Carnot refrigerator A has a 21 % higher coefficient of performance than Carnot refrigerator B. The temperature difference between the hot and cold reservoirs is 30 % greater for B than for A.

Question

Carnot refrigerator A has a 21 % higher coefficient of performance than Carnot refrigerator B. The temperature difference between the hot and cold reservoirs is 30 % greater for B than for A.

If the cold-reservoir temperature for refrigerator B is 150 K, what is the cold-reservoir temperature for refrigerator A?

Answer 1

To solve this problem, we first need to understand the concept of the coefficient of performance (COP) for Carnot refrigerators and the relationship between temperature differences and COP.

The coefficient of performance (COP) for a Carnot refrigerator is defined as the ratio of the desired output (in this case, the amount of heat removed from the cold reservoir) to the required input (the work done on the refrigerator).

COP = Heat Removed / Work Done

For Carnot refrigerators, the COP is given by the formula:

COP = Tc / (Th - Tc)

where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir.

Now, let's analyze the information provided in the question:

1. Carnot refrigerator A has a 21% higher COP than Carnot refrigerator B.
Mathematically, this can be written as:
COP(A) = 1.21 * COP(B)

2. The temperature difference between the hot and cold reservoirs is 30% greater for B than for A.
Mathematically, this can be written as:
Th(B) - Tc(B) = 1.30 * (Th(A) - Tc(A))

From the given information, we know that Tc(B) = 150 K. We need to find Tc(A).

Let's solve these equations to find the value of Tc(A):

Using the first equation:
COP(A) = 1.21 * COP(B)

Substituting the formulas for COP(A) and COP(B):
Tc(A) / (Th(A) - Tc(A)) = 1.21 * (Tc(B) / (Th(B) - Tc(B)))

Substituting Tc(B) = 150 K:
Tc(A) / (Th(A) - Tc(A)) = 1.21 * (150 K / (Th(B) - 150 K))

Using the second equation:
Th(B) - Tc(B) = 1.30 * (Th(A) - Tc(A))

Substituting Th(B) = Tc(B) + 150 K:
(Tc(B) + 150 K) - Tc(B) = 1.30 * (Th(A) - Tc(A))

Simplifying the equation:
150 K = 1.30 * (Th(A) - Tc(A))

Now we have two equations with two unknowns (Tc(A) and Th(A)). We can solve these equations simultaneously to find the value of Tc(A):

Tc(A) / (Th(A) - Tc(A)) = 1.21 * (150 K / (Tc(B) + 150 K))

150 K = 1.30 * (Th(A) - Tc(A))

To solve these equations, we need to know the value of Tc(B). Since Tc(B) is not given in the question, we cannot determine the value of Tc(A) without this information.