The average thermal conductivity of the walls (including
windows) and roof of a house in Figure P11.32 is
4.8 � 10�4 kW/m� °C, and their average thickness is
21.0 cm. The house is heated with natural gas, with a heat
of combustion (energy released per cubic meter of gas
burned) of 9300 kcal/m3. How many cubic meters of gas
must be burned each day to maintain an inside temperature of 25.0°C if the outside temperature is 0.0°C?
Disregard radiation and energy loss by heat through the
ground.
To solve this problem, we can use the formula for heat transfer through a material:
Q = (k * A * ΔT) / L
where:
Q is the heat transferred (in Watts)
k is the thermal conductivity of the material (in kW/m・°C)
A is the surface area of the material (in m²)
ΔT is the temperature difference between the inside and outside (in °C)
L is the thickness of the material (in m)
First, let's calculate the surface area of the walls and roof of the house. Assuming the house is a rectangular prism, we can calculate the surface area using the formula:
A = 2 * (length * width + length * height + width * height)
Given the dimensions in the figure, let's assume the length = 10m, width = 6m, and height = 4m.
A = 2 * (10 * 6 + 10 * 4 + 6 * 4) = 212 m²
Next, let's calculate the temperature difference:
ΔT = inside temperature - outside temperature = 25.0°C - 0.0°C = 25.0°C
Now, let's calculate the volume of gas burned each day to maintain the inside temperature:
Q = (k * A * ΔT) / L
Q = (4.8 * 10^4 kW/m・°C * 212 m² * 25.0°C) / (0.21 m)
Q = 2,545,714.29 kW
Since the energy released per cubic meter of gas burned is given as 9300 kcal/m^3, we need to convert the units to kW:
1 kcal = 4.184 kJ
1 J = 1 W・s
So, 9300 kcal/m^3 = (9300 * 4.184) kJ/m^3 = 38815.2 kJ/m^3 = 38.8152 MJ/m^3 = 38.8152 * 10^3 kW・s/m^3
Now, let's calculate the volume of gas burned each day:
Volume of gas burned = Energy required / Energy released per cubic meter
Volume of gas burned = Q / (38.8152 * 10^3 kW・s/m^3)
Volume of gas burned = 2,545,714.29 kW / (38.8152 * 10^3 kW・s/m^3) = 65.54 m^3
Therefore, approximately 65.54 cubic meters of gas must be burned each day to maintain an inside temperature of 25.0°C when the outside temperature is 0.0°C.
To determine the amount of gas that must be burned each day, we need to calculate the heat transfer from the house to the outside.
First, calculate the temperature difference between the inside and outside:
ΔT = inside temperature - outside temperature
ΔT = 25.0°C - 0.0°C
ΔT = 25.0°C
Next, calculate the heat transfer rate (Q) from the house using the formula:
Q = (k * A * ΔT) / d
Where:
k = thermal conductivity (4.8 x 10^4 kW/m·°C)
A = surface area of the walls and roof
ΔT = temperature difference (25.0°C)
d = average thickness (21.0 cm)
We need to convert the thickness from cm to meters:
d = 21.0 cm / 100
d = 0.21 m
Now, we need to find the surface area of the walls and roof. Since the figure is not provided, we cannot calculate the exact area, but we can assume a value. Let's assume the surface area is 100 m².
Substitute the values into the formula:
Q = (4.8 x 10^4 kW/m·°C) * (100 m²) * (25.0°C) / (0.21 m)
Calculate Q:
Q = 571,428.57 kW
Now, we need to convert the heat transfer rate from kilowatts to calories:
Q = 571,428.57 kW * 859.85 kcal/kW
Q = 491,520,000 kcal
Next, we need to calculate the volume of gas burned per day:
Gas volume = Q / energy released by 1 cubic meter of gas
Conversion from kcal to cal:
1 kcal = 1000 cal
The energy released by 1 cubic meter of gas is given as 9300 kcal/m³, which is equivalent to 9300 x 1000 cal/m³ = 9,300,000 cal/m³.
Gas volume = (491,520,000 cal) / (9,300,000 cal/m³)
Calculate the gas volume:
Gas volume = 52.80 m³
Therefore, approximately 52.80 cubic meters of gas must be burned each day to maintain an inside temperature of 25.0°C when the outside temperature is 0.0°C (disregarding radiation and energy loss through the ground).