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
Institute for Chemical Process Engineering,
the Technical University of Clausthal,
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
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:
A component mass conservation
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:
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:
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:
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:
The source is:
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
, , , ;
- equation of artificial state; - coefficient of artificial compressibility; , , - stress tensor components, Pa; – effective viscosity , .
Using Mac Cormak's algorithm we can write
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 ;
- column cross area, m2;
Sw - active area of the plate, m2;
T – temperature, K;
- step by time, s;
u - longitudinal velocity , m s-1 ;
- velocity component parallel to the wall, m s-1 ;
- velocity vector, m s-1;
v - transversal velocity, m s-1 ;
- velocity component normal to the wall, m s-1 ;
x, y - longitudinal and transversal co-ordinates, m;
- spacing in x;
- spacing in y;
- volume fraction of phase; heat transfer coefficient, W m-3 K-1;
- heat transfer coefficient accounting finite mass fluxes , W m-3 K-1;
- equimolar mass transfer matrix, s-1 ;
- the finite flux mass transfer coefficients matrix , s-1 ;
- energy dissipation, W m-3 ;
- coefficient, ;
- total interphase mass transfer term, kg s-1 m-3 ;
- volume fraction of vapour phase;
- dynamic viscosity, Pa s;
- kinematic viscosity, m2 s ;
- coefficient of turbulence viscosity, m2 s ;
- density, kg m-3 ;
- surface tension, N m-1;
- mean molecular shearing stress, Pa;
- mean turbulent shearing stress, Pa;
- shearing stress, Pa ;
- matrix of correction factors, ;
- correction coefficient, ;
- matrix, ;
L - liquid;
G - gas;
i,j - component;
s - interface;
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