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5SIMPLIFIED MATHEMATICAL MODEL OF THE CONTACT MEMBRANE
DISTILLATION PROCESS
A. Zhulynskyi
doctor of engineering L. Ladieva
doctor of engineering, associate professor department of automation of chemical plants National Technical University of Ukraine "Kyiv Polytechnic Institute" Kyiv, Ukraine
ABSTRACT
A mathematical model for the process control contact membrane distillation. The mathematical model takes into account heterogeneous porous membrane structure and reflects the change in membrane permeability over time and along the longitudinal channel membrane. The method and algorithm of the initial distribution coefficient of permeability of the membrane. To determine the initial distribution coefficient chosen optimality criterion, which provides necessary motive power process. Have decided to task formulated using variational method.
Keywords: membrane distillation, semipermeable membrane, mathematical model, the membrane permeability coefficient, methods of optimal control theory
1. Introduction
The offered mathematical model is one of the convective heat and mass transfer for description of the process of contact membrane distillation (CMD), which does not include convective heat conduction or diffusion with consideration of the efficiency of heat and mass [1].
In order to develop control systems of the process of CMD, the mathematical model of the process was created. It considers and takes into account the temperature distribution in both longitudinal and transverse directions channels [2]. The model was calculated with stratified method. Researches of temperature changes in the transverse direction have shown that the impact of temperature polarization decreases with increasing of the turbulence in the flow channels and is of few degrees only (2 - 5 C).
At high flow speed we can achieve an intensive heat transfer from fluid volume to membrane surface. That is why steam flow depends on diffusion characteristics of the membrane. For industrial membrane modules temperature change along the canal is essential and must be considered.
2. Objective statement
There was a task of developing of a mathematical model of the process CMD which would observe the changes that have a dominant influence on the output parameters of the process. Those impacts which aren't counted are given in parametric form by changing the coefficients of the time recorded effects.
It is assumed that the temperature of the solution in the core flow and the one on the surface of membrane are the same. The speed in all positions of the cross-sectional flow is considered as constant. The influence of temperature and concentration polarization is neglected.
Depending on several factors, transferring of the material through the membrane, as already noted, is carried out primarily by the following mechanisms of transfer: free-molecular (knudsen's diffusion), molecular (normal diffusion), and viscous (mass) flow. Depending on changes in pore diameter, microstructure characteristics, the thickness of the membrane, the contribution of each mechanism varies as well.
In already existing mathematical models of steam flow through the membrane is given with the consideration of the effective diffusion coefficients that were calculated for the average characteristics of the membrane [3,4], such as porosity, pore's effective radius and tortuosity factor. You must have an accurate data of the micro-structural parameters of porous body. But the possibility of an accurate assessment of important morphological characteristics of the membrane is very limited. Besides, the fact that the pore size of the membrane which is defined in various ways, for example, by blister's point, mercury pore- symmetry or by penetrating gases will have a different value.
Due to the braking action of the membrane's solid skeleton, the coefficients of diffusion in these cases are significantly lower for an unlimited amount of liquid. Polymer membranes, which are used in membrane distillation, are heterogeneous systems, consisting of two phases - a polymer matrix and an aggregate of pores. The pores are characterized by heterogeneity of shape, size, and orientation in space. Parameters of membrane are changing over time. Concentration polarization and salt formation on the membrane surface are one of the most important problems in the desalination of mineralized natural water or seawater. The concentration polarization is the increasing of concentration of solute at the membrane surface in contact with the solution to be processed. As a result, concentration polarization affects the productivity of the process of separation. Because of preferred solvent transfer through the membrane solute concentration near the surface increases which leads to a number of undesirable consequences. Driving force ÂP decreases and so the productivity reduces. When the limit of solubility or gel on the membrane surface sediments occurs, which significantly reduces the partial pressure of the solvent from giving side of the membrane and causes a reduction in productivity, which, as a rule, the more substantial is, the higher initial permeability of the membrane would be. High concentrations leads to partial or total destruction of the active layer of the membrane, its pollution and poisoning, which is a mean destruction of hydrophilic-hydrophobic balance of the surface layer of the
membrane and change of its porosity. The above factors lead to a significant deterioration in membrane productivity up to a total loss of semi permeable properties.
3. The main material research
To manage the process successfully, it is important to anticipate the membrane pollution. Which means that defects may occur then, the surface of the membrane may change and that will affect its work and shelf life.
From the foregoing it follows that the actual structure of the porous membranes for membrane distillation is a very complicated heterogeneous system that led to the need to consider a change in membrane permeability overtime and along the longitudinal channel membrane. In general, the transfer mechanism vapor through the membrane under the pressure gradient is given in the form [5]:
Jr = Kp ( x, t )[ Pp ( bp ,ep )-Pd ( 0, ed )
(i)
S Pp
c
d6p (x, t)
d t
+ WPrS Ppcp
dÔp ( s, t )
dx
+
(2)
+Kp ( x, t ) d ( Pp - Pd ) r (e) = 0
)
for the initial and boundary conditions
eP (t)|t=0 =ePo. (X)
0p (xt)|x=o =0P (t) (3)
where Wpx - is a speed of solution in the longitudinal direction of the channel, m/s;
S=lyd, d - channel area and width, m2, m;
cP - density and heat capacity of the solution, kg / m3, J / (kg * K);
r - latent heat of vaporization, J / kg.
The dynamics equation of the solvent in distillate channel has the following form:
p ^+Wd dd
- Kp ( x, t ) d ( Pp - P„ ) r (û) = 0
for the initial and boundary conditions
(*,t)|t=0 =^do (x), ^d (x, t)|x=0 (t)
(4)
(5)
^ the speed of distillate in the longitudinal
where Kp (x, t) stands for permeability coefficient that is represented by a variable length of the channel and time, may consider membrane changing [kg/(m2s Pa)];
Pd , Pd - stand for partial vapor pressure of the solvent on both sides of the membrane, [N/m2];
©P, ©d - are temperature solution and distillate, [K]; bP is the concentration of dissolved salts, [kg / kg]. It can be written, assuming that that the flow of solvent through the membrane is caused by capillary forces at low partial pressure difference. If the membrane is considered as idea, or so to say that the dissolved component passes through the membrane and selectivity factor 9 = 1, then there will be only distillate on the cold side of the membrane, that is why Pd =Pd( ©d). The partial vapor pressure of the solvent from the hot side of the membrane will depend not only on the temperature ©P, but on the concentration bP as well. The dependence of the saturated vapor pressure solvent from the concentration bP for the diluted solution is given by the approximate Raul's law.
For elementary volume solution for supply channel based on the heat balance equation compound
lx=0
where W
c
direction, m / s;
pd, cd - density and heat capacity of distillate kg / m3, J / (kg * K).
Similar to equation (4), considering temperature of distribution along the length of solution channel, composed equation that includes concentration distribution in the longitudinal direction. Dynamic equation for concentration of solute, according to material balance for a vapor, represented as:
d
(SWPxpP (1-bp))dx- k(x,t)p d • dx(1-b - b' ) =
dx
d
=Tt ( Spp (1-bp )dx ),
(6)
where bp - the salt concentration in the solution, kg / kg; (1-b) - the concentration of diffusing substance in the membrane pores meniscus from a hot side of the membrane;
bn' - the concentration of diffusing substance in the meniscus membrane pores from a cold side.
Yet solved equations took into account that the density of the solution is changed over time, depending on temperature and concentration of the solution change. To determine the permeability coefficient, represented variable by a length of the channel and time, where used methods of optimal control theory. Optimality criterion that provides motivity of the process where chosen to determine the initial distribution coefficient:
k x
I (•) = / Í'
'dxdt ^ min
0 0
(7)
provided that h = (6P -6d) - A#min > 0 and limitations as equalities.
To convert constraints as inequality
0D(X,t)~dd (X,t)-A6>min >0 , that related with implementation process CMD, introduced a new variable in limitation at the form of equality:
dx3 dt
= \eB ( x,t )-ed ( x,t )-A0mn ]2 •H (eD ( x,t )-ed ( x,t )-A^min )
x^
( 0 )=
x
(8)
H [ h ( t )]
H [ h ( x, t )] =
where function:
- modified Heaviside step 0, dB ( x, t )-dd ( x, t )-A0mn >0
K, dB (x,t)-ed ( x,t )-A0mn <0
(9)
lk ^x 1
I mod = \\e{dB ~9d ]dxdt + - Sx- ( tk )
00 2
(10)
membrane permeability coefficient
provides minimum of functional (10).
The problem is solved by a variational method. The new parameter assessment defined as:
Kp+1 = Kp
k-
6K.
x3 ( h )
3 V k / is a direct way to include restrictions in the form of functional inequalities
tk L
(11)
The multiplier S is a penalty function. mod minimized so that the affected limitation area changes poorly or not at all.
The problem to solve mathematical models (1) - (5), reduced to an optimization problem of finding the unknown parameter
Kp (t)
, that
where K - the step of gradient procedure; L - Lagrangian. Calculations showed that on the first iterations functional decreases rapidly, but then when approaching the optimum process slows down. Gradient method to minimize the functional guarantees convergence to a minimum, at each iteration requires an individual solution for Cauchy problem and conjugated systems. A large number of iterations and slow convergence at the edge of the minimum can be attributed to flaws. To increase the speed of convergence is possible to choose the optimal size of the step or use the second order search procedures or coupled directions.
Initial data at which the calculations are carried written in the table 1.
Table 1.
The data for the implementation of algorithm of membrane permeability coefficient search
Parameter Identification of parameter Unit of measurement Value
The length of the module LX m 0.5
The width of the channel d m 0.01
The height of the channel lP m 0.006
The density of the solution P kg/m3 1010.34
Solution heat capacity cP J/(kg K) 4077
The value of the input disturbances flow solution A©P BX K 20
step of gradient procedure K - M0-5
As a result, found distribution membrane permeability "solution temperature at the entrance - the concentration of the coefficient KP (pic. 1). Received transient process for the channel solution in output module" (pic.2 ).
Pic. 1. The distribution coefficient of permeability to the membrane along the length of the module
Pic. 2. The transition process through the channel " solution temperature at input - solution concentration at the output of module"
4. Summary
A mathematical model for the management of process contact membrane distillation, which takes into account heterogeneous porous membrane structure and reflects the change in membrane permeability over time and along the longitudinal channel membrane. This model has a high conformity to the processes and the reduction of computational costs.
Designed method and algorithm of the initial distribution of membrane permeability coefficient. Based on the implementation of computational experiments found that the algorithm provides estimates of the permeability coefficient by 14 - 15 iterations.
References
[1] Bryk M. T. Membrane distillation / M. T. Bryk, Nigmatullin // Chemical Reviews. - 1994. - №12(63). - C. 1114 - 1129.
[2] Zhulynskyi A. The mathematical description of process of the concentration of the solution by method of contact-membrane distillation based on temperature-polarization. /A.Zhulynskyi, L.Ladieva// The advanced science. Open access journal, June 2014, Volume 2014 Issue 7, P. 49-52.
[3] Uhrozov V. V. Mathematical modeling of process contact membrane distillation in the flow module / V. V. Uhrozov // Theoretical Foundations of Chemical Engineering. - 1994. -T.4. - P. 375 - 380.
[4] Vandyshev A.B. Modeling high membrane apparatus to obtain high-purity hydrogen / A.B. Vandyshev, V.M. Makarov, L.L. Muravyov and etc. // Theoretical Foundations of Chemical Engineering. - 1996. - T.30. -S.554- 559.
[5] Ladieva L. R. Research throughput of membrane in the membrane distillation / L. R. Ladieva, A. Zhulynskyi // Materials of the international Conference of Control «Automatics -2002». - Donets'k. : 2002. - T1. - P. 218.
