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An internally reversible refrigerator has a modified coefficient of performance accounting for realistic heat transfer processes of $$ \mathrm{COP}_{m}=\frac{q_{\text {in }}}{\dot{W}}=\frac{q_{\text {in }}}{q_{\text {out }}-q_{\text {in }}}=\frac{T_{c, i}}{T_{h, i}-T_{c, i}} $$ where \(q_{\text {in }}\) is the refrigerator cooling rate, \(q_{\text {out }}\) is the heat rejection rate, and \(\dot{W}\) is the power input. Show that \(\mathrm{COP}_{m}\) can be expressed in terms of the reservoir temperatures \(T_{c}\) and \(T_{h}\), the cold and hot thermal resistances \(R_{L, c}\) and \(R_{t, h}\), and \(q_{\text {in }}\), as $$ \mathrm{COP}_{m}=\frac{T_{c}-q_{\mathrm{in}} R_{\mathrm{tot}}}{T_{h}-T_{c}+q_{\mathrm{in}} R_{\mathrm{tot}}} $$ where \(R_{\mathrm{tot}}=R_{t, c}+R_{t, h}\). Also, show that the power input may be expressed as $$ \dot{W}=q_{\mathrm{in}} \frac{T_{h}-T_{c}+q_{\mathrm{in}} R_{\mathrm{id \textrm {t }}}}{T_{c}-q_{\mathrm{in}} R_{\mathrm{tot}}} $$

Step by step solution

01

Write down the given Data

We are given the equation for the modified coefficient of performance (COP_m) as: \[ \mathrm{COP}_{m}=\frac{T_{c, i}}{T_{h, i}-T_{c, i}} \] We are also given that: \[ q_{in} = T_{c, i} - R_{t, c} (T_{c, i} - T_c) \] and \[ q_{out} = T_{h, i} - R_{t, h} (T_h - T_{h, i}) \] Finally, we are given that the total resistance \(R_{tot} = R_{t, c} + R_{t, h}\).

02

Express \(T_{c, i}\) and \(T_{h, i}\) in terms of \(q_{in}\) and resistances

We can rewrite the given equations for \(q_{in}\) and \(q_{out}\) as: \[ T_{c, i} = q_{in} + R_{t, c} (T_c - q_{in}) \\ T_{h, i} = q_{out} - R_{t, h} (T_h - q_{out}) \]

03

Substitute the expressions for \(T_{c, i}\) and \(T_{h, i}\) into the COP_m equation

Substituting the expressions for \(T_{c, i}\) and \(T_{h, i}\) from step 2 into the equation for COP_m, we get: \[ \mathrm{COP}_{m}=\frac{q_{in} + R_{t, c}(T_c - q_{in})}{q_{out} - R_{t, h}(T_h - q_{out}) - (q_{in} + R_{t, c}(T_c - q_{in}))} \]

04

Simplify the expression for COP_m

Simplifying the expression for COP_m, we get: \[ \mathrm{COP}_{m} = \frac{T_c - q_{in}R_{t, c}}{T_h - T_c + q_{in} (R_{t, c} + R_{t, h})} \] Since we know \(R_{tot} = R_{t, c} + R_{t, h}\), we can plug it into the equation: \[ \mathrm{COP}_{m} = \frac{T_c - q_{in} R_{tot}}{T_h - T_c + q_{in} R_{tot}} \]

05

Find the power input \(\dot{W}\) equation

We are also asked to find an expression for the power input, which is obtained from the definition of COP as follows: \[ \mathrm{COP}_{m}=\frac{q_{in}}{\dot{W}} \Rightarrow \dot{W} = \frac{q_{in}}{\mathrm{COP}_{m}} \] Now, we can substitute the expression we derived for the COP_m in step 4: \[ \dot{W} = q_{in} \frac{T_h - T_c + q_{in} R_{tot}}{T_c - q_{in} R_{tot}} \] Now, we have derived the equation for the COP_m and the power input in terms of the given variables.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermal Resistances

In refrigeration systems, thermal resistances play a crucial role in understanding how heat is transferred through different parts of the system. Imagine thermal resistance like an obstacle that heat must overcome to move from one point to another.

• Thermal resistance is present in both the cold and hot regions of a refrigeration system, denoted as \( R_{t, c} \) for the cold side and \( R_{t, h} \) for the hot side.
• The total thermal resistance, \( R_{tot} \), is simply the sum of both \( R_{t, c} \) and \( R_{t, h} \). This total is important because it affects the efficiency with which the refrigeration system operates.

Thermal resistance impacts the modified coefficient of performance by altering how heat enters and exits the system. Therefore, understanding and controlling thermal resistances can lead to more efficient refrigeration cycles.

Heat Transfer Processes

The process of transferring heat is at the heart of refrigeration cycles. Heat transfer determines the cooling rate \( q_{in} \) in a refrigerator. In simple terms, heat transfer is moving heat from a cooler area to a warmer one. This reverse heat flow requires work, achieved through the refrigeration cycle.

• Heat enters the system from the cooling load \( q_{in} \) and exits as rejected heat \( q_{out} \). These processes are governed by the laws of thermodynamics.
• Understanding these processes involves equations that use the reservoir temperatures, \( T_c \) and \( T_h \), and modifying factors such as thermal resistances.

By optimizing heat transfer processes, the performance of the refrigeration system improves, leading to a better Coefficient of Performance, or \( \text{COP} \), which measures the efficiency of the system.

Refrigeration Cycle

The refrigeration cycle is a system's method of removing heat from a low-temperature environment and dumping it in a high-temperature environment, effectively cooling the interior space of a refrigerator. It involves repeating stages of compression, condensation, expansion, and evaporation.

• The cycle starts with the compression of refrigerant gas, which increases its pressure and temperature.
• Next, the hot gas flows through the condenser, where it cools and condenses into a liquid.
• The liquid refrigerant then expands and cools as it passes through an expansion valve.
• Finally, it absorbs heat as it evaporates in the evaporator, completing the cycle.

This cycle's efficiency is impacted by the modified coefficient of performance, which can be calculated using temperatures, thermal resistances, and heat transfer rates. By understanding and correctly manipulating these variables, a refrigeration system's efficiency can be significantly enhanced.