Air-Cooling Coil Submerged in an Ice/Water Thermal-Storage Tank: 17-Phys-B6 Applied Thermodynamics and Heat Transfer, Undated paper, Q6

Question 6: Air-Cooling Coil Submerged in an Ice/Water Thermal-Storage Tank (10 marks)

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.

Check: the 0.01 kg/s figure is read as the TOTAL air flow through the array, divided evenly among the 10 parallel tubes (0.001 kg/s each) — the reading that makes both this part's Reynolds number and part (b)'s multi-month melt time land in a physically sensible range for a seasonal ice-storage system; a single tube carrying the full 0.01 kg/s would imply roughly 10× the cooling duty and a correspondingly shorter (roughly one-month) ice supply.

Given. Air is cooled inside ten parallel copper tubes whose OUTSIDE surface sits in a 0°C ice/water bath — a huge effective outside heat-transfer coefficient (latent-heat boiling/melting) compared with the air-side film, so the tube wall is well approximated as isothermal at $T_o=0\,{}^{\circ}\text{C}$.

Given data
Quantity	Symbol	Value
Tube inside diameter	$D$	50 mm
Number of tubes	$N$	10
Total air flow rate	$\dot m_{total}$	0.01 kg/s (0.001 kg/s per tube)
Air inlet / exit temperature	$T_{m,i}/T_{m,o}$	$24\,{}^{\circ}\text{C}\ /\ 14\,{}^{\circ}\text{C}$
Bath temperature	$T_\infty$	$0\,{}^{\circ}\text{C}$ (ice/water)
Tank volume, ice fraction	$V,\ f_{ice}$	10 m³, 80%
Ice density, latent heat of fusion	$\rho_i,\ h_{sf}$	920 kg/m³, $3.34\times10^5$ J/kg

Find. (a) the tube length $L$ per tube; (b) the time to melt all the ice.

Approach. Get the air-side film coefficient from an internal-flow convection correlation at the tube's Reynolds number (checking laminar vs. turbulent first), then size $L$ from the isothermal-wall exponential temperature-decay relation. For part (b), the SAME per-tube duty (an energy balance on the air, not the convection correlation) gives the total heat extracted from the bath, which melts the ice at a rate set by its latent heat.

Per-tube flow and air properties at the mean bulk temperature ($19\,{}^{\circ}\text{C}=292$ K). $$\dot m_{tube}=0.01/10=0.001\text{ kg/s};\quad \mu=1.816\times10^{-5}\text{ Pa}\cdot\text{s},\ k=0.02580\text{ W/m}\cdot\text{K},\ Pr=0.708$$ $$Re=\frac{4\dot m_{tube}}{\pi D\mu}=\frac{4\times0.001}{\pi\times0.05\times1.816\times10^{-5}}$$ $$\boxed{Re=1402\ (\text{laminar}, Re<2300)}$$
Combined entry-length Nusselt number (Hausen correlation, isothermal wall) — solved together with $L$ since the Graetz number depends on $L$. Starting from the fully-developed value $Nu_\infty=3.66$ and iterating $$Nu=3.66+\frac{0.065(D/L)RePr}{1+0.04\left[(D/L)RePr\right]^{2/3}},\qquad L=\frac{-\ln\!\left(\dfrac{T_{m,o}-T_\infty}{T_{m,i}-T_\infty}\right)\dot m_{tube}c_p}{h\,\pi D}$$ converges (13 iterations) to $$\boxed{Nu=5.45,\quad h=2.81\text{ W/m}^2\cdot\text{K},\quad L=1.23\text{ m per tube}}$$ The thermal entry length ($\approx0.05\,Re\,Pr\,D\approx2.5$ m) is comparable to this tube length, confirming the flow is NOT fully developed and the simpler $Nu=3.66$ estimate ($L\approx1.83$ m) would over-size the coil.
Part (a) result and per-tube heat duty. $$q_{tube}=\dot m_{tube}c_p(T_{m,i}-T_{m,o})=0.001\times1006\times(24-14)$$ $$\boxed{q_{tube}=10.06\text{ W per tube}}$$
Part (b) — total cooling duty and ice mass. $$\dot Q_{total}=N\,q_{tube}=10\times10.06=\boxed{100.6\text{ W}}$$ $$m_{ice}=V\,f_{ice}\,\rho_i=10\times0.80\times920=\boxed{7360\text{ kg}}$$
Part (b) — melt time. $$t=\frac{m_{ice}h_{sf}}{\dot Q_{total}}=\frac{7360\times3.34\times10^5}{100.6}$$ $$\boxed{t=2.44\times10^7\text{ s}=283\text{ days}\ (\approx9.3\text{ months})}$$

Question 6 — results
Quantity	Value
Reynolds number (per tube)	1402 (laminar)
Film coefficient $h$	2.81 W/m²·K
Required tube length $L$	1.23 m
Total cooling duty (all 10 tubes)	100.6 W
Time to melt all the ice	283 days

Topic. Internal laminar convection with thermal entry effects, coupled to a latent-heat thermal-storage energy balance.

Key relations. Reynolds number $Re=4\dot m/(\pi D\mu)$; combined entry-length Nusselt correlation (Hausen); isothermal-wall exit-temperature relation $(T_{m,o}-T_\infty)/(T_{m,i}-T_\infty)=\exp(-h\pi DL/\dot mc_p)$; latent-heat melt time $t=m_{ice}h_{sf}/\dot Q$.

Why this works. Checking $Re$ BEFORE reaching for a correlation is what avoids the classic slip of applying Dittus–Boelter (turbulent-only) to a flow that is clearly laminar here; and because the required tube length turns out comparable to the thermal entry length, the fully-developed $Nu=3.66$ shortcut is not accurate enough on its own — the entry-corrected Hausen form is the one that gives a self-consistent (converged) length.

Common pitfall. Applying a turbulent correlation without checking $Re$ first; and using the TOTAL 0.01 kg/s (rather than the per-tube 0.001 kg/s) inside a SINGLE tube's Reynolds number and energy balance, which would badly overstate both the required length and the melt rate.

Source. Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed., Ch. 8 (internal flow, combined entry length); Ch. 6 (property evaluation at the mean bulk temperature).

Question 6 of 8: Air-Cooling Coil Submerged in an Ice/Water Thermal-Storage Tank

Question 6: Air-Cooling Coil Submerged in an Ice/Water Thermal-Storage Tank (10 marks)