You have a kiln where theinterior is a cube with an edge length of 30 cm. The walls of the kiln are made of an insulating refactory, and are 10 cm thick. The thermal conductivity of the refractory is .06 W/mK. The interior of the kiln is at a temperature of 1250 degrees C, and the kiln is surrounded by still air at 35 degres C. For these conditions:
a) Estimate the surface temperature of the vertical walls next to the still air in degrees C
b) Estimate the heat flux through the vertical walls in W/m^2
To estimate the surface temperature of the vertical walls next to the still air, we can use the concept of steady-state heat transfer through the kiln walls. The heat transfer equation in this case is:
Q = (k * A * ΔT) / d
Where:
- Q is the heat flux (W/m^2), which is the rate of heat transfer per unit area,
- k is the thermal conductivity of the refractory (0.06 W/mK),
- A is the surface area of the kiln walls (taking into account all four walls),
- ΔT is the temperature difference between the kiln interior and the still air outside the kiln, and
- d is the thickness of the kiln walls (10 cm).
a) To estimate the surface temperature of the vertical walls next to the still air, we need to find the temperature difference (ΔT) in degrees Celsius.
ΔT = T_interior - T_air
Where:
- T_interior is the temperature of the kiln interior, given as 1250 degrees Celsius,
- T_air is the temperature of the still air outside the kiln, given as 35 degrees Celsius.
ΔT = 1250 - 35 = 1215 degrees Celsius
Now, let's calculate the surface temperature (T_surface) using the rearranged heat transfer equation:
T_surface = (Q * d) / (k * A) + T_air
Since we don't know the heat flux (Q) or the surface temperature (T_surface) yet, we can leave them as variables for now.
b) To estimate the heat flux through the vertical walls, we can rearrange the heat transfer equation as follows:
Q = (k * A * ΔT) / d
However, we need the surface area (A) to calculate the heat flux. Since the kiln is a cube with an edge length of 30 cm, each wall has an area of (30 cm * 30 cm) = 900 cm^2. Since there are four walls, the total surface area (A) is:
A = 4 * 900 cm^2 = 3600 cm^2 = 0.36 m^2 (converting from cm^2 to m^2).
Now, we can substitute the values into the heat transfer equation:
Q = (0.06 W/mK * 0.36 m^2 * 1215 degrees Celsius) / (0.1 m)
Simplifying:
Q = 2.196 W
Therefore, the estimated heat flux through the vertical walls is approximately 2.196 W/m^2.
To find the surface temperature (T_surface), we can substitute the calculated heat flux (Q) into the rearranged equation:
T_surface = (2.196 W * 0.1 m) / (0.06 W/mK * 0.36 m^2) + 35 degrees Celsius
Simplifying:
T_surface ≈ 131.37 degrees Celsius
So, the estimated surface temperature of the vertical walls next to the still air is approximately 131.37 degrees Celsius.