or work into heat and include heat engines such as internal combustion engines
in ... vaporized gasoline in an internal combustion engine so that the device ...
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: U f = Ui . In other words, there is no net change in the internal energy after each cycle:
!U = U f " 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
!"
#W , Qin
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
!=
( "Qex ) "W Qin + Qex = = 1" Qin Qin 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
!W
! Qex
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 quasistatic 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 " dT = ! dV , T CV # T
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 lowRc density gas with CV = const , T and V satisfy TV v = 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"
Tc . Th
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 QTq s flowing into a system in an isothermal process at temperature T is simply given by
QTq s = T!S , where !S is a change in the entropy of the system after the isothermal process: !S = S(T ,V f ,n) " S(T, V i ,n) = S(T ,P f , 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 QTq s . 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
Qinqs = Th !Sh , where !Sh is a change in the entropy of the substance in this isothermal process, while the quasistatic heat !Qexqs that the substance in the engine discards to a colder reservoir at Tc is
!Qexqs = !T c "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
("Q ) = 1 " ("T #S ) = 1" T #S = 1" qs ex qs in
Q
c
Th #Sh
c
c
h
Th #Sh
= 1"
Tc . Th
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 ! = "W q s Qinqs , we can calculate the output work of a Carnot cycle by # T & !W q s = "CarnotQinqs = %% 1 ! c (( Th )Sh = (Th ! Tc ))Sh . Th ' $
For example, for a low-density gas with CV = const , we find
7 #V & !W q s = ( Th ! Tc ) "Sh = nR(Th ! Tc ) ln %% 3 (( $ V2 ' qs so that to increase the output work !W , we must increase the temperature difference Th ! Tc and/or the volume ratio V 3 V 2 .
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 = !T c "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 Qexqs 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 q s = Qinqs + Qexqs = (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 Qinqs of the process while the area underneath the lower portion of the curve for the cyclic process represents the exhaust heat !Qexqs of the process. As
! W q s = Qinqs + Qexqs , qs the area enclosed by the curve for the cyclic process represents the work output ! W . 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
! W qs
qs !Qex
S
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 qs As we have found in Sec.9.1, the quasi-static heat QV flowing into a system in an isochoric process at volume V is simply given by
QVq s = !U , where !U is a change in the internal energy of the system after the isochoric process: !U = U( T f ,V,n ) " U (Ti ,V,n) = U(T f ,P f , 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 qs quasi-static heat QV . 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 ,V 1 ,n) to state (T2 ,V2 ,n) . For this process, we find
T1V1 R c v = T2 V2 R cv . (ii) A quasi-static isochoric ignition of the gas proceeds from state (T2 ,V2 ,n) to state (T3 ,V 2 , n) . In this process, an input heat Qinqs flows into the gas and is given by Qinqs = !U = CV (T3 " T2 ) .
(iii) A quasi-static
(T4 ,V1 ,n) .
adiabatic expansion of the gas proceeds from state (T3 ,V 2 , n) to state
For this process, we find
T4 V1 R cv = T3 V2 R c v . (iv) A quasi-static isochoric exhaust of the gas proceeds from state (T4 ,V1 ,n) back to the initial state (T1 ,V 1 ,n) . In this process, an exhaust heat !Qexqs flows out of the gas and is given by
!Qexqs = ! "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"
( "Q ) = 1 " C (T qs ex qs in
" T1 ) T " T1 . = 1" 4 CV (T3 " T2 ) T3 " T2 V
Q
Subtracting T1V1 R c v = T2 V2 R cv from T4 V1
R cv
4
= T3 V2
R cv
, we obtain
" V % R cv T4 ! T1 = $$ 2 '' (T3 ! T2 ) # V1 &
so that # V & R cv T4 " T1 T T !Otto = 1" = 1" %% 2 (( = 1" 1 = 1" 4 . T3 " T2 T2 T3 $ V1 '
The efficiency of the Otto cycle is determined solely by the compression ratio V 2 V 1 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 # V & R cv # 1& 2 5 # 1& 6 5 # 1& 6 5 3 !Otto = 1" %% 2 (( = 1 " % ( = 1 " % ( = 1" % 1 " ( ) = 0.6 , $ 8' $ 2' $ 2' 5 $ V1 ' a
where we have used (1 ! x ) " 1 ! ax for x
