第 4 章 理想气体的热力过程Chapter 4 Thermodynamic processes of Ideal Gas

4.1 基本热力过程 Basic thermodynamic processes--- p.134,p.313

4.2 多变过程 Polytropic process ---P.135

4.3 活塞式压气机的理论压缩过程 Theoretical compression process ---P.336

4.4 多级压缩中间冷却 Multistage Compression with inter-cooling--- P.33

7~P343

A. It is to determine the change in the properties of working fluid during a process;

( 在于确定在一个过程中工质状态参数的变化量 )B. It is to determine the amount of heat, work interaction between the system and its surroundings during a process.

( 确定在一个过程中系统与外界所交换的功量和热量的多少 )

2. 本章研究的对象 The object of this chapter

Reversible processes of Ideal Gases in closed systems

( 闭口系统中理想气体的可逆过程 )

1. 本章的研究任务The task of this chapter

Tch pTcu v

pdvdTcq v

pdvdTcTds v

dvv

RdT

T

cds v

1

2

1

2 lnlnv

vR

T

Tcs v

vdpdTcTds p

vdpdTcq p

dpp

RdT

T

cds p

1

2

1

2 lnlnp

pR

T

Tcs p

1

2

1

2 lnlnp

pR

v

vcs p

pdvw vdpwt Tdsq

3. 所采取的步骤 The procedures adopted

2

22

1

11

T

vp

T

vpPv=RT

4.1 Basic Thermodynamic Process (4.1 基本热力过程 )

1. 等容过程Isochoric-----Constant Volume process

For a constant volume process, the addition or removal of heat will lead to a change in the temperature and pressure of the gas, as shown on the two graphs above.

Substitute into

2

22

1

11

T

vp

T

vpCv

2

2

1

1

T

p

T

p

(1) 过程方程 (Process equation)

pdvdTcq v dTcqdu v

dTcdh p

(2) 内能、焓及熵的变化量

The change in Internal Energy, Enthalpy and Entropy

The amount of heat added to a closed system during a constant volume process equals to the increase in internal energy.

定容过程中加入闭口系统的热量等于系统的内能的增加量

Entropy Change To find the Entropy change, start with the expression derived f

rom the first law, replacing dU using the definition of specific heat at constant volume and using the definition of entropy

(3) Work done and Heat Transferred( 功量和热量 )Applying the first law of thermodynamics to the process Replacing δW with the reversible work

since the volume is constant dV = 0

duq

dTcTds v1

2lnT

Tcs v

WQdU

PdVQdU

dUQ 0W

using the definition of the specific heat at constant volume

to replace dU in the first law.

The technical work done

dTmCQ V

2

1

T

T V dTCmQ

2

1

)( 21

p

p

t ppvvdpw

2. 等压过程 Isobaric-----Constant Pressure  

(1) 过程方程 Process equation

2

22

1

11

T

vp

T

vp

Cp

2

2

1

1

T

v

T

v

For a constant pressure process, the addition or removal of heat will lead to a change in the temperature and volume of the gas, as shown on the two graphs above. Substitute into

The amount of heat added to a closed system during a constant pressure process equals to the increase in enthalpy.

dTcdu v dTcdh p

pdvdTcq v )( pvddTcq v

dhq

(2) 内能、焓及熵的变化量The change in Internal Energy, Enthalpy and Entropy

The amount of heat added to a closed system during a constant pressure process equals to the increase in enthalpy.

pdvdTcq v )( pvddTcq v

dhq

（ 3 ） Work done and Heat Transferred

dHQ

2

1

T

T PdTCmQ

During a constant pressure process, heat is added or removed and the temperature and volume changes. The volume at the end of the process can be found using the ideal gas law and the work calculated from

2

1

0p

p

t vdpw

3. 等温压缩和膨胀Isothermal Compression and Expansion

For a constant temperature process, the addition or removal of heat will lead to a change in the volume and pressure of the gas, as shown on the two graphs above.

Substitute into

2

22

1

11

T

vp

T

vp

CT

2211 vpvp

(1) 过程方程 (Process equation)

(2) 内能、焓及熵的变化量 The change in Internal Energy, Enthalpy and Entr

opy

0 dTcdu v 0 dTcdh p

pdvdTcq v pdvq

pdvTds T

pdvds

v

Rdvds

1

2lnv

vRs

2

1lnp

pRs

(3) 功量和热量Work Done and Heat Transferred

pdvdTcq v

pdvwq

In an isothermal process, the temperature is constant. Applying the first law of thermodynamics to this closed process

For an ideal gas, the internal energy is a function of temperature only, and since the temperature is constant, then dU is zero and

using the ideal gas law and integrating between the start and end of the process

During an isothermal process, the work done by the system is equal to the heat added to the system, and all the work is technically usable.

2

1ln2

1

2

1p

pRTdp

p

RTvdpw

p

p

p

p

t

2

1

1

2 lnln2

1p

pmRT

V

VmRT

V

dVmRTWQ

V

V

4. 绝热过程 Adiabatic Process

(1) Process equation ( 过程方程 )

or

Cpvk

Quasi-static, adiabatic process for an ideal gasδq = cvdT + pdv and δq = cpdT – vdp then cvdT = -pdv and cp dT =vdp therefore then

finally, we arrive at the very useful expression

from which it can also be shown that

111

22 k

k

vp

vp0lnln

1

2

1

2 p

p

v

vk

dv

dp

p

v

c

c

v

p p

dp

v

dvk or

When the temperatures at the start and end of the process are known, the pressure is calculated from

1

1

2

1

2 )( k

k

T

T

p

p

(2) The change in Internal Energy, Enthalpy and Entropy ( 内能、焓及熵的变化量）Entropy ChangeThere is no heat transfer to or from the gas and the process is reversible so that

1

2

1

1

2 )( k

v

v

T

T

When the temperatures at the start and end of the process are known, the work done is calculated from

(3) Work Done and Heat Transferred( 功量和热量 )

1

)(

)(1

1

)(

21

2211

2

1

2

1

12

k

TTR

vpvpk

dvv

pvpdvw

TTcpdvw

k

k

v

)(1

)(1

)(

)(

21

1221

2

1

2

1

TTRk

k

TTRk

TTR

pvdpdvw

vdpw

t

t

0Q

kwwt

5. 多变过程Polytropic Process

Many processes can be approximated by the law:

where, P Pressure, v Volume, n an index depending on the process type.

n=0, results in P=constant i.e. isobaric process.

n=infinity, results in v=constant i.e. isochoric process.

n=1, results in P v=constant, which is an isothermal process for a perfect gas.

n=k, which is a reversible adiabatic process for a perfect gas.

Polytropic processes are internally reversible. Some examples are vapors and ideal gas in many non-flow processes, such as:

Some polytropic processes are shown in figure below:

The initial state of working fluid is shown by point 0 on the P-V diagram. The polytropic state changes are:

•0 to 1= constant pressure heating, •0 to 2= constant volume heating, •0 to 3= reversible adiabatic compression, •0 to 4= isothermal compression, •0 to 5= constant pressure cooling, •0 to 6= constant volume cooling, •0 to 7= reversible adiabatic expansion, •0 to 8= isothermal expansion.

When the temperatures at the start and end of the process are known, the pressure is calculated from

(2) 内能、焓及熵的变化量The change in Internal Energy, Enthalpy and Entropy Entropy ChangeThere is no heat transfer to or from the gas and the process is reversible so that

1

1

2

1

2 )( n

n

T

T

p

p 1

2

1

1

2 )( n

v

v

T

T

When the temperatures at the start and end of the process are known, the work done is calculated from

(3) Work Done and Heat Transferred( 功量和热量 )

1

)(

)(1

1

21

2211

2

1

2

1

n

TTR

vpvpn

dvv

pvpdvw

pdvw

n

n

)(1

)(1

)(

)(

21

1221

2

1

2

1

TTRn

n

TTRn

TTR

pvdpdvw

vdpw

t

t

nwwt

)(1

)(1

)(1

1

)()(

12

1212

2112

TTRn

kn

TTn

RTT

k

Rn

TTRTTcq

pdvdTcq

v

v

29

4.3 活塞式压气机的工作过程

Working Process of gas piston compressor

1. Theoretical processes of single-staged piston compressor

( 理想单级活塞式压气机的工作原理 )

let the initial pressure of the gas is ( 令气体初态压力为 ) the final pressure is ( 终态压力为 ) ，

pressure ratio is ( 升压比 )

1p

2p1p

2p

1

2

p

p

（ 1 ） b-1:charge stroke ( 吸气冲程 ) ， 吸气量增大，吸入气体的状态（ P,T) 不变 ;

(3) 2- a: discharge process ( 排气过程 ), 气缸内的气体减少，气体状态不变化

(2) 1- 2:compression stroke( 压缩冲程 ) ， 吸气量不变，压力升高 .

2. Analysis on theoretical work consumed by these processes ( 压气机的理论耗功量分析 )

取活塞右行一次的吸气量为控制质量，并忽略动能差及位能差进行分析。

吸气过程：气体推动活塞移动，做推挤功 ；

压缩过程：气体向活塞做膨胀功

排气过程：气体向活塞做推挤功

2

1

pdv

22vp

11vp

压缩过程功 : 指 1- 2 ，闭口系的膨胀功，是可逆过程；压气机耗功：指 a- 1- 2- b, 全过程是开口系，包括流动功。气体在全过程中所做的总功为：

3. 三种不同压缩过程的比较(Comparison of three kinds of compression processes)

Adiabatic process ( 绝热过程 ) ： 压缩过程进行的很快，热量来不及释放；

Isothermal process ( 等温过程 ) ：压缩过程中，理想的冷却条件，压缩产生的热量可及时排出；

Polytropic process ( 多变过程 ) ：采用了一定的冷却措施，但压缩期间 , 温度仍继续升高。

2

1

22

2

1

11 vdpvppdvvpw

]1)[(1

)(1

1

2

211122

k

k

g p

pTR

k

kvpvp

k

k

(1) Work consumption ( 耗功量）If it is isothermal compression, then若压缩过程为可逆定温过程，则：

If it is isentropic compression, then若压缩过程为可逆绝热压缩，则 :

st,c ww

1

21

1

211Tt,c ln

p

plnvww

p

pTRp g

If it is polytropic compression, then 若压缩过程为可逆多变过程，则 :

nt,c ww ]1)[(1

)(1

1

1

211122

n

n

g p

pTR

n

nvpvp

n

n

stW ,nt,Tt, WW

sT ,2n2,T2, TT

（ 2 ） Discharge Temperature ( 排气温度）

4. 余隙容积的影响The influence of residual volume of inter-space.

（ 1 ） The influence on discharge volume( 对排气量的影响）Residual Volume ( 余隙容积）： V3

We note that if the pressure ratio is too high, then it can not work normally.( 压缩压力不能太高，压力升高，效率下降，在极限情况下容积效率可达零 ）

p

V

1

2

2’

2”

The discharge volume will decrease . ( 余隙存在将使排气量减少）（ 2 ） The influence on compression work （对耗功量的影

响）

There is no influence on compression work.( 对耗功量无影响 )

)1(13

4

31

3

31

41

V

V

VV

V

VV

VVv

]1)[(1

]1))[((1

]1)[(1

]1)[(1

1

1

21

1

1

2411

1

4

344

1

1

211

n

n

n

n

n

n

n

n

c

p

pVp

n

n

p

pVVp

n

n

p

pVp

n

n

p

pVp

n

nw

31

3

VV

Vc

]1)[(1

1

1

2 nv p

pcLet then

Volume efficiency is defined as

Multistage compression with intercooler is especially attractive when a gas is to be compressed to very high pressures.

1. two stage compression （两级压缩）

intercooler

4.4 Multistage Compression with inter-cooling （多级压缩、中间冷却）

Assumed that the compression process 1-2 and 3-4 are with the same polytropic index n ( 假设 1-2 和 3-4 的多变指数相等。 )　

　

Assumed ( 假设 )

To minimize the total compression work, the optimal pressure ratio can be determined by ( 为了使压缩过程的耗功量最小 , 可以采用如下方法确定最佳中间压力）

We obtain

]1)[(1

]1)[(1

1

3

43

1

1

21

n

n

gn

n

gc p

pTR

n

n

p

pTR

n

nw

13 TT

3

4

1

2

p

p

p

p 02

dp

dwc

That is, to minimize the compression work during two stage compression, the pressure ratio across each stage of the compressor must be the same.

（各级压缩比都相等时多级压缩最省功。）

When this condition is satisfied, the compression work at each stage becomes identical.

( 当满足上述条件时，每一级压缩的耗功量都相等）

,, compcomp WW

2. Z-Stage Compression (n 级压缩）

1

1

1

2

p

p

p

p z

z

Optimal compression ration is

3. Isentropic Efficiency and Isothermal Efficiency ( 等熵效率和等温效率）

nc

scs W

W

worcompressorActual

workconpressorIsentropic

,

，

Isentropic efficiency is defined as the ratio of the work input is required to raise the pressure of a gas to a specified value in an isentropic manner to the actual work input.

Isothermal Efficiency ( 等温效率）

nc

scT W

W

worcompressorActual

workconpressorIsothermal

,

，

Isothermal efficiency is defined as the ratio of the work input is required to raise the pressure of a gas to a specified value in a reversible isothermal manner to the actual work input.

Exercise ( 练习 ) Draw the following polytropic process of air

on p-v and T-s diagram.

(1) Pressure rises, temperature increases and with heat rejection

(2) Working medium expands and the temperature drops with heat rejection at the same time

Exercise ( 练习 )(3)Expansion process with n=1.6,please judge

the sign of

(4)Compression process with n=1.3,please judge the sign of

uwq ,,

uwq ,,

Draw the following process on the T-s Diagram.