A commercial heat pump has CP=3 when the indoor temperature is 20°C and the outdoor temperature is 6°C. How much work is required to operate it if 3*10^6J of heat must be transferred to the room each hour?
To calculate the work required to operate the heat pump, we need to use the equation:
Work = Qh - Qc
where Qh is the amount of heat transferred from the hot reservoir (outdoor environment) to the system (the room) and Qc is the amount of heat transferred from the system to the cold reservoir (the outdoor environment).
Given that the heat transferred to the room each hour (Qh) is 3*10^6J, we need to find the amount of heat transferred from the system to the cold reservoir (Qc).
To find Qc, we can use the formula:
Qc = Qh - W
where W is the work done by the heat pump.
We are also given the coefficient of performance (COP) of the heat pump, which is defined as the ratio of Qh to the work done. In this case, the COP is given as COP = 3.
Using the COP equation, we can rewrite Qh as:
Qh = COP * W
Now let's substitute the given values into the equations:
Qh = 3 * W
Qc = Qh - W
We know that the temperature of the indoor environment (the room) is 20°C and the outdoor environment is 6°C. The temperature difference, ΔT, is the difference between these two values:
ΔT = 20°C - 6°C = 14°C
The formula for Qh in terms of ΔT is:
Qh = Cp * m * ΔT
where Cp is the specific heat capacity of air and m is the mass of the air.
Since we don't have the values for Cp and m, we can't calculate Qh directly. However, we can use the temperature difference and the known value of Qh to find the value of W.
Let's assume that the mass of air is constant for simplicity.
Now we can rewrite Qh as:
Qh = Cp * m * ΔT = 3 * 10^6J
We also know that Cp = 3.
Substituting the values, we get:
3 * 10^6J = 3 * m * 14°C
Simplifying, we find:
m = (3 * 10^6J) / (3 * 14°C)
m = 10^6J / 14°C
Now we can substitute the value of m into our equation for Qh:
Qh = Cp * m * ΔT = 3 * 10^6J
3 * 10^6J = 3 * (10^6J / 14°C) * 14°C
3 * 10^6J = 3 * 10^6J
Since the values are equal, we can conclude that our assumption was correct.
Therefore, the work required to operate the heat pump is:
W = Qh - Qc = Qh - (Qh - W) = Qh - Qh + W
W = 3 * 10^6J - 3 * 10^6J + W
Simplifying, we find:
W = W
Therefore, the work required to operate the heat pump is the same as the heat transferred to the room, which is 3 * 10^6J.
To determine the amount of work required to operate the heat pump, we need to use the equation:
Work = Heat transferred / Coefficient of Performance (CP).
Given that the heat transferred is 3*10^6 J and the CP is 3, we can substitute these values into the equation:
Work = 3*10^6 J / 3.
Simplifying the expression, we find:
Work = 1*10^6 J.
Therefore, the amount of work required to operate the heat pump is 1*10^6 J.