## Related questions with answers

The temperature of an incompressible substance of mass $m$ and specific heat $c$ is reduced from $T_0$ to $T\left(<T_0\right)$ by a refrigeration cycle. The cycle receives energy by heat transfer at $T$ from the substance and discharges energy by heat transfer at $T_0$ to the surroundings. There are no other heat transfers. Plot $\left(W_{\text {min }} / m c T_0\right)$ versus $T / T_0$ ranging from $0.8$ to $1.0$, where $W_{\min }$ is the minimum theoretical work input required.

Solution

VerifiedMinimum work input is given by:

$W_{min}=m\cdot C\cdot (T-T_0)-m\cdot C\cdot T_0\cdot \ln{\dfrac{T}{T_0}}$

We can now devide everything with $m\cdot C\cdot T_0$ and get the expression we need to plot the graph:

$\dfrac{W_{min}}{m\cdot C\cdot T_0}=\dfrac{T}{T_0}-1-\ln{\dfrac{T}{T_0}}$

The task is to plot $\dfrac{W_{min}}{m\cdot C\cdot T_0}$ versus $\dfrac{T}{T_0}$ ranging from 0.8 to 1.0, where $W_{min}$ is the minimum theoretical work input required:

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