CHEE 221: Chemical Processes and Systems
Module 4.
Energy Balances without Reaction
Part c: Calculating Changes in Enthalpy
(Felder & Rousseau Ch 8)
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.
•
Ĥ with change in P (at constant T and phase) (F&R 8.2)
•
Ĥ with change in T (at constant P and phase) (F&R 8.3)
•
Ĥ 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
CHEE 221
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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 Û and Ĥ 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 (Table B2)
– phase transition temperatures  Tm, Tb (at which data are often listed,
e.g. latent heats) (Table B1)
– latent heats  heat of vapourization, heat of melting (Table B1)
CHEE 221
F&R Ch 8.1b
3
Hypothetical Process Paths
Calculate the enthalpy change as 1 kg of ice at 0C is transformed to
superheated steam at 400C at 10 bar.
H 2O(s)
0C, 1 atm
Hˆ  ?
H2O(v)
400C, 10 atm
We could use the steam tables and find (Remember: final‐initial):
Ĥ  3264  ( 334)  3612 kJ/kg  - 334 kJ/kg is the enthalpy of ice
How would we calculate the enthalpy change if we didn’t have the steam
tables?
CHEE 221
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Construction of the Hypothetical Process Path
Our chosen
reference state
Hˆ 1  Hˆ m at the
normal melting point
(Tm ) of H2O (Table B.1)
H2O(s)
(0C, 1 atm)
Hˆ  ?
true path
Ĥ1
H2O(v)
(400C, 10 atm)
Ĥ 5
Hˆ 5  0
( A reasonable assumption
for small changes in P)
H2O(v)
(400C, 1 atm)
H2O(l)
(0C, 1 atm)
100
Hˆ 2 

H O
C p 2 ( l ) dT
0
(C p expressions are
found in Table B.2)
400
Ĥ 4
Ĥ 2
H2O(l)
(100C, 1 atm)
Ĥ 3
Hˆ 4 

H O( v )
Cp 2
dT
100
(C p expressions are
found in Table B.2)
H2O(v)
(100C, 1 atm)
Hˆ 3  Hˆ v at the
normal boiling point
(Tb ) of H2O (Table B.1)
Hˆ  Hˆ final  0  Hˆ 1  Hˆ 2  Hˆ 3  Hˆ 4  Hˆ 5
CHEE 221
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Changes in H and U with P (constant T, phase)
Ideal Gases:
– By definition, Uˆ  0 (molecules don’t interact, so changing P doesn’t
change the internal energy)
– H = U + PV but PV = nRT = 0 (constant T)  Hˆ  0
Non‐Ideal Gases:
– Changes in internal energy and enthalpy are small, provided P is small
(< 5 atm) 
Uˆ  Hˆ  0
– For steam, use tabulated values
In a problem, state that
changes in U and H with
Liquids and Solids:
respect to pressure are
small and will be
– Uˆ  0
neglected (except for
– Hˆ  PVˆ (but still very small)
steam tables).
CHEE 221
F&R Ch 8.2
6
Does H change with P? Steam Enthalpy Diagram
Use the steam tables to
determine the state (liquid,
vapour or mixture of the two;
saturated or supersaturated) and
approximate temperature (no
need to use extrapolate) of 1 kg
of water at 1 bar with the
following enthalpies (relative to
liquid water at the triple point)
a)
b)
c)
d)
e)
100 kJ
419 kJ
1500 kJ
2676 kJ
3000 kJ
Conclusion: Enthalpy is not a strong function of pressure below 10 bar
CHEE 221
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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).
Hˆ v (T , P ) : liquid  gas
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).
Hmelting
CHEE 221
Hˆ m (T , P ) : solid  liquid
F&R Ch 8.4a
8
Changes in U and H with T (constant P, phase):
Sensible Heat
Sensible heat refers to heat that must be transferred to raise or
lower the temperature of a substance without change in phase.
Hvapourization
Hmelting


CHEE 221
H
2) Sensible heat of liquid,
(Tmelting  Tvapourization)
H
3) Sensible heat of gas,
(Tvapourization  Tfinal)

Tinitial
1) Sensible heat of solid,
(Tinitial Tmelting)
H
Tfinal
F&R Ch 8.3a
9
Changes 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:
Q =  U (closed system; must
be kept at constant volume)

Uˆ   Uˆ
Cv (T )   lim

T 0 T   T



V
T2

Uˆ  C v (T )dT
T1
Slope = Cv = heat capacity at constant
volume
CHEE 221
Ideal gas: exact
Solid or liquid: good approximation
Nonideal gas: valid only if V constant
10
Changes with T (constant P, phase): Sensible Heat
Open System‐‐Find  H. Enthalpy, like internal energy, also depends strongly
on temperature.
˙ =  H˙
(open system; calculate at constant pressure)
Q

Hˆ   Hˆ
C p (T )   lim

T 0 T   T
T2

Hˆ  C p (T )dT



P
Ideal gas: exact
Solid or liquid: good approximation
Nonideal gas: exact only if P constant
Cp = heat capacity at constant pressure
T1
Liquids and Solids: Cp  Cv
Ideal Gases: Cp = Cv + R
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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
cal
J
or
mol K
g C
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”)
units:
Values of coefficients a, b, c, and d are given in Table B.2.
CHEE 221
F&R Ch 8.3b
12
Heat Capacity Calculations – Integration
Cp
T2

molkJC  a  bT  cT 2  dT 3

 kJ 
2
3
Ĥ 

a

bT

cT

dT
dT

 mol  T
1
b
c
d
 a( T2  T1 )  ( T2 2  T12 )  ( T2 3  T13 )  ( T2 4  T14 )
2
3
4
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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 – T12)  (T2 – T1)2
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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 Cp
equation.
• Interpolation may be required.
• The reference state of these gases is: 1 atm and 25C.
• 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.
CHEE 221
F&R Ch 8.3b
15
Putting it all together: Hypothetical Process Path
reference
state
Hˆ 1  Hˆ m at the
normal melting point
(Tm ) of H2O (Table B.1)
H2O(s)
(0C, 1 atm)
Hˆ  ?
true path
H2O(v)
(400C, 10 atm)
Ĥ 5
Ĥ1
Hˆ 5  0
( A reasonable assumption
for small changes in P)
H2O(v)
(400C, 1 atm)
H2O(l)
(0C, 1 atm)
100
Hˆ 2 

H O
C p 2 ( l ) dT
0
(C p expressions are
found in Table B.2)
400
Ĥ 4
Ĥ 2
H2O(l)
(100C, 1 atm)
Ĥ 3
Hˆ 4 

H O( v )
Cp 2
dT
100
(C p expressions are
found in Table B.2)
H2O(v)
(100C, 1 atm)
Hˆ 3  Hˆ v at the
normal boiling point
(Tb ) of H2O (Table B.1)
Hˆ  Hˆ final  0  Hˆ 1  Hˆ 2  Hˆ 3  Hˆ 4  Hˆ 5
CHEE 221
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Calculating enthalpy changes: Examples
1. Calculate the increase in specific enthalpy that occurs when
acetone(v) is heated from 25 C to 100 C.
2. A stream of nitrogen flowing at a rate of 1kg/min is heated
from 50C to 200C. Calculate the heat that must be
transferred.
3. Fifteen kg/min of air is cooled from 400C to 30C. Calculate
the required heat removal rate.
4. Estimate the increase in specific enthalpy when H2O(v) is
heated from 300 C to 450 C.
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Constructing a Process Pathway: 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 diagram
2. Perform all material balance calculations
3. 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.)
CHEE 221
F&R Ch 8.1c, 8.3d, 8.4c
19
Procedure for Energy Balance Calculations (cont’d)
6. Calculate all required values of Ûi or Ĥi and insert values into table
7. 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
F&R Ch 8.1c, 8.3d, 8.4c
20
Example 1: F&R 8.3‐5
A stream containing 10% CH4 and 90% air by volume is to be
heated from 20C to 300C. 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
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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
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Example 3: 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 ºC and 5 atm.
Shaft work is delivered to the system at a rate of 30 kW to achieve the
compression.
5.184 mol air/s
0.058 mol MeOH(v) /s
150 °C 1 atm
0 °C 5 atm
5.760 mol/s
0.10 mol MeOH(v) /mol
0.90 mol air/mol
0 °C 5 atm
Q
0.518 mol MeOH(l) / s
Ws
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?
CHEE 221
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