Expressing Compressor Capacity with Enthalpy Change, Specific Volume, Flow Rate & Efficiency

In this article, we take an in-depth look at an equation that describes the capacity of a compressor as a function of vital parameters such as stroke displacement, clearance volumetric efficiency, specific volume & refrigerant enthalpy change across the evaporator. It highlights the importance of mass/volume flow in expressing the capacity of the refrigeration system & is a crucial link to garner a deeper understanding of the vapor compression cycle.

The refrigerant handling capacity of a compressor can be expressed as below

Let’s investigate each of the parameters in depth to draw more insights

Displacement rate of compressor

This is quite simply, the volume of the stroke uncovered by the compressor per unit time during operation. It is dependent on the size of the bore, stroke length, number of cylinders & speed of rotation. As the compressor executes the suction stroke, the gaseous refrigerant begins to flow into its cylinders. Hence the volume available for this refrigerant to settle in, and the speed at which the compressor executes its discharge stroke are factors that affect capacity. A compressor that accommodates a greater amount of refrigerant while executing a suction stroke or discharging the same amount at a high rate yields higher capacity.

Volumetric efficiency

In general terms, it is a number that compares the volume of refrigerant that can ideally fill up the entire compressor cylinder to the quantity that actually fills it up. Losses that occur upon flow across valves or ports in the suction line, behavior of suction gases as they come in contact with the warm regions of the compressor, & the residue gases that undergo expansion during the suction stroke, are some of the major reasons why volumetric efficiency is never 100%.

Lets imagine a reciprocating compressor executing a suction stroke as described in the figure below.

There exists a small clearance between the piston & the top dead center of the cylinder that holds a tiny volume (Vc) of refrigerant even after the discharge stroke has been executed. This residual gas will be at a higher pressure than the refrigerant in the suction line. Therefore, during the next suction stroke of the compressor, the residual gas begins to expand & undergoes a reduction in pressure. It is observed that no amount of suction gas can flow into the compressor until the pressure of the residual gas becomes lower than the suction pressure, this happens at V1. The more stroke length required to expand the residual gas below the suction pressure, lesser the effective volume available to suck the fresh suction gas into the cylinder. This could explain why manufacturers try to keep clearance volume at a minimum. The suction stroke is completed as the piston sweeps a volume of V2.  

The volumetric efficiency can be expressed as

Simplifying the equation further

X constitutes the proportion of the clearance volume as a comparison to the effective volume.

Now, assuming the expansion from Vc to V1 to be isentropic

Refrigerant Enthalpy Change

The refrigerant's enthalpy change across the evaporator is directly proportional to the compressor capacity as evident from the equation. Enthalpy is the summation of the internal energy & the expansion work of the refrigerant.

H = U + PV

For a refrigerant at a particular temperature & pressure, its molecules will be in a state of movement that is either translational, rotational or vibrational. The kinetic energy of these molecules averaged out across the whole system of the refrigerant is how we define its temperature at that particular state. Also, there is potential energy stored in the form of chemical or nuclear bonds between atoms as well as the force fields that exist within the molecules. Therefore, the energy required for the refrigerant’s molecules to be at that particular state by virtue of its kinetic & potential energy, is accounted for by the internal energy (U).

PV is the work done by the refrigerant to occupy space for itself. It could be imagined as the expansion of the gas to accommodate itself. It is based on pressure, i.e., the force that the molecules exert per unit area on a container, & V is the space that the molecules of gas occupy at a particular state. Combining the two, we could imagine it to be the outward push by the refrigerant occupy its required space at a given state.

As the refrigerant enters the evaporator as a liquid & gas mixture, it begins to absorb heat from the refrigerated space. The absorbed heat raises the internal energy of the refrigerant as the molecules now start exhibiting greater movement. The refrigerant also begins expanding & hence the proportion of PV begins to rise as well. Observe the PH chart for R-134a, for a system working between -5°C & 45°C evaporating & condensing. Let’s analyze the refrigerant properties at the suction pressure corresponding to -5°C.

At saturation pressure for a saturation temperature of -5°C, the liquid enthalpy is 194.27 kJ/kg. Of this energy, only 0.0978% is used up for the refrigerant to occupy space whereas the rest is utilized as internal energy. As the refrigerant changes phase from liquid to gas, an increase in enthalpy is observed with the proportion of energy used for expansion rising to 5.0924% & the vapor enthalpy rising to 396.08 kJ/kg. The raise in percentage of energy utilized for PV work suggests that gas occupies more space in the evaporator than liquid. The gain in enthalpy of the refrigerant flowing through the evaporator corresponds to a greater heat absorbed from the room & hence a greater capacity that the compressor must handle.

Specific Volume

The capacity of the compressor has an inverse relationship to the specific volume of refrigerant. The greater the space occupied by the refrigerant at a particular suction temperature & pressure, lesser molecules fill up the compressor cylinder during its suction stroke. Varying suction line conditions affect the space occupied by the refrigerant molecules. The image below represents the difference between two systems of evaporating temperatures of -35 °C & -5°C. It is evident that the higher pressure condition of the -5°C condition causes lower specific volume & a higher compressor capacity. Precisely the reason why compressors have higher capacity at higher evaporating temperatures.

Article mentioned below goes into this concept at more detail.

References

• Coolpack: Saturated properties for R134a & software for plotting the PH chart https://www.ipu.dk/products/coolpack/
• Yunus A. Cengel , Michael A. Boles. Thermodynamics, An Engineering Approach. Fifth Edition.
• Genetron: Software saturated properties & internal energy references Saturated properties for R134a