Question

A refrigerator operates on the ideal vapor compression refrigeration cycle with \(\mathrm{R}-134 \mathrm{a}\) as the working fluid between the pressure limits of 120 and 800 kPa. If the rate of heat removal from the refrigerated space is \(32 \mathrm{kJ} / \mathrm{s}\), the mass flow rate of the refrigerant is \((a) 0.19 \mathrm{kg} / \mathrm{s}\) \((b) 0.15 \mathrm{kg} / \mathrm{s}\) \((c) 0.23 \mathrm{kg} / \mathrm{s}\) \((d) 0.28 \mathrm{kg} / \mathrm{s}\) \((e) 0.81 \mathrm{kg} / \mathrm{s}\)

Short answer

Question: In an ideal vapor compression refrigeration cycle using R-134a as the working fluid, the pressure limits are 120 kPa and 800 kPa, and the rate of heat removal from the refrigerated space is 32 kJ/s. Determine the mass flow rate of the refrigerant. Answer: To determine the mass flow rate of the refrigerant, follow the provided step-by-step solution. First, calculate the enthalpy values at each state point, then calculate isentropic enthalpies for states 2 and 4. Next, calculate the heat transfer in the evaporator using the enthalpy values. Finally, apply the First Law of Thermodynamics to find the mass flow rate: \(\dot{m} = \frac{32\,\text{kJ/s}}{Q_{evap}}\) The mass flow rate will be in kg/s.

Step-by-step solution

Calculate the enthalpy values at each state point

To determine the enthalpy values, begin by identifying the states of the refrigerant at each point in the cycle. Since the cycle is ideal, we can say that points 1 and 3 are saturated states (1 being saturated liquid and 3 being saturated vapor). Utilizing a thermodynamic property table or software for R-134a, obtain the following enthalpy values: \(h_{1} = h_{f} (P_{1} = 120 \,\text{kPa})\), \(h_{3} = h_{g} (P_{3} = 800 \, \text{kPa})\) Make sure you use the correct units (in kJ/kg) for the enthalpy values.

Calculate isentropic enthalpy for state 2

Since the compression process is adiabatic and isentropic, we have: \(s_1 = s_2\) Using the property table or software, find \(s_1 = s_f (P_1 = 120 \, \text{kPa})\). Then, with the entropy value and the pressure at state 2 (\(P_2 = 800 \, \text{kPa}\)), find the enthalpy at state 2 (\(h_2\)) using interpolation or software. Make sure to use kJ/kg for enthalpy units.

Calculate the isentropic enthalpy for state 4

During the expansion process, it's also adiabatic and isentropic. Therefore: \(s_3 = s_4\) Find \(s_3 = s_g (P_3 = 800\, \text{kPa})\). Then, find the enthalpy at state 4 (\(h_4\)) with the entropy value and the pressure at state 4 (\(P_4 = 120\, \text{kPa}\)) using interpolation or software. Make sure to use kJ/kg for enthalpy units.

Calculate the heat transfer in the evaporator, \(Q_{evap}\)

Now, we can determine the heat transfer in the evaporator using the enthalpy values calculated in Steps 1 and 4: \(Q_{evap} = h_1 - h_4\) Make sure to use kJ/kg as the units for the heat transfer.

Calculate the mass flow rate of the refrigerant

Finally, we can determine the mass flow rate of the refrigerant by applying the First Law of Thermodynamics: \(\dot{m} = \frac{\dot{Q}_{in}}{Q_{evap}}\) Given that the rate of heat removal from the refrigerated space is \(\dot{Q}_{in} = 32\,\text{kJ/s}\), plug \(Q_{evap}\) from Step 4 and the given \(\dot{Q}_{in}\) value into the equation above, then solve for the mass flow rate, \(\dot{m}\): \(\dot{m} = \frac{32\,\text{kJ/s}}{Q_{evap}}\) The mass flow rate will be in kg/s. Compare the calculated mass flow rate with the provided options to determine the correct answer.

Key concepts

Enthalpy in Thermodynamics

When studying refrigeration cycles, such as the vapor compression refrigeration cycle used in refrigerators and air conditioners, the concept of enthalpy becomes critical. Enthalpy is a measure of the total energy of a thermodynamic system, including internal energy and the energy required to make room for the system by displacing its environment.

Think of enthalpy as a way to quantify heat content. In the context of the refrigeration cycle, enthalpy values at various points indicate how much energy is added to or removed from the refrigerant as it circulates through the system. For instance, when a refrigerant absorbs heat from the area being cooled (the evaporator), its enthalpy increases. Conversely, when it discharges heat (the condenser), its enthalpy decreases.

Using thermodynamic tables or software to find enthalpy values, like in the exercise's Step 1, is typical. These tables list the properties (including enthalpy) of substances like refrigerants at different temperatures and pressures. To optimize understanding, it's useful to visualize the refrigeration cycle on a pressure-enthalpy (P-h) diagram, showing how enthalpy changes during the process.

Isentropic Process

An isentropic process is a theoretical process that is both adiabatic (no heat transfer) and reversible. In this idealized process, entropy remains constant. Entropy is a measure of a system's disorder; in an isentropic process, this disorder doesn't change.

In the refrigeration cycle, compressing and expanding the refrigerant ideally occur in an isentropic manner. During compression (from State 1 to State 2 in the exercise), the refrigerant is compressed without gaining or losing heat, and thus, the entropy remains the same. The same holds for expansion (from State 3 to State 4).

To solve for various states in an ideal vapor compression cycle, the condition of isentropic compression and expansion is used to find the enthalpy values at those states (as shown in the steps of the exercise). Understanding this concept is key to analyzing energy efficiency and performance in real-world applications, even though perfect isentropic processes don't occur outside theoretical scenarios.

First Law of Thermodynamics

At the heart of analyzing energy systems, like refrigeration cycles, is the First Law of Thermodynamics. This law, often called the law of energy conservation, states that energy cannot be created or destroyed, only transferred or transformed from one form to another.

In practical terms, within a vapor compression cycle, the First Law helps us determine the work done by the compressor and the heat exchanged in the evaporator and condenser. During Step 5 of the exercise, the First Law is employed to calculate the mass flow rate of the refrigerant by relating the heat removed from the refrigerated space to the enthalpy change in the evaporator.

Making sense of this law allows students to comprehend energy flow within refrigeration systems and can be applied to many other areas in physics and engineering, reinforcing the idea that while energy can shift between kinetic, potential, thermal, and other forms, the total energy in a closed system remains constant.