ChE 201

ChE 201ChE 201

Ch 8Chapter 8Balances on Nonreactive Processes

Energy Balance Equation

10/21/2012 ChE 201/[email protected] 2

Chapter 8

• 8.1 State properties and hypothetical process pathspaths

– State property - depends only on the initialState property depends only on the initial and final condition, not on the path

– Advantage of using state property • Hypothetical process path can be used for the

calculation of true processcalculation of true process.


State Variables

Travelling from Dhaka to Chittagong: - distance by odometer = not a state variable


- reading of an altimeter = state variable

•Example) Ice, -5 degree C, 1 atm Vapor 300 deg. C, 5 atm

True path

Ice ( 5 o C 1 atm) Vapor (300 o C 5 atm)

p ) , g , p g ,

Ice (-5 .C, 1 atm) Vapor (300 C, 5 atm)

ΔH1

I (0 C 1 ) ΔH6Ice (0 o.C, 1 atm) ΔH6ΔH2

Vapor (300 o.C, 1 atm)∑=

=Δ6

1

ˆi

iHH)

Liquid (0 o.C, 1atm)

ΔH3 ΔH5

Liquid (100 o.C, 1 atm) Vapor (100 o.C, 1 atm)ΔH4


Processes for Changes in energy for:

1. Changes in P at constant T, no phase change

2 Ch i T t t t P h h2. Changes in T at constant P, no phase change

3. Phase changes at constant P and T

4. Mixing of two liquids at constant T and P

5 Dissolving of gas/solid in a liquid at const T and P5. Dissolving of gas/solid in a liquid at const T and P

6. Chemical reaction at constant T and P (Chapter 9)


8.2 Changes in P at const. T

• Solid and Liquid• Solid and Liquid – U is nearly independent of P – H is only dependent on V*delta(P)

• Gases – U and H are independent of P for ideal gases.

f l ( l id l )– for low pressure (nearly ideal gas)– For real gases ,

• Tabulated H data


Tabulated H data • Thermodynamic relation

8.3 Changes in T at const volume or pressure

• 8.3a Sensible Heat and Heat Capacities • Sensible Heat– Heat required to raise temperature of q p

a substance• from 1st law of thermodynamics,

for closed system UQ Δfor closed system for open system

UQ Δ=

HQ Δ=


• Heat capacity at constant volume

UUdUd ⎞⎛ ∂⎫⎧ ˆˆ

VTv T

UdTUd

dTUdC ⎟

⎠⎞

⎜⎝⎛∂∂

==⎭⎬⎫

⎩⎨⎧

=→Δ

lim0

∫=Δ 2

1

T

T vdTCU

– Ideal gas : exact– Solid or liquid : good approximation – Real gas : valid only if V is constant


Heat Capacity at Constant Pressure• Heat capacity at constant pressure

HHdHd⎟⎞

⎜⎛ ∂

⎬⎫

⎨⎧ ˆˆ

liP

Tp TH

dTHd

dTHdC ⎟

⎠⎞

⎜⎝⎛∂∂

==⎭⎬⎫

⎩⎨⎧

=→Δ

lim0

∫2T

∫=Δ 2

1T pdTCH

Ideal gas : exact– Ideal gas : exact

– Solid or liquid : good approximationReal gas : valid only if P is constant– Real gas : valid only if P is constant

PVdTCHT

T p Δ+=Δ ∫ ˆ2


T p∫1

8.3b Heat Capacity Formulas and Mean Heat Capacitiesp

• Data for heat capacities 32 dTcTbTaC p +++=

values of a,b,c,d are tabulated:Table B.2 (F & R)Perry’s Handbook (pp. 2-161 to 2-186)“Properties of Gases and Liquid”Properties of Gases and Liquid

CC ≈Simple Relationship

For liquids and solids vp CC ≈

RCC vp +=

For liquids and solids ,

For ideal gases ,


vpg ,

Heat Capacities



• Mean Heat Capacity 12

2

1

)(ˆˆ dTTCHHC

T

T p=

−=

∫1212 TTTT

C p −−

TCH Δ=Δ ˆ TCH pΔΔ

Integration is replaced by simple multiplication

)()()()()(ˆ1212 1 refTprefTp TTCTTCTTH

s−−−=→Δ

(thi ti i lid b H i t t t )(this equation is valid because H is state property)


Steam Tables

Reference State: Triple point, 0.01 oC and


and

Steam Table






8.3c Estimation of Heat Capacities

Estimation Techniques when there are no data for Cp– Kopp’s Rule: simple method for estimating heat capacityKopp s Rule: simple method for estimating heat capacity

for solid and liquid at near 20 oC. For Example, (Cp)Ca(OH)2 = (Cp)Ca + 2 (Cp)O + 2 (Cp)H

(Cp values for atoms are given in Table B 10)(Cp values for atoms are given in Table B.10)

- Heat capacity of a mixture

– Other sources • Perry’s Handbook• Properties of Gases and Liquids


• Properties of Gases and Liquids


• Enthalpy change of mixtures • Rule 1 : assume additivity• Rule 1 : assume additivity

∑= )(TCyC piipm

l 2 d l l l h l

∫=Δ 2

1

T

T pmm dTCH

• Rule 2 : For dilute solution, neglect enthalpy change of the solute.


8.4 Phase Change Operation

• Heat duties of phase changes are substantial.• Latent Heats• Latent Heats

– Latent Heat of Vaporization • Liquid-Vapor Transition• Liquid Vapor Transition

– Latent Heat of Fusion• Solid-Liquid Transition q

– Latent Heat of Sublimation• Solid-Vapor Transition

• Principles can be also extended to solid-solid phase changes.


8.4a Latent Heats

• Latent Heat : heat required for a given phase change at fixed T (or P)change at fixed T (or P).– for pure component, F = 1– Specifying T or P is enough Spe y g o s e oug

• Standard heat of phase change : Latent heat at 1 atm. (Appendix B.1) ( pp )

For Closed systems, Q m U= Δ

For liquids and solids U HΔ ≈ ΔFor liquids and solids, m mU HΔ ≈ Δ

For gases, ( )m m mU H PV H RTΔ = Δ − Δ ≈ Δ −


8.4b Estimation and Correlation of Latent Heats

• See Perry’s Handbook or “Properties of Gases and Liquids”and Liquids

• Trouton’s Rule : Accuracy (+- 30 %) ˆ 0 088 ( ) Nonpolar liquidsH T KΔ

Chen’s Equation : Accuracy ( + 2 %)

0.088 ( ), Nonpolar liquidsˆ 0.109 ( ), Water, low m.w. alcohols

v b

v b

H T K

H T K

Δ =

Δ =

• Chen’s Equation : Accuracy ( +- 2 %) [ ]

)/(071log0297.00327.0)/(0331.0

)/(ˆ 10 ccbbv TT

PTTTmolkJH

−+−

=Δ)/(07.1 cb TT


• Estimation Equation of Heat of Fusionˆ ( / ) 0.0092 ( ), metallic elementsˆ ( / ) 0.025 ( ), inorganic compoundsˆ ( / ) 0 050 ( ) organic compounds

m m

m m

H kJ mol T K

H kJ mol T K

H kJ mol T K

Δ =

Δ =

Δ = ±

• Estimation of heat of vaporization from vapor pressure data

( / ) 0.050 ( ) ,organic compoundsm mH kJ mol T KΔ = ±

pressure data* ˆ(ln )

(1/ )vHd p

d T RΔ

=( )

ˆ1plot of ln p* vs. slope vHT R

Δ→ =


• Watson correlation : heat of vaporization for other temperatureother temperature

38.0

2)(ˆ)(ˆ ⎥⎤

⎢⎡ −

Δ=ΔTT

THTH c

8 4c Energy Balances Involving Phase

112 )()( ⎥

⎦⎢⎣ −

Δ=ΔTT

THTHc

vv

• 8.4c Energy Balances Involving Phase Changes – See example 8 4-4See example 8.4 4


Not to forget!!!!!

• Appendix B: Table B.1–Selected Physical Property Data– Heat of fusion @ Tm & 1 atm@ m

– Heat of vaporization @ Tb & 1 atm– Heat of formation @ 250C & 1 atm

Heat of combustion @ 250C & 1 atm– Heat of combustion @ 250C & 1 atm

• Appendix B: Table B.2---- Heat Capacities– State of the component – liquid or gas– Form of equation – 1 or 2Form of equation 1 or 2– Temperature unit – 0C or K– Unit of CP


Example 8.4-4


Solution: Example 8.4-4




Problem 8.22








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