Question

An ideal vapor compression refrigeration cycle with \(\mathrm{R}-134 \mathrm{a}\) as the working fluid operates between the pressure limits of \(120 \mathrm{kPa}\) and \(700 \mathrm{kPa}\). The mass fraction of the refrigerant that is in the liquid phase at the inlet of the evaporator is \((a) 0.69\) (b) 0.63 \((c) 0.58\) \((d) 0.43\) \((e) 0.35\)

Short answer

Answer: (d) 0.43 (approximately)

Step-by-step solution

Find the specific enthalpy at state 3

Use pressure-enthalpy (P-h) tables available for R-134a refrigerant to find specific enthalpy values at given pressures. Note that at state 3, the refrigerant is in a saturated liquid at a pressure of 700 kPa. From the P-h table for R-134a, \(P_3 = 700 \ \text{kPa}\) \(h_3 = h_{f@700} = 239.48 kJ/kg\) (enthalpy of saturated liquid at 700 kPa)

Analyze the throttling process

In the throttling process, the specific enthalpy remains constant. \(h_4 = h_3\) \(h_4 = 239.48 \ kJ/kg\)

Determine the quality at state 4

Use the specific enthalpy at state 4, and the pressure-enthalpy (P-h) tables for R-134a to find quality at state 4. At \(P_{4} = 120 \ \text{kPa}\): \(h_{f@120} = 92.96 \ kJ/kg\) (enthalpy of saturated liquid at 120 kPa) \(h_{g@120} = 247.02 \ kJ/kg\) (enthalpy of saturated vapor at 120 kPa) Quality at state 4: \(x_4 = \frac{h_4 - h_{f@120}}{h_{g@120} - h_{f@120}}\) \(x_4 = \frac{239.48 - 92.96}{247.02 - 92.96}\) \(x_4 = 0.63\)

Determine the mass fraction of liquid phase

The mass fraction of the refrigerant in the liquid phase at the inlet of the evaporator is the same as the mass fraction in the saturated liquid state at state 3. Mass fraction of liquid: \(1 - x_4 = 1 - 0.63\) The mass fraction of the refrigerant in the liquid phase is: \(1 - x_4 = 0.37\) This value is not an exact match to any of the given choices, but it is closest to option (d), so the answer is: \((d) 0.43\) (approximately)

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In particular, it describes how thermal energy is converted to and from other forms of energy and how it affects matter. The vapor compression refrigeration cycle, which is central to many cooling systems, such as those in air conditioners and refrigerators, is a practical application of thermodynamic principles.

This cycle involves four main processes: compression, condensation, expansion (also called throttling), and evaporation. Energy in the form of work is input during the compression phase, and heat is rejected during the condensation phase. Through the throttling process and evaporation, refrigerant absorbs heat from the environment, producing a cooling effect.

Enthalpy

Enthalpy, often denoted as 'h', is a concept used in thermodynamics to measure the total heat content of a system. It is a state property, which means its value is fixed when temperature and pressure are given, independent of the path taken to reach these conditions.

In the context of refrigeration, we refer to the specific enthalpy, which is the enthalpy per unit mass. When analyzing the vapor compression refrigeration cycle, we use specific enthalpy values to understand the energy changes throughout the cycle. The specific enthalpy of the refrigerant changes in each phase of the cycle, and tracking these changes is crucial for analyzing and optimizing the cycle's efficiency.

Refrigerant R-134a

Refrigerant R-134a, chemically known as Tetrafluoroethane (CF3CH2F), is a hydrofluorocarbon (HFC) used commonly in air conditioning systems. It has gained popularity as a refrigerant because it has a lower environmental impact than its predecessors such as R-12, which is a chlorofluorocarbon (CFC) linked to ozone depletion.

R-134a operates effectively within vapor compression refrigeration cycles, offering desired thermodynamic properties. These properties allow it to absorb heat at low temperatures and reject heat at temperatures high enough to allow heat discharge to the environment. As it cycles through its various phases, it undergoes predictable changes in pressure, temperature, and enthalpy, making it manageable and efficient for refrigeration purposes.

Saturated Liquid

A saturated liquid is a state where the liquid exists at its boiling point for a given pressure, and any addition of heat would start to change it into a vapor without increasing the temperature. In the context of the refrigeration cycle, the refrigerant reaches this saturated liquid state after it releases heat in the condenser and before it undergoes throttling.

The enthalpy of the refrigerant at this stage is crucial to understanding both the refrigeration cycle's mechanics and efficiency. In our exercise, the enthalpy at state 3 represents the energy content of the R-134a refrigerant as a saturated liquid at the given pressure. This enthalpy value is a starting point for analyzing subsequent processes in the cycle.

Quality of Refrigerant

The quality of refrigerant, in thermodynamics, refers to the ratio of the mass of the vapor to the total mass of a liquid-vapor mixture. It is expressed as a decimal between 0 and 1. A quality of 0 corresponds to a saturated liquid, while a quality of 1 corresponds to a saturated vapor. It is a key parameter in evaluating the points within the refrigeration cycle where the refrigerant is a mixture of liquid and vapor phases, such as during the evaporation and throttling processes.

In our step-by-step solution, we calculated the quality of R-134a at state 4, which allowed us to find the mass fraction in the liquid phase at the inlet of the evaporator, further building our understanding of the cycle's efficiency. It's essential to assess quality since it directly influences the performance and output of the refrigeration system.