UA\times LMTD=2850\times92.2=262.7\text{ kW}$ — matches within rounding.
Question 7 — results
QuantityValue
Overall coefficient $U$7916 W/m²·°C
(i) Sodium outlet temperature $T_o$532.4 K (259.3°C)
(ii) Heat transfer rate $\dot Q$262.5 kW