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Modeling and calculation of temperature-concentration polarisation in the membrane distillation process (MD)
245
Desalination, 93 (1993) 245-258 Elsevier Science Publishers B.V., Amsterdam
Modeling and calculation of temperatureconcentration polarisation in the membrane distillation process (MD) S.P. Agashichev and A.V. Sivakov Department of Chemical Engineering, Mendeleev Institute, Miusskaja Square 9, Moscow A-190 125190 (Russia)
SUMMARY
A model for a membrane distillation process in a plate-and-frame unit has been developed. It is based on a mass and energy balance equation for hydrodynamic, temperature and concentration boundary layers. The model takes into account energy interdependence between flow in feed and in permeate channels. A model taking into consideration temperature-concentration polarisation (TCP) predicts temperature and concentration values at the membrane surface. The model consists of an analytical equation and permits simulation or analysis of the influence of various factors to permeate flux.
SYMBOLS = const in cont. profile approximation (a=-5) : ; f,F G I L
1 n
= = = = = = =
membrane width thermal capacity diffusion coefficient flow section mass flux specific enthalpy membrane channel length = L/n = length of interval = number of open pore at the unit surface membrane
001 l-9164/93/$06.00
0 1993 Elsevier Science Publishers B.V.
All rights reserved.
246 = = = = = = = = = = = = = = = = = = = = = = =
pressure membrane pore radius temperature longitudinal liquid velocity transverse flux component velocity heat flux constituents concentration physical distance coordinate distance between the channel walk membrane thickness hydrodynamic layer thickness temperature layer thickness concentration layer thickness the first dimensionless distance the second dimensionless distance coordinate the third dimensionless distance coordinate thermal conductivity density fluid viscosity porosity tortuosity factor (heat) Prandtl number Pr = CCJX (diffusion) Prandtl number Pr* = dpD
Subscripts 1 2 a e cross conv long c dif K mol m
= = = = = = = = = = = =
flow in feed (hot) channel flow in permeat (cool) channel entrance section exit section transverse flux constituents convective constituent longitudinal constituent result constituent diffusive constituent Knudsen constituent molecular constituent membrane constituent
INTRODUCTION
Membrane distillation (MD) is a relatively non-conventional process. It is being developed for the purification of water effluents and for removal of water from solutions to effect concentration. The process relies on a nonwetted hydrophobic membrane through which water vapour is driven by a chemical potential gradient. It has a wide potential application. MD-based techniques have been drawing scientific and commercial attention due to
247
their ecological benefit and low capital and operation costs. It could be used for desalination purification and solution concentration by means of low grade (low potential) heat. The MD-based techniques have a wide potential interest and prospects especially for the Gulf region. It could be used for solving some ecological and industrial problems. Potential application of the MD method has been discussed by many authors [l-8]. Lack of a good model and calculation methods have hampered the practical implementation of MD. Both concentration polarisation and temperature polarisation should be taken into consideration in modelling the MD process. It is the combined phenomena that determine the driving force of the MD process, that in turn determines the transmembrane flux. The underlying concepts of this paper were adopted from [l-8]. The previous paper of our research team [9] proposed the submodel and calculated a technique for the TP-phenomenon influence on the MD process. This paper describes modelling of combined temperature and concentration polarisation (TCP) phenomena in a symmetric channel of plate-and-frame unit for the MD process.
MODELLING
Using mass-and energy balance equations for hydrodynamic, temperature and concentration boundary layers, we obtained the temperature and concentration distribution along the membrane surface. The main heat and transport constituents can be seen from Fig. 2. The model consists of some submodels describing mass- and heat-transfer constituents. The submodels describe transfer phenomena in both channels and in membrane space. For the conjugate solution to be obtained the hydraulic, temperature and concentration submodels should be introduced into the comprehensive model. Mutual energy influence between flow both in hot and in cool channels should also be taken into account.
MAIN ASSUMPTIONS OF THE MODEL
The symmetric plate-and-frame channel of the MD unit is taken into consideration. Balance equations were written for the control volume. (elementary parallel-piped). The intersection line of the control volume with the figure plane is shown in Fig. 1. Both “heat” and “cool” channel can be represented as conjugate system of elementary volumes. The intersection line of the elementary foundation with the figure plane is the section "abed" .
248
I d
2 .P.
..I
c
Fig. 1. Control volume of the MD channels.
There is no mass- and heat-transfer across symmetry planes. The intersections of the symmetry plane with the figure plane are shown by segment “ad” “bc” (Fig. 2). The model is based on the next assumption: 1. There are incompressible, laminar, continuous flows both in “hot” and in “cool’ channels. 2. Velocity, temperature and concentration variation within hydrodynamic, temperature and concentration boundary layers are taken into consideration in the model. 3. There is a concave meniscus from both ends of membrane pores. Liquid evaporates from the hot membrane side meniscus and condensates at the cool membrane side. 4. There are water-vapour air mixtures within the free porous space in the membrane. 5. In Fig. 2 the first interval ranges from Z=O to Z=6,; the second from Z=8, to Z=6,; the third from Z=6, to Z=A,=h,. The first interval belongs to concentration, temperature and hydraulic boundary layer. In the first interval there is Z-variation of temperature, velocity and concentration. The second interval lies between concentration boundary layer and temperature layer boundary. Within the second interval there is Z-variation of both temperature and velocity while concentration is constant. The third interval lies between temperature boundary layer and hydrodynamic boundary layer. Within the third interval only velocity varies while temperature and concentration remain constant. 6. For solving the equations the transfer from physical distance coordinate-Z to dimensionless distance coordinates has been made. The first dimensionless coordinate r=z/6, is used to describe the velocity profile. The velocity profile is approximated by the function.
249
.
Fig. 2. Membrane distillation process. The main heat and mass-flux constituents: channel; 2, cool channel; m, membrane.
1, hot
(1) The velocity variation occurs within the viscous boundary layer 6, (Fig. 2). 7. The second dimensionless coordinate 8 =q (P,)’ is used to describe the temperature profile T=f(8). The scale unit of B-coordinate is temperature boundary layer thickness l/(P,.)’ in T-coordinate (Fig. 3). The temperature profile is approximated as follows:
Fig. 3. Concentration, temperature, and hydrodynamic boundary layers in the original and in the dimensionless coordinate systems.
43- f1d.f= sin 4
-
GM
(5” 1
(2)
8. The third dimensionless coordinate 4 = (P,*)1’31(P,) ’ is used to describe the concentration profile C=fl$). The scale unit of the &coordinate is concentration boundary layer thickness (P,)“I(P,*)1’3 in the previous 8 coordinate (Fig. 3). The concentration profile is approximated as follows:
X4- xl = e44 %M
-
(3)
3
9. The relationship between temperature, concentration boundary layers and viscous boundary layer was adopted from [lo]
(4)
251 THE LONGITUDINAL
HEAT CONSTlTUENTS
e,
lo,,gconv
The convective heat transfer expression across the entrance control plane “ap” (Fig, 1) was presented in [9]. There are two longitudinal subconstituents (the first one within temperature boundary layer z < S,, the second one-between temperature- and hydrodynamic layer boundary (6, < z < 6,). The equation for the longitudinal convective heat constituent Q110ng,.onV was written as follows:
Qllullgumv =
z=8wpwzIzdF = pcbh,
/
WI
zi()
THE LONGITUDINAL
MASS CONSTITUENTS
G, l0ngconv
a similar approach was used as for For expressing the G1 l,,ng,OnV
Q 1longconvin [9]). Convective mass-transfer through entrance plane “ap” (Fig. 1) can be expressed as follows: G llonsconv =
sz=4v
wz xz dF = ~z~wZ~zdF
z&)
+ s,“‘“‘”wzxzdF I
(a)
The first term on the right hand of Eqn. (6) means longitudinal mass constituent within concentration boundary layer (from z=O to z=&. It can be determined as follows: G llongconv =
s‘=” z_o
wz xz dF
(7)
For the solution of the Eqn. (7) to be obtained, the third dimensionless coordinate-+ was used where
(8)
252
5 = x1
+
(xIM
-
x1
)e
a6
(9)
dF = b dZ = b h,d e/(P;)”
{x1 + (xIy
- x1
(10)
bh,W
(11)
)eao> P,’ l/3
11
Solving Eqn. (ll), the third dimensionless coordinate-$ was used. The second term on the right-hand side of Eqn. (6) means longitudinal mass constituents-G 1longconv** between concentration boundary layer (2=&J and the boundary of hydrodynamic layer (z=BW=hr)
G lag+*
=
s
z=aww,xJybcdF I
&j
(12)
where (13)
dF = b h, cf4 /(P;)”
(14)
while %=x1 const. Having integrated Eqns. (11) and (12), an equation for longitudinal mass constituent Gllongconvwas obtained.
253 THE CROSS-CHANNEL
HEAT CONSTITUENT
There are some cross- channel heat constituents towards and opposite membrane surface direction. They are: heat conductivity constituent (Q,x>; heat-convective constituent (G IconV);heat consitituents by means of diffusive mass fluxes (Q&. They can be expressed as follows:
Q 1A
=
Qldif
-
=
L
(5
D
(16)
-hf)tifN
X1M --+ %
(
CROSS-CHANNEL
(c4t*
-c&t,,)&
I
MASS CONSTITUTENT-
There are convective G, coIIvCmM and diffusive G, diffcrosssubconstituents of mass flux. G lcmlv-salt G lamv-4
= Xl
=
(Pl
&
&
-
3)
(19)
(20)
wif
wx,, - x1)
G ldif-salt
= -
(21)
fM 8,
GIds-4
=
D
($4
-
x1df4)
fM/ ax
The result cross-channel mass-constituent G,, cToBs can be expressed as follows: G 1CQUW
=
%cimlt
+G lamv-sq - G,,,
+ G,,,
(23)
254 TRANSMEMBRANE MASS CONSTlTUENTS
Assuming the membrane to be impermeable to salt, in this case the transmembrane mass flux consist only of water. There are two main mechanisms for porous space transport. They are convective and diffusive. Convective mass flux through membrane porous space GMconV.Depending upon the relationship between the mean free molecular path and the pore size, two main diffusive models can be used. (the molecular diffusion model and the Knudsen diffusion model). In some cases there are both mechanisms at the same time. Diffusion coefficient for complex transfer D*
Determination of diffusion coeffkients was given in [ 111. The diffusion constituent of transmembrane mass flux GMdifcan be written as: MD’
G Mdif
=
RT
(P,, - P2M) XAM
Taking into account the Claudius-Clapeyron Eqn. (26) [lo], the temperature difference (AT = Tt, - AT,,) can be introduced instead of the pressure difference (AI’ = P,, - P&. Convective transmembrane mass flux can be expressed using the Pouseille equation. AP = PA IM(t,,
GMcalv =
AP bA,
(26)
- t&/RT2
$I2 @.f.
&pap
For the AF to be expressed through AT in (27), the Claudius-Clapeyron Eqn. (26) is used. The resulting transmembrane mass constituent GM, results as follows:
G MC
=
%onv +%if
(28)
255 TRANSMEMBRANE
HEAT CONSTITUENT
The resultant transmembrane heat-constituent consists of the following subconstituents: heat conductivity flux through membrane matrix QMh; heat conductivity of water-air mixture in porous space QMxag;convective heat flux through porous space aq. vapour QMnVq;comprehensive diffusive heat flux through porous space QMdSq. These subconstitutuents were described in detail [9].
QM=+¶ =
MD* RP
%f -p2M) fM e I&_ xAM
(32)
The resultant transmembrane heat constituent QMc is summarized from the following subconstituents: (33)
LONGITUDINAL
FLUX VELOCITY AND TRANSMEMBRANE
FLUX
Longitudinal velocity is varied along the channels because of the transmembrane flux. Digital value at each length interval was determined from the mass balance equation.
For the comprehensive model equations to be solved the expression for the transmembrane water flux VI should be evaluated. It was obtained on the base of the mass-balance equation for water in membrane and in channels (%mas aq = GMcq) Eqns. (28) and (23).
256
0 START
TlMa=TlMa(min
initial) 1
t Xltla=XlHa~min
Qlconv
%?l~(eq.la);
Glconv(l9,20);Gldif QZa(eq.7.8.9) .
initial)
(eq.17); aq(21);Gldif
Qldifteq.18) salt(22);GlZ(23)
TZa(vo~.calc)=f
TPHasf(TlMa;PlZ) Q2along T2a
Ieq-29-33)
convteq.
1s)
(voI)H(Q2along;T2Ma)
I
0 END
Fig. 4. Flowsheet of the computer model.
(eq.15)
257 CALCULATION TECHNIQUE
The calculation technique was elaborated on the basis of the described model (Eqns. (l-34)). For solving the system equations, the following is to be used: l l l
l
the initial hydrodynamic W, (inlet),IV, (inlet); the initial temperature and concentration Tt, T2 (inlet; XI, X, (inlet); channels geometry: length L; width b; heights 2Ir1, 2h2 membrane parameters: pore size RPr; quantity of open pores per unit square n; membrane thickness Am; thermal conductivity of membrane material h; parameter of 1iquid media: density p; viscosity p; thermal conductivity h; thermal capacity C.
The filtration length L is subdivided into equal length intervals 1= (L/n). The calculation on each length interval is carried out by a technique of sequential approximation. At first one adopts a digital value of the membrane surface temperature TIMaand concentration XIMa and iterate to the temperature in volume of “cool” channel T2ccdcjto Tzcfactj.The calculation of surface hot temperature TIMais carried till TzMatcalcl = T2,,~f8cq. Having calculated the longitudinal heat flux in exit section (QtJ, the hot surface membrane temperature (TIM2 in exit section was calculated. The sequential calculation of digital values at the next length intervals was carried out, having calculated the first interval values. All exit values of the foregoing length interval (values at the outlet section of the interval) are adopted to entrance values of the next length interval (values of the inlet section of the interval). The flowsheet of computer model is shown in Fig. 4.
CONCLUSIONS
The model is used for analysis and simulation of the membrane distillation process in semipermeable plate-and-frame channel. The model enables determination of the size of the model, the physical properties of the fluid, the temperature and concentration in both channels.
258 REFERENCES 1 2 3 4 5 6 7 8 9
10 11
K. Schneider and T.J. van Gassel, Chem. Ing. Techn., 56 (1984) 514-521. E. Drioli and W. Yonglie, Desalination, 53 (1985) 339-346. G.S. Sarti, C. Gostoli and S. Matulli, Desalination, 56 (1985) 277-286. J.M. Smith and H.C. van Ness, Introduction to Chemical Engineering Thermodynamics, 3rd ed., McGraw-Hill, New York, 1975. F. Belluci, E. Drioli, F.S. Gacto, D.G. Mita, N. Pagiluca, and D. Tomadacis, J. Membr. Sci., 7 (1980) 169. F. Belluci, J. Membr. Sci., 9 (1981) 285-301. R.W. Schofield, A.G. Fane and C.J.D. Fell, J. Membr. Sci., 33 (1987) 299-313. H. Schlichting, Grenzschicht-Theorie, funfte Auflage, Verlag G. Braun, Karlsruhe, 1974. S.P. Agashichev, A.A. Hakobjan and Yu. I. Dytnersky, Modelling and Calculating of Temperature Polarisation in Membrane Distillation Process, Ab. N 196, In: Seventh Symp. on Separation Science and Technology for Energy Application, Knoxville, TN, October 20-24, 1991. F. Daniels and R.A. Alberty, Physical Chemistry, 4th edn., Wiley, New York, 1975. T.K. Scherwood, R.L. Pigfordand C.R. Wilke,Mass Transfer,McGraw-Hill, New York, 1975.
Desalination, 93 (1993) 245-258 Elsevier Science Publishers B.V., Amsterdam
Modeling and calculation of temperatureconcentration polarisation in the membrane distillation process (MD) S.P. Agashichev and A.V. Sivakov Department of Chemical Engineering, Mendeleev Institute, Miusskaja Square 9, Moscow A-190 125190 (Russia)
SUMMARY
A model for a membrane distillation process in a plate-and-frame unit has been developed. It is based on a mass and energy balance equation for hydrodynamic, temperature and concentration boundary layers. The model takes into account energy interdependence between flow in feed and in permeate channels. A model taking into consideration temperature-concentration polarisation (TCP) predicts temperature and concentration values at the membrane surface. The model consists of an analytical equation and permits simulation or analysis of the influence of various factors to permeate flux.
SYMBOLS = const in cont. profile approximation (a=-5) : ; f,F G I L
1 n
= = = = = = =
membrane width thermal capacity diffusion coefficient flow section mass flux specific enthalpy membrane channel length = L/n = length of interval = number of open pore at the unit surface membrane
001 l-9164/93/$06.00
0 1993 Elsevier Science Publishers B.V.
All rights reserved.
246 = = = = = = = = = = = = = = = = = = = = = = =
pressure membrane pore radius temperature longitudinal liquid velocity transverse flux component velocity heat flux constituents concentration physical distance coordinate distance between the channel walk membrane thickness hydrodynamic layer thickness temperature layer thickness concentration layer thickness the first dimensionless distance the second dimensionless distance coordinate the third dimensionless distance coordinate thermal conductivity density fluid viscosity porosity tortuosity factor (heat) Prandtl number Pr = CCJX (diffusion) Prandtl number Pr* = dpD
Subscripts 1 2 a e cross conv long c dif K mol m
= = = = = = = = = = = =
flow in feed (hot) channel flow in permeat (cool) channel entrance section exit section transverse flux constituents convective constituent longitudinal constituent result constituent diffusive constituent Knudsen constituent molecular constituent membrane constituent
INTRODUCTION
Membrane distillation (MD) is a relatively non-conventional process. It is being developed for the purification of water effluents and for removal of water from solutions to effect concentration. The process relies on a nonwetted hydrophobic membrane through which water vapour is driven by a chemical potential gradient. It has a wide potential application. MD-based techniques have been drawing scientific and commercial attention due to
247
their ecological benefit and low capital and operation costs. It could be used for desalination purification and solution concentration by means of low grade (low potential) heat. The MD-based techniques have a wide potential interest and prospects especially for the Gulf region. It could be used for solving some ecological and industrial problems. Potential application of the MD method has been discussed by many authors [l-8]. Lack of a good model and calculation methods have hampered the practical implementation of MD. Both concentration polarisation and temperature polarisation should be taken into consideration in modelling the MD process. It is the combined phenomena that determine the driving force of the MD process, that in turn determines the transmembrane flux. The underlying concepts of this paper were adopted from [l-8]. The previous paper of our research team [9] proposed the submodel and calculated a technique for the TP-phenomenon influence on the MD process. This paper describes modelling of combined temperature and concentration polarisation (TCP) phenomena in a symmetric channel of plate-and-frame unit for the MD process.
MODELLING
Using mass-and energy balance equations for hydrodynamic, temperature and concentration boundary layers, we obtained the temperature and concentration distribution along the membrane surface. The main heat and transport constituents can be seen from Fig. 2. The model consists of some submodels describing mass- and heat-transfer constituents. The submodels describe transfer phenomena in both channels and in membrane space. For the conjugate solution to be obtained the hydraulic, temperature and concentration submodels should be introduced into the comprehensive model. Mutual energy influence between flow both in hot and in cool channels should also be taken into account.
MAIN ASSUMPTIONS OF THE MODEL
The symmetric plate-and-frame channel of the MD unit is taken into consideration. Balance equations were written for the control volume. (elementary parallel-piped). The intersection line of the control volume with the figure plane is shown in Fig. 1. Both “heat” and “cool” channel can be represented as conjugate system of elementary volumes. The intersection line of the elementary foundation with the figure plane is the section "abed" .
248
I d
2 .P.
..I
c
Fig. 1. Control volume of the MD channels.
There is no mass- and heat-transfer across symmetry planes. The intersections of the symmetry plane with the figure plane are shown by segment “ad” “bc” (Fig. 2). The model is based on the next assumption: 1. There are incompressible, laminar, continuous flows both in “hot” and in “cool’ channels. 2. Velocity, temperature and concentration variation within hydrodynamic, temperature and concentration boundary layers are taken into consideration in the model. 3. There is a concave meniscus from both ends of membrane pores. Liquid evaporates from the hot membrane side meniscus and condensates at the cool membrane side. 4. There are water-vapour air mixtures within the free porous space in the membrane. 5. In Fig. 2 the first interval ranges from Z=O to Z=6,; the second from Z=8, to Z=6,; the third from Z=6, to Z=A,=h,. The first interval belongs to concentration, temperature and hydraulic boundary layer. In the first interval there is Z-variation of temperature, velocity and concentration. The second interval lies between concentration boundary layer and temperature layer boundary. Within the second interval there is Z-variation of both temperature and velocity while concentration is constant. The third interval lies between temperature boundary layer and hydrodynamic boundary layer. Within the third interval only velocity varies while temperature and concentration remain constant. 6. For solving the equations the transfer from physical distance coordinate-Z to dimensionless distance coordinates has been made. The first dimensionless coordinate r=z/6, is used to describe the velocity profile. The velocity profile is approximated by the function.
249
.
Fig. 2. Membrane distillation process. The main heat and mass-flux constituents: channel; 2, cool channel; m, membrane.
1, hot
(1) The velocity variation occurs within the viscous boundary layer 6, (Fig. 2). 7. The second dimensionless coordinate 8 =q (P,)’ is used to describe the temperature profile T=f(8). The scale unit of B-coordinate is temperature boundary layer thickness l/(P,.)’ in T-coordinate (Fig. 3). The temperature profile is approximated as follows:
Fig. 3. Concentration, temperature, and hydrodynamic boundary layers in the original and in the dimensionless coordinate systems.
43- f1d.f= sin 4
-
GM
(5” 1
(2)
8. The third dimensionless coordinate 4 = (P,*)1’31(P,) ’ is used to describe the concentration profile C=fl$). The scale unit of the &coordinate is concentration boundary layer thickness (P,)“I(P,*)1’3 in the previous 8 coordinate (Fig. 3). The concentration profile is approximated as follows:
X4- xl = e44 %M
-
(3)
3
9. The relationship between temperature, concentration boundary layers and viscous boundary layer was adopted from [lo]
(4)
251 THE LONGITUDINAL
HEAT CONSTlTUENTS
e,
lo,,gconv
The convective heat transfer expression across the entrance control plane “ap” (Fig, 1) was presented in [9]. There are two longitudinal subconstituents (the first one within temperature boundary layer z < S,, the second one-between temperature- and hydrodynamic layer boundary (6, < z < 6,). The equation for the longitudinal convective heat constituent Q110ng,.onV was written as follows:
Qllullgumv =
z=8wpwzIzdF = pcbh,
/
WI
zi()
THE LONGITUDINAL
MASS CONSTITUENTS
G, l0ngconv
a similar approach was used as for For expressing the G1 l,,ng,OnV
Q 1longconvin [9]). Convective mass-transfer through entrance plane “ap” (Fig. 1) can be expressed as follows: G llonsconv =
sz=4v
wz xz dF = ~z~wZ~zdF
z&)
+ s,“‘“‘”wzxzdF I
(a)
The first term on the right hand of Eqn. (6) means longitudinal mass constituent within concentration boundary layer (from z=O to z=&. It can be determined as follows: G llongconv =
s‘=” z_o
wz xz dF
(7)
For the solution of the Eqn. (7) to be obtained, the third dimensionless coordinate-+ was used where
(8)
252
5 = x1
+
(xIM
-
x1
)e
a6
(9)
dF = b dZ = b h,d e/(P;)”
{x1 + (xIy
- x1
(10)
bh,W
(11)
)eao> P,’ l/3
11
Solving Eqn. (ll), the third dimensionless coordinate-$ was used. The second term on the right-hand side of Eqn. (6) means longitudinal mass constituents-G 1longconv** between concentration boundary layer (2=&J and the boundary of hydrodynamic layer (z=BW=hr)
G lag+*
=
s
z=aww,xJybcdF I
&j
(12)
where (13)
dF = b h, cf4 /(P;)”
(14)
while %=x1 const. Having integrated Eqns. (11) and (12), an equation for longitudinal mass constituent Gllongconvwas obtained.
253 THE CROSS-CHANNEL
HEAT CONSTITUENT
There are some cross- channel heat constituents towards and opposite membrane surface direction. They are: heat conductivity constituent (Q,x>; heat-convective constituent (G IconV);heat consitituents by means of diffusive mass fluxes (Q&. They can be expressed as follows:
Q 1A
=
Qldif
-
=
L
(5
D
(16)
-hf)tifN
X1M --+ %
(
CROSS-CHANNEL
(c4t*
-c&t,,)&
I
MASS CONSTITUTENT-
There are convective G, coIIvCmM and diffusive G, diffcrosssubconstituents of mass flux. G lcmlv-salt G lamv-4
= Xl
=
(Pl
&
&
-
3)
(19)
(20)
wif
wx,, - x1)
G ldif-salt
= -
(21)
fM 8,
GIds-4
=
D
($4
-
x1df4)
fM/ ax
The result cross-channel mass-constituent G,, cToBs can be expressed as follows: G 1CQUW
=
%cimlt
+G lamv-sq - G,,,
+ G,,,
(23)
254 TRANSMEMBRANE MASS CONSTlTUENTS
Assuming the membrane to be impermeable to salt, in this case the transmembrane mass flux consist only of water. There are two main mechanisms for porous space transport. They are convective and diffusive. Convective mass flux through membrane porous space GMconV.Depending upon the relationship between the mean free molecular path and the pore size, two main diffusive models can be used. (the molecular diffusion model and the Knudsen diffusion model). In some cases there are both mechanisms at the same time. Diffusion coefficient for complex transfer D*
Determination of diffusion coeffkients was given in [ 111. The diffusion constituent of transmembrane mass flux GMdifcan be written as: MD’
G Mdif
=
RT
(P,, - P2M) XAM
Taking into account the Claudius-Clapeyron Eqn. (26) [lo], the temperature difference (AT = Tt, - AT,,) can be introduced instead of the pressure difference (AI’ = P,, - P&. Convective transmembrane mass flux can be expressed using the Pouseille equation. AP = PA IM(t,,
GMcalv =
AP bA,
(26)
- t&/RT2
$I2 @.f.
&pap
For the AF to be expressed through AT in (27), the Claudius-Clapeyron Eqn. (26) is used. The resulting transmembrane mass constituent GM, results as follows:
G MC
=
%onv +%if
(28)
255 TRANSMEMBRANE
HEAT CONSTITUENT
The resultant transmembrane heat-constituent consists of the following subconstituents: heat conductivity flux through membrane matrix QMh; heat conductivity of water-air mixture in porous space QMxag;convective heat flux through porous space aq. vapour QMnVq;comprehensive diffusive heat flux through porous space QMdSq. These subconstitutuents were described in detail [9].
QM=+¶ =
MD* RP
%f -p2M) fM e I&_ xAM
(32)
The resultant transmembrane heat constituent QMc is summarized from the following subconstituents: (33)
LONGITUDINAL
FLUX VELOCITY AND TRANSMEMBRANE
FLUX
Longitudinal velocity is varied along the channels because of the transmembrane flux. Digital value at each length interval was determined from the mass balance equation.
For the comprehensive model equations to be solved the expression for the transmembrane water flux VI should be evaluated. It was obtained on the base of the mass-balance equation for water in membrane and in channels (%mas aq = GMcq) Eqns. (28) and (23).
256
0 START
TlMa=TlMa(min
initial) 1
t Xltla=XlHa~min
Qlconv
%?l~(eq.la);
Glconv(l9,20);Gldif QZa(eq.7.8.9) .
initial)
(eq.17); aq(21);Gldif
Qldifteq.18) salt(22);GlZ(23)
TZa(vo~.calc)=f
TPHasf(TlMa;PlZ) Q2along T2a
Ieq-29-33)
convteq.
1s)
(voI)H(Q2along;T2Ma)
I
0 END
Fig. 4. Flowsheet of the computer model.
(eq.15)
257 CALCULATION TECHNIQUE
The calculation technique was elaborated on the basis of the described model (Eqns. (l-34)). For solving the system equations, the following is to be used: l l l
l
the initial hydrodynamic W, (inlet),IV, (inlet); the initial temperature and concentration Tt, T2 (inlet; XI, X, (inlet); channels geometry: length L; width b; heights 2Ir1, 2h2 membrane parameters: pore size RPr; quantity of open pores per unit square n; membrane thickness Am; thermal conductivity of membrane material h; parameter of 1iquid media: density p; viscosity p; thermal conductivity h; thermal capacity C.
The filtration length L is subdivided into equal length intervals 1= (L/n). The calculation on each length interval is carried out by a technique of sequential approximation. At first one adopts a digital value of the membrane surface temperature TIMaand concentration XIMa and iterate to the temperature in volume of “cool” channel T2ccdcjto Tzcfactj.The calculation of surface hot temperature TIMais carried till TzMatcalcl = T2,,~f8cq. Having calculated the longitudinal heat flux in exit section (QtJ, the hot surface membrane temperature (TIM2 in exit section was calculated. The sequential calculation of digital values at the next length intervals was carried out, having calculated the first interval values. All exit values of the foregoing length interval (values at the outlet section of the interval) are adopted to entrance values of the next length interval (values of the inlet section of the interval). The flowsheet of computer model is shown in Fig. 4.
CONCLUSIONS
The model is used for analysis and simulation of the membrane distillation process in semipermeable plate-and-frame channel. The model enables determination of the size of the model, the physical properties of the fluid, the temperature and concentration in both channels.
258 REFERENCES 1 2 3 4 5 6 7 8 9
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