Question

Consider a two-stage cascade refrigeration system operating between the pressure limits of \(1.2 \mathrm{MPa}\) and \(200 \mathrm{kPa}\) with refrigerant-134a as the working fluid. The refrigerant leaves the condenser as a saturated liquid and is throttled to a flash chamber operating at 0.45 MPa. Part of the refrigerant evaporates during this flashing process, and this vapor is mixed with the refrigerant leaving the low-pressure compressor. The mixture is then compressed to the condenser pressure by the high-pressure compressor. The liquid in the flash chamber is throttled to the evaporator pressure and cools the refrigerated space as it vaporizes in the evaporator. The mass flow rate of the refrigerant through the lowpressure compressor is \(0.15 \mathrm{kg} / \mathrm{s}\). Assuming the refrigerant leaves the evaporator as a saturated vapor and the isentropic efficiency is 80 percent for both compressors, determine \((a)\) the mass flow rate of the refrigerant through the high-pressure compressor, \((b)\) the rate of heat removal from the refrigerated space, and \((c)\) the COP of this refrigerator. Also, determine \((d)\) the rate of heat removal and the COP if this refrigerator operated on a single-stage cycle between the same pressure limits with the same compressor efficiency and the same flow rate as in part ( \(a\) ).

Short answer

Question: Calculate the mass flow rate of the refrigerant through the high-pressure compressor, the rate of heat removal from the refrigerated space, the COP of the two-stage cascade system, and the rate of heat removal and COP for the single-stage cycle. Please follow the provided step-by-step solution and use the refrigeration thermodynamic cycle and properties of the refrigerant-134a, along with the given pressure limits and conditions to solve the problem. Refer to the refrigerant-134a table for property values.

Step-by-step solution

List of Known Parameters

Pressure limits: \(P_H = 1.2 MPa\), \(P_L = 200 kPa\); Flash chamber pressure: \(P_{FC} = 0.45 MPa\); Throttle to FC: saturated liquid; Refrigerant: R-134a; Low-pressure compressor flow rate: \(\dot{m}_{LPC} = 0.15 kg/s\); Evaporator: saturated vapor; Isentropic compressor efficiency: \(η_c = 80\%\) Step 2: Find enthalpy and entropy at state points

State Point Enthalpies and Entropies

We need to find the state properties at points (1, 2, 3, 4, 5, and 6) shown in the diagram. We will use the refrigerant-134a table to find the enthalpy values. Using the pressures, look up the enthalpy and entropy of saturated vapor/liquid at each point within the system. Step 3: Calculate enthalpy values at points 2s and 5s after-isentropic compression

Isentropic Enthalpy Values

We need to find the enthalpy values at points 2s and 5s after isentropic compression. Applying an isentropic process, \(s_2^s = s_1\) and \(s_5^s = s_4\). Using this relation along with the pressure values, \(P_2 = P_5 = P_H\), find the enthalpy values at the isentropic conditions, \(h_2^s\) and \(h_5^s\), from the refrigerant-134a table. Step 4: Calculate actual enthalpy values at points 2 and 5 using isentropic efficiency

Actual Enthalpy Values at Points 2 and 5

Now, use isentropic efficiency for both compressors to calculate actual enthalpy values at points 2 and 5: For the high-pressure compressor, \(η_c = \frac{h_5^s - h_4}{h_5 - h_4}\). Solve for \(h_5\). For the low-pressure compressor, \(η_c = \frac{h_2^s - h_1}{h_2 - h_1}\). Solve for \(h_2\). Step 5: Calculate mass flow rate in high-pressure compressor

Mass Flow Rate in High-Pressure Compressor

Apply mass and energy balance across the flash chamber to calculate the mass flow rate of refrigerant in the high-pressure compressor (\(\dot{m}_{HPC}\)). We can use, \(\dot{m}_{HPC} = \dot{m}_{LPC}(H_1 - H_3)/(H_4 - H_3)\). Step 6: Calculate heat removal rate from the refrigeration space

Heat Removal Rate

Calculate the rate of heat removal from the refrigeration space: \(\dot{Q}_{in} = \dot{m}_{LPC}(h_1 - h_6)\). Step 7: Calculate the COP of the two-stage cascade system

COP of the Two-Stage Cascade System

We can calculate the COP of the cascade system using \(\mathrm{COP} = \frac{\dot{Q}_{in}}{\dot{W}_{net}}\), where \(\dot{W}_{net} = \dot{m}_{HPC}(h_5 - h_4) + \dot{m}_{LPC}(h_2 - h_1)\). Step 8: Calculation for single-stage cycle

Single-Stage Cycle Calculations

To determine the rate of heat removal and COP for the single-stage cycle, repeat steps 3-4 and 6-7 for a single-stage cycle assuming a saturated vapor and compressor efficiency of 80% and a flow rate of 0.15 kg/s. Calculate the enthalpy and entropy values for the required state points and use the same formulas for heat removal rate and COP. After performing the steps above, we will have the required mass flow rate of the refrigerant through the high-pressure compressor, the rate of heat removal from the refrigerated space, the COP of the two-stage cascade system, and the rate of heat removal and COP for the single-stage cycle.

Key concepts

Refrigerant-134a Properties

In the realm of refrigeration systems, refrigerants such as Refrigerant-134a are critical. R-134a, a hydrofluorocarbon (HFC), replaced older chlorofluorocarbons like R-12 due to its lower ozone depletion potential. Properties of R-134a vary with temperature and pressure, influencing its behavior in refrigeration cycles. For instance, at the condenser, R-134a exits as a saturated liquid, a state where it is fully condensed but about to evaporate. Conversely, it leaves the evaporator as a saturated vapor, where it has fully vaporized but is about to start condensing. These properties, readily found in tables or through thermodynamic software, are essential for understanding refrigeration cycles and are used in enthalpy and entropy calculations.

Isentropic Compression

Isentropic compression is a term used to describe an ideal compression process where entropy remains constant. In real-world applications, compressors don't achieve perfect isentropy due to inefficiencies, but they can approach it. In this case, the compressors are given an isentropic efficiency of 80%, indicating they are 80% as effective as an ideal compressor. By comparing the actual compression process to an idealized isentropic one, we can determine the work input required and the state of the refrigerant after the compression. This comparison requires knowing the enthalpies before and after the process, taking into account the isentropic efficiency.

Coefficient of Performance (COP)

The coefficient of performance (COP) is a measure of a refrigeration system's efficiency, defined as the ratio of heat removal rate from the refrigerated space to the work input. It is a critical parameter and tells us how effectively the refrigerator uses energy. A higher COP indicates a more efficient system. In the context of this problem, the COP is affected by factors such as the compression process's isentropic efficiency and the thermophysical properties of the refrigerant. Calculating the COP requires careful energy balance and understanding of the refrigeration cycle steps.

Mass Flow Rate

Mass flow rate refers to the amount of mass passing through a cross-section of a system per unit time and is a vital variable in thermal systems. For the given problem, understanding and calculating the mass flow rate—which is given for the low-pressure compressor and must be determined for the high-pressure one—is fundamental. The mass flow rate influences the system's capacity and efficiency, determining the rate of heat absorption in the evaporator and rejection in the condenser. The mass flow rates for different sections of the cycle are related through mass balance and reflect the conservation of mass principle in the system.

Enthalpy Calculations

Enthalpy, a measure of heat content in a system, is crucial for analyzing thermal cycles. Enthalpy calculations serve as a basis for determining energy changes during refrigeration processes. They rely on accurate state point information derived from the refrigerant's properties and the given operational details, such as isentropic efficiency and pressure levels. With this exercise focusing on both isentropic and actual compression stages, it showcases how enthalpy values are altered due to real-life inefficiencies, impacting the calculation of COP and other performance indicators.

Energy Balance

Energy balance is a fundamental concept that ensures all energy entering, leaving, and being consumed within a system is accounted for. It's the backbone of the first law of thermodynamics and is applied within each component of the refrigeration cycle to relate mass flow rates, enthalpy, and work input/output. In understanding this problem's cascade refrigeration system, applying energy balance helps calculate various important quantities such as the mass flow rate in the high-pressure compressor and the overall work input, which in turn is needed to calculate the system's COP.

Heat Removal Rate

The heat removal rate, often measured in watts, quantifies the amount of heat energy extracted from the refrigerated space per unit of time. It's a direct indicator of a refrigeration system's ability to maintain the desired temperature within a space or product. Calculating this parameter involves using the mass flow rate of the refrigerant, along with its enthalpy change, across the evaporator where heat is absorbed. This concept ties in with energy balance, as we can deduce the system's cooling performance and can be compared between systems, like the two-stage cascade against a single-stage system mentioned in the problem.