Question

To defog the rear window of an automobile, a very thin transparent heating element is attached to the inner surface of the window. A uniform heat flux of \(1300 \mathrm{~W} / \mathrm{m}^{2}\) is provided to the heating element for defogging a rear window with thickness of \(5 \mathrm{~mm}\). The interior temperature of the automobile is \(22^{\circ} \mathrm{C}\), and the convection heat transfer coefficient is $15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. The outside ambient temperature is \)-5^{\circ} \mathrm{C}$, and the convection heat transfer coefficient is $100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. If the thermal conductivity of the window is \)1.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, determine the inner surface temperature of the window.

Short answer

Based on the provided information and using heat transfer concepts, the inner surface temperature of the rear window of an automobile is calculated to be approximately 12.1°C.

Step-by-step solution

Determine Heat Transfer through the Window

First, we need to determine the heat transfer through the window. We are given the heat flux and can thus calculate the total heat transfer by multiplying the heat flux by the area of the window. It is given by: $$Q =q''A$$ Here, we are only asked to find the inner surface temperature of the window, so we don't need to find the exact value for the heat transfer Q. The ratios will remain constant for the steps we're going to calculate.

Calculate Heat Transfer by Conduction

To find the inner surface temperature of the window, we first need to determine the heat transfer by conduction that occurs through the window. This can be calculated using Fourier's law: $$q''_{cond} = \frac{k(T_i - T_o)}{L}$$ Where \(q''_{cond}\) is the conductive heat flux, \(k\) is the thermal conductivity \((1.2 \ \mathrm{W / m \cdot K})\), \(T_i\) is the inner surface temperature of the window, \(T_o\) is the outer surface temperature of the window, and \(L\) is the thickness of the window \((5 \times 10^{-3} \ \mathrm{m})\).

Calculate Heat Transfer by Convection (Interiors and Exteriors)

Next, we need to determine the heat transfer by convection that occurs on both the inside and outside of the window. This can be calculated using Newton's law of cooling: $$q''_{conv,i} = h_i(T_{in} - T_i)$$ $$q''_{conv,o} = h_o(T_o - T_{amb})$$ Where \(q''_{conv,i}\) is the convective heat flux at the inner surface, \(q''_{conv,o}\) is the convective heat flux at the outer surface, \(h_i\) is the convection heat transfer coefficient for the interior \((15 \ \mathrm{W / m^2 \cdot K})\), \(h_o\) is the convection heat transfer coefficient for the exterior \((100 \ \mathrm{W / m^2 \cdot K})\), \(T_{in}\) is the interior temperature of the automobile \((22^\circ \ \mathrm{C})\), and \(T_{amb}\) is the outside ambient temperature \((-5^\circ \ \mathrm{C})\).

Combine the Equations and Solve for Inner Surface Temperature

At steady-state, the heat transfer through the convection processes equals the heat transfer through the conduction process. Therefore: $$q''_{cond} = q''_{conv,i} + q''_{conv,o}$$ Substituting the equations for heat transfer from steps 2 and 3 into the above equation and solving for the inner surface temperature, we get: $$\frac{k(T_i - T_o)}{L} = h_i(T_{in} - T_i) + h_o(T_o - T_{amb})$$ We know that \(q''_{cond} = q''\), so we get: $$\frac{k(T_i - T_o)}{L} = q''$$ Now, we substitute this back into the equation and solve for the inner surface temperature: $$q'' = h_i(T_{in} - T_i) + h_o \left(\frac{Lq''}{k} + T_i -T_{amb} \right)$$ Plugging in the given values and solving for \(T_i\), we find that the inner surface temperature of the window is approximately: $$T_i = 12.1^\circ \ \mathrm{C}$$