APPLICATION OF EQUATION-SOLVER EQUATRAN-GFOR SEPARATION PROCESS ENGINEERING

Katsumi YOKOYAMA* and Goro OGUCHI
Omega Simulation Co., Ltd.
2-20-9, Nishi-waseda, Shinjuku-ku, Tokyo, 169-0051 JAPAN*Corresponding author:
Fax:81-3-3208-4911, E-mail:katsumi@omegasim.co.jp

ABSTRACTThis paper briefly explains the functions of the equation solver called EQUATRAN-G andthen evaluates the feasibility and practicality of the software with three actual examples(equilibrium flash calculation, Teiele-Geddes method for distillation and batch distillation)
applied in the distillation field of separation process engineering.

INTRODUCTION
The conventional way to calculate the process analysis and design in chemicalengineering was to create individual programs by using programming languages such as
FORTRAN and BASIC. However, the rapid growth of personal computers and the advent of
the packaged software designed for easy mathematical calculations have made it possible toperform such calculations employing only equations without programming.
FUNCTIONS
Equation Solving Functions
EQUATRAN-G enables the numerical calculations of simultaneous linear and nonlinear equations, ordinary differential equations including high-order and non-linear types,
optimization and least-squares methods such as non-linear equations. Those equations canbe entered in any order, and there is no need to modify the equations or transpose them.
Calculation procedures are automatically selected and applied. To solve simultaneous nonlinear equations, for instance, the iterative method is generally necessary. EQUATRAN-G isable to identify whether the simultaneous equations are linear or non-linear and select a
calculation method to solve them. It allows users to specify variable names as well as
equations that are used to check the convergence. As for ordinary differential equations, thehigh-order equations can be written by simply describing one statement that specifies integral
calculations, and this is done without having to modify the equations. For optimization
calculations, users need to specify only independent and evaluation variables.
EQUATRAN-G can also be used to solve compound-complex problems with a combinationof ordinary differential equations and linear/non-linear equations, the simultaneous non-linear
equations involved in the calculation of an optimization problems, etc. More complex
problems, such as multiple integrals, two-point boundaries value and mini-max problems,parameter identification in dynamic simulation, can be solved with user functions that aredescribed by EQUATRAN-G.
Model Descriptions
EQUATRAN-G features a variety of functions for easy model descriptions asfollows:
Array Variables -- One and two-dimensional array variables are used.
Built-in Functions -- Thirty-six different kinds of functions, including logarithmic,
trigonometric and exponential functions, etc. are incorporated.
Tables -- When a diagram or a table is given to show the relationships between variables, they
are defined by using a TABLE statement and described in equations like a function.
Conditional Equations -- The equations where expressions on the right side change depending
on certain conditions can be described.
User Functions and Macros – Modules are available for the description of large-scale models.
Graphing and Report Creating Functions
A variety of graphing functions are available for use in the science and technology
field, including semi-logarithmic and logarithmic scale graphs, interpolation by spline curves,
curve-fitting by linear, quadratic and cubic equations, etc. Graphs are easily created by theAutoSetup function, and data can be clearly and concisely displayed on a graph by employing
makeup functions. Reports can be easily created since calculation results are represented in
the report-form text where data can be entered in any format.
APPLICATION FOR DISTILLATION

Example 1 Equilibrium Flash Calculation
Problem
The heated ternary liquid mixture of benzene,
toluene and xylene with the mole fractions of 0.4,0.4 and 0.2, respectively are fed continuously intoa vapor-liquid separator at the feed rate of F=100kgmol/h. The temperature of the separator is
t=118°C and the pressure is P=1140 mmHg.
Calculate the composition and the flow rate of thevapor and liquid phases under these conditions
assuming that the mixture is an ideal solution.

Vapor flow rate : V
Composition : y
Feed rate : F
Composition : z

Pressure : P
Temperature : tLiquid flow rate : L
Composition : x

31: /* Antoine constants (p[mmHg],t[C]) */32:/* Benzene, Toluene and Xylene */33: A = ( 6.90565, 6.95464, 7.00988)
34: B = ( 1211.03, 1344.80, 1462.27)
35: C = ( 220.790, 219.482, 215.105)
36:
37: EQ: SUM(y-x) = 0
38:
39: RESET L#50 [0,100] BY EQ
40:
41: OUTPUT L,V,x,y,K

Results/* Equilibrium Flash Calculation */Equations
Material balance<<L +V = F
Lxi + Vyi = Fzi

(1)
(2)

yi = Ki xi(3)o

Ki = pi / P

å(y − x ) = 0i= 56.35664 : Liquid flow rate[kgmol/h]

V= 43.64336 : Vapor flow rate [kgmol/h]y(6)i(i=1,2,…, N)

Source text
1: /* Equilibrium Flash Calculation */2:
3:
4: LOCAL N = 3 /* Number of components */5: VAR F = 100"Feed rate[kgmol/h]"..
6:,t = 118"Temperature[C]"..
7:,P = 1140 "Pressure[mmHg]"..
8:,L"Liquid flow rate[kgmol/h]"..
9:,V"Vapor flow rate [kgmol/h]"..
10:,z(N) = ( 0.4, 0.4, 0.2 ) ..
11:
"Feed mole fraction[-]"..
12:
,x(N)
"Liquid mole fraction [-]"..
13:,y(N)"Vapor mole fraction [-]"..
14:
,K(N)
"K value[-]"..
15:,p(N)"Vapor pressure[mmHg]"..
16:,A(N)"Antoine constants A"..
17:
,B(N)
"Antoine constants B"..
18:,C(N)"Antoine constants C"
19:
20: /* Material balance */21: L + V = F
22: L*x + V*y = F*z
23:
24: /* Liquid vapor equilibrium */25: y = K*x
26: K = p/P
27:
28: /* Antoine equation p[mmHg], t[C] */29: LOG10(p) = A - B/(t+C)
30:=1) 0.5433351

K(5)=1) 0.2889993(4)

Vapor pressure(Antoine equation)
olog pi = Ai − Bi / (t + Ci )Summation of mole fraction>>

Lx

Liquid vapor equilibrium

Calculation Results: Liquid mole fraction
2) 0.4344453: Vapor mole fraction
2) 0.3555209=: K Value

1) 1.880057

2) 0.8183329[-]

3) 0.2765554[-]

3) 0.101144[-]

3) 0.3657279

Example 2 Teiele-Geddes Method forDistillation
Problem
Calculate the vapor composition, liquidcomposition and temperature at each stage of thedistillation column for separation of pentane andhexane. Constant molar overflow is assumed ateach stage. The structure of the distillation
column and the operating conditions are listed
below.
Structure of distillation column
Total number of stages N = 10
Feed stage
Nf = 7(The top is the first stage.）
Operating conditions
Reflux ratio
R = 3 [-]
Distillate flow rate
D = 4.8 [kgmol/h]Feed flow rate
F = 10.0 [kgmol/h]Feed mole fraction of pentane
zf = 0.5 [-]q-value (Liquid ratio of feed) q = 0.1 [-]Operating pressure
Ptotal =760 [mmHg]

Source text
OmittedResultsEquations
Material balance

Thiele-Geddes Method for Distillation

dx1 i= −( D + L) x1 i + Vy2 i
(1)
dt
dx
U j j i = Lx j −1 i − Vy j i + Lx j i + Vy j +1 idt
U180

Liquid phase
Vapor phase
Temperature70(0.660

0.4

50

0.2

40030
10

Temperature

Mole fraction

[-]

0.8

[C]

1.0UN)( j = 2,3,L, N − 1)(2)

dxN i= LxN − 1 i − Vy N i + DxN idt(3)

Total material balance
3456789V = D+ LdU N= −DdtL
R=DStage(5)(6)

Summation of mole fractionåx = 1
åy =1
ji

(7)

ji

(8)

ii

Source text
OmittedResults
Top mole fraction
Bottom mole fraction
Top temperature
Bottom temperature

Batch Distillation
1.0

110

0.8

100

0.6

90

0.4

80

0.270[-]

This is a case of stripping the methanol from asolution containing 22.3 mol% of methanol in thewater by the batch distillation column. Calculate
the changes in the distillate composition of thecolumn with five ideal stages. (The condenser andreboiler are regarded as one stage.) It is assumed
that constant molar overflow is expected within thecolumn and the initial condition is supposed to be
the distillation that starts after a sufficiently longtime of total reflux operation. The condenser is atotal condenser.
Operation Conditions
Reflux ratio
R=3
Flow rate of distillate D = 2 [kgmol/h]Feed mole fraction of methanol
xF = 0.2 [-]Liquid Holdup
Condenser
U1 = 2 [kgmol]
Each stage
Uj = 0.1 [kgmol]
Reboiler
UN = 20 [kgmol]

Liquid mole fraction

Example 3 Batch Distillation
Problem(4)0[C]2Temperature

160
012Time34[h]CONCLUSION
Compared with conventional programming languages, EQUATRAN-G is easy to use,
can modify the mathematical models more easily because of the use of equations and canenhance productivity. EQUATRAN-G can also be applied in a variety of areas ranging frombalance calculations (e.g., equilibrium flash calculations) to dynamic simulations (e.g., batch
distillation calculations).