Question

A solid plate, with a thickness of 15 cm and a thermal conductivity of 80 W/m·K, is being cooled at the upper surface by air. The air temperature is 10°C, while the temperatures at the upper and lower surfaces of the plate are 650 50 and 60°C, respectively. Determine the convection heat transfer coefficient of air at the upper surface and discuss whether the value is reasonable or not for force convection of air.

Answer 1

To solve this problem we will use the two principles that are visible according to the phenomena described in the problem: Heat transfer by conductivity and Heat transfer by convection.

This thermal transfer will be equivalent and with it we can find the value asked.

Note: We will assume that the temperature value at the plate surface is: 60 ° C (For the given value of 650 50)

For Thermal Transfer by Conduction

$Q_{cn} = -kA \frac{\Delta T}{\Delta x}$

$Q_{cn} = -kA \frac{T_1-T_2}{L}$

Where,

k = Thermal conductivity

A = Cross-sectional Area

$T_2$ = Temperature of the bottom surface

$T_1$= Temperature of the top surface

L = Length

Replacing we have that

$Q_{cn} = -(80W\cdot K)(A)\frac{50\°C-60\°C}{15cm\frac{1m}{100cm}}$

$Q_{cn} = 5333.33A$

For Thermal Transfer by Convection

$Q_{cv} = hA(T_1-T_{\infty})$

Where,

h = Convection heat transfer coefficient

$T_{\infty}$= Surrounding temperature

A = Surface Area

Replacing we have that

$Q_{cv} = hA(50\°C-10\°C)$

$Q_{cv} = 40hA$

Since the rate of heat transfer by convection is equal to that given by conduction we have to:

$Q_{cn}=Q_{cv}$

$5333.33A = 40hA$

$h = 133.33W/m^2\cdot K$

It is stated that the typical values of forced convection of gases lies in the range of $(25-250)W/m^2\cdot K$. The obtained value is reasonable for forced convection of air.