LESSON - 13 REFRIGERATORS OR HEAT PUMPS, THE CLAUSIUS STATEMENT AND PERPETUAL MOTION MACHINE OF SECOND KIND

13.1. REFRIGERATORS OR HEAT PUMPS

The second cycle we were not able to complete was the one that involved the impossibility of transferring heat directly from a low-temperature body to a high-temperature body. This can be done with a refrigerator or heat pump.

 13.1.1. Refrigerators

A vapor-compression refrigeration cycle, which involves four main components: a compressor, a condenser, an expansion valve and an evaporator, is shown in Fig. 13.1(a). The working fluid is the refrigerant, such as Freon or Ammonia, which goes through a thermodynamic cycle while flowing through the components of the refrigerator.

The refrigerant undergoes the following processes to complete a cycle as it flows through different components of refrigerator:

Evaporator: The heat is transferred to the refrigerant in evaporator from the refrigerated space (low-temperature reservoir) as the refrigerant’s pressure and temperature are lower than the refrigerated space. 

Compressor: Work is done on the refrigerant in the compressor during pressure and temperature increase.

Condenser:  Heat is transferred from the refrigerant to atmospheric air (high-temperature reservoir) in the condenser, as the refrigerant’s pressure and temperature are higher than atmosphere air.

Throttling valve: The pressure drop occurs as the refrigerant flows through the throttle valve or capillary tube.

Thus refrigerator is a device that operates in a cycle, that requires work input and accomplishes the objective of transferring heat from a low-temperature body (refrigerated space) to a high-temperature body (atmospheric air). The principle of refrigerator is shown in Fig. 13.1(b).

Fig. 13.1(a). A simple refrigeration cycle.

Fig. 13.1 (b).   Principle of refrigeration.

The "efficiency" of a refrigerator is expressed in terms of coefficient of performance (COP). In the case of a refrigerator the objective (i.e., energy sought) is the heat transferred from the refrigerated space to refrigerant ‘Q_L’  and the energy that costs is the work ‘W’ on compressor.

 Thus the COP_refrigerator     

13.1.2. Heat pump It is another device that transfers heat from a low-temperature medium to a high-temperature medium (Fig. 13.2). Refrigerators and heat pumps operate on the same cycle but differ in their objectives.  The objective of a: refrigerator is to maintain the refrigerated space at a low temperature by removing heat from the refrigerated space (low-temperature reservoir), and discharging this heat to a atmosphere (higher-temperature reservoir). heat pump, however, is to maintain a heated space at a high temperature. This is accomplished by absorbing heat from a cold atmospheric air in winter or well water (low-temperature reservoir), and supplying this heat to a house (high-temperature medium).

Fig. 13.2. Heat pump.

The "efficiency" of a heat pump is also expressed in terms of the coefficient of performance (COP). In the case of a heat pump the objective (i.e., energy sought) is Q_H (heat transferred to heated space from refrigerant) and the energy that costs is the work W.

 Therefore,   COP_{Heat Pump}  

Can a refrigerator work as a heat pump?

The answer is yes. In winter, an ordinary refrigerator placed in the window of a house with its door open to the cold outside air will function as a heat pump since it will absorb heat from outside and reject this heat into the house through the condenser coils behind it.

Similarly, in winter, the air conditioner having its evaporator outside and condenser inside the house will act as a heat pump.

The comparision of COPs of heat pump and refrigerator reveals that the COP_{Heat Pump} = COP_Refrigerator + 1    for fixed value of  and  .

Problem13.1: For heating a house in winter it requires 204820 kJ/h of heat. The work input of 28424 kJ/h is needed by heat pump for its operation which absorbs heat from cold air outside in winter to supply heat to the house. Calculate the COP of heat pump and heat abstracted from outsides. What will be COP of heat pump when it serves to cool the house in the summer requiring same 204820 kJ/h of heat rejection?

Solution: Refer fig. 13.3

 Given: Work input, W = 28424 kJ/h;            Heat supplied by heat pump, Q_H = 20484 kJ/h

(a) Determine the  COP of heat pump:    Formula: COPHeat Pump =      Answer: COPHeat Pump =  = 7.205   (b) Determine heat abstracted from outside, QL:    Formula: Work input, W = QH – QL   or   QL = QH − W      Answer: QL = QH - W = 204820 − 28424 = 176396 kJ/h

Fig. 13.3: Schematic of problem 13.1

(c) Determine COP of heat pump when it serves to cool the house in the summer, COP_refrigerator

    Formula: COP_refrigerator = 

    Answer: COP_refrigerator =  = 6.205

These results suggested that the same device has two values of COP depending upon the uses.

13.2. STATEMENT OF THE SECOND LAW OF THERMODYNAMICS

 On the basis of discussion we considered for refrigerator we are now ready to state the Clausius Statement of Second Law.

13.2.1. The Clausius Statement

The Clausius statement is second classical statement of the Second Law of Thermodynamics after defining the Kelvin-Planck statement of the second law of thermodynamics. It is related to refrigerators or heat pumps. It is expressed as follows:

“It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a cooler body to a hotter body”.

Fig. 13.4 (a).   Impossible cycle as it  violates Second Law (Clausius Satement).

Fig. 13.4 (b). Possible cycle as it satisfies Second  Law (Clausius Satement).

This statement is related to the refrigerator or heat pump, and in fact states that it is impossible to construct a refrigerator that operates without an input of work to transfer heat from a cooler body to a hotter body as shown in Fig. 13.4(a). The only alternative is that some work must be done on the system from the surrounding as shown in Fig. 13.4(b). This also implies that the coefficient of performance is always less than infinity.

Thus the    

13.3. OBSERVATIONS REGARDING KELVIN-PLANCK AND CLAUSIUS STATEMENTS

In regard to Kelvin-Planck statement and Clausius statement of the Second Law of Thermodynamics, three observations can be made.

13.3.1.  First observation

The first is that both Kelvin-Planck and Clausius statements are negative statements:

It is impossible to "prove" a negative statement. However, like every other law of nature, we can say that the Second law of thermodynamics is based on experimental evidence. Every relevant experiment that has been conducted has either directly or indirectly verified the second law, and to date, no experiment has ever been conducted that contradicts the Second law. The basis of the Second law is therefore experimental evidence.

13.3.2.  Second observation

A second observation is that both Kelvin-Planck and Clausius statements are equivalent:

Two statements are equivalent; therefore either of the statement can be used as the expression of the Second law of thermodynamics. Any device that violates the Kelvin-Planck statement also violates the Clausius statement and vice versa or any device that obeys the Kelvin-Planck statement also obeys the Clausius statement and vice versa. This can be proved as follows.

13.3.2.1. Proof that a device that violates the Clausius statement also violates the Kelvin-Planck statement.

Consider the heat engine–heat pump combination as shown in Fig. 13.5 (a), working between the same high temperature and low temperature reservoirs. The device at the left in this figure is a heat pump that requires no work and transfers heat, ‘Q_L’ from low temperature to high temperature reservoir, and thus violates the Clausius statement. Let us assume that in the device on right side ‘Q_H’ heat which is grater than Q_L, be transferred from the high-temperature reservoir to the heat engine, and  let the heat engine reject the ‘Q_L’ heat as it does an amount of work W (which equals (Q_H - Q_L).

(a)	(b)

Fig. 13.5. Demonstration of violation of the Clausius statement implies a violation of the Kelvin-Planck statement.

Since the net heat transfer to the low-temperature reservoir is zero, the heat engine, the heat pump and the low-temperature reservoir can be viewed as a device as shown in Fig. 13.5 (b) that operates in a cycle and produces no effect other than doing certain amount of work and the exchange of heat with a single reservoir. This is clear violation of the Kelvin Planck statement. Thus, a violation of the Clausius statement results in the violation of the Kelvin-Planck statement. 

13.3.2.2. Proof that a device that violates the Kelvin-Planck statement also violates the Clausius statement.

The complete equivalence of these two statements is established when it is also shown that a violation of the Kelvin-Planck statement implies a violation of the Clausius statement. This can also be shown in a similar manner with the help of Fig. 13.5 (b). 

(a)	(b)

Fig. 13.5. Demonstration of violation of the Kelvin-Planck statement implies violation of the Clausius statement.

Therefore from the above proofs it is clear that the Kelvin-Planck and Clausius statements are equivalent expressions of the Second law of thermodynamics.

13.3.3.  Third observation

A third observation is that usually the second law of thermodynamics has been stated as the impossibility of constructing a Perpetual Motion machine of Second Kind.

13.3.3.1. Perpetual Motion machine of Second Kind (PMM-II)

A device that violated the second law (Kelvin-Planck Statement) could be made into a perpetual motion machine of the second kind as follows:

Consider the heat engine–heat pump combination as the power plant of a ship as shown in Fig. 13.6 (a).

In Heat pump: An amount of heat Q_L is transferred from the ocean to a high-temperature body by means of a heat pump. The work required is W′  (= Q_H − Q_L)  and the heat transferred to the high-temperature body is Q_H.

In Heat engine: Let the Q_H heat be transferred to a heat engine and does the same amount of work W (= Q_H), Thus the ship engine violates the Kelvin-Plank statement of the Second law of thermodynamics. Of this work an amount of work W′  (= Q_H - Q_L) is required to drive the heat pump, leaving the net work W_net (= Q_L) available for driving the ship.

Fig. 13.6 (a): A perpetual-motion machine of the second kind

Since the net heat transfer to the high-temperature reservoir is zero, the heat engine, the heat pump and the high-temperature reservoir can be viewed as a ship engine as shown in Fig. 13.6 (b) that operates in a cycle and produces no effect other than doing QL amount of work with exchange of heat QL with a single reservoir (Ocean). This is clear violation of the Kelvin Planck statement of Second law of thermodynamics. Thus, we have a perpetual motion machine of second kind in the sense that work is done by utilizing freely available sources of energy from ocean or atmosphere.

Fig. 13.6 (b) .Perpetual motion machine of second kind