22-Mec-A1 Applied Thermodynamics and Heat Transfer · May 2018
Question 6 of 8: Water cooled in a tube at constant wall temperature
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format: National Examination 16-Mec-A1, 3 hours, open book. Eight questions of equal value: 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).
Reference texts: Çengel & Boles, Thermodynamics: An Engineering Approach (9th ed., McGraw-Hill) — first-law cycle analysis, wet-region steam properties, the ideal regenerative Rankine cycle with a closed feedwater heater, the air-standard Diesel cycle, and isentropic-efficiency compressor analysis; Çengel & Ghajar, Heat and Mass Transfer (6th ed.) and Incropera, DeWitt, Bergman & Lavine, Fundamentals of Heat and Mass Transfer (8th ed., Wiley) — steady conduction with internal generation in a composite cylinder, internal-flow convection at constant wall temperature, combined convection–radiation surface balances, and the LMTD method for counterflow heat exchangers. Water/steam properties are IAPWS-consistent; air is treated as an ideal gas with constant specific heats.
Question 6 — Water cooled in a tube at constant wall temperature (Part B, equal value)
Given. Water cooled in a constant-wall-temperature tube; properties at the mean bulk temperature $T_b=(40+6)/2=23\ ^\circ\text{C}$: $c_p=4182\ \text{J/kg}\cdot\text{K}$, $k=0.607\ \text{W/m}\cdot\text{K}$, $\mu=9.3\times10^{-4}\ \text{Pa}\cdot\text{s}$, $Pr\approx6.4$.
Quantity
Value
Mass flow, $\dot m$
0.01 kg/s
Tube diameter, $D$
1 cm = 0.01 m
Wall temperature, $T_s$
0 °C
Inlet / outlet water temp
40 °C / 6 °C
Flow condition
fully developed (hydrodynamically & thermally)
Find. the total heat-transfer rate and the required tube length.
Figure 6 — Water flows through a thin-walled tube held at 0 °C by the ice bath; its bulk temperature decays exponentially from 40 °C toward the wall temperature, reaching 6 °C at the outlet.
Approach. Get the duty from the bulk-temperature drop, classify the flow (laminar) to set $Nu=3.66$ and hence $h$, then use the constant-wall-temperature exponential-decay relation to solve for length.
Heat-transfer rate (energy balance on the water).
$$\dot Q=\dot m\,c_p(T_i-T_o)=0.01(4182)(40-6)=1422\ \text{W}$$
$\dot Q\approx 1.42$ kW
Flow regime. The Reynolds number for round-tube flow is
$$Re_D=\frac{4\dot m}{\pi D\mu}=\frac{4(0.01)}{\pi(0.01)(9.3\times10^{-4})}=1.37\times10^{3}<2300$$
so the flow is laminar.
Convection coefficient (fully developed, constant $T_s$). For laminar fully-developed flow at constant wall temperature $Nu_D=3.66$:
$$h=\frac{Nu_D\,k}{D}=\frac{3.66(0.607)}{0.01}=221\ \text{W/m}^2\text{}\cdot\text{K}$$
Required length (exponential decay toward $T_s$). With constant wall temperature,
$$\frac{T_o-T_s}{T_i-T_s}=\exp\!\left(-\frac{h\,\pi D L}{\dot m c_p}\right)\;\Rightarrow\;L=\frac{\dot m c_p}{h\,\pi D}\ln\!\frac{T_i-T_s}{T_o-T_s}$$
$$L=\frac{0.01(4182)}{221\,\pi(0.01)}\ln\!\frac{40-0}{6-0}=\frac{41.82}{6.94}\,(1.897)=11.4\ \text{m}$$
$L\approx 11.4$ m
Check — long tube.
The 11.4 m length is genuinely large: laminar water flow gives a modest $h\approx220\ \text{W/m}^2\text{}\cdot\text{K}$, so a long surface is needed to pull the stream from 40 °C down to 6 °C. Using $Nu=3.66$ is appropriate because the problem states the flow is fully developed both hydrodynamically and thermally; a developing-flow correlation would give a somewhat higher average $h$ and a shorter tube.
