AP3290_chapter_3_I_10

Thermodynamics Y Y Shan

AP3290 26

Chapter 3 Heat engines and the Second law of thermodynamics

3.1 Heat capacity (or specific heat)

Heat capacity,(also known as the specific heat) is the amount of heat energy per unit

mass required to raise the temperature by one degree. More heat energy is required to increase

the temperature of a substance with high specific heat capacity than one with low specific heat

capacity.

The specific heat of water is 1 calorie/gram °C = 4.186 joule/gram °C which is higher

than any other common substance. As a result, water plays a very important role in temperature

regulation. The specific heat per gram for water is much higher than that for a metal. The

above relationship does not apply if a phase change is encountered, because the heat added or

removed during a phase change does not change the temperature.

3.1.1 Heat capacity at constant volume and constant pressure

More scientifically in thermodynamics, the heat capacity C should be mathematically

defined as the ratio of a small amount of heat δQ added to the body, to the corresponding small

increase in its temperature dT under some specific conditions or constraint:

x

xdT

QC

=

δ

x refers to the constraint or conditions under which heat capacity is measured. The most

common and conditions are under constant volume and constant pressure.

, i.e., the most important heat capacities are heat capacity at constant volume V,

V

VdT

QC

=

δ


AP3290 27

and heat capacity at constant pressure P,

P

PdT

QC

=

δ

When heat capacity is measured at constant volume, from the first law:

1-eq3

and ,

thatso ,0 where

,

VVV

VT

U

dT

dU

dT

QC

dUQ

PdVW

WQdU

∂

∂=

=

=

=

=−=

+=

δ

δ

δ

δδ

When heat capacity is measured at constant pressure, from the first law:

2-eq3

that so

PPPP

PT

VP

T

U

dT

PdVdU

dT

QC

PdVdUWdUQ

WQdU

∂

∂+

∂

∂=

+=

=

+=−=

+=

δ

δδ

δδ

If we define a state function:

PVUH +=

PPP

PT

H

dT

PdVdU

dT

QC

PdVdUdH

∂

∂=

+=

=

+=

δ

and

pressure,constant fconditiono under the then

The defined function H is namely enthalpy.


AP3290 28

3.1.2 The CP~CV relationship for an ideal gas

In general, for any system, a relationship between CV and CP can be obtained by using

the first law, and it can be simplified after applying the second law

(a) Internal energy for an ideal gas and Joule’s law

The state function U is the function of temperature and volume, the two state coordinates of

gases, i.e.

),( VTUU =

From partial differential calculus,

dVV

UdT

T

UVTdU

TV

∂

∂+

∂

∂=),(

It can be derived that (see math appendix later)

tcoefficien sJoule' called is where

3-eq3

i.e. ,

U

U

V

T

UVT

V

T

V

TC

V

U

V

T

T

U

V

U

∂

∂

∂

∂−=

∂

∂

∂

∂

∂

∂−=

∂

∂

Now look at the Joule’s experiment, in 1845, Joule performed an experiment where a

gas at high pressure inside a bath at the same temperature was allowed to expand into a larger

volume.

Two vessels, labeled A and B, are immersed in an insulated tank containing water. One

thermometer is used to measure the temperature of the water in the tank. The two vessels A

and B, being thermally equilibrium with the bath, are connected by a tube, the flow through


AP3290 29

which is controlled by a stop. Initially, A contains gas at high pressure, while B is nearly empty.

The stop is removed so that the vessels are connected and the final temperature of the bath is

noted.

The fact that the temperature of the bath was unchanged at the end of the process

indicates that the temperature of gas does not change either ( 0=∆T ) when its volume

changes ( 0≠∆V ), i.e .

,0=∂

∂

V

T

From eq3-3, we obtain

0=

∂

∂

TV

U eq.3-4

This concludes that the internal energy of an ideal gas is volume independent, it is the function

of temperature alone, U=U(T,V)=U(T). The equation above is called Joule's law which stated

as (∂U/∂V)t = 0

(b) Ideal gas’s )(VP

CC − value and VP

CC / value

It can be proved that, for ideal gas,

nRCCVP

=− eq3-5

Derivation:

nRCC

P

nRPC

dT

dVPC

dT

QC

PdVdTCQtherefore

dTCPdVQWQdUFirst

dTClawsJouledTC

dVV

UdT

T

UVTdU

VP

V

P

V

P

P

V

V

VV

TV

=−

=⋅+=

+=

=

+=

=−=+=

=+=

∂

∂+

∂

∂=

so nRT),PV (using ,

:

:law

) '(0

),(

δ

δ

δδδ

The ratio VP

CC / is defined as: γ=VP

CC / , so that


AP3290 30

nRC

nRC

V

P

1

1

1

−=

−=

γ

γ

γ

The most common values being γair = 1.4 for air, and γmonoatomic gas = 1.66 .

3.2 Heat engines

(a) Thermodynamic cycle: Thermodynamics began as the study of the efficiency of heat

engines. An engine will be defined as a machine which at the end of a complete cycle,

consisting several thermal processes, can convert some heat energy into useful mechanical

work. The process can then be repeated another cycle and so on. As we know that work done

in a thermal process is not a state function, meaning it depends on the path (which curve you

consider for integration from state 1 to 2). For a system in a cycle which has states 1 and 2, the

work done depends on the path taken during the cycle. If, in the cycle, the movement from 1 to

2 is along A and the return is along C, then the work done is the lightly shaded area. However,

if the system returns to 1 via the path B, then the work done is larger, and is equal to the sum of

the two areas

The right figure above shows a typical indicator diagram as output by an automobile engine.

The shaded region is proportional to the work done by the engine, and the volume V in the x-

axis is obtained from the piston displacement, while the y-axis is from the pressure inside the

cylinder. The work done in a cycle is given by W, where


AP3290 31

(b) Thermal efficiency of a heat engine

Thermal efficiency ( ) is a performance evaluation of a thermal device such as an

car engine, or a furnace,. The input, Hin

QQ or , to the device is heat energy (or the heat

energy of a fuel that is consumed) . The desired output is mechanical work, .

The thermal efficiency of a heat engine is the percentage of heat energy that is transformed

into work, the efficiency is defined as

,

When expressed as a percentage, the thermal efficiency must be between 0% and 100%.

Due to inefficiencies such as friction, heat loss, and other factors, thermal efficiencies

are typically much less than 100%. For example, a typical gasoline automobile engine

operates at around 25% thermal efficiency, and a large coal-fueled electrical generating

plant peaks at about 36%

(c) Schematic expression of a heat engine in thermodynamics

For a heat engine, the heat energy comes from a high temperature reservoir, TH, (such

as fuel burning or boiling water for creating steaming gas). This heat energy absorbed

from the high temperature reservoir, QH, will be partly converted to mechanical work, Wout,

and some heat energy released to low temperature reservoir, TH, (such as cooling water or

cooling fan). Schematically, it is often shown as:


AP3290 32

3.3 Analysis of thermodynamic cycles (or engine cycles)


AP3290 33

A thermodynamic cycle is a series of thermodynamic processes (usually consists four)

which returns a system to its initial state, so that the change of the total internal energy is

zero: 0=∆U . A thermodynamic cycle is a closed loop on a P-V diagram

A typical P-V diagram for a thermodynamic cycle represents a heat engine. The cycle

consists of four states (1, 2, 3, 4) and four thermodynamic processes (lines 1-2, 2-3, 3-4, 4-

1). The area enclosed by the loop is the work (W) done by the process:

14433221 →→→→+++== ∫ WWWWpdVW

out

The first law of thermodynamics dictates that the work is equal to the balance of heat (Q)

transferred into the system over any cycle:


AP3290 34

in

out

in

out

th

outinout

outin

out

Q

Q

Q

W

QQW

QQQ

WQU

−=

=

−=

−≡

=−=∆

1

:efficiency The

thatso obsorbedheat net where

0

thη

η

3.3.1 Analysis of the typical four processes in a heat engine cycle

Each heat engine proces can be described on a PV diagram. Besides constant pressure,

volume and temperature processes, a useful process is the adiabatic process where no heat

enters or leaves the system.

(i) Constant volume process (isometric or isochoric process)

The first law gives: 0 , ==+=∆ WQWQU


AP3290 35

(ii) Constant pressure process(isobaric process)

The first law gives: VPTnCWQUP

∆−∆=+=∆

(iii) Isothermal Process: the result of expansion gives the work expression below.

The first law gives: WQU +=∆


AP3290 36

For ideal gas, internal energy is only temperature dependent, so that for isothermal process:

∫=−=

=+

=∆

PdVWQ

WQ

U

0

e therefor,0


AP3290 37

(iv) Adiabatic process: one in which no heat is gained or lost by the system. The first law

of thermodynamics with Q=0 shows that all the change in internal energy is in the form of

work done.

PdVdWdU

VPWWQU

−==

∆−==+=∆

form aldifferenti itsor

In an adiabatic process, the P-V cure of an ideal gas has an exponential form of

heats. specifice of ratio theis constant, is where

7-eq.3

or

γ

γ

γ

K

KPV

KVP

=

= −

This is called the adiabatic condition of ideal gas. The derivation can be shown in the

following:


AP3290 38

KconstPV

V

dV

P

dP

PdVVdPVdPPdVPdV

VdPPdVdTCfromsteps

CCCnRstep

VdPPdVPVdnRdTsonRTPVionStateEquatstep

PdVdTCfromsteps

dTCdUslawfromJoulestep

adiabaticdQPdVdWdWdQdUfirstlawstep

V

VVP

V

V

==

=+

=+⇒−+=−

−+=

−=−=

+===

−=

=

=−==+=

−

−

.

obtained becan gas idealfor condition adiabatic the8,-equation3 solve :step6

condition. adiabatic theofequation aldifferenti thecalled is This

8--eqn3----------- 0

0)1)(( :b a, from :step5

-(b)----------------- )1)((:4,3

)1(:4

)(,,::3

-(a)------------------------- :2,1

:':2

),0(,::1

1

1

γ

γ

γγ

γ

γ

Documents

AP3290_chapter_3_I_10