Design of Vapor-Compression Refrigeration Cycles

Goal

We want to design a vapor-compression refrigeration cycle to absorb heat from a cool environment and reject it to a warm environment. The design is to be based upon the ideal vapor-compression refrigeration cycle, with four components: a cooler (where we reject the heat), a throttle, a heater (where we absorb the heat), and a compressor.

Basics of Vapor-Compression Refrigeration Cycles

The general idea
The challenge in refrigeration (and air conditioning, etc.) is to remove heat from a low temperature source and dump it at a higher temperature sink. Compression refrigeration cycles in general take advantage of the idea that highly compressed fluids at one temperature will tend to get colder when they are allowed to expand. If the pressure change is high enough, then the compressed gas will be hotter than our source of cooling (outside air, for instance) and the expanded gas will be cooler than our desired cold temperature. In this case, we can use it to cool at a low temperature and reject the heat to a high temperature.
diagram of vcrs


Vapor-compression refrigeration cycles specifically have two additional advantages. First, they exploit the large thermal energy required to change a liquid to a vapor so we can remove lots of heat out of our air-conditioned space. Second, the isothermal nature of the vaporization allows extraction of heat without raising the temperature of the working fluid to the temperature of whatever is being cooled. This is a benefit because the closer the working fluid temperature approaches that of the surroundings, the lower the rate of heat transfer. The iso- thermal process allows the fastest rate of heat transfer.

More details

An ideal refrigeration cycle looks much like a reversed Carnot heat engine or a reversed Rankine cycle heat engine. The primary distinction being that refrigeration cycles lack a turbine, using a throttle instead to expand the working fluid. (Of course, a turbine could be incorporated into a refrigeration cycle if one could be designed to deal with liquids, but the useful work output is usually too small to justify the cost of the device.)

The cycle operates at two pressures, Phigh and Plow, and the statepoints are determined by the cooling requirements and the properties of the working fluid. Most coolants are designed so that they have relatively high vapor pressures at typical application temperatures to avoid the need to maintain a significant vacuum in the refrigeration cycle.

The T-s diagram for a vapor-compression refrigeration cycle is shown below.

Description of Cycle Stages


We will examine each state  point and component in the refrigeration cycle where design assumptions must be made, detailing each assumption. As we can see from the example design constraints, very few numbers need be specified to describe a vapor-compression refrigeration cycle. The rest of the assumptions are determined by applying reasoning and background knowledge about the cycle. The two principle numerical design decisions are determining P high and T low, at the cooler outlet and the compressor inlet.

Cooler (Condenser) inlet (S1)


This state need not involve any design decisions, but it may be important to come back here after the cycle has been solved and check that T2, which is the high temperature of the cycle, does not violate any design or safety constraints. In addition, this is as good a place as any to specify the working fluid.
Cooler (Condenser): Heat Rejection (CLR1)
The cooler (also known as the condenser) rejects heat to the surroundings. Initially, the compressed gas (at S1) enters the condenser where it loses heat to the surroundings. During this constant-pressure process, the coolant goes from a gas to a saturated liquid-vapor mix, then continues condensing until it is a saturated liquid at state 2. Potentially, we could cool it even further as a subcooled liquid, but there is little gain in doing so because we have already removed so much energy during the phase transition from vapor to liquid.


Cooler (Condenser) outlet (S2)

We cool the working fluid until it is a saturated liquid, for reasons stated above. An important design question arises at this state: how high should the high pressure of the cycle be?
We choose P high so that we can reject heat to the environment. P high is the same as P2, and P2 determines the temperature at state S2, T2. (T2 is just the saturation temperature at P high). This temperature must at least be higher than that of the cooling source, otherwise no cooling can occur.

However, if T2 is too high (that is, higher than the critical temperature TC for the working fluid), then we will be beyond the top of the saturation dome and we will lose the benefits of the large energy the fluid can reject while it is being cooled. Furthermore, it is often impractical and unsafe to have very high-pressure fluids in our system and the higher P2 we choose, the higher T1 must be, leading to additional safety concerns. To find an applicable pressure, use the saturation tables to find a pressure which is somewhere between the saturation pressure of the warm air yet still in the saturation region.

For reference, TC for our four working fluids are given below.

Throttling (THR1)

The high-pressure, saturated liquid is throttled down to a lower pressure from state S2 to state S3. This process is irreversible and there is some inefficiency in the cycle due to this process, which is why we note an increase in entropy from state S2 to S3, even though there is no heat transfer in the throttling process. In theory, we can use a turbine to lower the pressure of the working fluid and thereby extract any potential work from the high pressure fluid (and use it to offset the work needed to drive the compressor). This is the model for the Carnot refrigeration cycle. In practice, turbines cannot deal with the mostly liquid fluids at the cooler outlet and, even if they could, the added efficiency of extracting this work seldom justifies the cost of the turbine

Heater (Evaporator): Heat Absorption (HTR1)


The working fluid absorbs heat from the surroundings which we intend to cool. Since this process involves a change of phase from liquid to vapor, this device is often called the evaporator. This is where the useful "function" of the refrigeration cycle takes place, because it is during this part of the cycle that we absorb heat from the area we are trying to cool. For an efficient air conditioner, we want this quantity to be large compared to the power needed to run the cycle.
The usual design assumption for an ideal heater in a refrigeration cycle is that it is isobaric (no pressure loss is incurred from forcing the coolant through the coils where heat transfer takes place). Since the heating process typically takes place entirely within the saturation region, the isobaric assumption also ensures that the process is isothermal.

Compressor Inlet (S4)

Where do we want S4?
Typically, we want state S4 to be right at the saturated vapor side of the saturation dome. This allows us to absorb as much energy from the surroundings as possible before leaving the saturation dome, where the temperature of the working fluid starts to rise and the (now non-isothermal) heat transfer becomes less efficient.
Of course, we would get the same isothermal behavior if we were to start the compression before the fluid was completely saturated. Further, there would seem to be a benefit in that state point S1 (see Figure 1) would be closer to the saturation dome on the P-high isobar, allowing the heat rejection to be closer to isothermal and, therefore, more like the Carnot cycle.

It turns out that, for increased efficiency, we can choose S4 such that S1 is on the saturation dome, instead of outside of it in the superheat region. Figure 4 shows the T-s diagrams for two refrigeration cycles, one where S4 is a saturated vapor and the other (in light green) where S4 has been moved further into the saturation dome to allow S1 to be a saturated vapor.

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