1 10. Heat devices: heat engines and refrigerators (Hiroshi Matsuoka)

In this chapter, we will discuss how heat devices work. Heat devices convert heat into work or work into heat and include heat engines such as internal combustion engines in automobiles and steam engines in power plants as well as refrigerators and air conditioners. We will idealize processes in these devices to be quasi-static so that we can calculate the quasi-static heat flowing into a system during these processes in terms of changes in state variables such as the internal energy, enthalpy, and entropy of the system.

Heat devices based on cyclic processes

Each heat device is based on a cyclic process of a substance such as a gas mixture of air and vaporized gasoline in an internal combustion engine so that the device including the substance inside returns to the same macroscopic state after each cycle and by repeating the same cyclic process, it continuously converts heat into work or work into heat.

A special feature of the cyclic process is that the device returns to the same macroscopic state after each cycle so that the value of the internal energy of the device at the end of the cycle is the same as that at the beginning of the cycle: Uf =Ui . In other words, there is no net change in the

internal energy after each cycle: !U =Uf "Ui = 0 .

The first law applied to a cyclic process

Applying the first law of thermodynamics to the cyclic process in the heat device, we then find

!U = Q +W = 0 ,

where Q is the total amount of heat that flows into the device in each cycle while W is the total amount of work that is done on the device in each cycle. We have therefore obtained a very useful relation between Q and W: !W = Q .

10. 1 Heat engines

A heat engine is a heat device that converts heat into work. Clearly, it would be ideal if we could construct a heat engine that could absorb some heat from a heat source and convert it completely into a work output to the outside. It turns out that we cannot construct such a heat

2 engine. As we will discuss later, this statement of the impossibility of a “perfect” heat engine is nothing but “the second law of thermodynamics.”

Efficiency of a heat engine

Any real heat engine must therefore discard some energy as heat to the outside so that a part of the input heat must be wasted as the “exhaust” heat discarded to the outside. Note that the input heat Qin is positive or Qin > 0 as it flows into the engine, whereas the “exhaust” heat Qex is negative or Qex < 0 as it flows out of the engine. The net total amount of heat Q for each cycle of the engine is then the sum of Qin and Qex :

Q =Qin +Qex .

Note that as Qex < 0 , we find Q < Qin . As the total amount W of work done on the engine is related to Q by !W = Q , the work done by the engine !W is related to Qin by !W = Q = Qin +Qex <Qin .

To improve engine, we must then aim at increasing the “efficiency” ! defined by ! "

#WQin

,

which measures how much of the input heat is converted to the work output. In other words, ! measures the efficiency of the conversion of the input heat into the work output. As !W <Qin , the efficiency of the engine must be always less that 1, ! <1 , and the closer to 1 ! gets, the more efficient the engine gets. To improve a heat engine, we must therefore understand what determines the efficiency of the heat engine so that we can find a way to increase its efficiency toward 1.

The efficiency ! can be also expressed in terms of Qin and Qex as

3

! ="WQin

=Qin +QexQin

= 1""Qex( )Qin

so that we can calculate the efficiency directly from Qin and Qex . This equation indicates that to increase the efficiency, we must make the discarded heat !Qex smaller compared to the input heat Qin .

Qin

!Qex

!W

Efficiency is not everything

Clearly, the efficiency of a heat engine is not the only factor that dictates the design of the particular engine for a particular purpose, which requires, for example, a particular size for work output. Once we have incorporated all the specifications required for the particular engine, we then try to maximize its efficiency. Ideal heat engines based on quasi-static cyclic processes

Applying the second law of thermodynamics, we will show later that the efficiency of a heat engine reaches a maximum value when the cyclic process for the heat engine becomes completely quasi-static so that the quasi-static cyclic process provides an upper limit for the efficiency of the hat engine. We will therefore examine quasi-static cyclic processes as idealized models for heat engines.

10.1.1 The Carnot cycle: an idealized model for a heat engine

As an idealized model for a heat engine, a Carnot cycle is a quasi-static cyclic process in which a substance inside the engine exchanges heat with two heat reservoirs at two different temperatures, Tc and Th , where Tc < Th . These heat reservoirs are much larger than the engine so that their heat capacities are much larger than that of the substance inside the engine (recall that a heat capacity of a system is extensive so that it is proportional to the mole number of the system) and any heat exchange between the substance and the reservoirs hardly changes the temperatures of the reservoirs.

4 In the Carnot cycle, the substance absorbs heat from the hotter reservoir through a quasi-

static isothermal process at temperature Th and discards heat to the colder reservoir through another quasi-static isothermal process at temperature Tc . Following the isothermal process at temperature Th , the substance decreases its temperature to Tc through a quasi-static adiabatic process before it goes through the quasi-static isothermal process at temperature Tc followed by another quasi-static adiabatic process that increases the temperature back to Th .

Temperature changes as volume changes in a quasi-static adiabatic process To construct a Carnot cycle for a given substance, we must be able to change the temperature

of the substance by changing its volume during a quasi-static adiabatic process. As shown in Sec.9.4, along a quasi-static adiabatic process, an infinitesimal change dT in the temperature is related to an infinitesimal change dV in the volume through 1

TdT = ! "

CV# T

dV ,

which indicates that as long as ! " 0 , the temperature of the substance changes as its volume is changed so that by either expanding or compressing the volume, we can change the temperature from a given value to any other value we like.

For example, as we have found in Sec.9.4, along a quasi-static adiabatic process of a low-density gas with CV = const , T and V satisfy TV R cv = const so that we can sketch a Carnot cycle with a low-density gas on the TV diagram as shown below. Both the quasi-static process from state 1 to state 2 and the quasi-static process from state 3 to state 4 are adiabatic.

5

The universal efficiency for the Carnot cycles between reservoirs at Tc and Th

The Carnot cycle occupies a special place in the study of heat engines, because the value of its efficiency ! depends only on Tc and Th so that all the Carnot cycles that operate with two heat reservoirs at Tc and Th share the same “universal” efficiency:

!Carnot = 1"

TcTh

.

To derive this universal efficiency, we need to express the quasi-static heat flowing into a system in an isothermal process in terms of an entropy change in the system. Quasi-static heat and entropy change along a quasi-static isothermal process

As we have found in Sec.9.3, the quasi-static heat QTqs flowing into a system in an isothermal

process at temperature T is simply given by QT

qs = T!S ,

where!S is a change in the entropy of the system after the isothermal process: !S = S T ,Vf ,n( ) " S T,Vi ,n( ) = S T ,Pf , n( ) " S T, Pi ,n( ) , so that if we can find the initial and final values of the entropy S through an analytical expression for S or in a numerical table for S in some reference handbook, we can find the quasi-static heat QTqs .

Deriving the universal efficiency for the Carnot cycles between reservoirs at Tc and Th

6

In a Carnot cycle, the quasi-static heat Qinqs that the substance in an engine absorbs from a

hotter reservoir at Th is then Qin

qs = Th!Sh ,

where !Sh is a change in the entropy of the substance in this isothermal process, while the quasi-static heat !Qex

qs that the substance in the engine discards to a colder reservoir at Tc is

!Qexqs = !Tc"Sc ,

where !Sc is a change in the entropy of the substance in this isothermal process. In contrast, along the quasi-static adiabatic processes, there is no heat flowing into the substance in the engine so that the entropy of the substance remains constant and there is no change in the entropy of the substance in these processes.

As the Carnot cycle is a cyclic process, the entropy of the substance must return to its initial value after the cycle so that the net change in the entropy after the cycle is zero:

0 = !S = !Sh + !Sc ,

from which we obtain ! "Sc = "Sh . We then find the efficiency of the Carnot cycle to be

!Carnot = 1""Qex

qs( )Qinqs =1 "

"Tc#Sc( )Th#Sh

= 1"Tc#ShTh#Sh

= 1"TcTh

.

This expression for the efficiency is universal for all the Carnot cycles that operate between two heat reservoirs at temperatures, Tc and Th . The output work of a Carnot cycle

As the efficiency of a cyclic process is defined by ! = "Wqs Qinqs , we can calculate the output

work of a Carnot cycle by

!W qs = "CarnotQinqs = 1 ! Tc

Th

#

$ % %

&

' ( ( Th)Sh = Th ! Tc( ))Sh .

For example, for a low-density gas with CV = const , we find

7

!W qs = Th ! Tc( )"Sh = nR Th ! Tc( ) ln V3V2

#

$ % %

&

' ( (

so that to increase the output work !W qs , we must increase the temperature difference Th ! Tc and/or the volume ratio V3 V2 . The Carnot cycle in the TS diagram and its universal efficiency

We can understand the above derivation of the efficiency of the Carnot cycles between two heat reservoirs at Tc and Th by sketching a Carnot cycle in “the TS diagram” shown below.

In this diagram, the quasi-static input heat Qinqs = Th!Sh corresponds to the sum of the two shaded

areas below the line from state 2 to state 3, while the quasi-static discarded heat !Qexqs = !Tc"Sc

corresponds to the area of the shaded area below the line from state 1 to state 4. The efficiency of the cycle depends on the ratio between Qex

qs and Qinqs , which is equal to the ratio between the

two rectangular areas whose heights along the T-direction are Tc and Th , respectively. A crucial point here is that the entropy changes along the two isothermal processes are equal in magnitude: !Sh = !Sc . This point is also the reason why the efficiency of the Carnot cycles between two

heat reservoirs at Tc and Th is universal despite that the values of entropy at states 1 through 4 may vary from one Carnot cycle to another.

The shaded area enclosed by the Carnot cycle in this TS diagram also corresponds to the work output because

!W qs =Qinqs +Qex

qs = Th ! Tc( )"Sh .

8 Input heat, exhaust heat, and work output in the TS diagram

In general, for a quasi-static cyclic on the TS diagram shown below, the area underneath the upper portion of the curve for the cyclic process represents the input heat Qin

qs of the process while the area underneath the lower portion of the curve for the cyclic process represents the exhaust heat !Qex

qs of the process. As !W qs =Qin

qs +Qexqs ,

the area enclosed by the curve for the cyclic process represents the work output !W qs . Therefore, for a given upper portion of the curve for the cyclic process, the smaller the area underneath the lower portion of the curve for the cyclic process becomes, the higher the efficiency gets.

T

S

!Qexqs

!W qs

10.1.2 The Otto cycle: an idealized model for an internal combustion engine As an idealized model for a cyclic process in an internal combustion engine, we will examine

an Otto cycle of a low-density gas or a mixture of air and vaporized gasoline, for which we assume:

(i) the ideal gas law is a good approximation for the equation of state of the gas. (ii) the heat capacity at constant volume of the gas is constant: CV = const .

The Otto cycle consists of two quasi-static adiabatic processes and two quasi-static isochoric or isovolumic processes. Before we describe the Otto cycle in more detail, we need to remind us how we can find the quasi-static heat flowing into a system during a quasi-static isochoric process.

9 Quasi-static heat in an isochoric or isovolumic process: V = const

As we have found in Sec.9.1, the quasi-static heat QVqs flowing into a system in an isochoric

process at volume V is simply given by QV

qs = !U ,

where!U is a change in the internal energy of the system after the isochoric process: !U =U Tf ,V,n( ) "U Ti ,V,n( ) =U T f ,Pf , n( ) "U Ti ,Pi , n( ) , so that if we can find the initial and final values of the internal energy U through an analytical expression for U or in a numerical table for U in some reference handbook, we can find the quasi-static heat QV

qs .

The Otto cycle of the low-density gas The Otto cycle of the low-density gas consists of the following four quasi-static processes:

(i) A quasi-static adiabatic compression of the gas proceeds from state T1,V1,n( ) to state T2 ,V2 ,n( ) . For this process, we find

T1V1

R c v = T2V2R cv .

(ii) A quasi-static isochoric ignition of the gas proceeds from state T2 ,V2 ,n( ) to state T3,V2, n( ) .

In this process, an input heat Qinqs flows into the gas and is given by

Qin

qs = !U = CV T3 " T2( ) . (iii) A quasi-static adiabatic expansion of the gas proceeds from state T3,V2, n( ) to state

T4 ,V1,n( ) . For this process, we find T4V1

R cv = T3V2R c v .

(iv) A quasi-static isochoric exhaust of the gas proceeds from state T4 ,V1,n( ) back to the initial

state T1,V1,n( ) . In this process, an exhaust heat !Qexqs flows out of the gas and is given by

!Qex

qs = !"U = !CV T1 ! T4( ) = CV T4 ! T1( ) . We can now sketch this Otto cycle on the TV diagram as shown below.

10

The efficiency of the Otto cycle

The efficiency of the Otto cycle is then given by

!Otto = 1""Qexqs( )Qinqs =1 "

CV T4 " T1( )CV T3 " T2( )

= 1"T4 " T1T3 " T2

.

Subtracting T1V1

R c v = T2V2R cv from T4V1

R cv = T3V2R c v , we obtain

T4 ! T1 =V2V1

"

# $ $

%

& ' '

R cv

T3 ! T2( )

so that

!Otto = 1" T4 " T1T3 " T2

= 1" V2V1

#

$ % %

&

' ( (

R c v

= 1" T1T2

= 1" T 4T3

.

The efficiency of the Otto cycle is determined solely by the compression ratio V2 V1 so that to improve its efficiency we must decrease the compression ratio. The compression ratio for an automobile engine is typically 1 8 , and for air, cv ! 5 2( )R so that

!Otto = 1" V2V1

#

$ % %

&

' ( (

R cv

=1 " 18# $ % & ' ( 2 5

= 1 " 12# $ % & ' ( 6 5

= 1" 1 " 12

# $ %

& ' ( 6 5

)35

= 0.6 ,

where we have used 1 ! x( )a " 1 ! ax for x << 1 . The output work of the Otto cycle

As the efficiency of a cyclic process is defined by ! = "Wqs Qinqs , we can calculate the output

work of the Otto cycle by

11

!W qs = "OttoQinqs = CV 1 !

V2V1

#

$ % %

&

' ( (

R c v) * +

, +

- . +

/ + T3 ! T2( ) .

so that to increase the output work !W qs , we must increase the temperature difference T3 ! T2 and/or decrease the compression ratioV2 V1 . The Otto cycle in the TS diagram and its efficiency compared with that of the Carnot cycle

The efficiency of the Otto cycle is also less than that of the Carnot cycle between reservoirs at T1 and T3 as !Otto = 1"

T1T2

= 1" T 4T3

<1 " T1T3

= !Carnot,

which can be seen in the sketch of an Otto cycle in “the TS diagram” shown below.

SUMMARY FOR SEC.10.1

1. For a heat engine that receives an input heat Qin and does some work !W on the outside

while discarding an exhaust heat !Qex , we can calculate its efficiency by

! ="WQin

=Qin +QexQin

= 1""Qex( )Qin

.

2. In a Carnot cycle, a substance absorbs heat from a hotter reservoir through a quasi-static

isothermal process at temperature Th and discards heat to a colder reservoir through another quasi-static isothermal process at temperature Tc . Following the isothermal process at Th , the substance decreases its temperature to Tc through a quasi-static adiabatic process before it goes through the quasi-static isothermal process at Tc followed by another quasi-static

12 adiabatic process that increases the temperature back to Th . The efficiency !Carnot of the Carnot cycle depends only on Th and Tc by the following universal equation:

!Carnot = 1"

TcTh

3. An Otto cycle of a low-density gas is an idealized model for a cyclic process of a mixture of

air and vaporized gasoline in an internal combustion engine and consists of two quasi-static adiabatic processes each connected with two quasi-static isochoric processes at two volumes V1 and V2 (V1 > V2 ). Its efficiency depends on the compression ratio V2 V1 as

!Otto = 1" V2V1

#

$ % %

&

' ( (

R cv

,

where cv is the molar heat capacity at constant volume for the gas.