Advance Chemical Engineering Thermodynamics

By Dr.Dang Saebea

Volumetric Properties of Pure Fluids

Part II

PVT behavior of pure substances

• the critical point

the sublimation curve

the fusion curve

the vaporization curve

the triple point

Figure: A pressure-temperature diagram

Super critical regoin

The virial equations• The compressibility factor (Z) :

• Two Forms

- The parameters B’, C’, D’, etc. are virial coefficients, accounting of interactions between molecules.

The methods of statistical mechanics allow derivation of the virial equations and provide physical significance to the virial coefficients

The only equation of state proposed for gases having a firm basis in theory.

PVZ

RT

2 3(1 ...)PV

Z B P C P D PRT

2 3(1 ...)

PV B C DZ

RT V V V Reciprocal volume

Reciprocal pressure

1

2

3

Ideal gas• No interactions between molecules.

U = U (T)

Z = 1; PV = RT

The equation of state:

An internal energy of an ideal gas depends on temperature only:

4

5

2 3(1 ...)PV

Z B P C P D PRT

2 3(1 ...)

PV B C DZ

RT V V V

• Gases at pressure up to a few bars may often be considered ideal and simple equations then apply.

)(TCdT

dU

T

UC V

VV

)(TCRCRdT

dU

dT

dH

T

HC PV

PP

Implied property relations for an ideal gas The definition of heat capacity for ideal gas

The definition of enthalpy for ideal gas

( ) ( )H U PV U T RT H T 7

8

The heat capacity at constant pressure

6

vdU C dT

pdH C dT

The heat capacity at constant pressure

vU C dT

pH C dT

There are four different situations for ideal gas

1. Isobaric: the gas is held at a constant pressure

2. Isochoric: the gas is held at a constant volume

3. Isothermal: the gas is held at a constant temperature

4. Adiabatic: No heat flows in or out of the gas

dTCdWdQ V PdVdW

PdVR

CVdP

R

CdQ PV

Mechanically reversible closed-system process, for a unit mass or a mole:

Equation for process calculation for ideal gases

+

VdQ C dT PdV

p

dPdQ C dT RT

P

V

dVdQ C dT RT

V

with P = RT/V

with V = RT/P

with T = PV/R

dVdW RT

V

dPdW RdT RT

P

dW PdV

9

11

13

10

12

14

• Isothermal process

Equation for process calculation for ideal gases

• Isobaric process P const.

• Isochoric (Constant-V) process V const.

15

0U H

By Eq. 9, 10 2 2

1 1

ln lnV P

Q W RT RTV P

(const T)

16By Eq. 11, 12 pQ H C dT (const P)

vdU C dT pH C dT

vdU C dTpH C dT

17By Eq. 11, 12 vQ U C dT (const V)

• Adiabatic process; Constant heat capacities

Equation for process calculation for ideal gases

By Eq. 9

p

v

C

C

By Eq. 11 /

2 1

1 2

vR CT V

T V

/

2 2

1 1

pR CT P

T P

By Eq. 18 and 19/

2 1

1 2

p vC CP V

P V

18 19 20

Definition

.PV const 1 .TV const (1 )

.TP const

21 22 23

For adiabatic closed system

1p

v v

C R

C C

1v

RC

vW U C T

2 1 2 2 1 1

1 1

RT RT PV PVW

24

251 1

1 1 2 1 2

1 1

1 11 1

PV P RT PW

P P

26

Equation for process calculation for ideal gases

• Polytropic process

A model of some versatility with constant. defined as a process represented by the empirical equation.

.PV const 1 .TV const (1 )

.TP const

27 28 29

1

1 2

1

11

RT PW

P

1

1 2

1

( )1

( 1)( 1)

RT PQ

P

30

31

The work of an irreversible process is calculated:– First, the work is determined for a mechanically

reversible process.– Second, the result is multiple or divided by an

efficiency to give the actual work.

Irreversible process

Example1

•  

Air is compressed from an initial condition of 1 bar and 25°C to a final state of 5 bar and 25 °C by three different mechanically reversible processes in a closed system.

(a) heating at constant volume followed by cooling at constant pressure; (b) isothermal compression;

(c) adiabatic compression followed by cooling at constant volume. Assume air to be an ideal gas with the constant heat capacities, CV = (5/2)R and CP = (7/2)R. Calculate the work required, heat transferred, and the changes in internal energy and enthalpy of the air in each process.

Example 2

An ideal gas undergoes the following sequence of mechanically reversible processes in a closed system:(1) From an initial state of 70°C and 1 bar, it is compressed adiabatically to 150 °C. (2) It is then cooled from 150 to 70 °C at constant pressure.(3) Finally, it is expanded isothermally to its original state.Calculate W, Q, ΔU, and ΔH for each of the three processes and for the entire cycle. Take CV = (3/2)R and CP = (5/2)R. If these processes are carried out irreversibly but so as to accomplish exactly the same changes of state (i.e. the same changes in P, T, U, and H), then different values of Q and W result. Calculate Q and W if each step is carried out with an efficiency of 80%.

Example 3

Example 4

Find the equation for the adiabatic expansion/compression of a binary mixture of ideal gases, with composition xA, xB. Use a closed system without kinetic and potential effects.

The End