The door of a domestic refrigerates has an area of 1.7 m2 & it basically consists of a thin metal sheet with a 25 mm thick layer of insulation on the inside. The thermal conductivity of this insulation is 0.25 W / m-deg & heat transfer on each side of the door is 10 W / m2-deg. Determine the heat flow rate through the door & the temperature of the metal sheet. The refrigerated chamber & the room are at 0° C & 20° C respectively. Neglect thermal resistance due to the sheet metal.
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To determine the heat flow rate through the door, we can use the formula for heat transfer:
Q = A * ΔT / R
Where:
Q is the heat flow rate
A is the area of the door
ΔT is the temperature difference between the refrigerated chamber and the room
R is the total thermal resistance
First, let's calculate the thermal resistance:
R = (L1 / k1) + (L2 / k2)
Where:
L1 is the thickness of the insulation
k1 is the thermal conductivity of the insulation
L2 is the thickness of the metal sheet
k2 is the thermal conductivity of the metal sheet
Given:
L1 = 25 mm = 0.025 m
k1 = 0.25 W / m-deg
L2 = 0 (neglecting the resistance due to the metal sheet)
k2 = unknown (to be determined)
Using these values, we can calculate the total thermal resistance:
R = (0.025 / 0.25) + (0 / k2)
R = 0.1 + 0
R = 0.1
Now, let's calculate the temperature difference:
ΔT = T1 - T2
Where:
T1 is the temperature of the refrigerated chamber = 0° C = 273 K
T2 is the temperature of the room = 20° C = 293 K
ΔT = 273 - 293
ΔT = -20
Since the temperature difference is negative, we will consider its magnitude:
ΔT = |-20| = 20
Now, substitute the values into the formula for heat flow rate:
Q = (1.7 * 20) / 0.1
Q = 34 / 0.1
Q = 340 W
Therefore, the heat flow rate through the door is 340 W.
To find the temperature of the metal sheet, we can rearrange the formula for heat transfer:
Q = A * ΔT / R
Solve for ΔT:
ΔT = Q * R / A
Substituting the values:
ΔT = 340 * 0.1 / 1.7
ΔT = 34 / 1.7
ΔT = 20
Since ΔT = T1 - T2, we can find T2:
T2 = T1 - ΔT
T2 = 273 - 20
T2 = 253 K
Converting back to Celsius:
T2 = 253 - 273
T2 = -20 °C
Therefore, the temperature of the metal sheet is -20 °C.