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Systematic Methods for Chemical Process Design (Draft August 22, 1996) Lorenz T. Biegler
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algorithms for both modular and equation based process simulation modes and also provides
some background on nonlinear programming theory. Moreover, these concepts also help to
set the stage for the advanced optimization approaches presented in Part IV of this text.
Practical examples and case studies are used to highlight all of the concepts presented in thenext three chapters, and these are often motivated from industrial applications. From these we
hope to motivate both the complexity of the applications and the effectiveness of the modeling
and solution strategies. As a result, these three chapters set the stage for an understanding of
modern computer aided simulation and optimization tools in process engineering.
7.1 Introduction
The development of mass and energy balance models is a basic component upon which
process evaluation and design decisions need to be made. As in Chapter 2, we consider the
candidate flowsheet model as a large set of nonlinear equations that describe
a) connectivity of the units in the flowsheet through process streams
b) the specific equations of each unit, which are described by conservation laws aswell as constitutive equations for that unit.
c) underlying data and relationships that relate to physical properties and which serveas building blocks for each unit operation model.
In this chapter we focus on topics b) and c) and present a more detailed representation of the
unit operations models. To do this, we reconsider the approach taken in Chapter 2. In that
chapter we decoupled the relations between the mass balance, temperature and pressure
specifications and the energy balance. This allowed us to execute the mass balance first,
specify the temperature and pressure levels by assuming saturated output streams, and then
calculate the energy balance and energy duties once temperatures and pressures were fixed.
These calculations were made possible by assuming:
• ideal behavior in phase equilibrium• relative volatilities 'nearly' independent of temperature
• ideal behavior for energy balances
• noninteracting components in unit operations (except for reactors)
• fixed conversion reactor models
• simplifications in applying shortcut calculations
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equilibrium relations, a more complex flash model results than was addressed in Chapter 2.
Here we derive this model and discuss direct and nested solution strategies for these flash
models. The fourth section extends the flash model to equilibrium staged separations. In
particular, we discuss the derivation of distillation models along with methods for their
solution. It should be noted that this model easily extends beyond conventional distillationcolumns to cover complex column configurations. Again, direct and nested modes for the
solution of these models will be discussed. Extensions to other equilibrium stage separation
operations such as absorption and extraction will also be outlined.
The fifth section deals with unit models that are less detailed than the ones described above
and include transfer and exchange operations carried out by pumps, compressors and heat
exchangers of various types. We retain the motivation of design calculations and assume that
sizing and costing can be done once the mass and energy balance is fixed. Consequently, the
mass and energy balance models themselves will be largely unaffected by geometric
considerations. In this section we also consider reactor models briefly, with the same set of
assumptions. The least section summarizes the chapter and presents some future directions
for flowsheet modeling. These address some of the shortcomings exhibited by the models in
this chapter but at the expense of more computationally intensive models.
7.2 Thermodynamic Options for Process Simulation
This section provides a brief summary of thermodynamic relationships that are required forthe formulation of nonideal, equilibrium-based process models. Clearly, treatment of this
broad area will be incomplete and somewhat superficial, as a large (and burgeoning) literature
is devoted to this topic. Instead, we consider a qualitative description of physical property
models that are available in current process simulators. Supporting these models, one finds a
tremendous amount of effort devoted to the construction and verification of physical property
data banks, based on careful experimentation. The models themselves are based on concepts
of solution thermodynamics as discussed, for example, in Smith and VanNess (1987) and
VanNess and Abbott (1982). A summary of thermodynamic options is presented in Reid et al
(1987) and exhaustive details of the physical property options can be found in the user
manuals of most process simulators. Built on top of this are robust numerical procedures for
the calculation of thermodynamic and transport properties. Nevertheless, within a process
simulator, this is often presented to the user simply as a set of options, often with few
guidelines (or knowledge of the consequences) for their selection.
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In this section there is no attempt at providing a complete survey of these options, just a basic
understanding of these relationships. We start by concentrating on thermodynamic
calculations that support nonideal phase equilibrium, through chemical potentials and
fugacities, and then continue with applications to the calculation of other thermodynamic
quantities, especially partial molar enthalpies and volumes. Once covered, thesethermodynamic and physical property calculations provide the basic building blocks for the
detailed unit operations models which follow.
Phase Equilibrium
Phase equilibrium is determined when the Gibbs free energy for the overall system is
minimization. Here, underlying relationships for phase equilibrium are derived from a
minimization of the Gibbs free energy of the system. Given a mixture of n moles with NC
components. If we have equilibrium between NP phases and nip moles for each component
i in phase p, this can be expressed by the following problem:
Min n G = ΣΣ nip µip
s.t. Σp nip = ni, i = 1, ...NC
nip ≥ 0
where ni is the total number of moles for component i, G is the Gibbs energy per mole of the
system and the chemical potential of component i is defined by
µi = [∂(n G)/ ∂ni ] with T, P, and n j (j≠i) constant
For nonempty phases, the solution of this optimization problem is given by equality of thechemical potentials across phases, i.e.:
µi1 = µi2 ...µi,NP i = 1,...NC
To describe the chemical potential, we define a mixture fugacity for each of these phases and
components according to:
d µip = RT d ln f ip
and integrating from the same initial condition (say, µ') for all phases gives:
µip - µ' = RT ln (f ip /f')
Simplifying this expression shows that the mixture fugacities must also be the same in all
phases:
f i1 = f i2 ...f i,NP i = 1,...NC
Confining ourselves to vapor-liquid equilibrium (VLE), we now specialize the fugacities to
particular cases. For the vapor phase, we introduce a fugacity coefficient defined by:
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φi = f iv / (yi P)
where yi is the mole fraction of component i in the vapor mixture and P is the total pressure.
For the liquid phase, we define an activity ai as well as the activity coefficient γ i according to:
γ i = ai / xi = f il / (xi f il0)
where f il0 is the pure component fugacity. This pure component fugacity is further defined
by:
f il0 = f il(T, P, xi = 1) = Pi0(T) φi (xi = 1, Pi0, T) exp[ ∫ P
Pi0
Vil(T, P) /RT dp]
where the exponential of the volume integral in this expression is known as the Poynting
correction factor. Equating the mixture fugacities in each phase now leads to a reasonably
general expression:
φi yi P = γ i xi f 0il for i = 1, NC
and we can define K values, Ki = γ i f 0il / (φi P), that will be used for the flash calculations in
the next section.
As was assumed in Chapter 2, there are a number of simplifications that can be made to the
above expressions for the ideal case:
• for an ideal solution in the liquid phase, the activity coefficient γ i = 1.
• for an ideal solution in the vapor phase, the mixture fugacity f iv = f iv0
• for a mixture of ideal gases in the vapor phase, φi = 1• for negligible liquid molar volumes or for low pressures, the volume integral is
negligible and the Poynting factor is unity.
The nonideal cases can be characterized by violations of the above simplifications. Violation
of the first assumption is the most common and we frequently expect nonideality in the
liquid phase. The second assumption is valid for most chemical systems up to moderate
pressure levels and we will not consider any modifications of this assumption in this text.
The third and fourth assumptions are valid for low to medium pressures. In considering
nonideality in phase equilibrium, we first consider nonideality in the liquid phase when the
third and fourth assumptions are valid. Then we consider higher pressure systems where
nonidealities need to be considered for the vapor phase as well.
Liquid Activity Coefficient Models
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Departures from ideality can be represented by defining departure functions or excess
thermodynamic quantities. For molar Gibbs free energy we define:
G = Gid + GE
or
GE /RT
= G/RT - G
id /RT = Σ xi ln(f il /f il
0) - Σ xi ln xi
GE /RT = Σ xi ln(f il /(xif il0)) = Σ xi ln γ i
where Gid is the molar Gibbs free energy for the ideal system and GE is the excess molar
Gibbs free energy. The activity coefficient can also be treated as a partial molar quantity of
GE /RT:
ln γ i = [∂(n GE /RT )/ ∂ni ] with T, P, and n j (j≠i) constant
and after some manipulation, we can obtain, for component i, a direct relationship between
GE /RT and ln γ i from the following equation:
ln γ i = GE /RT + ∂(GE /RT )/ ∂xi - Σk xk ∂(GE /RT )/ ∂xk
Example
For a binary system, consider the simplest excess function, the two suffix Margules model,
G E /RT = A x1 x2. What are the activity coefficients for this model?
Applying the expression:
ln γ i = GE /RT + ∂(GE /RT )/ ∂xi - Σ xk ∂(GE /RT )/ ∂xk
leads to
ln γ 1 = A x1 x2 + A x2 - 2(A x1 x2) = A x2 (1 - x1) = A x22
ln γ 2 = A x1 x2 + A x1 - 2(A x1 x2) = A x1 (1 - x2) = A x12
The Margules model, however, applies only to nearly ideal systems with molecules of similar
sizes. Similarly, other models derived before computer simulation tools were developed
(e.g., regular solution theory and the van Laar equations) have relatively simple forms and
are largely restricted to nonpolar, hydrocarbon mixtures; these are less widely used than
current methods.
For process simulation, the popular liquid activity coefficient models estimate
multicomponent activity using only binary interactions among molecules. This assumption is
valid for nonelectrolyte mixtures where there are only short range (two body) interactions in
the mixture. A great advantage to this approach is that relatively little data are needed to model
complex mixtures accurately. Current liquid activity coefficient models include the Wilson
equation:
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GE /RT = -Σi xi ln(Σ j x j Λij )
with binary parameters Λij, and the NRTL (non random two liquid) equation:
GE /RT = Σi xi [(Σ j τ ji G ji x j )/(Σk Gki xk )]
with related binary parameters τ ji and G ji that can be derived from simpler forms. Both
models have parameters that often need to be estimated from experimental data, although
Reid et al (1987) discuss approximations to these parameters that yield reasonable results. Of
these two models, the Wilson equation is more accurate for homogeneous mixtures and it is
computationally the least expensive of all of the methods in this section. However, it is
functionally inadequate to deal with equilibrium between two liquid phases (LLE) or with
two liquids and a vapor phase (VLLE). The NRTL equation must be used in this case.
The UNIQUAC (Universal Quasi Chemical) model also handles vapor liquid and liquid-
liquid phase equilibrium. It is mathematically more complicated than NRTL but requires
fewer adjustable parameters, which are also less dependent on temperature. In addition, this
model is applicable to a wider range of components. The UNIQUAC model is given by:
GE /RT = Σi xi ln(Φi /xi) + (ζ /2) Σi qi xi ln(θi / Φi) - Σi qi xi ln(Σ j θ j τ ji)
where ζ is typically set to 10 and all of the parameters except τ ji are calculated from pure
component properties. The first two terms in this model represent combinatorial
contributions due to differences in size and shape of the molecule mixtures and are based
only on pure component information. The last term is a residual contribution to the excess
molar Gibbs energy, is based on energy interactions between molecules, and requires binary
interaction parameters τ ji. As a result the activity coefficient can be represent by both parts as:
ln γ i = ln γ iC + ln γ iR
A further extension of these models is given by group contribution methods. Here the models
contain parameters that characterize interactions between pairs of structural groups in the
molecule (e.g., methyl, -OH, ketone, olefin). This information can then be used to predict
activity coefficients in molecules with similar structural groups, for which data may not be
available. This essentially describes the UNIFAC (UNIQUAC Functional-group Activity
Coefficient) model, which starts with the UNIQUAC equations and retains the combinatorial
(or pure component) parts. Here the residual activity coefficient is substituted with a linear
combination of group residual activity coefficients:
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Here the id superscript deals with the pure component quantities using ideal mixing rules.
From the properties of thermodynamic partial derivatives, the excess Gibbs energies
presented above can be used directly for the following excess properties:
VE = (∂GE / ∂P)T
∆SE
= -(∂GE / ∂T)P
∆HE = ∆GE + T ∆SE
or
∆HE /RT = -T(∂[GE /RT]/ ∂T)P
Example
Find the excess quantities for the UNIQUAC model, assuming all of the parameters are
temperature and pressure independent.
The UNIQUAC model is given by:
GE /RT = Σi xi ln(Φi /xi) + (ζ /2) Σi qi xi ln(θi / Φi) - Σi qi xi ln(Σ j θ j τ ji)Using the above relations the excess quantities are:
VE = (∂GE / ∂P)T = 0
∆HE = ∆GE + T ∆SE = 0 or
∆HE /RT = -T(∂[GE /RT]/ ∂T)P = 0
∆SE = -(∂GE / ∂T)P
= -R[Σi xi ln(Φi /xi) + (ζ /2) Σi qi xi ln(θi / Φi)
- Σi qi xi ln(Σ j θ j τ ji)]
Therefore all of the thermodynamic options that were developed for phase equilibrium can be
extended directly to calculation of enthalpies, densities and entropies. In the next sections, we
will describe where these quantities are needed.
Implementation in Process Simulators
This section describes only a small fraction of physical property options that are available tothe user within current process simulation tools. The above survey avoids giving a long list
of options but should give the reader an appreciation of the breadth of models available for
physical property estimation. Currently used models are not mathematically simple nor are
they inexpensive to calculate, although these have been automated so that they can be
accessed easily. Nevertheless, their selection and use should not be done carelessly, nor
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should this aspect of process simulation be taken for granted. It is therefore hoped that this
section provides some background and guidelines for proper selection of these options.
The primary application for these nonideal models is on phase equilibrium calculations (also
referred to as flash calculations) as these are the basic building block for thermodynamics-based unit operations models. These models also apply directly to energy balances and other
process calculations. Moreover, in terms of numbers of equations and fraction of
computational effort, calculating these properties represents a significant part (up to 80%) of
the simulation and modeling task. In the remaining sections of this chapter we will develop
more detailed models based on thermodynamic concepts and we will see how they interact
with the physical property calculations described in this section.
7.3 Flash Calculations
In process simulation programs, flash calculations represent the most frequently invoked and
most basic sets of calculations. A flash calculation is required to determine the state of any
process stream following a physical or chemical transformation. This occurs after the
addition or removal of heat, a change in pressure or a change in composition due to reaction.
In this section we consider the derivation of the nonideal flash problem and two common
approaches for its solution. Unlike Chapter 2, we make no simplifications in the model to
allow for a simplified solution procedure. Consequently, the solution of this model requires
the numerical algorithms developed in Chapter 8.
Derivation of Flash Model
Consider the phase separation operation represented in Figure 7.1 with the same notation as
in Chapter 2.
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V, yi
L, xi
F, zi
fk = F zi
vi = V yi
li = L xi
Q
Figure 7.1: Flash Unit
In Chapter 2, we developed a linear split fraction model for this unit based on the molar
flows for NC components i in the feed, vapor and liquid streams, f i, vi and li, respectively.
Here we assume that the state of the feed stream is completely defined so that we know the
inlet flowrate, mole fractions (zi) and enthalpy. By defining the mole fractions as xi = li /(Σi
li) and yi = vi /(Σi vi) we obtain a minimal set of mass balances:
f i = vi + li, i = 1, ... NC
equilibrium equations:
γ i(l, T) f 0i(T, P) = (φi(v, T) P), i = 1, ... NC
and an enthalpy balance:
F Hf (f, T, P) + Q = V Hv (v, T, P) + L Hl (l, T, P)
which gives us (2NC + 1) equations for the (2 NC + 3) variables, v i, li, T, P and Q. As in
Chapter 2, we therefore have two degrees of freedom to specify the flash problem.
However, when a phase disappears, a model derived from mass balances in molar flows
leads to undefined compositions for dewpoint and bubble point conditions. Moreover, since
nonlinear phase equilibrium relations are composition dependent, we now develop a slightlydifferent flash model in terms of total flows and mole fractions. Following the minimal
description above, the mass balance over the unit is given by:
zi F = Vyi + Lxi, i = 1, ... NC
and an enthalpy balance yields,
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F Hf + Q = V Hv(y, T, P) + L Hl (x, T, P).
Equilibrium expressions are given by:
yi = Ki xi , i = 1, ... NC
with physical property definitions from section 7.2 used to define the K values:
Ki = γ i(x, T) f 0i(T, P) / (φi(y, T) P), i = 1, ... NC
We therefore have 3 NC + 5 variables (yi, Ki, xi, L, P, Q, T, V) and only 3 NC + 1
equations so far. Note that we have not specified any conditions yet on the conditions of xi
and yi (e.g., that mole fractions sum to one). Interestingly, this choice needs to be made
carefully since spurious roots are introduced even with some obvious choices.
Because neither liquid nor vapor mole fraction is specified we can include an overall mass
balance:
F = L + V
Now consider the simpler case where T and P are specified. This decouples the enthalpy
balance and allows Q to be calculated once the mass balance is solved. Combining the overall
mass balance equation with the component mass balance and equilibrium expressions above
leads to the following relations for the mole fractions:
xi =zi
1 + Ki − 1( )V
F
yi =Kizi
1 + Ki − 1( )V
F
Now we need an additional specification on either set of mole fractions to obtain a model
with the required two degrees of freedom.
Consider the two obvious choices: ∑xi = 1 or ∑yi = 1. Now for the first choice we have:
∑i xi = ∑i [F zi /( F + (Ki - 1)V)] = 1
and the flash model is trivially satisfied for every flash problem if we set xi = zi and V = 0.
Similarly, if we use ∑yi = 1 we find that:
∑yi = ∑ [F Ki zi / (F + (Ki - 1)V)] = 1
and the flash model is trivially satisfied for every flash problem if we set yi = zi and V = F.
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Clearly, either equation leads to spurious solutions (at the feed composition) that are
completely unrelated to the true solution of the flash problem. To eliminate the trivial
solutions we consider an alternate specification due to Rachford and Rice ( J. Petrol.
Technol., 4, 10 (1952)). By taking the difference of ∑xi = 1 and ∑yi = 1 we have:
∑yi - ∑xi = 0.
Note that this new specification, along with the overall mass balance, still leads to the correct
specifications on the mole fractions. Applying this condition to the relations for the mole
fractions leads to:
∑yi - ∑xi =∑ [F (Ki - 1)zi /( F + (Ki - 1)V)] = 0
and we see that the above spurious roots cannot solve this equation. In fact, x i = zi or xi = zi
are allowable solutions only under the (quite appropriate) condition that K i = 1 and we have
an azeotropic mixture.
Strategies for Flash Calculations
The flash model can be given concisely by:
zi F = Vyi + Lxi, i = 1, ... NC
yi = Ki xi , i = 1, ... NC
Ki = γ i(x, T) f 0i(T, P) / (φi(y, T) P), i = 1, ... NC
F = V + L
∑yi - ∑xi = 0
F Hf + Q = V Hv(y, T, P) + L Hl (x, T, P).
and now leaves two degrees of freedom to be specified. While many alternatives are possible
for design calculations, flash calculations are often solved for degrees of freedom chosen
among the variables (V/F, Q, P and T).
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The simplest case is given by the (P, T) flash since this requires no iteration for the enthalpy
balance. For this case, the flash problem can be solved by:
TP Flash Calculation Sequence
1) For fixed zi (make sure ∑zi = 1) and F, specify T, P. Proceed if between
bubble and dew points. (For composition dependent nonidealities, provide an initial
guess for xi and yi)
2) guess V/F
3) calculate Ki = γ i(x, T) f 0i(T, P) / (φi(y, T) P)
4) calculate xi = zi /(1 + (Ki - 1)V
F) and yi = Ki xi
5) evaluate the implicit relation ψ (V/F) = ∑xi - ∑yi. If ψ (V/F) is zero (or within a
small tolerance), STOP. Else, go to 6.6) Update the guess for V/F and go to 3.
Example of TP flash
Consider a mixture of 40 mol % methanol, 20 mol % propanol and 40 mol % acetone.
Perform a TP flash calculation at 1 atm and 343 K (70 C).
For ease of demonstration we model this mixture with the two suffix Margules equation,
estimated from Holmes and van Winkle (1970) and Reid et al (1987). For a ternary mixture,
we have:
GE /RT = -0.0753 x1 x2 + 0.6495 x1 x3 + 0.557 x2 x3
with activity coefficients given by:
ln γ 1 = -0.0753 x22 + 0.6495 x32 +0.0171 x2 x3
ln γ 2 = -0.0753*x12 + 0.557x32 - 0.1678 x1x3
ln γ 3 = 0.6495x12 +0.557*x22 + 1.2818 x2 x1
Using the Antoine constants the vapor pressures are given by ln Pi0 = Ai - Bi /(Ci + T) with
Pi0 in mm Hg, T in K and the following data:
Methanol Propanol AcetoneAi 18.5874 17.5439 16.6513Bi 3626.55 3166.38 2985.07Ci -34.29 -80.15 -35.93
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Since the phase equilibrium occurs at low pressure, the activity coefficients represent the only
source of nonideality and the K values are given by Ki = γ i Pi0 / P. Applying the flash
calculation sequence given above with a secant method for V/F, and starting from V/F = 0.5,
we obtain convergence to a tolerance of 10-6 for ψ (V/F) in 20 iterations. For this mixture at
this temperature and pressure, V/F = 0.8622 and the compositions, K values and activitycoefficients are given by:
yi xi Ki γ iMethanol 0.4109 0.3320 1.237 1.0065
Propanol 0.1549 0.4821 0.321 1.0006
Acetone 0.4342 0.1859 2.336 1.5011
TP specifications are most common for narrow boiling mixtures, where all of the
components have boiling points in a narrow range, such as benzene and toluene. Here V/F
can vary between zero and one with a small range in temperature. This case is common for
mixtures separated by distillation. On the other hand, for wide-boiling mixtures (such as air
and water) the TP specification in the flash calculation sequence works poorly as the
equilibrium temperature varies widely for small changes in V/F. These mixtures are
commonly separated by absorption and the specifications (V/F, T) and (V/F, P) are used.
Otherwise, the algorithm is similar to the flash calculation sequence presented above. Also,
for these cases, note that the enthalpy balance is not needed in the iteration loop.
Finally, when a specification on the heat input, Q, is made (as in an adiabatic flash), then an
enthalpy balance is imposed and needs to be incorporated into the flash algorithm. Often the
enthalpy balance is treated by guessing the temperature, say, and solving the TP flash in an
inner loop. The enthalpy is then calculated, matched to the heat input and the temperature is
reguessed in the outer loop. This calculation sequence makes the flash calculation much more
time consuming. Alternatively, all of the equations flash model can be solved simultaneously
using the Newton or Broyden method developed in Chapter 8. With this simultaneous
approach both wide and narrow boiling mixtures can be handled in a straightforward way.
However, for all of these methods, nonideal thermodynamic routines need to be called
frequently and this increases the computational expense.
Example of PQ flash
Consider a liquid feed mixture of 40 mol % methanol, 20 mol % propanol and 40 mol %
acetone at 373 K and 10 atm. Perform an adiabatic flash calculation at 1 atm.
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Using the physical property information from the previous example and heat capacities in
Reid et al (1987), we note that ∆HE = 0 and that ideal enthalpy relations can be chosen for
both liquid and vapor phases. The vapor and liquid enthalpies can therefore be calculated
from the relations developed in Chapter 2:∆Hv (T,y) = Σi yi (H0
f,i + ∫ TT0 C0
pi(τ) dτ)∆HL(T,x) = Σi xi (H0
f,i + ∫ TT0 C0
pi(τ) dτ - ∆Hivap(T))
∆Hivap(T) = ∆ Hi
vap(Tb) [(Tic - T)/(Ti
c - Tib)]0.38
Cpi(T) = ai + bi T + ci T2 + di T3
based on a reference temperature of 298 K and with the following data for heat capacities in
cal/gmol-K.
Methanol Propanol Acetone
ai 5.052 0.59 1.505bi 1.694d-2 7.942d-2 6.224e-2ci 6.179e-6 -4.431e-5 -2.992e-5
di -6.811e-9 1.026e-8 4.867e-9
∆ Hivap 8426. 9980. 6960.
Tib 337.8 370.4 329.4
Tic 512.6 536.7 508.1
The initial liquid feed enthalpy is -6.331 kcal/gmol and starting from a guess of 343 K, weexecute the TP flash algorithm as the inner loop. In an outer loop we match the specific
enthalpy for the liquid and vapor streams to the feed enthalpy and reguess the temperature.
The adiabatic flash calculation converges in about five outer iterations to a temperature of
334.6 K with V/F = 0.1781. The results of the adiabatic flash are given below:
yi xi Ki γ iMethanol 0.3877 0.4027 0.963 1.0877
Propanol 0.0520 0.2321 0.224 1.0381
Acetone 0.5603 0.3652 1.534 1.2906
Inside-out method for flash calculations
The flash calculation sequences developed above suffer from two drawbacks:
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• they are designed either for wide boiling or narrow boiling mixtures and perform poorly
for the opposite case
• they require frequent calls to evaluate nonideal thermodynamic functions, especially when
the enthalpy balance needs to be incorporated in the flash calculation.
To address these concerns, Boston and Britt (1978) developed an 'inside-out' algorithm thatgreatly accelerates the solution of flash problems. In an outer loop, this approach matches the
nonideal physical property equations to simplified expressions for K values and enthalpies
(similar to those used in Chapter 2) and then uses these expressions to solve the flash
equations in an inner loop. The solution of these equations is then used to update the
simplified expressions and the procedure terminates once the simplified expressions match
the actual nonideal ones in the outer loop.
To illustrate the advantages of the inside-out algorithm, we consider the PQ flash with the
flash equations given above. Boston (1980) further suggests the following simplifications for
the inner loop:
Ki = αi Kb
ln(Kb) = A + B (1/T - 1/T*)
H'v = C + D(T - T*)
H'l = E + F(T - T*)
where the parameters A, B, C, D, E, F and αi are available for matching with the nonideal
expressions for K values and enthalpies (H'v and H'l computed on a mass basis). Kb is an
average K value that is based on a geometric weighting of component K values. As inChapter 2, αi represents the relative volatilities and H'v0 and H'l0 are the ideal gas
enthalpies (on a mass basis) with reference temperature T*. To handle both wide and narrow
boiling mixtures in the inner loop, Boston and Britt define an artificial iteration variable, R ≡
Kb / (Kb + L/V). This variable captures the dominance of temperature or V/F for wide and
narrow boiling mixtures, respectively, and eliminates the need for separate algorithms for
these systems. This is because R can now vary widely both for large changes in T (wide
boiling) and L/V (narrow boiling). Now once the parameters (A, B, C, D, E, F, αi) are fixed
from the outer loop, we can derive the following relations through the substitution of the
flash equations and the simplified expressions:
zi F = f i = Vyi + Lxi.
Using yi = Ki xi and by defining Ki = αi Kb, we have:
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f i = (VKi + L) xi = (αiVKb + L) xi
Dividing by (VKb + L) and substituting for R yields:
f i /(VKb + L) = (αiVKb + L) xi /(VKb + L)
f i /(VKb + L) = (αiR + 1-R) xi
We now define a new set of variables:
pi ≡ xi (VKb + L) = xi L/ (1-R)
= f i / (αiR + 1-R).
Note that the pi
are determined only from R and quantities specified in the outer loop. From
the summation and equilibrium equations we can recover:
L = (1-R) Σ pi
V = F - L
Kb = (Σ pi / Σ αipi)
xi = pi /(VKb + L)
yi = αiKb xi
T = ((ln Kb - A)/ B + 1/T*)-1
Using R as the iteration variable, the flash calculation is completed by checking the simplified
enthalpy balance. The Boston-Britt algorithm can be summarized by the following calculation
sequence:
Inside Out Calculation Sequence
1) Initialize A, B, C, D, E, F, αi
2) Guess R
3) Solve for pi, Kb, T, L, V, xi and yi using the above equations
4) Convert flow rates to a mass basis and evaluate simplified mass enthalpies for the
balance equation:
ψ (R) = H'f + Q/F' + (L'/F') (H'l(x, T, P) - H'v (y, T, P)) - H'v (y, T, P).
5) If ψ (R) is within a zero tolerance, go to 6). Else, update the guess for R and go to
step 3).
6) At first pass, obtain new values of A, B, C, D, E, F, αi by comparing with
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nonideal expressions. Thereafter, update only A, C, E, and αi by using
Broyden's method to match these parameters with the nonideal expressions.
Boston and Britt prefer a mass basis for the enthalpy balance to avoid insensitivity to R
(through L/F) when (Hv - Hl) are close to zero on molar terms. This algorithm converges
much more quickly than the algorithms developed above and has been incorporated as the
standard flash algorithm in commercial process simulators. While this derivation deals only
with the PQ flash formulation, several other cases can be derived (see Exercise 4).
To demonstrate this algorithm, Boston and Britt solved a wide variety of nonideal systems
including narrow and wide boiling systems, and with Wilson, UNIQUAC, NRTL and
Equation of State options. Typical experience on these examples was less than 6 outer
interations (where physical property evaluations are required). Finally, numerical
experiments have shown that the above algorithm often can deal with composition dependent
K values even though the simplified expressions are not a function of xi. For highly nonideal
cases, however, Boston (1980) suggests a modification that makes the simplified K values
composition dependent and makes the algorithm more robust.
7.4 Distillation calculations
Distillation is perhaps the most detailed and well modeled unit within a process simulator,
since it is can often be represented accurately by an equilibrium stage model. The distillation
column can be modeled as a coupled cascade of flash units and we now consider the detailed
phase equilibrium behavior on each tray as well as mass and energy balances among trays.
Also, the thermodynamic models and flash algorithms considered in the previous sections
therefore have an important influence on the calculation of this unit. In this section we
construct a detailed equilibrium stage model for a conventional column and briefly discuss
methods to solve these models. The section concludes with a small example to illustrate these
concepts.
In contrast to the shortcut models in Chapter 2, we now consider a more detailed tray by tray
model that extend from the flash calculations in the previous section. Shortcut models are not
suitable for detailed modeling because of the assumption of constant relative volatility for all
trays and equimolar overflow. Clearly this assumption can be violated for nonideal systems,
especially with azeotropes. Moreover, even for nearly ideal systems, shortcut models are
based on the concept of key component specifications. However, if we choose different key
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(and distributed, 'between key') components we can obtain significantly different results for
the mass balance. As a result, the shortcut approach for distillation is only approximate at
best.
1
2
V
D
L
B
Fj
NT-1
NT
PVj
PLj
PVj
PLj
PVj
PLj
V j
V j+1L j
L j-1
M jF j
PVj
PLj
Tray j
Figure 7.2 Schematic of Distillation Column Model
Consider the conventional distillation column shown in Figure 7.2. The model of this
distillation column consists of indices, j, for each of the NT trays and NC components i. As
seen from the figure, there is a cascade of trays starting with a reboiler for vapor boilup at the
bottom and a vapor condenser at the top. Each tray has a liquid holdup (M j) and a much
smaller vapor holdup with liquid and vapor mole fractions are given by xij and yij
respectively. Each tray has vapor and liquid flowing from it (L j and V j) and is connected to
streams above and below. Possibilities at every tray also exist for a vapor or liquid feed (F j)
as well as liquid or vapor products (PL j or PV j). Enthalpies are calculated for each of these
streams (Hv or Hl, based on tray temperature, T j); equilibrium expressions relate yij to xij on
each tray. The column pressure is usually specified (P j) for each tray although a more
complex model can be incorporated that considers tray hydraulics and pressure drops across
each tray. Similarly heat sources and sinks (Q j) can be included for each tray. The distillation
model for Figure 7.2 is given by:
Mass balance
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F j zij + L j-1 xi, j-1 + V j+1 yi, j+1 - (PL j + L j) xij - (V j + PV j) yij = 0
i = 1,... NC, j = 1, ...NT
Equilibrium expressions
yij = Kij xij
Kij = K(T j, P j, xij)
Summation equations
Σi xij = 1 Σi yij = 1 j = 1, ...NT
Heat balance
F j HF j + L j-1 Hl, j-1 + V j+1 Hv, j+1 - (PL j + L j) Hlj - (V j + PV j) Hvj + Q j= 0, j = 1,
...NT
Hlj = h(T j, P j, x j), Hvj = H(T j, P j, y j), HF j = HF(Tf , Pf , z j)
These Mass, Equilibrium, Summation and Heat (MESH) equations form the standard model
for a tray by tray distillation model. Note that the thermodynamic properties (K values and
specific enthalpies) are expressed as implicit functions that require the physical property
models in section 7.2. For the condenser the balance equations are further simplified to:
Mass balance
V1 yi, 1 - (DL + L0) xiD - DV yiD = 0 i = 1,... NC
Summation equations
Σi xiD = 1 Σi yiD = 1
Equilibrium expressions
yiD = KiD xiD
KiD = K(TD, PD, xD)
Heat balance
V1 Hv,1 - (DL + L0) HlD - DV HvD - Qcon= 0
HlD = h(TD, PD, xD), HvD = H(TD, PD, yD)
and similarly the reboiler equations are given by:
Mass balance
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LN-1 xi, N-1 - BL xiB - (VN + BV) yiB = 0
i = 1,... NC, j = 1, ...NT
Equilibrium expressions
yiB = KiB xiB
KiB = K(TB, PB, xB)Summation equations
Σi xiB = 1 Σi yiB = 1
Heat balance
LN-1 Hl, N-1 - BL HlB - (VN + BV) HvB + Qreb= 0, j = 1, ...NT
HlB = h(TB, PB, xB), HvB = H(TB, PB, yB)
For the reboiler and condenser, the Summation and Equilibrium equations are dropped if the
overhead and bottom products, D and B, are single phase. The combined systems consists of
(NT + 2)(2NC + 3) + 2 equations and (NT + 2)(3NC + 5) + 3 variables. After specifying
the number of trays, feed tray location, and the feed flowrate, composition and enthalpy (NT
(NC +1) variables) only NT+1 degrees of freedom remain. A common specification for the
MESH system is to fix the pressures on the trays and the reflux ratio, R = L0 /D.
Many algorithms have been invented to solve the MESH system of equations. In fact, Taylor
and Lucia (1995) observe that since the late 1950s at least one new distillation algorithm has
been published in almost every year. Early methods were devoted to developing
decompositions of the MESH equations by fixing a subset of variables and solving for theremaining ones in an inner loop.
For instance, if the temperatures and flowrates are fixed, one can solve for the compositions
componentwise using the linearized Mass and Equilibrium equations in an inner loop. In the
outer loop, the temperatures and flowrates are adjusted using the Summation and Heat
equations. In this scheme, pairing the temperatures with the energy balance leads to the
'sumrates' method, applicable for wide boiling mixtures suitable for absorption. On the other
hand, pairing the flowrates with the Heat balance leads to the 'bubble point' method, more
suited to narrow boiling mixtures. A simplification of the 'bubble point' approach occurs in
the case of equimolar overflow where the flowrates are fixed (by specifying the reflux ratio)
and the tray temperatures are determined by the Summation equations. Here the equimolar
overflow assumption is based on heats of vaporization that are assumed the same for all
components. In this case the Heat balance is redundant and is deleted.
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Solving the Summation and Heat equations simultaneously for the temperatures and
flowrates in the outer loop was proposed about in the early 70's and this leads to algorithms
appropriate for both wide boiling and narrow boiling mixtures. However, a nonlinear
equation solver (see Chapter 8) is required for this case. Decomposition strategies for the
MESH equations often lead to fast algorithms for conventional distillation columns. Fornonideal systems with composition dependent K values, however, the Equilibrium equations
become nonlinear in x, which leads to additional computational difficulty and expense.
Moreover, additional design specifications such as product purity must be imposed as an
outer loop for these algorithms.
A more direct way to deal with these difficulties is to apply Newton-Raphson methods to the
total set of MESH equations. This approach was first suggested in the mid 60's and is now
perhaps the most popular method for distillation. Moreover, the Newton approach leads to
coordinated strategy for solving a general class of nonideal separation problems. This
approach can be summarized for distillation by combining the MESH equations and the
vector of variables into a large set of nonlinear equations and variables, f(w) = 0. Linearizing
these equations about a current point wk at which the variable vector is specified, we have:
f(wk ) + (∂f/ ∂w) (wk+1 - wk) = 0
with w chosen as the next estimation for iteration k+1. This value is determined from the
solution of the linear equations:
(∂f/ ∂w) (wk+1 - wk) = (∂f/ ∂w) ∆w = -f(wk )
Solving the linear equations requires evaluation of the Jacobian matrix, (∂f/ ∂w), using the
partial derivatives from the MESH equations. By grouping the MESH equations according to
each stage, the Jacobian matrix becomes block tridiagonal and can be factorized with
computational effort that is directly proportional to the number of trays. Moreover, the
simultaneous Newton approach easily allows the addition of design specifications without
imposing an outer loop for the column calculation. Also, the approach is extended in a
straightforward manner to deal with complex column configurations including heat loops and
pumparounds, bypass streams and multiply coupled columns.
Nevertheless, there are a few drawbacks to this simultaneous approach. One difficulty comes
from obtaining derivatives from the physical property equations for the K values and the
equilibrium expresions, especially if ∂K/ ∂x ≠ 0. For highly nonideal systems, accurate
derivatives are a necessity for good performance. Fortunately, most process simulators now
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incorporate analytic partial derivatives for these calculations. The Newton method also
requires good initialization procedures and these are often problem dependent and require
some skill on the user's part. Automatic initialization strategies generally are based on
obtaining good starting points for the Newton method by using simple shortcut calculations
or initial application of the decoupling strategies used by earlier distillation algorithms.Nevertheless, even with these intuitively helpful strategies, current distillation algorithms can
encounter difficulties, especially for highly nonideal systems.
Finally, inside-out concepts have also led to popular and fast distillation algorithms. Similar
to the inside-out flash algorithm, this approach removes the composition dependence for the
K values and enthalpies and solves these simplified MESH equations in an inner loop. As
discussed above, this calculation is much easier than direct solution of the MESH equations.
Again, these simplified quantities are compared with the detailed thermodynamics in an outer
loop and convergence occurs when the simplified properties match with the rigorous ones.
As with the flash algorithm, Boston and coworkers demonstrated this approach on a wide
variety of equilibrium staged systems including absorbers and distillation columns. This
approach can be significantly faster than the decoupled algorithms or the direct Newton
solvers. Moreover, for systems that are only mildly nonideal, the inside-out strategy is less
sensitive to a good problem initialization.
Example
To illustrate the formulation and solution of the MESH equations we consider the separationof Benzene, Toluene and O-Xylene. Here the problem formulation is modeled in GAMS; the
component mixture is nearly ideal and for illustration purposes, we define γ i = 1 so that the K
values are given by Pi0(T)/P. Similarly, the vapor and liquid enthalpies were calculated using
the ideal enthalpy relations given above and developed in Chapter 2. For this separation we
have a bubble point feed at 1.2 atm and a flowrate of 50 kg-mols/h. The feed composition is
xB = 0.55, xT = 0.25, xO = 0.20 and the feed temperature is therefore 390.4 K. We specify
the number of trays at 40 (including the condenser and reboiler) and the column pressure at 1
atm. (For simplicity we assume no pressure drops through the column). Also, we specify the
feed tray location to be the tenth tray below the condenser. Setting up the MESH equations
for this column and accounting for these equations, we need an additional specification for
this column and for this we specify the reflux ratio (R = L1 /D). For this example we perform
a parameteric study of the reflux ratio to study its effect on the column performance.
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Figures 7.3, 7.4 and 7.5 show the composition and temperature profiles for this column for
reflux ratios specified at R = 0.5, 1.0 and 2.0. In all cases note that the profiles are
nondifferentiable at the feed point and otherwise they remain fairly constant for trays 10
(immediately below the feed) through tray 30. In fact, these trays can be removed without
severely affecting the column performance. For the benzene profiles the purity increasessubstantially as the reflux ratio increases. For the lowest reflux ratio, xB = 0.899. For a
reflux ratio of one it becomes xB = 0.975 and for the highest reflux ratio the distillate is
almost pure benzene (xB = 0.999).
5 04 03 02 01 000.0
0.2
0.4
0.6
0.8
1.0
xb/r=0.5
xb/r=1
xb/r=2
Tray number
B e n z e n e
m o l e
f r a c t i o n s
Figure 7.3 Benzene Composition Profiles for Different Reflux Ratios
For the middle component, toluene, mole fractions above the feed decrease with increasing
reflux ratio. Below the feed the mole fractions rise steadily and then suddenly dip down due
to the mass balance in the reboiler. The bottoms mole fractions increase with increasing
reflux ratio. The benzene mole fraction in the bottom stream remains fairly constant at about
0.04. The o-xylene profile, not shown here, is obtained by difference of the benzene and
toluene profiles. Finally, the temperature profiles decrease with increasing reflux ratio and
approach the boiling point of benzene in the condenser (354 K) .
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5 04 03 02 01 000.0
0.2
0.4
0.6
0.8
xt/r=0.5
xt/r=1
xt/r=2
Tray number
T o l u e n e
m o l e
f r a c t i o n s
Figure 7.4 Toluene Composition Profiles for Different Reflux Ratios
5 04 03 02 01 00
350
360
370
380
390
400
t/r=0.5
t/r=1
t/r=2
Tray number
T r a y
t e m p e r a t u r e s
Figure 7.5 Temperature Profiles for Different Reflux Ratios
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Note that from this example the product purities were not specified directly. If this problem
were extended to an optimization framework (see Chapter 9), inequality constraints could be
specified for these purities. However, while equations that define the top and bottom purities
can be added easily to the MESH equations, adjusting the remaining column specifications is
not always easy. For instance, in the above example the number of trays and the feed traylocation were fixed and these are discrete variables. To satisfy the purities, only the reflux
ratio (and possibly overhead pressure) could be varied but this may not give enough freedom
to satisfy the specifications. Instead, we have solved this example in a simulation rather than
design mode with reflux directly specified. By avoiding direct purity specifications we end
up with a more time-consuming design procedure, but also avoid convergence failures that
occur from unreachable specifications.
Also, as can be seen from this small example, even simple distillation systems can lead to
large, nonlinear systems of the MESH equations. Solving these with the above algorithms
needs to be done with care and a good understanding of the design or simulation problem.
For large columns it is not unusual to encounter columns with several thousand nonlinear
equations. To reduce the size of these systems, the composition and temperature profiles in
the column (e.g., in Figures 7.3, 7.4 and 7.5) can be approximated with lower order
polynomials, rather than an evaluation at each tray. By choosing interpolation points for these
lower order approximations, one can write interpolating equations similar to the MESH
equations, but at far fewer points than the number of trays. For units with large numbers of
trays (such as superfractionators with over 100 trays) this approach can significantly reducethe problem size and computational burden. This approach is known as collocation and a
related approach for solving differential equations will be presented in Chapter 19.
This brief summary only gives a sketch of available methods for distillation units. More
detailed discussion of nonideal distillation behavior is covered in Chapter 12 and systematic
methods for the synthesis of separation sequences is described in Chapter 14. The methods
that were outlined above can also be extended readily to more complex systems such as three
phase distillation and reactive distillation. For three phase distillation, the MESH equations
need to extended to cover an additional liquid phase and sufficiently general thermodynamic
models (e.g., NRTL or UNIQUAC) need to be selected. On the other hand, the three phase
problem is fraught with additional numerical difficulties. For these problems the Gibbs
energy minimization (implicitly solved on each tray) contains local solutions and
consequently, nonunique solutions and singular points abound for systems described by the
MESH equations. Moreover, trivial solutions (with xi converging to the feed composition)
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can occur for poor initialization of the compositions. Thus, while three phase variations have
been developed for the above algorithms, some work still remains in the development of
reliable and robust methods. Reactive distillation operations can also be formulated by
augmenting the Mass and Heat equations in the MESH system with the appropriate reaction
terms. As with multiphase distillation, reactive distillation frequently exhibits more complexnonlinear behavior along with solution multiplicities. Doherty and coworkers have
investigated these nonideal systems and have analyzed their behavior with geometric
approaches. This approach will be discussed in greater detail in Chapter 14.
Finally, an equilibrium stage model for an absorption of distillation operation is only an
approximation to the actual behavior of these systems. The above models ignore mass and
heat transfer effects and also do not consider important features such as the column
geometry, influences of flows, and transport characteristics on trays. In the past these have
been handled by overall column and tray efficiencies and these can easily be incorporated into
the MESH equations. However, for systems far removed from equilibrium behavior, these
efficiencies represent a crude approximation at best. More recently, mass transfer models
have been developed to describe these systems more accurately. Taylor and coworkers
discuss mass transfer or rate-based models made up of the MERQ (mass, equilibrium, rate
and energy) equations. These models, however, require additional transport properties for
both mass and heat transfer characteristics, in addition to the phase equilibrium models. Also,
uncertain parameters such as interfacial area between phases must be estimated.
Nevertheless, rate-based models are already being introduced in commercial applications andtheir success will spur further development of better mass transfer models and more complete
physical property data banks.
7.5 Other unit operations
In Chapter 2, several additional unit operations models were described for evaluating a
candidate flowsheet. These include simple units such as mixers and splitters, transfer units
such as valves, pumps and compressors, energy exchangers that include a variety of heat
exchanger models and process reactors. For design purposes, the conceptual models for
these units remain largely unchanged from the descriptions in Chapter 2, except possibly for
process reactors. However, the solution of these unit models is often complicated by the
substitution of more detailed thermodynamic relations, developed in section 7.2. Below we
provide a brief summary of these extensions.
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Wb = f (p2-p1)/(ρ ηp ηm)
where ηp and ηm are the pump and motor efficiencies described in Chapter 3. This V ∆ p
work is added to the stream energy and thus specifies the molar enthalpy of the outlet
stream. From this relation, the temperature is calculated using the nonideal enthalpy models
outlined in section 7.2. Additional detailed sizing and costing can then be applied once the
stream conditions and the work requirements of these units have been calculated. These
additional calculations are beyond the scope of this chapter.
Compressors and Turbines
P1, T1 P2, T2
W
P1, T1 P2, T2
W
P2 > P1, T2 > T1 P2 < P1, T2 < T1
TurbineCompressor
f f
Mass and energy balances for compressors and turbines are usually made for preliminary
design by a direct specification of the outlet pressure or pressure change. For an isentropic
compressor, the ideal compression work can be calculated from the change in enthalpy of the
stream, calculated by holding the entropy constant. The temperatures and enthalpies are
calculated after the iterative calculation:
∆S(T1, P1, f) = ∆S(T2, P2, f)
where f is the molar flowrate. The theoretical work is then given by:
WT = [HV (P2, T2, f) - HV (P1, T1, f)]
where the entropies and enthalpies are calculated using nonideal thermodynamic models. The
actual work is then calculated using adjustments from isentropic behavior through an
isentropic efficiency and a motor efficiency, both specified by the user so that: Wb = WT / ηm ηs for the compressor and Wb = ηm ηs WT for the turbine. Additional detailed sizing
and costing can then be applied once the stream conditions and the work requirements of
these units have been calculated. These are beyond the scope of this chapter.
Heat Transfer Equipment
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For heat exchangers, the simplest units are those with three process temperatures specified
and the fourth is calculated by closing the energy balance.
Heat
Exchanger
FA, T1
FB, T4 FB, T3
FA, T2
For a countercurrent, shell and tube heat exchanger, for instance, with T1, T2 and T3
specified, this is given by:
Q = H(FA, T2, P2) - H(FA, T1, P1) = H(FB, T3, P3) - H(FB, T4, P4)
and T4 is solved for iteratively from the nonideal enthalpy balance. Sizing equations for
these heat exchangers can be found from the following equation:
Q = UA ∆Tlm
where Q is the heat duty, known from the energy balance, A is the required area, the logmean temperature (∆Tlf ) is given by :
∆Tlm = [(T1 - T4) - (T2 - T3)]/ ln{(T1 - T4)/(T2 - T3)}
and the overall heat transfer coefficient, U, is often specified by the user. Should phase
changes occur between the inlet and outlet, a more accurate sizing of the exchanger requires
a partitioning into multiple exchangers for the subcooled, two-phase and superheated
portions. The boundaries of these partitions are determined by finding the bubble and dew
points of the multiphase stream. More detailed heat exchanger calculations can be performed
through the following models.
If no phase changes occur in the heat exchanger:
• the overall heat transfer coefficient can be calculated by estimating tube and shell
side resistances with heat transfer correlations and combining them.
• geometry of the heat exchanger can be dealt with simply by calculating an
appropriate geometric factor to give: Q = F U A ∆Tlm. This approach covers a
wide range of exchanger calculations and is often used for multiple pass shell
and tube heat exchangers with shell side baffles.
If phase changes occur in the heat exchanger:
• more detailed (and time consuming) calculations need to be made by calculating
the internal temperature profiles directly. Here a multipoint boundary value
problem is formulated and the shell and tubeside differential equations are
integrated along the length of the heat exchanger. Nonideal enthalpies also need
to be calculated within the solution of the differential equations.
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Modern process simulators allow for all of these options and therefore permit the calculation
of quite detailed heat exchanger designs. See Welty et al. (1984) for a survey of models.
Fortunately, for many preliminary designs (especially in processes where raw material
conversion to product dominates the design objective), simple heat exchanger models are
adequate to determine an accurate mass and energy balance and also to give a goodapproximation for the area requirements for heat exchange.
Reactor Models
In process simulators, reactor models are often greatly simplified. A major reason for this
lies in the fact that physical properties are almost entirely based on thermodynamic concepts
and there is no general database for reaction kinetics. Moreover, for many new and even
existing processes, the reaction kinetics are simply not known, or are too difficult to obtain,
at the design stage.
fIN, TIN
ReactorfR, TR
As a result, simplified models are often used within flowsheeting tools. However, far more
can be done to exploit the character of the process when the reactor performance is modeled
accurately. Process synthesis approaches that deal with reactor networks are discussed later
in Chapters 13 and 19. Nevertheless, for process simulation, reactor models can be
classified into the following three types:
• stoichiometric reactors
• equilibrium-based reactors
• specific kinetic models
The simplest types, stoichiometric reactors are similar to the linear reactor models that were
described in Chapter 2. Here we specify the molar conversion of the NR parallel reactions in
advance. This requires that for each reaction r, we define a limiting component l(r), andnormalized stoichiometric coefficients γ
r,k = (Cr.k /Cr,l(r)), r = 1, NR for each component k,
where the coefficients Cr,k appear in the specified reactions. Defining the fraction converted
per pass based on limiting reactant as ηr, r=1,NR, gives us:
f Rk = f INk + ∑r=1
NR
γ r,k ηr f INl(r)
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With an outlet pressure specification and nonideal thermodynamic models, an energy
balance can also be completed for stoichiometric reactors according to:
QR = ∆H(TR,f R) - ∆H(TIN,f IN)
and this allows us either to specify the outlet temperature and calculate the appropriate
reactor heat duty, or specify this heat duty (say, adiabatic) and calculate the outlettemperature.
Equilibrium reactor models provide a better description for many industrial reactors and still
allow thermodynamic calculations that are compatible with process simulation databases.
For a single reaction,
aA + bB --> cC + dD
the equilibrium conversions can be given directly from:
(f C)c (f D)d / [(f A)a (f B)b] = K = exp(-∆Grxn(T)/RT)
where f i is the fugacity of component i and where, for the above reaction, ∆Grxn is given
by:
∆Grxn = (c∆Gf,C + d∆Gf,D) - (a∆Gf,A + b∆Gf,B)
and ∆Gf,i are the free energies of formation that can be evaluated as a function of
temperature. For gas phase reactions at 'low' pressure, the fugacity can be replaced by the
partial pressures and the expression becomes:
(PC)c (PD)d / [(PA)a (PB)b] = Kor in terms of mole fractions:
(yC)c (yD)d / [(yA)a (yB)b] = K P(a+b-c-d)
where P is the total pressure of the system.
Example
Consider the water gas reaction:
CO + H 2O <==> CO2 + H 2
at a pressure of 5 atm and a temperature of 600 K. What is the equilibrium concentration.
The Gibbs energy of reaction can be determined by:
∆Gf,CO2 = -94.26 kcal/gmol ∆Gf,CO = -32.81 kcal/gmol
∆Gf,H2 = 0. kcal/gmol ∆Gf,H2O = -54.64 kcal/gmol
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Finally, specific kinetic models are sometimes incorporated within process simulations. The
most common models are the ideal reactor models such as plug flow reactors (PFRs) and
continuous stirred tank reactors (CSTRs). For a reactor stream with an inlet concentration c0
and flowrate F0, the PFR equation is given by:
d(Fc)/dV = R(c, T), c(0) = c0where c is the vector of molar concentrations, V is the reactor volume and R(c) is the vector of
reaction rates. For continuous stirred tank reactors (CSTR), the outlet concentration is given by:
F c - F0 c0 = V R(c, T)
Note that for both reactors, the vector of reaction rates (reaction rate for each species) needs
to be specified. This task is frequently left up to the user, if kinetic expressions are available
for the reacting system. Moreover, these equations also require thermodynamic models for
the calculation of enthalpies for the energy balance around the reactor. As with the
stoichiometric models, this is necessary to determine the temperatures for a given heat load
specification, or vice versa.
Of course, there are many more detailed reactor models that could be developed. However,
these are considerably more computationally intensive and are usually used for 'off-line'
studies, rather than integrating them directly into the flowsheet. More detail on these reactor
models and their role in reactor network synthesis is presented in Chapters 13 and 19.
7.6 Summary and Future Directions
This chapter provides a concise summary of detailed unit operations models frequently used
in computer-aided process design tools. These process simulation tools are essential for the
analysis and evaluation of candidate flowsheets. In the next chapter we continue the
discussion of process simulation by describing the overall calculation strategy for the
simulation of a process flowsheet. In particular, we will present and describe the algorithms
needed to solve the process models given in this chapter. Moreover, we will discuss the
integration of these models to simulate the entire flowsheet.
At the present time, most detailed unit operations for preliminary process design are based
on thermodynamic models. Consequently, section 7.2 was devoted to a concise overview of
these models for nonideal process behavior. The motivating problem for this discussion was
phase equilibrium and this allows us to present nonidealities both in the liquid and the gas
phases. Popular thermodynamic models include Equation of State (EOS) models for
hydrocarbon mixtures and liquid activity coefficient models for nonideal, nonelectrolyte
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solutions. For the liquid activity coefficient models, model parameters frequently need to be
determined from VLE or VLLE data; in the absence of these data, group contribution
methods using the UNIFAC model have been very successful. The nonidealities that are
described by these models can also be used directly in calculations of specific volumes,
enthalpies and entropies. However, we note that the nonideal models in this section need tochosen with care because:
• they are far more complicated than ideal models and incur a much greater
computational cost for process calculations
• they are defined for specific mixtures and often yield highly inaccurate results if
not selected appropriately.
To develop the unit operations models, we note that the states of process streams are
determined entirely by their thermodynamic properties. These properties and nonideal
models for them are also considered sufficient for many of the unit operations in preliminary
design. Separations are usually assumed to consist of equilibrium stage models, with
efficiencies used to determine the actual column capacities. Simple mixing and splitting
operations are similar to those developed in Chapter 2, except that now nonideal models are
used to complete the energy balance. Similarly, transfer operations, including heat
exchangers, pumps and compressors are altered slightly to accommodate nonideal
thermodynamic models. These modifications are adequate to determine a reasonably
accurate mass and energy balance for a candidate process flowsheet. Nevertheless, detailed
sizing and costing for these units have not been covered in this chapter. Instead, forpreliminary design we will rely on the simplified strategies developed in Chapter 4. To
develop more detailed designs, there is a wealth of literature devoted to each unit operation
and its coverage is clearly beyond the scope of this text. The reader is instead encouraged to
consults the unit operations texts at the end of this chapter.
Finally, there are a number of research advances related to unit operations modeling that are
starting to be incorporated in process simulation tools. Certainly, more detailed reactor
models have been incorporated into process flowsheets whenever the need arises and a good
kinetic model is available for a specific process. In addition, mass transfer models are
becoming well developed for absorption and nonideal distillation processes. These models
are essential when tray efficiencies cannot adequately describe deviations from an
equilibrium model. In fact, several process simulators have already incorporated these rate
based models as standard models.
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As a longer term horizon, there are numerous advances in molecular dynamics and statistical
mechanics that are leading to important breakthroughs in physical property modeling when
no experimental data are available. While these methods are still too computationally
intensive to incorporate directly within a process simulator, they are becoming useful in
filling in the gaps present for many nonideal model parameters. As a result these approacheswill also play a bigger role in the development of future process simulation strategies.
References and Further Reading
This chapter provides only a brief description of modeling concepts and elements used in
process design. As a result, it is necessarily incomplete for all of these elements. For each
section, a broad literature exists and this needs to be consulted for relevant details of the
process models and their application to a particular design problem. An incomplete list of
survey references is given below.
Further information of on thermodynamic models, flash calculations and their use in process
simulation can be found in:
• Smith, J. M., and H. C. van Ness, Introduction to Chemical EngineeringThermodynamics, McGraw-Hill, New York (1987)
• van Ness, H. C. and Michael M Abbott, Classical thermodynamics of nonelectrolyte solutions: with applications to phase equilibria, McGraw-Hill,New York (1982)
• Fredenslund, A., Peter Rasmussen, and Jurgen Gmehling. Vapor-liquid equilibria using UNIFAC : a group contribution method , Elsevier Scientific,New York (1977)
• Reid, R.C., J. M. Prausnitz and B. E. Poling, The properties of gases and liquids, McGraw-Hill, New York (1987)
• Hirata, M. and S. Ohe, "Computer aided data book of vapor liquid equilibria"American Elsevier, New York (1975)
• Gmehling, J. and U. Onken, "The Dortmund Data Bank: A computerized systemfor retrieval, correlation, and prediction of thermodynamic properties of mixture"DECHEMA (1988)
Further details on the inside-out method can be found in:
• Boston, J. F. and H. I. Britt, "A radically different formulation for solvingphase equilibrium problems," Comp. and Chem. Engr., 2, p. 109 (1978)
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• Boston, J. F., "Inside-out algorithms for multicomponent separation processcalculations," in Computer Applications to Chemical Engineering, Squires andReklaitis (eds.), ACS Symposium Series #124, p. 135 (1980)
Reviews and detailed descriptions for distillation modeling for process simulation can be
found in
• Taylor, R. and A. Lucia, "Modeling and Analysis of MulticomponentSeparation Processes," in FOCAPD IV, Biegler and Doherty (eds.), AIChESymp. Ser #304, p. 19 (1995)
• Wang, J. C. and Y. L.Wang, "A Review on the Modeling and Simulation of Multistaged Separation Processes," Proc. FOCAPD, Mah and Seider (eds.),Engineering Foundation, Vol. II, p. 121 (1980)
Finally, there are several standard texts for unit operations models:
• J.M. Coulson and J.F. Richardson, Chemical Engineering: Vol. 2 - Unit Operations,Pergamon Press, Oxford (1968)
• Geankoplis. C. J., Transport processes and unit operations, Allyn and Bacon,Boston, (1978)
• Henley, E. J., and J. D. Seader, Equilibrium stage separation operations inchemical engineering, Wiley, New York (1981)
• McCabe, W.L., J. C. Smith and P. Harriott, Unit operations of chemical
engineering, McGraw-Hill, New York (1992)
• Perry's Chemical Engineers' Handbook. Don W. Green (ed.) McGraw-Hill,New York (1984)
• Welty, J.R., C. E. Wicks and R. E. Wilson, Fundamentals of Momentum, Heat and Mass Transfer, Wiley, New York (1984)
Further description of the unit models and physical property options can be found in the
documentation itself for the process simulators themselves. Three useful references are:
• ASPEN Plus User's Guide• HYSIM User's Guide• Pro/II User's Manual
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Exercises
1. For the multicomponent two suffix Margules model, derive the expressions for activity
coefficients used for the methanol, propanol, acetone example.
2. Derive the equation for the fugacity coefficients used in the equation of state models.
3. Simplify the flash equations and the TP flash algorithm to develop bubble and dewpoint
algorithms. Find the bubble and dewpoints for the 40 mol % methanol, 20 mol % propanol
and 40 mol % acetone system at one atm.
4. Fill in the steps in the derivation of the inside-out algorithm. Show that for a TP flash,
that the Boston-Britt model is related to the PQ flash algorithm presented in this chapter.
5. Using the GAMS case study model as a guide, solve the benzene, toluene, o-xylene
column for 30 trays with stage 15 as the feedtray location. Vary this location and comment
on the change in the distillate composition for a reflux ratio of 5.
6. Resolve the benzene, toluene, o-xylene column example and plot the liquid and vapor
flowrates using a reflux ratio of 5. Is equimolar overflow a good assumption for this
system?
7. Modify the MESH equations to deal with an equimolar overflow assumption. How are
the equations simplified?
8. Apply the shortcut models developed in Chapters 2 and 3 to the benzene, toluene, o-
xylene column example. Which specifications would you make to compare this model to the
tray-to-tray model?
9. Resolve the example with the equilibrium reaction:
CO + H 20 <==> CO2 + H 2
and show that the Gibbs free energy minimization yields the same result as for the relation:
(f C)c (f D)d / [(f A)a (f B)b] = K = exp(-∆Grxn(T)/RT)
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10. For the water gas shift example, show that the temperature correction for ∆Grxn is not
negligible for a temperature change from 298 K to 600 K. Resolve Example 7.6 with this
correction.