Question

Refrigerant-134a enters the compressor of a refrigeration system as saturated vapor at \(0.14 \mathrm{MPa}\), and leaves as superheated vapor at \(0.8 \mathrm{MPa}\) and \(60^{\circ} \mathrm{C}\) at a rate of \(0.06 \mathrm{~kg} / \mathrm{s}\). Determine the rates of energy transfers by mass into and out of the compressor. Assume the kinetic and potential energies to be negligible.

Short answer

The rate of energy transfer by mass into the compressor is 2.529 kJ/s, and the rate of energy transfer by mass out of the compressor is negligible.

Step-by-step solution

State the specific enthalpy values for the inlet and outlet conditions

From the refrigerant-134a tables, determine the specific enthalpy at the inlet condition (saturated vapor at \(0.14\mathrm{MPa}\)) and outlet condition (superheated vapor at \(0.8\mathrm{MPa}\) and \(60^{\circ}\mathrm{C}\)). At the inlet, the refrigerant is saturated vapor at \(0.14\mathrm{MPa}\). From the tables, we find that: \(h_{in} = h_{g}|_{0.14\mathrm{MPa}} = 252.02\mathrm{kJ/kg}\) At the outlet, the refrigerant is superheated vapor at \(0.8\mathrm{MPa}\) and \(60^{\circ}\mathrm{C}\). From the tables, we interpolate to find that: \(h_{out} = h(T=60^{\circ}\mathrm{C}, P=0.8\mathrm{MPa}) = 294.17\mathrm{kJ/kg}\)

Apply the First Law of Thermodynamics to the compressor

The First Law of Thermodynamics, considering conservation of mass and energy for the compressor, is expressed as: \(\dot{m}(h_{out} - h_{in}) = \dot{Q} - \dot{W}\) where \(\dot{m}\) is the mass flow rate, \(h_{in}\) and \(h_{out}\) are specific enthalpies at the inlet and outlet, \(\dot{Q}\) is the heat transfer rate, and \(\dot{W}\) is the work transfer rate. In this problem, heat transfer is assumed to be negligible, so we have: \(\dot{m}(h_{out} - h_{in}) = -\dot{W}\)

Calculate the work transfer rate

Using the given mass flow rate \(\dot{m}\) and the specific enthalpies found in Step 1, calculate the work transfer rate \(\dot{W}\): \(\dot{W} = -\dot{m}(h_{out} - h_{in})\) \(\dot{W} = -0.06\mathrm{kg/s}(294.17\mathrm{kJ/kg} - 252.02\mathrm{kJ/kg})\) \(\dot{W} = -2.529\mathrm{kJ/s}\) The negative sign indicates that work is being done on the compressor.

Determine the rates of energy transfers by mass

Since we assumed that the kinetic and potential energies are negligible and heat transfer is also negligible, the rate of energy transfer by mass is equal to the work transfer rate found in Step 3: Rate of energy transfer by mass into the compressor: \(-2.529\mathrm{kJ/s}\) Rate of energy transfer by mass out of the compressor: \(0\mathrm{kJ/s}\) since heat transfer is assumed to be negligible. In conclusion, the rate of energy transfer by mass into the compressor is \(2.529\mathrm{kJ/s}\), and the rate of energy transfer by mass out of the compressor is negligible.

Key concepts

Enthalpy

Enthalpy, represented by the symbol h, is a measure of the total energy content within a system. It combines internal energy—which accounts for the microscopic energy due to the molecular motion and interactions—with the product of pressure and volume energy, associated with the work required to establish the system's volume.

Understanding enthalpy is critical in the study of thermodynamics, especially when dealing with refrigeration compressors, as it helps to track the energy changes during the refrigeration cycle. As seen in the given exercise, the specific enthalpy values (i.e., enthalpy per unit mass) at various pressure and temperature conditions are used to determine the amount of work performed by or on the system.

The enthalpy values for refrigerant-134a (h_{in} and h_{out}) are obtained from standard tables at specified conditions. An increase in enthalpy from the inlet to the outlet of the compressor ( h_{out} > h_{in}) implies that energy is added to the refrigerant, mainly in the form of work done on the gas to compress it. This is a key concept in understanding how refrigeration systems operate.

First Law of Thermodynamics

The First Law of Thermodynamics, also known as the law of conservation of energy, dictates that energy cannot be created or destroyed, only transformed from one form to another or transferred between systems. In the context of a refrigeration compressor, this law is applied to ensure that the energy balance accounts for all the work and heat interactions.

Mathematically, the First Law for a control volume like a compressor is expressed as (dot{m})(h_{out} - h_{in}) = dot{Q} - dot{W}. Here, dot{Q} denotes the heat transfer rate—which is often negligible in adiabatic compression as assumed in our exercise—and dot{W} is the work transfer rate. Specifically in this exercise, the First Law simplifies to dot{m}(h_{out} - h_{in}) = -dot{W} since heat transfer is not considered, revealing that the work done on the system (compressor) is directly related to the change in enthalpy and mass flow rate.

Applying this principle is essential for identifying the energy requirements of the compressor and, by extension, the refrigeration system as a whole. In industrial applications, understanding this law aids in estimating the energy efficiency and in designing compressors that adequately match the requirements of the refrigeration cycle.

Mass Flow Rate

Mass flow rate, denoted by dot{m}, is defined as the amount of mass passing through a cross-section of a system per unit time. Its unit of measurement is kg/s. In refrigeration systems, it represents how much refrigerant flows through the compressor over time, and it is a critical factor in the performance of the refrigeration cycle.

The exercise provides the mass flow rate of refrigerant-134a as 0.06 kg/s. Combined with the specific enthalpy change within the compressor, mass flow rate contributes to calculating the work dot{W} completed by the compressor, using the relationship identified by the First Law of Thermodynamics.

Higher mass flow rates generally imply more significant refrigeration effects, but they also require compressors to perform more work to handle the increased flow of refrigerant. Optimizing mass flow rate is crucial for the design and operation of an efficient refrigeration system, balancing the cooling capacity with the power consumption of the compressor.