Exit Gas Temperature in a Ceramic-Lined Duct: 22-Mec-A1 Applied Thermodynamics and Heat Transfer, December 2016, Q6

Question 6: Exit Gas Temperature in a Ceramic-Lined Duct (equal value)

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.

Quantity	Value
Duct side / length	0.3 m / 30 m
Wall / ceramic thickness	2 mm steel, 38 mm ceramic
Inside / outside coefficients	$h_i=100$, $h_o=10$ W/m²·°C
Conductivities	$k_\text{steel}=25$, $k_\text{ceramic}=0.2$ W/m·°C
Gas flow / $c_p$ / inlet	1.5 kg/s / 1100 J/kg·°C / 800 °C
Surroundings	20 °C

Figure 6 — Duct-wall build-up (inner film → 2 mm steel → 38 mm ceramic → outer film). The ceramic and the outer film dominate the resistance; the gas cools along the 30 m run following a single-stream exponential decay.

Approach. Add the four series resistances (inside film, steel, ceramic, outside film) to get the overall $U$, then march the single gas stream along the duct with the exponential temperature-decay relation.

Overall coefficient. On a per-unit-area (thin-wall) basis, $$\frac1U=\frac1{h_i}+\frac{t_s}{k_s}+\frac{t_c}{k_c}+\frac1{h_o}=0.01+0.00008+0.19+0.10=0.300\ \text{m}^2\text{}\cdot\text{°C/W}$$ ==**$U\approx3.33$ W/m²·°C** (the ceramic, 0.19, and outer film, 0.10, dominate).==
Conductance of the duct. Inner surface area $A_s=4(0.3)(30)=36$ m², so $UA_s=3.33\times36=120$ W/°C.
Single-stream temperature decay. With $\dot m c_p=1.5\times1100=1650$ W/°C, $$T_\text{out}=T_\infty+(T_\text{in}-T_\infty)\,e^{-UA_s/\dot m c_p}=20+(800-20)\,e^{-120/1650}$$ $$=20+780\,e^{-0.0727}=20+725.3$$ ==**$T_\text{out}\approx745$ °C.**==

Check

Thin-wall plane resistances on the inner area are used; carrying the true logarithmic/area corrections for the square section (outer perimeter 1.52 m vs inner 1.2 m) lowers the exit temperature by only ~1 °C (to ≈ 744 °C). The gas cools just 55 °C over 30 m because the ceramic makes the wall a good insulator — the duct loses little of its thermal energy.

Quantity	Result
Overall coefficient $U$	≈ 3.33 W/m²·°C
Duct conductance $UA_s$	≈ 120 W/°C
Exit bulk gas temperature	≈ 745 °C

Concept & Theory

Topic. Single-stream internal flow losing heat through a multilayer wall to a fixed ambient.

Key relations. Series resistance $1/U=1/h_i+\sum t/k+1/h_o$; single-stream decay $\dfrac{T_\text{out}-T_\infty}{T_\text{in}-T_\infty}=e^{-UA_s/\dot m c_p}$.

Why this works. With a constant outer environment and constant $U$, an energy balance on a differential length integrates to an exponential approach of the bulk temperature toward the ambient, governed by the number of transfer units $UA_s/\dot m c_p$. Here that group is small (0.073), so the gas barely cools.

Common pitfall. Treating the duct wall (not the surroundings) as the sink and using an LMTD between gas and wall, or omitting the outside convective film — which here is 33 % of the total resistance.

Source. Incropera et al., Fundamentals of Heat and Mass Transfer §8.3 (the energy balance and constant-surroundings exponential temperature variation).

Question 6 of 8: Exit Gas Temperature in a Ceramic-Lined Duct

Question 6: Exit Gas Temperature in a Ceramic-Lined Duct (equal value)