Final answer:
The temperature of the reservoir to which the heat pump rejects heat is 500 K and the rate of heat rejection from the heat pump is 166.5 kW.
Step-by-step explanation:
To find the temperature of the reservoir to which the heat pump rejects heat, we start by calculating the Carnot efficiency of the heat engine. The Carnot efficiency is given by:
Ec = 1 - Tc/Th
where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir. Substituting the given values, we have:
Ec = 1 - 300/1000 = 0.7
Next, we calculate the coefficient of performance (COP) of the heat pump. The COP is given by:
COP = 1/(1 - Ec)
Substituting the value of Ec we just calculated, we have:
COP = 1/(1 - 0.7) = 3.33
The COP is also equal to the ratio of heat extracted from the cold reservoir (Qc) to the work input (W) to the heat pump.
In this case, we are given that the heat extracted from the cold reservoir is twice the heat rejection rate of the heat engine, and the rate of heat supply to the engine is 50 kW. Therefore, Qc = 2 * 50 kW = 100 kW.
We can now use the COP formula to solve for the rate of heat rejection from the heat pump (Qh):
Qh = COP * W = 3.33 * 50 kW = 166.5 kW.
Finally, to find the temperature of the reservoir to which the heat pump rejects heat, we use the equation:
COP = 1 - Tc/Th2
where Th2 is the temperature of the reservoir. Rearranging the equation, we have:
Th2 = Tc / (1 - COP) = 300 / (1 - 3.33) = 500 K
Therefore, the temperature of the reservoir to which the heat pump rejects heat is 500 K and the rate of heat rejection from the heat pump is 166.5 kW.