CHEE 221 1

Energy Balances. F&R Chapter 8

How do we calculate enthalpy (and internal energy) changes when we don’t have tabulated data (e.g., steam tables) for the process species?Basic procedures to calculate enthalpy (and internal energy changes) associated with the following processes are covered in Chapter 8 (no Reaction): Remember, 3 pieces of information set the thermodynamic state of matter: P, T and Phase.

• with change in P (at constant T and phase) (F&R 8.2)

• with change in T (at constant P and phase) (F&R 8.3)

• with change in phase (at constant T and P) (F&R 8.4)

To keep track of our calculations, we will summarize enthalpies in an Inlet-Outlet Enthalpy Table. Sections not covered in Ch 8 include 8.3c, 8.3e, 8.4b, 8.4d, 8.4e, 8.5

H

H

H

CHEE 221 2

Hypothetical Process Paths

To calculate enthalpy changes, we need to construct a hypothetical process path, with:– A starting point: your defined reference state (phase, T and P)– An end point: the conditions of the stream of interest (inlet or

outlet)

Since and are state properties (values are dependent only on the state of the species (phase, T and P) and not how they got there), any convenient process path from a reference state to a process state can be chosen

The ideal process path will allow you to make use of: – sensible heats heat capacities– phase transition temperatures Tm, Tb (at which data are often

listed, e.g. latent heats)– latent heats heat of vapourization, heat of melting

U H

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Hypothetical Process Paths

Calculate the enthalpy change as 1 kg of ice at 0C is transformed to superheated steam at 400C at 10 bar.

We could use the steam tables and find (Remember: final-initial):

How would we calculate the enthalpy change if we didn’t have the steam tables?

H2O(s)

0C, 1 atm atm 10 C,400OH (v)2

?ˆ H

ice ofenthalpy the is kJ/kg 334- kJ/kg 3612334)(3264 H

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Hypothetical Process Path

54321ˆˆˆˆˆ0ˆˆ HHHHHHH final

1H 5H

3H

?ˆ H

4H2H

H2O(s)

(0C, 1 atm)H2O(v)

(400C, 10 atm)

H2O(l)

(0C, 1 atm)

H2O(l)

(100C, 1 atm)H2O(v)

(100C, 1 atm)

H2O(v)

(400C, 1 atm)

true path

B.1) (Table OH of )(point melting normal

the at

2m

m

T

HH ˆˆ1

B.2) Table in found are sexpression

100

0

p

OHp

C

dTCH l

(

ˆ )(22

B.1) (Table OH of )(point boiling normal

the at

2b

v

T

HH ˆˆ3

B.2) Table in foundare sexpression

400

100

p

OHp

C

dTCH v

(

ˆ )(24

P) in changes small forassumption reasonable A (

0ˆ5 H

reference state

CHEE 221 5

Changes in H and U with P (constant T, phase)

Ideal Gases:– By definition, (molecules don’t interact, so changing P

doesn’t change the internal energy)– H = U + PV but PV = nRT = 0 (constant T)

Non-Ideal Gases:– Changes in internal energy and enthalpy are small, provided P

is small (< 5 atm) – For steam, use tabulated values

Liquids and Solids: –– (but still very small)

0ˆ U

0ˆ H

0ˆ UVPH ˆˆ

0ˆˆ HU

In a problem, state that changes in U and H with respect to pressure are small and will be neglected (except for steam tables).

CHEE 221 6

Changes in U and H with T (constant P, phase)-Sensible HeatSensible heat refers to heat that must be transferred to raise or lower the temperature of a substance without change in phase.

Hmelting

Hvapourization

Tinitial Tfinal

1) Sensible heat of solid, H (Tinitial Tmelting)

2) Sensible heat of liquid, H (Tmelting Tvapourization)

3) Sensible heat of gas, H (Tvapourization Tfinal)

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Changes in U and H with T (constant P, phase)-Sensible Heat

Closed System--Find U . The quantity of sensible heat required to produce a temperature change in a system can be determined from the appropriate form of the first law of thermodynamics:

2

1

)(ˆ

ˆˆlim)(

0

T

Tv

VTv

dTTCU

TU

TUTC

Slope = Cv = heat capacity at constant volume

Ideal gas: exactSolid or liquid: good approximationNonideal gas: valid only if V constant

Q = U (closed system; must be kept at constant

volume)

CHEE 221 8

Changes in U and H with T (constant P, phase)-Sensible Heat

Open System--Find H. Enthalpy, like internal energy, also depends strongly on temperature.

2

1

)(ˆ

ˆˆlim)(

0

T

Tp

PTp

dTTCH

TH

THTC

Ideal gas: exactSolid or liquid: good approximationNonideal gas: exact only if P constant

Cp = heat capacity at constant pressure

Liquids and Solids: Cp Cv

Ideal Gases: Cp = Cv + R

Q = H ˙˙ (open system; calculate at constant pressure)

CHEE 221 9

Heat Capacity Formulas

Heat capacity – the amount of heat required to raise the temperature of one mole or one gram of a substance by one degree Celsius without change in phase

– units:

If Cp were constant, our job would be easy: H = CP (T2-T1)But, heat capacities are functions of temperature and are expressed in polynomial form:

Cp = a + bT + cT2 + dT3 (Form “1”)or,

Cp = a + bT+ cT-2 (Form “2”)

Values of coefficients a, b, c, and d are given in Table B.2.

C gcal or

K molJ

CHEE 221 10

Notes Regarding Table B.2

Be sure you use the correct functional form– Cp = a + bT + cT2 + dT3 (Form 1) or Cp = a + bT+ cT-2 (Form 2)

Temperature units are sometimes K and sometimes C

Positive exponent in table heading means you use negative exponent in the expression– E.g., if a x 103 = 123.0 a = 123.0 x 10-3

Be careful when you integrate! (T22 – T1

2) (T2 – T1)2

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Heat Capacity Calculations – Integration

32 dTcTbTaC p C molkJ

)TT(d)TT(c)TT(b)TT(a

dTdTcTbTaHT

T

41

42

31

32

21

2212

32

432

2

1

molkJ

CHEE 221 12

Specific Enthalpies of Gases – Table B.8

Table B.8 lists specific enthalpies (kJ/mol) of selected gases (mainly combustion products, i.e. this table might be useful when solving energy balances for combustion problems) as a function of temperature.

• Can be used to estimate H changes as an alternative to integrating the Cpequation.

• Interpolation may be required.

• The reference state of these gases is: 1 atm and 25C.

• Use this table as you would for the steam tables, however, note that for H2O, the units and reference state are different than the steam tables.

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Phase Changes (at constant T and P) – Latent Heat

Phase changes occur from the solid to the liquid phase, and from the liquid to the gas phase, and the reverse. The specific enthalpy change (heat) associated with the phase change at constant T and P is known as the latent heat of the phase change (i.e., latent heat of vapourization or simply heat of vapourization).

:),(ˆ PTHm

:),(ˆ PTHv liquid gas

solid liquid

Hmelting

Hvapourization

Table B.1 reports these two latent heats for substances at their normal melting and boiling points (i.e., at a pressure of 1 atm).

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Hypothetical Process Path

54321ˆˆˆˆˆ0ˆˆ HHHHHHH final

1H 5H

3H

?ˆ H

4H2H

H2O(s)

(0C, 1 atm)H2O(v)

(400C, 10 atm)

H2O(l)

(0C, 1 atm)

H2O(l)

(100C, 1 atm)H2O(v)

(100C, 1 atm)

H2O(v)

(400C, 1 atm)

true path

B.1) (Table OH of )(point melting normal

the at

2m

m

T

HH ˆˆ1

B.2) Table in found are sexpression

100

0

p

OHp

C

dTCH l

(

ˆ )(22

B.1) (Table OH of )(point boiling normal

the at

2b

v

T

HH ˆˆ3

B.2) Table in foundare sexpression

400

100

p

OHp

C

dTCH v

(

ˆ )(24

P) in changes small forassumption reasonable A (

0ˆ5 H

reference state

CHEE 221 15

Examples Week 9

– Calculate the increase in specific enthalpy that occurs when acetone(v) is heated from 25 C to 100 C.

– A stream of nitrogen flowing at a rate of 1kg/min is heated from 50C to 200C. Calculate the heat that must be transferred.

– Fifteen kg/min of air is cooled from 400C to 30C. Calculate the required heat removal rate.

– Estimate the increase in specific enthalpy when H2O(v) is heated from 300 C to 450 C.

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Example 8.4-2

One hundred moles per second of liquid hexane at 25 ºC and 7 bars pressure is vaporized and heated to 300 ºC at constant pressure. Estimate the rate at which that must be supplied.

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Procedure for Energy Balance Calculations

1.Draw and completely label a process flow diagram2.Perform all material balance calculations3.Choose a reference state (phase, T, P) for each species involved

– If using enthalpy tables, use reference state used to generate table

– If no tables are available, choose one inlet or outlet condition as the reference state for the species

4.Construct an inlet-outlet enthalpy table– Columns for inlet and outlet amounts of each species along with

their corresponding Ûi or Ĥi values– Use a separate row for each phase of a species– Identify unknowns with variables (e.g., Ĥ1, Ĥ2, etc.)

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Procedure for Energy Balance Calculations

6.Calculate all required values of Ûi or Ĥi and insert values into table7.Calculate U or H (e.g., H=miĤi- miĤi)8.Write the appropriate form of the energy balance equation, remove any negligible term, and calculate any other terms in the energy balance equation (i.e., W, Ek, Ep) 9.Solve for the unknown quantity in the energy balance equation

– Typically solve for Q but sometimes asked to solve for a mass (mole) flow or occasionally a T.

CHEE 221 19

Example 1 F&R 8.3-5

A stream containing 10% CH4 and 90% air by volume is to be heated from 20C to 300C. Calculate the required rate of heat input in kilowatts if the flow rate of the gas is 2.00 x 103 litres (STP)/min.

CHEE 221 20

Example 2 F&R 8.1-1

Acetone (denoted as Ac) is partially condensed out of a gas stream containing 66.9 mole% acetone vapour and the balance nitrogen. Process specifications and material balance calculations lead to the flowchart shown below.

The process operates at steady-state. Calculate the required cooling rate.

CHEE 221 21

Example: Final Exam 2006

In the following process for condensing methanol vapour from air most of the entering methanol is liquefied in this steady-state process, with the remaining fraction exiting with the air stream. Both exit streams are at 0 °C and 5 atm. Shaft work is delivered to the system at a rate of 30 kW to achieve the compression.

Construct an inlet-outlet enthalpy table for the process, and calculate all unknown enthalpies. Identify the reference states selected for the components, and state all assumptions. What is the rate (kW) at which heat must be removed from the condenser?

Q Ws

0 °C 5 atm

0.518 mol MeOH(l) / s

0 °C 5 atm 5.760 mol/s 0.10 mol MeOH(v) /mol 0.90 mol air/mol

150 °C 1 atm

5.184 mol air/s 0.058 mol MeOH(v) /s

CHEE 221 22

Interactive Tutorial

Now is a good time to try Interactive Tutorial 5…