SHR Chapter 2
Chemical Engineering
Thermodynamics
A Brief Review
1
Thermo.key - February 7, 2014
“Rate” vs. “State”
Thermodynamics tells us how things “end up” (state) but not how (or how
fast) they “get there” (rate)
Limitations on getting to equilibrium:
• Transport (mixing): diffusion, convection
‣ example: why doesn’t more of the ocean evaporate to make higher humidity in the deserts? (heat transfer
alters dew point locally, atmospheric mixing limitations, etc.)
• Kinetics (reaction): the rate at which reactions occur are too slow to get to equilibrium
‣ example: chemical equilibrium says trees (and our bodies) should become CO2 and H2O (react with air)
‣ example: chemical equilibrium says diamonds should be graphite (or CO2 if in air)
Sometimes, mixing & reaction are sufficiently fast that equilibrium
assumptions are “good enough.”
• if we assume that we can use thermodynamics to obtain the state of the system, we avoid a lot
of complexity.
‣ do it if it is “good enough!”
• Most unit operations (e.g. distillation) assume phase equilibrium, ignoring transport limitations.
‣ good enough if things are mixed well and have reasonable residence/contact times.
2
Thermo.key - February 7, 2014
Some Terminology...
✓
◆
@N M
“Partial Molar” property M̄i ⌘
@Ni T,P,Nj
change in ℳ due to adding a differential amount
of species i. For ideal mixtures, M̄i = Mi
Gibbs-Duhem
Equation:
✓
@M
@P
◆
dP +
T,y
✓
“Residual” property M ⌘ M
R
how much ℳ deviates from
@M
@T
◆
P,y
M
ig
C
X
i=1
ℳ -
arbitrary thermodynamic
property (per mole)
Ni
-
moles of species i yi
-
mole fraction of species i
(liquid or gas)
Note at constant T and P,
yi dM̄i = 0
M
ig
=
M=
C
X
i=1
C
X
i=1
C
X
i=1
yi M̄i
yi dM̄i = 0
yi Mig
i
the ideal gas (ℳig ) behavior.
“Excess” property M ⌘ M
E
how much ℳ deviates from
the ideal solution behavior.
M
ideal
M
ideal
=
C
X
i=1
3
yi M i
Thermo.key - February 7, 2014
SHR §2.2
Phase Equilibrium
Gibbs Energy
Phase Equilibrium
is achieved when
G is minimized.
(thermodynamic property)
G = G(T, P, N1 , N2 , . . . , NC )
dG =
S dT + V dP +
i=1
Chemical Potential
(thermodynamic property,
partial molar Gibbs energy)
For multiple
phases in a
closed system,
Gsystem =
N
X
G(p)
p=1
dGsystem =
N
X
dG(p)
In phase equilibrium,
T(1) = T(2) = ⋯ = T(N) P(1) = P(2) = ⋯ = P(N)
dGsystem =
For a nonreacting
system, moles
are conserved.
(p)
dNi
=0
(1)
dNi
=
Problem:
µi ! 1
as P ! 0
(p)
dNi
dGsystem =
(p)
µi
=
(1)
µi
=
(2)
µi
Solution: Partial Fugacity
⇣µ ⌘
i
¯
fi = C exp
RT
C is a temperaturedependent constant.
4
N
X
" C
N
X
X⇣
p=2
p=2
p=1
To minimize G, dG=0. But each
term in the summation for dG is
independent, so each must be zero.
N
X
= ··· =
(N )
µi
i=1
@Ni
µi ⌘
p=1
p=1
N
X
◆
C ✓
X
@G
"
C
X
✓
dNi
P,T,Nj
@G
@Ni
(p)
◆
P,T,Nj
(p)
µi dNi
i=1
(p)
µi
(1)
µi
⌘
#
(p)
dNi
T,P
#
T,P
For phase equilibrium, the
chemical potential of any
species is equal in all phases.
For phase equilibrium, the fugacity
of any species is equal in all phases.
(p)
(1)
(2)
(N )
f¯i = f¯i = f¯i = · · · = f¯i
Thermo.key - February 7, 2014
SHR §2.2.1 & Table 2.2
Chemical Potential, Fugacity, Activity
Thermodynamic Quantity
Chemical potential
Partial fugacity
Fugacity coefficient of a
pure species
Fugacity coefficient of a
species in a mixture
Activity
Definition
µi ⌘
⇣
∂G
∂Ni
⌘
Description
gas and ideal solution
T,P,Nj
f i = C exp (µi/( RT ))
Partial molar free energy, gi
µi = gi
Thermodynamic pressure
f iV = yi P
fi ⌘ f i/P
Deviation of fugacity due to
f¯i = fi for a pure species
pressure
fiV ⌘
fiL ⌘
f iL/( xi P)
to pressure and composition
fiL = Pis/P
Relative thermodynamic
aiV = yi
pressure
aiL = xi
aiV/yi
Deviation of fugacity due to
giV = 1.0
aiL/xi
composition.
giL = 1.0
giV ⌘
For phase equilibrium, the chemical potential, fugacity,
activity (but not activity coefficient or fugacity
coefficient) of any species is equal in all phases.
=
o
iL xi fiL
fiL = Pis/P
Deviations to fugacity due
giL ⌘
f¯iL = ¯iL xi P
fiV = 1.0
f iV/(yi P)
ai ⌘ f i/ f io
Activity coefficient
Limiting case of ideal
f¯iV = ¯iV yi P
=
iV
o
yi fiV
5
fiV = 1.0
Pis vapor pressure of species i.
yi
vapor mole fraction of species i.
xi liquid mole fraction of species i.
¯iL
yi
¯
¯
fiL = fiV )
= ¯
xi
iV
we will use this on
the next slide...
Thermo.key - February 7, 2014
SHR §2.2.2 & Table 2.3
Phase Equilibrium - K-Values
Phase equilibrium ratio:
How much species i is
enriched in the vapor
Note: liquid-liquid
case equivalent is the
“partition coefficient”
Ki ⌘
Relative ↵ ⌘ Ki =
ij
volatility:
Kj
yi
xi
αij = 0
easy
(1)
KDi ⌘ xi /x(2)
i
¯iL
yi
Solve for yi/xi from Ki ⌘
= ¯
xi
iV
the previous slide:
o
iL fiL
=
=
iV P
yi/xi
Note: liquid-liquid case equivalent
is the “relative selectivity” βij.
yj/xj
ij
αij = 1
impossible
Concept: determine Ki from
thermodynamic models.
This gives us yi/xi.
iL iL
iV
Application
Ki = f̄iL/f̄iV
Benedict-Webb-Rubin-Starling, SRK, PR
SHR Table 2.3
mixtures from cryogenic to critical
Ki =
giL fiL/f̄iV
All mixtures from ambient to near critical.
Approximate forms
Ki = Pis/P
Raoult’s law (ideal)
Modified Raoult’s law
Poynting correction
Henry’s law
s
Ki = giL fiV
K =
⇣is⌘
Pi
P
Ideal solutions at low pressures
giL Pis
exp
Ki =
/P
⇣ ⌘R
1
RT
Nonideal liquid solutions near ambient pressure
P
Pis
viL dP
Hi
P
Nonideal liquid solutions at moderate pressures
and below Tc
Mix & match for different
species in a mixture
Equation
equations of state, Hydrocarbon and light gas
Activity coefficient
K Di
K Dj
αij = ∞
easy
Rigorous forms
Equation of state
⌘
Low to moderate pressures for species above Tc
6
Thermo.key - February 7, 2014
Example: Raoult’s Law (Ideal Mixtures)
Pis
Raoult’s law: Ki =
P
Need: Pis - saturation pressure for each species
Antoine equation: log10 Pis = A
Derivation of Raoult’s Law:
pi = xi Pis
pi = y i P
combine
to find
B
C +T
SHR Figure 2.3: Pis for a few species
partial pressure is the
weighted saturation pressure
(ideal liquid mixture)
Dalton’s law
(ideal mixture of gases)
Atmospheric Pressure
yi
Pis
=
= Ki
xi
P
Pis
Modified Raoult’s law: Ki = i
P
For Raoult’s law, Ki is independent of composition,
but varies exponentially with temperature!
7
Thermo.key - February 7, 2014
Liquid-Phase Activity Coefficients
Wilson 2-parameter
0 model1for mixtures:
ln
k
C
X
xj ⇤kj A
ln @
=1
C
X
j=1
For a binary system:
ln
ln
1
2
=
ln (x2 + x1 ⇤21 )
✓
PC
j=1 xj ⇤ij
!
◆
⇤12
⇤21
x1 + x2 ⇤12
x2 + x1 ⇤21
✓
◆
⇤12
⇤21
x1
+
x1 + x2 ⇤12
x2 + x1 ⇤21
ln (x1 + x2 ⇤12 ) + x2
=
i=1
xi ⇤ik
vjL
⇤ij =
exp
viL
ij
=
ji
ii
=
jj

(
ij
ii )
RT
⇤ij 6= ⇤ji
⇤ii = 1
• λij are functions of temperature
•
but not composition
Λij = Λij = 1 implies ideal
solution (γi = γj = 1)
Parameters λij or Λij are typically determined
from experimental data via regression.
Alternatively, if we can obtain γ at infinite dilution,
ln
ln
1
1
1
2
=1
ln ⇤12
⇤21
=1
ln ⇤21
⇤12
2 equations,
2 unknowns
Less accurate than regression.
8
Thermo.key - February 7, 2014
SHR Figure 2.4
Example: Graphical Methods
1. Locate appropriate species /mixture
2. Locate appropriate T and P.
3. Draw a straight line between the two
points to determine Ki.
NOTE: this can also be used to estimate
the bubble point for a pure fluid. (how?)
Example: at T=60 F and P=1 atm,
• Kpropane = 7
• Kisopentane = 0.64
9
Thermo.key - February 7, 2014
DePriester Chart
Empirical Correlation:
ln K =
a1
a2
b2
b3
+
+
a
+
b
ln
p
+
+
3
1
T2
T
p2
p
NOTE: T in °R and p in psia!
Compound
a
a
a
b
b
b
Methane
-292,860
0
8.2445
-0.8951
59.8465
0
Ethylene
-600,076.875
0
7.90595 -0.84677 42.94594
0
Ethane
-687,248.25
0
7.90694
49.02654
0
Propylene
-923,484.6875
0
7.71725 -0.87871 47.67624
0
Propane
-970,688.5625
0
7.15059 -0.76984
0
6.90224
Isobutane
-1,166,846
0
7.72668 -0.92213
0
0
n-Butane
-1,280,557
0
7.94986 -0.96455
0
0
Isopentane
-1,481,583
0
7.58071 -0.93159
0
0
n-Pentane
-1,524,891
0
7.33129 -0.89143
0
0
n-Hexane
-1,778,901
0
6.96783 -0.84634
0
0
-0.886
M. L. McWilliams, Chemical Engineering, 80(25), 1973 p. 138.
Chemical Engineering Progress
(AIChE),Vol. 74(4), pp. 85-86
10
Thermo.key - February 7, 2014
SHR §2.5.2
Thermo from Equations of State
fugacity
coefficient
V
vapor
"
Z P✓
1
= exp
v
RT 0

Z 1✓
1
P
= exp
RT v
(
partial fugacity ¯ = exp 1
iV
RT
coefficient
Z
1
V
"✓
RT
P
◆
RT
v
@P
@Ni
◆
dP
◆
#
dv
T,V,Nj
V =v
C
X
For Redlich-Kwong (and S-R-K) EOS:
ln Z + Z
RT
V
#
dV
1
ln Z
)
Z3
Z2 + A
P =
RT
v b
mixing rules:
B2 Z
a
v 2 + bv
C
X
i=1

(yi hoi ) + RT Z
A=
aP
R2 T 2
¯iL
yi
Ki ⌘
= ¯
xi
iV
bP
RT
B=
j=1
1
i=1
✓
3A
B
ln 1 +
2B
Z
◆
✓
◆
A
B
= exp Z 1 ln (Z B)
ln 1 +
B
Z
"
r
¯i = exp (Z 1) Bi ln (Z B) A 2 Ai
B
B
A
i=1

AB = 0
2
3
C
C
C
X
X
X
p
4
y i yj a i a j 5 b =
a=
y i bi
i=1
h=
Ni volume
B
Bi
B
!
✓
B
ln 1 +
Z
◆#
Note that these can be used in vapor or liquid phases, by
using appropriate values for Z, and using appropriate phase
mole fractions (yi for vapor, xi for liquid)
(because cubic EOS can approximate two-phase behavior.)
R-K:
S-R-K (much better):
RTi,c
bi = 0.08664
Pi,c
bi = 0.08664
RTi,c
Pi,c
⇥
2
R2 Ti,c
1 + f! 1
ai = 0.42748
Pi,c
2.5
R2 Ti,c
ai = 0.42748
Pi,c T 0.5
f! = 0.48 + 1.574!
11
0.176! 2
0.5
Ti,r
⇤2
ω - acentric factor
Thermo.key - February 7, 2014
Example: K-Values from SRK
Given a mixture of propane
(x=0.0166, y=0.3656) and
benzene (x=0.9834, y=0.6344)
at 300 K & 1 atm, find the Kvalue using SRK.
Redlich-Kwong (&SRK) parameters
T
P
ω
Propane
Benzene
369.8
562.2
4250
4890
0.149
0.209
Requires xi, yi. This is a problem since
we typically need Ki to
determine this ratio!
This is typically used within
another solver to determine
xi, yi (more soon).
Z3
Z2 + A
"
¯i = exp (Z
B
1)
¯iL
yi
Ki ⌘
= ¯
xi
iV
Bi
B
B2 Z
ln (Z
AB = 0
B)
A
B
2
r
Ai
A
Bi
B
!
✓
ln 1 +
B
Z
◆#
1. Calculate ai, bi and then Ai, Bi (tedious, but not difficult).
2. Calculate A and B using mixing rules.
3. Calculate Z in the vapor and liquid phases. This requires
us to solve the cubic equation and pick the appropriate
roots.
4. Calculate partial fugacity coefficient in vapor and liquid.
5. Calculate Ki.
12
Thermo.key - February 7, 2014
SHR §2.8
Heuristics for Property Model Selection
Aqueous
Solutions
Hydrocarbons
with or without
light gases
electrolytes and no
polar compounds?
Yes
Corresponding
States
No
Yes
Wide boiling
range?
with light gases?
Special
NRTL
No
Polar Compounds
Yes
liquid phase
separation?
SRK
(non cryogenic
T, all P)
PR
(all T, P)
No
No
BWRS
(all T, subcritical P)
PSRK
(see SHR
§2.7.1)
UNIFAC
Wilson
NRTL
Yes
NRTL
UNIQUAC
See also http://www.che.utah.edu/~sutherland/PropertySelection.pdf for a
much more extensive property selection flowchart from Professor Ring.
13
Thermo.key - February 7, 2014