Minimum Composite-Window Thickness for a Self-Cleaning Oven: 17-Phys-B6 Applied Thermodynamics and Heat Transfer, Undated paper, Q5

Question 5: Minimum Composite-Window Thickness for a Self-Cleaning Oven (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 printed opening line names the oven-air temperature as $T_a=400\,{}^{\circ}\text{C}$, but the SAME sentence a few words later states the wall and air temperature that is actually used throughout the rest of the problem is $T_w=T_a=500\,{}^{\circ}\text{C}$.

Given. A two-layer plastic window separates the $500\,{}^{\circ}\text{C}$ oven interior from $25\,{}^{\circ}\text{C}$ room air; convection AND radiation act in PARALLEL on the hot (inside) face (same driving $\Delta T$, so their coefficients simply add), while only convection acts on the cool (outside) face.

Given data
Quantity	Symbol	Value
Oven air / wall temperature	$T_a=T_w$	$500\,{}^{\circ}\text{C}$
Room air temperature	$T_o$	$25\,{}^{\circ}\text{C}$
Max. outside surface temperature	$T_{s,o}$	$50\,{}^{\circ}\text{C}$
Inside convection coefficient	$h_{ci}$	25 W/m²·K
Inside radiation coefficient	$h_r$	25 W/m²·K
Outside convection coefficient	$h_\infty$	25 W/m²·K
Thermal conductivity, layer A	$k_A$	0.18 W/m·K
Thermal conductivity, layer B	$k_B$	0.10 W/m·K
Thickness relation	$L_A$	$3L_B$

Composite plane wall: parallel convection+radiation on the hot face, series conduction through A then B, convection alone on the cool face.

Find. The minimum total window thickness $L_A+L_B$ that keeps $T_{s,o}\le50\,{}^{\circ}\text{C}$.

Approach. At the LIMITING design point $T_{s,o}=50\,{}^{\circ}\text{C}$ exactly, the outside convection alone fixes the heat flux; the same flux must cross the inside-convection resistance and both conduction resistances in series between the oven and that outside surface, which is one equation in the one unknown $L_B$ (via $L_A=3L_B$).

Heat flux set by the outside-surface limit. $$q''=h_\infty(T_{s,o}-T_o)=25\times(50-25)$$ $$\boxed{q''=625\text{ W/m}^2}$$
Total resistance needed between the oven and the outside surface. $$R''_{total}=\frac{T_w-T_{s,o}}{q''}=\frac{500-50}{625}$$ $$\boxed{R''_{total}=0.720\text{ m}^2\cdot\text{K/W}}$$
Subtract the inside (parallel convection+radiation) resistance. $$R''_{conv,in}=\frac{1}{h_{ci}+h_r}=\frac{1}{50}=0.020\text{ m}^2\cdot\text{K/W}$$ $$R''_{cond,budget}=0.720-0.020=\boxed{0.700\text{ m}^2\cdot\text{K/W}}$$
Solve the two-layer conduction budget for $L_B$ (with $L_A=3L_B$). $$\frac{L_A}{k_A}+\frac{L_B}{k_B}=\frac{3L_B}{0.18}+\frac{L_B}{0.10}=L_B\left(16.67+10.0\right)=0.700$$ $$\boxed{L_B=26.25\text{ mm}}$$ $$L_A=3L_B=\boxed{78.75\text{ mm}}$$

Question 5 — results
Quantity	Value
Thickness of B, $L_B$	26.25 mm
Thickness of A, $L_A$	78.75 mm
Minimum total thickness $L_A+L_B$	105.0 mm

Topic. Steady one-dimensional composite-wall conduction with combined convection+radiation on one face and convection alone on the other, solved as a resistance network.

Key relations. Series thermal resistance $R''=\sum L_i/k_i+1/h$; parallel surface coefficients simply add when both act over the SAME $\Delta T$ ($h_{combined}=h_{conv}+h_{rad}$).

Why this works. Fixing the design point exactly at $T_{s,o}=50\,{}^{\circ}\text{C}$ turns an inequality constraint into a single equality, so the heat flux is known immediately from the outside boundary condition alone (independent of the plastic layers); everything upstream of that surface is then just Ohm's-law-style resistances in series, one linear equation in one unknown once $L_A=3L_B$ is substituted.

Common pitfall. Adding $h_{ci}$ and $h_r$ as if they were resistances in SERIES (they act in parallel on the same surface, so their COEFFICIENTS add, not their resistances); and forgetting the inside convective/radiative resistance term entirely, which would under-design the window.

Source. Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed., Ch. 3 (composite plane-wall conduction, thermal resistance networks).

Question 5 of 8: Minimum Composite-Window Thickness for a Self-Cleaning Oven

Question 5: Minimum Composite-Window Thickness for a Self-Cleaning Oven (10 marks)