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Modelling of multicomponent reactive distillation
in a tray column
V.A. Danilov, A.G.
Laptev, S.V. Karpeev,
Department of Chemical Engineering,
Kazan State Technological University,
K. Marx, 68
420015 Kazan
Russia
danilov@dionis.kfti.kcn.ru
Alfons Vogelpohl,
Institute for Chemical Process Engineering,
the Technical University of Clausthal,
D-38678 Clausthal-Zellerfeld,
Germany
vogelpohl@itv.tu-clausthal.de
Abstract
It is suggested a
mathematical model of reactive distillation of multicomponent mixture in a
column with a bubble plate. Description of transfer processes in two phase
gas – liquid stream is based on two fluid model.
The model can be applied to design and to simulate separation capability of
an industrial column for multicomponent reactive distillation.
Keywords: multicomponent mass
transfer, heat transfer, chemical reaction, distillation
Introduction
Rectification
is well-known technique of multicomponent mixtures separation in chemical
industry. There are various methods of separation process modelling in a tray
column. Usage of material and heat balance equations is common to all methods.
Two principal methods of simulation for multicomponent distillation are differential
methods1-4 and integral methods5-15. “Differential”
methods1-4 start from the mass and heat balances of a differential
or finite element of the two phase film on the plate, and take into account
the rate of mass transfer (without chemical reaction).
Plate
by plate calculation (integral method) is traditional method of multicomponent
separation processes modelling in a column unit. It is based on iterative
solution of mass and energy balance equations. Chemical reactions are taken
into account by source terms. There are equilibrium and non - equilibrium
models of separation process in a column. Drawback of non equilibrium models5-12
is complication of heat and mass transfer driving forces calculation taking
into account flow structure. Further development of non equilibrium model
of reactive distillation by using cell model is shown by Krishna8-12
.
In accordance
with equilibrium model influence of hydrodynamics of two-phase flow on a contact
device and non-equilibrium mass and heat transfer are taken into account by
an efficiency of a tray13-20.
There
are various approaches for modelling transfer processes in two-phase flow.
Many simplified models for bubble tray efficiency (for rectification without
chemical reaction) calculation have been proposed:
plug flow
in liquid and gas phases21;
plug flow
in gas phase and complete mixing in liquid phase22,23;
diffusion
model in gas and liquid phases22,23
As is
known, Computational Fluid Dynamics (CFD) can be used for solving a wide range
of fluid flow problems and applications. Most commonly
used for gas liquid stream modelling is two fluid model24-26.
Complication of a problem is of necessity for closure
of mathematical description of the process by definition of the coefficients
of turbulent exchange and mass transfer in phases. The model is used for fluid
dynamics modelling in a bubble column27-29.
Solution of three-dimensional momentum transfer equations in both phases
of gas – liquid stream on sieve plate is given by Krishna30, which
has been obtained by using CFX-F3D commercial CFD package. While simultaneous
solution of momentum, mass and energy equations for both phases is a complicated
problem.
Modelling of transfer processes in two phase stream on a tray
The two-fluid conservation equations for adiabatic
two-phase flow24-26,30 are written as:
Mass conservation
, (1)
Momentum
conservation:
, (2)
A component mass conservation
(3)
Energy
conservation
, (4)
At the
given work it is suggested a method of simplification of transfer processes
description in two phase flow taking into account phases interaction peculiarity
on a plate. To simplify a model it is accepted uniform distribution of disperse
(gas) phase in two phase flow (
). Separation process is steady state. As is known, dispersing
gas into liquid flow results in continuous phase turbulization. Froth regime
of bubble plate is characterised with well developed turbulence in liquid
phase which gives full mixing along height (
, 
, 
). Turbulence in the bulk of liquid phase is accepted isotropic
(
) and thermal properties changing is insignificant.
The equations of momentum, mass and
energy transfer for liquid phase are written in two-dimensional form as:
, (5)
, (6)
, (7)
, (8)
, (9)
Peculiarity
of two phase motion on a bubble tray is gas (vapour) and liquid streams interaction
in cross flow. Since 
and gas velocity in
two phase flow is greater then liquid one therefore 
, 
. Moreover momentum source projection on OX and OY axes in equation
(4),(5) are equal to zero 
. It is well known that a component concentration fundamental
change in gas phase occurs in gas jet entering to liquid layer. The terms
evaluation in a component mass transfer equation in gas phase show
, 
, 
.
Taking
into consideration accepted presumptions equation of a component mass transfer
in gas phase takes on form:
or
, (10)
where 
- marked volume of two
phase flow 
; 
- column matrix of mass
flow rate transferred through interface in volume 
, 
; 
- column matrix of the
total mass flow rate in volume 
caused by chemical reactions, 
; 
- column matrix of component
concentration in gas flow entering to volume 
; 
- column matrix of component concentration in exit gas flow
leaving the volume 
; 
- gas mass flow rate
in volume 
, 
.
Similarly equation
of heat transfer in gas phase is written as:
, (11)
where 
- enthalpy of gas (vapour)
flow leaving the volume
; 
- enthalpy of gas flow
entering to volume 
;
- heat flux transferred through interface in volume 
, 
; 
– the total
heat flux, caused by chemical reactions in volume 
, 
.
Boundary conditions
for equation (1) - (5) are set as follow (Fig.1):
;
;
, 
, vL =0, 
, 
;
, 
, 
, 
, 
, 
.
Influence of dispersed
phase in equations (1)-(5) is taken into account by source terms and coefficients
of turbulence exchange.
Fig 1. Two dimensional
model of a bubble tray. (
-liquid flow path)
Determination of source
terms and the coefficients of turbulence exchange
An element of column matrix of mass
source 
is calculated as:
, (12)
The source 
is:
. (13)
The source
of mass caused by chemical reaction is defined as:
, 
, (14)
where 
- kinetic rate constant
of 
chemical reaction; 
- concentration
of p - component taking part in chemical
reaction; 
- kinetic power low coefficient; 
– number of chemical
reactions; 
– stochiometric coefficient.
The source of heat rt is:
, 
, (15)
The source
of heat caused by chemical reaction 
is:
, 
, (16)
where 
- thermal effect of
á chemical reaction.
Characteristics of turbulence exchange
, aT and
the volumetric coefficients of heat transfer are determined by Laptev’s model31,32.
As it was shown by Taylor and Krishna33 the matrix method of solving
multicomponent mass transfer problems requires formulation of the matrix function
by rearranging the binary correlations to 
and employing Sylvester’s
theorem. As correlation 
it is used Laptev’s
model31,32 for binary coefficients of mass transfer calculation.
Phase equilibrium with chemical reaction is calculated by well-know method34
.
Characteristics of turbulence exchange
, aT and
the volumetric coefficients of heat transfer are determined by Laptev’s model31,32
.
Solution
of transfer equations
The set of simultaneous equations (5)-
(7) is solved by Mac Cormak's method35,36. According to the method
the equations (5)-(7) are rewritten as
, (17)
where
, 
, 
, 
;
- equation of artificial state; 
- coefficient of artificial compressibility; 
, 
, 
- stress tensor
components, Pa; 
– effective viscosity
, 
.
Using
Mac Cormak's algorithm we can write
Predictor
(18)
Corrector
(19)
After each step variables 
can be found by decoding
column matrix U:
in the following way
, 
, 
.
Step by
time is limited by Courant’s condition35 :
,
where b- sound speed in medium.
To calculate the field
of pressure it is used a method of artificial compressibility35
: 
.
An obtained profile of velocity of
liquid phase on a tray is used to determine the field of concentration of
a component by numerical solution of multicomponent mass transfer equation
(8), simultaneously with material balance in gas phase (10) for a cross type
plate with allowance for chemical reactions. The field of temperature in liquid
phase on a tray is found by numerical solution of heat transfer equation (9)
simultaneously with equation of heat balance in gas phase (11) for a cross
type plate. An obtained profile of component concentration in each phase allows
to determine an efficiency of separation process. Analysis of the calculated
fields of velocity and concentration in phases can show the influence of various
factors on separation capability of a contact device. For a given field of
a component concentration it is calculated an element of efficiency column
matrix in phases by Murphry:
, 
,
where 
, 
- mean concentration
of i - component in liquid and gas flows entering to the tray; 
- concentration of i
- component in liquid at equilibrium with vapour entering to the tray; 
- concentration of i - component in gas at equilibrium with
liquid entering to the tray; 
- mean concentration
of i - component in liquid leaving the tray; 
- mean concentration
of i - component in gas leaving the tray.
Stagewise
calculation by material and heat balance equations allows to find concentration
and temperature fields along column height taking account a separation efficiency
of a tray: 
,
where 
- mean concentration
of i - component in vapour stream entering to p - tray; 
- concentration of i
- component in vapour stream at equilibrium with liquid entering to p - tray;
- mean concentration
of i - component in vapour stream leaving p -tray.
The developed model was used to simulate
reactive rectification of water (1)– acetic acid (2)- acetic anhydride (3)
in column with cap trays using Marek’s experimental data37. Chemical
reaction occurs in liquid phase:
Reaction if of the second order (in equation (14)
). The kinetic rate constant of chemical reaction37 (litre kg-mole-1 hour-1
) is
The field of velocity and water concentration in liquid
phase are given on Fig.2 and Fig.3. As is obvious from fig.2, complex liquid
motion on the tray leads to appearance of secondary flows near the wall.
The model
can be applied to design and simulate separation capability of an industrial
tray column for multicomponent reactive distillation38 .
Fig.1 The profile
of liquid velocity u/u0 on the tray.
(L = 581.5 kg/h; G =
620.3 kg/h; Dcolumn
= 0.6 m; hL = 0.011 m ;
= 1.36 10-3 m/s2).
Fig.2 The profile
of water concentration (CL,
water - CL0, water) 100 / CL0, water
in liquid phase on the tray
(L = 581.5 kg/h;
G = 620.3 kg/h; Dcolumn
= 0.6 m; hL = 0.011 m ;
DT = 1.36 10-3 m/s2;
CL0,water=
0.05854 mass.fraction; EL, water = 0.476, EG, water
= 0.626,
overall experimental
column efficiency 0.5 )
List of symbols
a - molecular temperature conductivity, m2
s-1;
aT - turbulence temperature conductivity,
m2 s-1;
C - column
matrix of component concentration, mass fraction;
- column matrix of component
concentration, mass fraction;
Cp - specific heat, J kg-1
K-1;
- matrix of molecular
diffusion, m2 s-1;
- diagonal matrix of
turbulence diffusion, m2 s-1;
- substantive derivative;
- enthalpy, J kg-1;
hc - clear
liquid height, m;
- acceleration due to
gravity, 
= 9.81, m s-2;
- matrix of overall volumetric mass transfer coefficients, 
, s-1 ;
- overall volumetric heat transfer coefficient
, W m-3 K-1;
- matrix of equilibrium
constants;
m – number of components.
N - normal direction to the wall;
- mass flux of i - component,
kg s-1 m-3;
- total mass flux, 
kg s-1 m-3;
P – pressure, Pa;
- mean molecular heat flux, W m-2 ;
- mean turbulent heat flux, W m-2;
- interphase a component
mass transfer term, kg s-1
m-3 ;
rch - interphase a component
mass transfer term caused by chemical reaction, kg s-1 m-3 ;
rp - interphase momentum transfer term, N m-3 ;
rt - interphase heat
transfer term, W m-3 ;
rch,t - interphase heat transfer term caused by chemical reaction, W m-3
;