Sizing a Shell-and-Tube Heat Exchanger — Number of Tubes and Passes: 17-Phys-B6 Applied Thermodynamics and Heat Transfer, Undated paper, Q8

Question 8: Sizing a Shell-and-Tube Heat Exchanger — Number of Tubes and Passes

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. A 1-shell-pass, N-tube-pass exchanger heats tube-side water using shell-side water as the hot fluid; the tube-side mass flow and the given velocity/density fix how many tubes must run in parallel, while the required heat-transfer AREA (from the duty and a corrected LMTD) fixes how many passes of the length-limited tubes are needed.

Given data
Quantity	Symbol	Value
Tube-side (heated) flow rate	$\dot m_t$	3.8 kg/s
Tube-side inlet / outlet temperature	$T_{t,i}/T_{t,o}$	$38\,{}^{\circ}\text{C}/55\,{}^{\circ}\text{C}$
Shell-side flow rate, inlet temperature	$\dot m_s,\ T_{s,i}$	1.9 kg/s, $94\,{}^{\circ}\text{C}$
Overall heat-transfer coefficient	$U$	1420 W/m²·K
Tube inside diameter	$d$	1.905 cm
Tube-side density, velocity	$\rho,\ V$	961 kg/m³, 0.386 m/s
Maximum length per pass	$L_{max}$	2.44 m

Find. The number of tubes $N$ and the number of tube passes.

Approach. Get the duty from the tube side, the shell-side outlet temperature from its own energy balance, then the counter-flow LMTD and its 1-shell/N-tube-pass correction factor $F$ to size the total heat-transfer AREA. Separately, the given velocity and density fix how many tubes must run in parallel to carry $\dot m_t$; dividing the required area by that many tubes' available length-per-pass (capped at 2.44 m) gives the number of passes.

Duty and shell-side outlet temperature. $$\dot Q=\dot m_tc_p(T_{t,o}-T_{t,i})=3.8\times4180\times(55-38)$$ $$\boxed{\dot Q=270.1\text{ kW}}$$ $$T_{s,o}=T_{s,i}-\frac{\dot Q}{\dot m_sc_p}=94-\frac{270{,}057}{1.9\times4195}$$ $$\boxed{T_{s,o}=60.1\,{}^{\circ}\text{C}}$$
Counter-flow LMTD and correction factor. $$\Delta T_1=T_{s,i}-T_{t,o}=94-55=39.0,\qquad \Delta T_2=T_{s,o}-T_{t,i}=60.1-38=22.1$$ $$LMTD_{cf}=\frac{\Delta T_1-\Delta T_2}{\ln(\Delta T_1/\Delta T_2)}=\boxed{29.76\,{}^{\circ}\text{C}}$$ $$P=\frac{T_{t,o}-T_{t,i}}{T_{s,i}-T_{t,i}}=0.304,\qquad R=\frac{T_{s,i}-T_{s,o}}{T_{t,o}-T_{t,i}}=1.993$$ Using the standard 1-2 shell-and-tube $F$-chart formula with these $P,R$: $$\boxed{F=0.878}$$
Required total heat-transfer area. $$A_{req}=\frac{\dot Q}{U\,F\,LMTD_{cf}}=\frac{270{,}057}{1420\times0.878\times29.76}$$ $$\boxed{A_{req}=7.28\text{ m}^2}$$
Number of tubes, from the velocity constraint. Each tube-flow-path carries $\dot m_{per\,tube}=\rho V(\pi d^2/4)$: $$\dot m_{per\,tube}=961\times0.386\times\frac{\pi(0.01905)^2}{4}=0.1057\text{ kg/s}$$ $$N=\frac{\dot m_t}{\dot m_{per\,tube}}=\frac{3.8}{0.1057}=35.9\ \rightarrow\ \boxed{N=36\text{ tubes}}$$
Number of passes, from the length limit. $A_{req}=N\times n_{pass}\times\pi d\times L_{pass}$ with $L_{pass}\le L_{max}=2.44$ m: $$n_{pass}\ge\frac{A_{req}}{N\,\pi d\,L_{max}}=\frac{7.28}{36\times\pi\times0.01905\times2.44}=1.38\ \rightarrow\ \boxed{n_{pass}=2\text{ (rounded up, even)}}$$ $$L_{pass}=\frac{A_{req}}{N\,n_{pass}\,\pi d}=\frac{7.28}{36\times2\times\pi\times0.01905}$$ $$\boxed{L_{pass}=1.69\text{ m}\ (<2.44\text{ m}\ \checkmark)}$$

Question 8 — results
Quantity	Value
Heat duty $\dot Q$	270.1 kW
Required area $A_{req}$	7.28 m²
Number of tubes $N$	36
Number of tube passes	2
Actual tube length per pass	1.69 m

Topic. Shell-and-tube heat-exchanger sizing by the LMTD correction-factor method, combined with a continuity constraint on tube count.

Key relations. Counter-flow LMTD; correction factor $F(P,R)$ for a 1-shell/N- tube-pass exchanger; sizing equation $\dot Q=UFA\,LMTD_{cf}$; continuity $\dot m=\rho VA_{cross-section}$.

Why this works. Two INDEPENDENT constraints pin down the two unknowns: the required total AREA comes purely from thermal duty and $U$ (nothing to do with individual tube geometry), while the number of PARALLEL tubes comes purely from continuity at the stated velocity (nothing to do with the thermal duty) — only the final step, dividing area by (tubes × passes × circumference), couples the two and reveals how many passes the length-limited tubes need.

Common pitfall. Rounding the tube count DOWN (35 tubes could not carry the full 3.8 kg/s at the stated velocity) or the pass count to an odd number (a 1-shell-pass exchanger needs an even tube-pass count so the tubes return to the same end); and omitting the correction factor $F$, which would understate the required area by about 12% here.

Source. Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed., Ch. 11 (LMTD method, shell-and-tube correction factors).

Question 8 of 8: Sizing a Sh​ell-and-Tube Heat Exchanger — Number of Tubes and Passes

Question 8: Sizing a Sh​ell-and-Tube Heat Exchanger — Number of Tubes and Passes

Question 8 of 8: Sizing a Shell-and-Tube Heat Exchanger — Number of Tubes and Passes

Question 8: Sizing a Shell-and-Tube Heat Exchanger — Number of Tubes and Passes