exergetic Cycle efficiency

The use of the cycle thermal efficiency is well understood. However, the previous section demonstrated that the maximum value of the thermal effi-ciency is not 100%, but rather the Carnot effieffi-ciency. The concept of exergy and the exergetic efficiency provides an alternate framework for determining the performance of a cycle.

The exergetic efficiency of a cycle is defined in the same way that it was for a device, Equation 3.83, which is repeated below:

η = exergy output (exergy task) exergy input (exergy resource)

x (3.101)

3.10.2.1 Power Cycles

Consider a power cycle operating between two thermal reservoirs as shown in Figure 3.11. This figure indicates that the low-temperature reservoir is at the dead-state temperature. If the dead state is considered as the atmo-sphere, then this depiction of the power cycle is accurate; the ultimate low-temperature sink is the atmosphere.

An alternative expression for the exergetic efficiency can be developed based on Figure 3.11. For a power cycle, the exergy task is the delivery of the cycle work. The exergy input is the exergy associated with the heat transfer from the high-temperature reservoir. Therefore, the exergetic efficiency of a power cycle can be written as

ηx,power= cycle

H

W XQ

(3.102) Applying the exergy balance equation (Equation 3.76) to the system bound-ary that encloses the complete power cycle results in

− − − des= 0

H 0 cycle

X_Q X_Q X_W X (3.103)

T_L= T₀ QL= Q₀ Q_H T_H

W_cycle

P

FIGURe 3.11

Power cycle operating between TH and the dead state.

There is no exergy associated with the heat transferred to the dead state, because it is at the dead-state temperature T₀. This can be proven with the following equation:

Substituting Equation 3.104 into Equation 3.103 and solving for the cycle work gives

= cycle= − des

cycle H

X_W W X_Q X (3.105)

Substitution of Equation 3.105 into Equation 3.102 results in η_x,power= −1 ^des

H

X

X_Q (3.106)

Equation 3.106 is an important equation, because it demonstrates that the maximum exergetic efficiency of a power cycle is 100% for the case of a reversible cycle with no exergy destruction. Unlike the thermal efficiency that is bounded by the Carnot efficiency at the upper limit, the exergetic efficiency does reach a maximum of 100% in the case of an ideal cycle with no irreversibility.

The exergy content of the heat transfer from the high-temperature reser-voir relative to the dead state at T₀ is

Substitution of Equation 3.107 into Equation 3.102 results in

η =

Equation 3.108 indicates that the exergetic efficiency of a power cycle operat-ing between TH and T0 is directly proportional to the thermal efficiency and inversely proportional to the Carnot efficiency. In addition, since the Carnot efficiency of a power cycle is less than 1, the exergetic efficiency of a power cycle must be larger than the thermal efficiency.

3.10.2.2 Refrigeration and Heat Pump Cycles

Consider a refrigeration cycle operating between a low-temperature source, TL, and the dead state, T0, as the sink. This cycle is shown in Figure 3.12. The exergetic efficiency of this cycle can be determined by applying Equation 3.101.

η =−

A negative sign must be used in the numerator of Equation 3.109, because the exergy output defined in Equation 3.101 is really an input to the refrigeration cycle. The exergy balance applied to the system boundary surrounding the cycle shown in Figure 3.12 results in

+ − des= 0

L cycle

X_Q X_W X (3.110)

Solving Equation 3.110 for the exergy transferred due to QL and substituting into Equation 3.109 reveals the following:

ηx,refrig= −1 des cycle

X

W (3.111)

Similar to what was seen for the power cycle, Equation 3.111 indicates that the exergetic efficiency of a refrigeration cycle has an upper limit of 100% in the case of a completely reversible cycle with no exergy destruction.

The exergy associated with the heat transfer from the low-temperature source is given by

=  −

Substituting Equation 3.112 into Equation 3.109 results in

η =

Following similar thinking, the exergetic efficiency of a heat pump cycle can also be developed. The result, shown in Equation 3.114, is left as an exercise for the reader.

Refrigeration cycle operating between TL and the dead state.

3.10.2.3 Significance of the Exergetic Cycle Efficiency

As demonstrated in Sections 3.10.2.1 and 3.10.2.2, the exergetic efficiency of any cycle is always bounded by 0 and 1 (0%–100%). This makes the exergetic cycle efficiency a much more intuitive index for cycle performance compared to the thermal efficiency. This fact, coupled with the definition of the exer-getic cycle efficiency, Equation 3.101, demonstrates the task/resource nature of the exergetic efficiency. High values of exergetic efficiency indicate that the exergy resource is well matched for the exergy task the cycle is providing.

EXAMPLE 3.15

Two refrigeration systems are being considered for a cold storage applica-tion: a standard vapor compression refrigeration system and an absorption refrigeration system. In both cases, the cold space must be maintained at –5°C.

To maintain this temperature, heat must be removed from the cold space at a rate of 35 kW. The vapor compression system requires an electrical input of 10 kW to accomplish the refrigeration. The absorption refrigeration system utilizes a 58 kW heat input at 50°C to drive the cycle instead of electricity. Both cycles sink to the atmosphere, which can be considered the dead state, at 22°C.

Determine the COP_C and exergetic efficiency of each refrigeration cycle.

SOLUTION: A sketch of the two refrigeration cycles is shown in Figure E3.15.

For the vapor compression system, the COP_C is given by

FIGURe e3.15

=  =35 kW= 10 kW 3.50

C,vap L

cycle

COP Q

W

The energy input to the absorption cycle is heat. Therefore, the COP_C is deter-mined by

=  =35 kW= 58 kW 0.60

C,abs L

in

COP Q

Q T_L=–5˚C

T_in=50˚C

T_L=–5˚C T_H=T₀= 22˚C T_H=T₀= 22˚C

R W˙_cycle = 10 kW R

Q˙_L = 35 kW Q˙_L = 35 kW

Q˙_in = 58 kW

Q˙₀ Q˙₀

Vapor compression system Absorption system

3.10.2.4 Energy/Exergy Conundrum

Example 3.15 reveals an engineering trade-off between thermal efficiency and exergetic efficiency. Recall that thermal efficiency is the ratio of energy sought to energy that costs. Therefore, systems with high thermal efficiency are less expensive to operate. On the other hand, systems with high exer-getic efficiency are making the best use of the exergy resource to perform a task. Therefore, designing a thermal system to have a high thermal

The calculated values of the COP_C for each cycle indicate that the vapor com-pression cycle is performing better, relative to energy. The higher COP_C of the vapor compression cycle indicates that more refrigeration can be accomplished per kW of energy input compared to the absorption cycle.

The exergetic efficiency of the vapor compression cycle can be found using Equation 3.113.

The exergetic efficiency of the heat-driven absorption cycle can be derived from Equation 3.101 and is given by

Although the vapor compression cycle is the better cycle at transporting and utilizing energy, it is the absorption cycle that is almost twice as efficient at utilizing the exergy input to achieve the refrigeration task. In this section, the exergetic efficiency was referred to several times as a task/resource index.

The task of the refrigeration cycle (independent of what type it is) is to move heat, Q_L. With the vapor compression system, this is accomplished with work, W_cycle. With the absorption system, the cooling is accomplished with a heat input, Q_in. The low value of exergetic efficiency for the vapor compression cycle indicates that the exergy input (work) is not well matched to the exergy output (heat). On the other hand, the exergetic efficiency of the absorption system is much higher because the exergy input is heat and the exergy task is movement of heat. Using heat to move heat is a much better exergetic solution than using work to move heat.

efficiency is good from the economic point of view, whereas designing a system to have high exergetic efficiency is good from the environmental (exergy resource) perspective. As demonstrated in Example 3.15, what is good for high thermal efficiency (minimum cost) is not necessarily good for the environment (exergy resource). This is the conundrum consumers and industry face.

For example, businesses are in business to make money. Therefore, it seems logical for a business to design systems that maximize thermal efficiency, thereby reducing operating costs. However, what impact does this practice have on the environment? Even if exergy is not a consider-ation in system designs, it is still being destroyed in the system! The prob-lem that industry faces is that designing to maximize exergetic efficiency may not result in a good profit stream to keep the company in business.

However, for an industry to gain various certifications, minimum levels of performance must be met. For example, the Energy Star program was created by the United States Environmental Protection Agency in 1992 under authority of the Clean Air Act. The Energy Star program is volun-tary, but minimum standards have been established that industries must meet for various types of systems to have a product worthy of the Energy Star certification. These minimum requirements are meant to minimize environmental impact by reducing air pollution. Consumers then have a choice between products that have achieved the Energy Star certification or products that do not carry the certification. Industries that certify their products through the Energy Star program are practicing good exergy conservation.

Consumers are faced with many energy/exergy decisions. For example, should you heat your home with electric resistance heating or air-cooled solar collectors? Electric resistance heating is 100% efficient in that all the electricity (work) entering the heater leaves as heat. However, this is a device that uses work to move heat. As seen in Example 3.15, this is not a very exergetically efficient method of heating. Of course, this also depends on where the electricity originates. If you live in a region where electrical energy is delivered from a hydroelectric plant, then you are relying on the hydrological cycle, which can be considered renewable. However, if you live in a region where electrical energy is delivered from a fossil fuel–fired power plant, then as the electricity is used to run the electric heater, the exergy resource (the fossil fuel) is being depleted. From an environmental point of view, you would be much better heating your home with the air-cooled solar collectors because of their high exergetic efficiency (heat mov-ing heat). Of course, the conundrum is that air-cooled solar collectors cost quite a bit more than electric resistance heaters.

The purpose of this presentation is not to solve the energy/exergy conun-drum. Instead, it is meant to provoke some high-level exergy-based thinking in the engineers who will shape the future of the world.