22-Mec-A1 Applied Thermodynamics and Heat Transfer · May 2015
Question 6 of 8: Uniformly Heated Water-Heater Tube
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper: National Examinations — 07-Mec-A1 Applied Thermodynamics and Heat Transfer, May 2015. Open-book, 3-hour paper. 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, all of equal value. Full worked solutions to all eight questions are given below.
Reference texts: Çengel & Boles, Thermodynamics: An Engineering Approach (9th ed., McGraw-Hill) — closed- and open-system energy balances, steam tables, vapour and gas power cycles 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) — composite-wall conduction, internal-flow and cross-flow convection correlations, natural convection with radiation, and the ε–NTU / LMTD-correction heat-exchanger methods. Freon-12 property data are taken from the appendix supplied with the exam; steam, air and water data from standard tables.
Given. Tube $D=0.010\ \text{m}$, $L=4\ \text{m}$; uniform wall flux $q''=1000\ \text{W/m}^2$; water inlet 10 °C at $\dot m=12\ \text{kg/hr}=3.33\times10^{-3}\ \text{kg/s}$. Water properties near the mean bulk (≈ 15 °C): $c_p=4186$, $k=0.593\ \text{W/m}\cdot\text{K}$, $\mu=1.17\times10^{-3}\ \text{Pa}\cdot\text{s}$, $\text{Pr}\approx8$.
Find. The total heat-transfer rate and the tube-wall temperature at the exit.
Figure 5 — Constant-heat-flux internal flow: the bulk temperature rises linearly along the tube while the wall runs a fixed offset $q''/h$ above it once the flow is thermally fully developed.
Approach. The total heat is the flux times the wetted area (fixing the bulk temperature rise by energy balance); the exit wall temperature is the exit bulk temperature plus the constant-flux offset $q''/h$, using the fully-developed laminar Nusselt number at the exit.
Total heat-transfer rate. $$\dot Q=q''\,(\pi D L)=1000\,\pi(0.010)(4)=\boxed{125.7\ \text{W}}.$$
Exit bulk temperature. Energy balance on the water:$$T_{b,o}=T_{b,i}+\frac{\dot Q}{\dot m c_p}=10+\frac{125.7}{3.33\times10^{-3}(4186)}=\boxed{19.0\ ^\circ\text{C}}.$$
Flow regime. $\text{Re}_D=\dfrac{4\dot m}{\pi D\mu}=\dfrac{4(3.33\times10^{-3})}{\pi(0.010)(1.17\times10^{-3})}=363$ — laminar. The thermal entry length $\approx0.05\,\text{Re}\,\text{Pr}\,D\approx1.5$ m, so by the 4 m exit the flow is fully developed.
Exit wall temperature. For fully-developed laminar flow with constant flux, $\text{Nu}_D=4.36$, so $h=\text{Nu}_Dk/D=4.36(0.593)/0.010=259\ \text{W/m}^2\text{K}$ and$$T_{s,o}=T_{b,o}+\frac{q''}{h}=19.0+\frac{1000}{259}=\boxed{22.9\ ^\circ\text{C}}.$$
Check
Near the inlet the developing thermal boundary layer gives a higher $h$ (larger $\text{Nu}$) and therefore a smaller wall–bulk gap; the exit, being fully developed, is where the coefficient is lowest and the wall runs hottest — 3.9 °C above the local bulk. The bulk rise (9 °C) is set purely by the energy balance and is independent of $h$.
