Problem HW4.1
Two reversible refrigeration cycles are arranged in series. The first cycle receives energy by heat transfer from a cold reservoir at temperature $T_C$ and rejects energy by heat transfer to a reservoir at an intermediate temperature $T$ greater than $T_C$. The second cycle receives energy by heat transfer from the reservoir at temperature $T$ and rejects energy by heat transfer to a higher- temperature reservoir at $T_H$.
Sketch the problem
a) Obtain an expression for the coefficient of performance of a single reversible refrigeration cycle operating directly between cold and hot reservoirs at $T_C$ and $T_H$, respectively, in terms of the coefficients of performance of the two cycles.
b) Repeat the solution for the case of two reversible heat pump cycles in series with same thermal reservoir temperatures $T_C$ and $T_H$.
Tips:
• Consult lecture example from lecture 15
• a) Express $\beta$, $\beta_1$ and $\beta_2$ in terms of $T_H$, $T_C$ and $T$ using their definitions. Then get expressions for $1/\beta_1$, $1/\beta_2$ and combine to get expression for $1/\beta$ and then for $\beta$.
• b) Do similar algebraic manipulations with the $\gamma$'s instead of the $\beta$'s.
