The emissivity of galvanized steel sheet, a common roofing material, i

Step 1 
Given: 
- steel sheet roof emissivity ${\displaystyle ϵ=0.13}$. 
- steel sheet roof absorptivity ${\displaystyle {\alpha }_s=0.65}$ 
- air temperature ${\displaystyle T_{\mathrm{\infty }}={16}^{\circ }C\dots =289K}$ 
- heat transfer rate ${\displaystyle G_s=750\frac{W}{m^2}}$ 
- natural convection ${\displaystyle h=7\frac{W}{m^2}K}$ 
Total heat transfer rate is the sum of radiation and convection heat transfer rates, multiplied by the absorptivity factor:
${\displaystyle \alpha G_s=q_{conv}^{ }+q_{rad}^{ }=hA(T_{\mathrm{\infty }}-T_m)+ϵ\sigma A(T_{sur}^4-T_s^4)}$ 
${\displaystyle \alpha G_s-h(T_s-T_{\mathrm{\infty }})-ϵ\sigma (T_s^4-T_{sur}^4)=0}$ 
${\displaystyle 0.65\cdot 750+7\cdot 289-7T_s-7.371\cdot {10}^{-9}{T}_s^4+7.371\cdot {10}^{-9}\cdot 289=0}$ 
${\displaystyle 2562-7T_s-7.371\cdot {10}^{-9}{T}_s^4=0}$ 
Use iteration method to solve for ${\displaystyle T_s}$: 
${\displaystyle T_s=350.17K\dots ={77.17}^{\circ }C}$ 
In this situation, the temperature of the roof's surface is too high for a cat to feel comfortable walking on. ${\displaystyle {77.17}^{\circ }C}$ which may even cause burns to its paws. 
Convection and radiation transfer rates together make up the total heat transfer rate, multiplied with absorptivity factor. Using the iterative method, you can use that equation to determine the temperature of the roof.

Step 1
Write the emissivity of common roofing material.
${\displaystyle ϵ=0.13}$
Write the absorptivity of solar radiation.
${\displaystyle {\alpha }_s=0.65}$
Write the temperature of the surrounding of the roofing material.
${\displaystyle T\mathrm{\infty }={16}^{\circ }C}$
${\displaystyle =16+273K}$
${\displaystyle =289K}$
Write the heat transfer rate of roofing material.
${\displaystyle G_s=750\frac{W}{m^2}}$
Write the convective heat coefficient of the surrounding.
${\displaystyle h=7\frac{W}{m^2}K}$
Step 2
In this problem, the heat transfer from the roofing material will be equal to the sum of heat due to convection and radiation.
Write the expression of heat transfer.
${\displaystyle \alpha G_s=h(T_s-T_{\mathrm{\infty }})+ϵ\sigma (T_s^4-T_{\mathrm{\infty }}^4)}$
Substitute the given values and make an expression of the surface temperature of the roof.
${\displaystyle \alpha G_s=h(T_s-T_{\mathrm{\infty }})+ϵ\sigma (T_s^4-T_{\mathrm{\infty }}^4)}$
${\displaystyle \left(0.65\right)\left(450\frac{W}{m^2}\right)=\left(7\frac{W}{m^2}K\right)(T_s-289K)+\left(0.13\right)(5.67\times {10}^{-8})(T_s^4-{289}^4)}$
${\displaystyle 7.371\times {10}^{-9}{T}_s^4-51.42+7T_s-2023-487.5=0}$
${\displaystyle 7.371\times {10}^{-9}{T}_s^4+7T_s-2561.92=0}$
Step 3
From the above expression, calculate the surface temperature of the roof.
${\displaystyle 7.371\times {10}^{-9}{T}_s^4+7T_s-2561.92=0}$
${\displaystyle T_s=350.16K}$
${\displaystyle =\left(350.16K\right)-273K}$
${\displaystyle T_s={77.16}^{\circ }C}$
The surface temperature of the roof is very high so it will not be comfortable for a cat to walk on the roof. The temperature for comfort ranges from 20 to ${30}^{\circ }C$.