The temperature of an electronic oven is 160 °C. The temperature at the outer surface of the oven in the kitchen is 50 °C. The oven (surface area = 1.6 m2) is insulated with material that has a thickness of 0.020 m and a thermal conductivity of 0.045 J/(s m C°). How much energy will be used to operate the oven over six hours?
To calculate the energy used to operate the oven over six hours, we need to determine the rate of heat transfer through the insulation and then multiply it by the time (six hours) to get the total energy.
First, we need to calculate the rate of heat transfer (Q) using Fourier's law of heat conduction:
Q = k * A * (θ1 - θ2) / d
where:
Q = rate of heat transfer
k = thermal conductivity
A = surface area of the oven
θ1 = temperature inside the oven
θ2 = temperature outside the oven
d = thickness of the insulation
Given:
k = 0.045 J/(s m C°)
A = 1.6 m^2
θ1 = 160 °C
θ2 = 50 °C
d = 0.020 m
Now, substitute the given values into the equation:
Q = 0.045 * 1.6 * (160 - 50) / 0.020
Q = 0.045 * 1.6 * 110 / 0.020
Q = 0.045 * 1.6 * 5500
Q = 396 J/s
The rate of heat transfer through the insulation is 396 J/s.
To calculate the total energy used over six hours, we need to convert the time from hours to seconds:
6 hours = 6 * 60 * 60 seconds
Now, multiply the rate of heat transfer by the time:
Total energy = Q * time
Total energy = 396 J/s * (6 * 60 * 60) s
Total energy = 396 J/s * 21600 s
Total energy = 8,553,600 J
Therefore, the energy used to operate the oven over six hours is 8,553,600 Joules.
To determine the energy used to operate the oven over six hours, you need to calculate the heat transfer through the insulation.
The formula for heat transfer is given by:
Q = (k * A * ΔT * t) / d
Where:
Q = Heat transfer (in Joules)
k = Thermal conductivity of the material (in J/(s m C°))
A = Surface area (in m^2)
ΔT = Temperature difference (in C°)
t = Time (in seconds)
d = Thickness of the material (in meters)
Let's calculate the energy used step-by-step:
1. Convert the thickness of the material to meters:
Thickness (m) = 0.020 m
2. Convert the time to seconds:
Time (s) = 6 hours * 60 minutes/hour * 60 seconds/minute = 21,600 s
3. Calculate the temperature difference:
ΔT = Temperature at the inner surface - Temperature at the outer surface
ΔT = 160 °C - 50 °C = 110 °C
4. Substitute the values into the heat transfer formula:
Q = (k * A * ΔT * t) / d
Q = (0.045 J/(s m C°) * 1.6 m^2 * 110 °C * 21,600 s) / 0.020 m
5. Calculate the energy used:
Q = (0.045 J/(s m C°) * 1.6 m^2 * 110 °C * 21,600 s) / 0.020 m
Q ≈ 20,376,000 J
Therefore, approximately 20,376,000 Joules (or 20.376 MJ) of energy will be used to operate the oven over six hours.