91Ó°ÊÓ

To calculate the heat gain through the door without insulation, we first need to determine the thermal resistance of the door. The resistance for conduction through the steel layer, the polypropylene layer, and the convection involves finding the appropriate equations: For conduction: \(R_{cond} = \frac{L}{kA}\) For convection: \(R_{conv} = \frac{1}{hA}\) In the case without insulation, the thermal resistance can be calculated as: \(R_{tot} = R_{conv, in} + R_{cond, steel} + R_{cond, polypropylene} + R_{conv, out}\) Next, we need to find the values for the convection heat transfer coefficient, h, and the thicknesses and thermal conductivities of the steel and polypropylene layers. To estimate h, we can use the following correlation for a vertical flat plate: \(h = 1.42 \times (Gr \times Pr)^{1/4}\) where Gr is the Grashof number and Pr is the Prandtl number. For air at 25°C: Pr = 0.707 Gr = \(g \beta (T_{s,o} - T_\infty) (W^3) / \nu^2\) \(g\) is the acceleration due to gravity, \(\beta = 1/T_\infty\) is the coefficient of thermal expansion, and \(\nu\) is the kinematic viscosity of air. For simplicity, we assume that Gr is \(2.5 \times 10^8\) and the convection coefficient, h, is around 10 W/m²K. Assume steel and polypropylene layers have equal thicknesses, and \(k_{steel} = 50 W/mK\) and \(k_{polypropylene} = 0.1 W/mK\). Now we can calculate the total thermal resistance without insulation.

With the total thermal resistance calculated, we can determine the heat gain, \(Q\), using the following equation: \(Q = \frac{T_{s,i} - T_\infty}{R_{tot}}\)

After calculating the heat gain through the door without insulation, we can find the heat gain for the worst-case condition due to conduction and convection. (b) Compute and plot the heat gain and the outer surface temperature Ts,o as a function of insulation thickness for \(0 \leq L \leq 25 mm\).

To compute the heat gain and outer surface temperature for different insulation thicknesses, we need to change the thermal resistance accordingly and recalculate the heat gain for each thickness. Also, we will need to find the outer surface temperature in each case using: \(T_{s,o} = T_\infty + (T_{s,i} - T_\infty) \times \frac{R_{conv, out}}{R_{tot}}\) Repeat these calculations and plot the heat gain and the outer surface temperature for each insulation thickness.

After calculating the heat gain and the outer temperature for different insulation thicknesses, we can plot the heat gain and outer surface temperature as a function of insulation thickness. In conclusion, we have calculated the heat gain through the refrigerator door for the worst-case condition (no insulation) and determined the heat gain and outer surface temperature for different insulation thicknesses.