VAPOR COMPRESSION REFRIGERATION CYCLES

Carnot Refrigeration Cycle (Carnot Heat Pump)

A refrigeration cycle is just a reversed heat engine cycle.

Therefore, heat is transferred from a low temperature level to a high temperature level. However, according to the second law of thermodynamics this cannot be accomplished without the use of external energy. The working mediums that are used in compression refrigeration systems are called refrigerants.

The following figure shows a typical Carnot refrigeration cycle:

1. The refrigerant evaporates at a constant temperature and pressure as heat is absorbed from the low temperature region in the evaporator.
2. The vapor leaving the evaporator is then compressed to a higher pressure and external work is required in this process.
3. Heat is rejected to the high temperature region in the condenser as the refrigerant condenses at constant temperature and pressure.
4. To complete the cycle, the liquid from the condenser is returned to its original state by an expansion process.

The overall process is represented on a T-S diagram in the following figure:

The coefficient of performance, COP is used to measure the performance of a refrigeration cycle. It is the ratio of the refrigeration obtained to the work required, i.e.,

The COP of a Carnot refrigeration cycle is given by:

A ton is a common unit used in practice to describe the refrigeration effect. One ton of refrigeration is the term used to refer to 12,000 Btu/h.

Therefore, a chiller or condensing unit with a cooling capacity of 60,000 Btu/h is said to have a capacity of 5 tons. One ton of refrigeration approximates to 3.5 kW of cooling.

As seen from the T-S graph of a Carnot refrigeration cycle, both the compression and expansion steps in the Carnot refrigeration occur within the two-phase region. The compression of a two-phase mixture, however, is very difficult in practice and is generally avoided. This problem can be eliminated by simply allowing the refrigerant to evaporate completely in the evaporator resulting in the production of a saturated vapor.

On the other hand, the fluid passing through the expander is mostly liquid and its specific volume is relatively low. Thus, the amount of work produced by the expander is not appreciable. For this reason, much less expensive and almost maintenance-free throttling expansion devices are preferred over expanders in practice.

The modified cycle and its representation on a T-S diagram are shown in the following two images:

The vapor compression refrigeration cycles can also be represented on a P-H diagram as shown:

Refrigerants

The design of a vapor compression refrigeration system is greatly influenced by the physical, thermodynamic and chemical properties of the refrigerant used.

The desirable properties for a refrigerant can be summarized as follows:

• Positive evaporating pressures: This prevents leakage of atmospheric air into the system during operation.
• Moderately low condensing pressures: This allows the use of light weight equipment on the high-pressure side of the system.
• Low freezing point.
• High latent heat of vaporization and relatively high critical temperatures: A high latent heat means a high refrigeration effect per kg of refrigerant circulated and low power cost for circulation.
• Low cost.
• Inertness and stability
• Not be toxic, irritating or flammable.

 

However, no single compound meets all these requirements.

Ammonia and sulfur dioxide were the early refrigerants for commercial use due to their high latent heats of vaporization. However, they have the obvious drawbacks of being highly toxic and corrosive, and with NH₃ being flammable.

Research efforts in the 1920’s led to the conclusion that small molecules having C-F bonds (fluorocarbons) were suitable choices, with dichlorodifluoromethane (CCl₂F₂) having the best properties. However, these compounds accumulated in the atmosphere and resulted in destruction of O₃ molecules in the ozone layer.

Hydrofluorocarbons (HFCs) were developed as alternatives to the ozone-depleting refrigerants. HFC-134a (1,1,1,2-tetrafluoroethane - CF₃CH₂F) has become the refrigerant of choice to replace CFC-12 in most refrigeration and auto air conditioning systems.

The Gaussian Integral

The Gaussian or Probability Integral is an essential concept in mathematics particularly in the fields of probability theory, statistics and quantum mechanics.

The Gaussian integral, closely related to the erf function, is the integral of the one-dimensional Gaussian function over (-∞, ∞).

The Gaussian Integral can also be defined as the integral of the exponential of -x² over the entire real line.

It can be calculated using the trick of combining two one-dimensional Gaussians:

In this case, Since the variable in the integral is a dummy variable i.e, it integrates out in the end, we can rename from x to y.

When switching to polar coordinates we get:

Example

Prove the following:

Solution

Convert the integral into the polar coordinates (r, θ) where x² + y² = r², 

and dxdy = rdrdθ:

Evaluating the integrals:

Therefore: