Question

Air enters the compressor of an ideal Brayton refrigeration cycle at $100\mathrm{kPa},270\mathrm{K}$. The compressor pressure ratio is 3 , and the temperature at the turbine inlet is $310\mathrm{K}$. Determine (a) the net work input, per unit mass of air flow, in $\mathrm{kJ}/\mathrm{kg}$. (b) the refrigeration capacity, per unit mass of air flow, in $\mathrm{kJ}/\mathrm{kg}$. (c) the coefficient of performance. (d) the coefficient of performance of a Carnot refrigeration cycle operating between thermal reservoirs at $T_C=270\mathrm{K}$ and $T_{\mathrm{H}}=310\mathrm{K}$, respectively.

Short answer

Net work input is calculated using the temperature rise, refrigeration capacity uses the heat absorbed, and COP is \frac{q_in}{w_net}. COP_{Carnot} = \frac{T_C}{T_H-T_C}.

Step-by-step solution

Identify Known Conditions

List the given values: - Inlet pressure, $P_1=100\text{kPa}$ - Inlet temperature, $T_1=270\text{K}$ - Compressor pressure ratio, $P_2/P_1=3$ - Turbine inlet temperature, $T_3=310\text{K}$

Isentropic Compression (1 to 2)

Use the isentropic relation for an ideal gas: \ T_2 = T_1 \left[ \frac{P_2}{P_1} \right]^{(\frac{\theta -1}{\theta})} \ Here, $\theta$ is the specific heat ratio (approx. 1.4 for air). Calculate $T_2$: \ T_2 = 270 \text{K} \left[ 3 \right]^{(\frac{0.4}{1.4})} \Calculate $T_2$.

Calculate Compressor Work (w_c)

The work input to the compressor is given by: \backslash[ w_c = c_p (T_2 - T_1) \backslash] where $c_p$ is the specific heat at constant pressure (\backslashapprox 1.005 \text{kJ}/\text{kg.K}).Calculate $w_c$.

Isentropic Expansion (3 to 4)

Use the isentropic relation for expansion: \backslash[ T_4 = T_3 \left( \backslashfrac{P_4}{P_3} \backslashright)^{(\frac{\theta -1}{\theta})} \backslash] Here, $P_4=P_1$ and $P_3=P_2$. Calculate $T_4$.

Calculate Turbine Work (w_t)

The work output from the turbine is given by: \backslash[ w_t = c_p (T_3 - T_4) \backslash]. Use the value of $T_4$ calculated in Step 4 to calculate $w_t$.

Calculate Net Work Input (w_net)

The net work input (per unit mass) is the difference between compressor work and turbine work: \backslash[ w_net = w_c - w_t \backslash]. Calculate $w_{n}et$.

Calculate Refrigeration Capacity (q_in)

Refrigeration capacity is the heat absorbed from the cold space: \backslash[ q_in = c_p (T_1 - T_4) \backslash]. Use the value of $T_4$ from Step 4.

Calculate Coefficient of Performance (COP)

The coefficient of performance for the cycle is: \backslash[ COP = \frac{q_in}{w_net} \backslash]. Use the values of $q_{i}n$ and $w_{n}et$ from Steps 6 and 7.

Calculate Carnot Coefficient of Performance (COP_Carnot)

For a Carnot cycle: \backslash[ COP_{Carnot} = \frac{T_C}{T_H - T_C} \backslash]. Use $T_C=270\text{K}$ and $T_H=310\text{K}$.

Key concepts

Isentropic Process

The isentropic process is a fundamental concept in thermodynamics, especially for cycles like the Brayton refrigeration cycle. In an isentropic process, entropy remains constant. This means that the process is both adiabatic (no heat transfer occurs) and reversible. For ideal gases, we often use the relations that tie temperature and pressure together.
Take for instance: $$T_2=T_1{\left(\frac{P_2}{P_1}\right)}^{\frac{\gamma -1}{\gamma }}$$ where $\gamma$ is the specific heat ratio.
In a Brayton cycle, there are two isentropic processes:

• Compression in the compressor (1 to 2)
• Expansion in the turbine (3 to 4)

Understanding these help us calculate other key parameters in the cycle.

Compressor Work

Compressor work refers to the energy required to compress the air in the cycle. It is a critical component since it is the work input needed to achieve refrigeration. The work done on the air during compression can be calculated using:
$$w_c=c_p(T_2-T_1)$$ Here, $c_p$ is the specific heat at constant pressure. Plugging in the values after calculating $T_2$ using the isentropic relation gives us the work done by the compressor. This energy is necessary to raise the pressure of the refrigerant.

Turbine Work

Turbine work is the energy extracted during the expansion process. In the Brayton refrigeration cycle, air expands isentropically in the turbine, producing work. This work output can be evaluated by:
$$w_t=c_p(T_3-T_4)$$ where $T_4$ is found using the isentropic relation for expansion. The energy extracted here partially offsets the energy input needed for compression, and it is integral in calculating the net work required by the cycle.

Coefficient of Performance

The coefficient of performance (COP) is a measure of the efficiency of a refrigeration cycle. It is the ratio of the refrigeration effect to the work input:
$$COP=\frac{q_{i}n}{w_{n}et}$$ where $q_{in}$ is the refrigeration capacity, which is the heat absorbed from the cold space:
$$q_{in}=c_p(T_1-T_4)$$ The net work input, $w_{net}$, is the difference between the work done by the compressor and the work produced by the turbine:
$$w_{net}=w_c-w_t$$ Understanding COP is crucial as it indicates the effectiveness of the refrigeration cycle. The higher the COP, the more efficient the cycle is.