22-Mec-A1 Applied Thermodynamics and Heat Transfer · May 2015
Question 8 of 8: Shell-and-Tube Heat Exchanger Sizing
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. Cold water (tube side) $\dot m_c=3.8\ \text{kg/s}$, 38 → 55 °C; hot water (shell side) $\dot m_h=1.9\ \text{kg/s}$ entering at 94 °C; $U=1420\ \text{W/m}^2\text{K}$; tube bore $D=1.905\ \text{cm}$, $\rho=961\ \text{kg/m}^3$, tube velocity $V=0.386\ \text{m/s}$; maximum length 2.44 m; $c_p\approx4180\ \text{J/kg}\cdot\text{K}$.
Find. The number of tubes and the number of tube passes that fit the 2.44 m length limit.
Figure 6 — One-shell-pass / two-tube-pass arrangement: the cold tube water traverses the shell twice, giving a correction factor $F$ applied to the counterflow LMTD.
Approach. Fix the duty and hot-water exit from energy balances, compute the counterflow LMTD and its 1-2 correction factor, size the required area, then get the tubes per pass from the velocity constraint and the number of passes from the length limit.
Duty and hot-side exit. $\dot Q=\dot m_c c_p(55-38)=3.8(4180)(17)=270\ \text{kW}$; the hot water leaves at $T_{h,o}=94-\dot Q/(\dot m_h c_p)=94-34.0=60.0\ ^\circ\text{C}$.
Corrected mean temperature difference. Counterflow ends $\Delta T_1=94-55=39$, $\Delta T_2=60-38=22$, so $\text{LMTD}=(39-22)/\ln(39/22)=29.7\ ^\circ\text{C}$. With $P=(55-38)/(94-38)=0.304$ and $R=(94-60)/(55-38)=2.0$, the 1-shell/2-tube factor is $F=0.877$, giving$$\Delta T_m=F\cdot\text{LMTD}=0.877(29.7)=26.0\ ^\circ\text{C}.$$
Tubes per pass from the velocity. Tube-side flow area $A_\text{flow}=\dot m_c/(\rho V)=3.8/(961\cdot0.386)=0.01024\ \text{m}^2$; each tube offers $\pi D^2/4=2.85\times10^{-4}\ \text{m}^2$, so$$N_\text{tube/pass}=\frac{0.01024}{2.85\times10^{-4}}=35.9\approx\boxed{36\ \text{tubes per pass}}.$$
Number of passes from the length limit. The total single-tube length needed is $A/(N\pi D)=7.31/(36\cdot\pi\cdot0.01905)=3.39\ \text{m}$. A single pass would exceed the 2.44 m limit, but two passes of $3.39/2=1.70\ \text{m}$ each fit comfortably:$$\boxed{2\ \text{tube passes},\ 36\ \text{tubes each (72 tubes total)},\ 1.70\ \text{m per pass}.}$$
Check
The two-tube-pass choice is exactly what makes the 1-2 correction factor $F=0.877$ (used above) self-consistent, and the resulting 1.70 m pass length sits under the 2.44 m ceiling with margin. Energy closes: hot side $1.9(4180)(94-60)=270\ \text{kW}=$ cold-side duty. A single pass (3.39 m) would violate the length limit, so two passes are the minimum that works.
