Diffusive Mass Transport in Periodic Boundary Conditions

International Journal of Engineering Practical Research (IJEPR) Volume 3 Issue 3, August 2014 www.seipub.org/ijepr doi: 10.14355/ijepr.2014.0303.01

Diffusive Mass Transport in Periodic Boundary Conditions A. F. Miguel*1 Geophysics Centre of Evora, University of Evora, Rua Romao Ramalho 59, 7000‐671 Evora, Portugal *1

afm@uevora.pt

Received 22th Jan. 2013; Accepted 19th Feb. 2013; Published 12th Aug. 2014 © 2014 Science and Engineering Publishing Company

Abstract Many problems of practical importance related with mass transport in finite porous materials take place This paper presents approximate analytical solutions for under periodic boundary conditions being the period mass transport in porous materials under periodic boundary being of few minutes to several months and years conditions. The influence of mass concentration gradients and temperature gradients (Soret effect) on mass flux is accounted. The significance of this study is double‐fold: the solutions provide an approach not only to predict mass and temperature distribution but also to evaluate the importance of the transfer coefficients in mass transport analysis; and the solutions serve to gauge the accuracy of any numerical simulation methods that can be developed for more complex cases.

Keywords Mass Transport; Analytical Solutions; Porous Materials

Introduction Mass transfer through a porous material has attracted the attention of several authors in view of potential applications. Such transfer is frequently encountered in filtration, biomaterials, soil sciences, etc. (Nield and Bejan, 2006; Chaves et al., 2005; Eaton et al., 1991).

Theoretically, it can be approached using a double‐ diffusive convection model (Nield and Bejan, 2006).

This model with the appropriated boundary conditions is usually solved using numerical techniques (see for example, Eaton et al. 1991), even in the simplest cases. There are important reasons to search for analytical solutions. Among others, analytical solutions give more physical insight into the nature of processes and are not dependent of the so‐ called numerical errors (Mikhailov, 1976; Minkowycz et al, 2006). Analytical solutions have been obtained for a variety of simple cases, mainly assuming an infinite or a semi‐infinite porous material (see for example, Carslaw and Jaeger, 1959; Segol, 1993). The state‐of‐art has been summarized in the book of Nield and Bejan (2006).

(Nield and Bejan, 2006). In the present investigation, approximate analytical solutions to mass transport equations in porous materials with periodic boundary conditions are obtained. As the boundary conditions are periodic, the steady solutions are also periodic and were Fourier analysed.

Formulation Consider that the mass flux through a porous material is due to mass concentration gradients and temperature gradients (Soret effect). According to Nield and Bejan (2006), the mass, momentum and energy conservation equations are:

t+U+J=0 (1) J=‐D‐DTT (2) fCf+ (1-smCsmT/t+ (fCfuT) + (effT) =0 (3) where U is the fluid velocity, T is the temperature, t is the time,  is the density,  is the porosity,  is the mass, D is the mass diffusivity due to mass concentration gradients, DT is the Soret coefficient of the porous material, C is the specific heat and  is the thermal conductivity. The subscript f means fluid, sm solid matrix and eff effective.

Assuming that the moisture moves along the x‐ direction only, substitution of Eq. (2) in (1) leads to

/t+u(/x)= (D /x)(/x)+(DT/x) (T/x)+D(/x2)+DT(T/x2) (4) Where u is the component of the vector velocity in the x‐direction.

If the enthalpy flow is neglected compared with energy storage, and the effective thermal conductivity

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is constant, Eq. (3) becomes

T/t=k(/x2) (5)

{[0/((Pe/2)2+n2)]exp(Pe/2){n[(Pe/2)cos(n)+n sin(n)] +(Pe/2)[(Pe/2) sin(n)‐n cos(n)]}} (11) where Pe is the Peclet number (uL/D)

with k=eff /fCf+(1-smCsm Equations (4) and (5) describe the one‐dimensional mass and energy fluxes through the porous material.

The quantity n is a positive root that satisfies the transcendental equation [n2‐(Pe/2)2] tan(n )‐(Pe/)n=0 (12)

Mass Transfer in a Porous Material with Constant Diffusivity Coefficients

The values of these roots can be found for example in Carslaw and Jaeger (1959).

Consider a finite porous material with thickness.

Assuming that the medium is isotropic and the diffusivity coefficients (D and DT) and the temperature gradient through the porous material are constants (physical assumptions), Eq. (4) yields

Mass Transfer in a Porous Material with No Constant Diffusivity Coefficients under Periodic Boundary Conditions

/t+u(/x)=D(/x2) (6)

According to Bear (1972) and Miguel (2000) D and DT could be dependent on the concentration and temperature, so Eq. (4) can be rewritten as

/t+u (/x) = D (/x2) +d1u (/x) +

The initial and boundary conditions are

(t=0)=o

0x(7.1) (x=0)=(D/u)(/x)(x=0)(7.2) (x=)=(D/u)(/x)(x=)(7.3) In order to find the appropriated solution for this problem, we now recall to the transformation of coordinate (Yosida, 1991)

=exp[(u/2D)x-(u/2D)2 Dt]

+d2(/x)2+(DT/)(T/x)2+DT(T/x2)

With

Let us to consider that the porous material is under periodic boundary conditions

(x=0)=aver,0+0* sin t (x=)=aver,+* sin (t+, ) T(x=0)=Taver,0+T0* sin (t+T,0) T(x=)=Taver,+T* sin (t+T, )

Applying this transformation to both equations leads us to

in which the initial and boundary conditions are given by

(t=0)=o exp(-u/2D)0x

(10.1)

 (x=0)=(2D/u)(/x)(x=0)(10.2)  (x=)=-(2D/u)(/x)(x=)(10.3)

This problem given by Eqs. (9)and (10) can be solved through separation of variables (Carslaw and Jaeger, 1959) or using a methodology based on the Laplace transform with respect to time (Miguel, 2014). The general solution can be written as 

θ= 

n=1

2exp (-(Pe/2ξ) 2 -α 2n )Dθ t  α n cos(α nx)+(Pe/2ξ) 2sin(α nx)  ( (Pe/2ξ)2 +α 2n )ξ+2(Pe/2ξ)2



 0 exp[( Pe / 2 )  x ][ n cos ( n x)  ( Pe / 2 ) sin( n x)]dx

Let us to assume that the time of response for the temperature is much shorter than for the mass concentration, i.e., the thermal diffusivity is much larger than the mass diffusivity (physical assumption).

Notice that the majority of situations with practical interest obey to this criterion. Thus, the temperature gradient can be considered constant but periodic (Eaton, 1991), and can be written as

T/xT/x=-1 {Taver,+T* sin (t+T,)--[Taver,0+T0*

sin (t+T,0)]}=-1 (Taver+Tsin sin t+ +Tcos cos t)=constant (15) with

Taver= Taver,‐Taver,0 Tsin= T* cos T,‐T0* cos T,0 Tcos= T* sin T,‐T0* sin T,0

2exp (-(Pe / 2 ) 2  α 2n )Dθ t 

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(14.1) (14.2) (14.3) (14.4)

Where  is the angular frequency and  the phase displacement. The subscript avers means time average.

0

2   n cos(α nx)  (Pe / 2 ) sin(α nx)    2 2 2 n 1 ( (Pe / 2 )  α n )  2(Pe / 2 )

d1u= [(D/) + (DT/)]T/x d2=D/

being the fluid velocity (u) constant.

(8)

/t=D(/x2) (9)

(13)

Thus,

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T/x2=0 (16) Use of this result in Eq. (13) provide us

/t=D(/x2)+d1u(/x)+d2(/x)2+ (17) +(DT/)(T/x)2

and the respective solution can be presented as (Yosida, 1991)

aver= 1x2+2x+aver,0

(25)

with

1=-1(-1aver-2)

with

0.5d222+(2-1Daver-0.5-1d2aver)2= = aver(2-1Daver+ d1u+-1d2aver)+ daver3 aver=aver,-aver,0

d1u=d1-u

-1

As the material is under periodic boundary conditions, the steady solution is also periodic and the Fourier analysis can be used. Considering only the first harmonic

aver+sin sin t+cos cos t

(18)

This allows us to express /t, /x and /x as

where Daver(/2) is the value of Dave is the middle of the porous material (x=/2).

The time dependent part of Eq. (23) can be divided in a part corresponding to the sin terms and another corresponding to the cos terms

/t=sin cos t - cos sin t (19) /x=aver/x+(sin/x)sint+(cos/x)cost (20) /x=aver/x +(sin/x) sin t+ +(cos/x) cos t(21)

{Daver(/2)(sin/x)+[daver1+2d2(aver/x)](sin/x)} sint=-[dsin1(aver/x)+dsin3+cos]sin t (26) {Daver(/2)(cos/x)+[daver1+2d2(aver/x)](cos/x)} cos t=-[dcos1(aver/x)+dcos3-sin]cos t (27)

The conductivity D is also periodic and the Fourier analysis can be used. Expanding D to the first harmonic we obtain

In order to develop the solution for the previous equations we need to consider that

DDaver+Dsin sin t+Dcos cos t (22) Where Daver is the mean of D over one time period.

Substitution of Eqs. (19) to (22) into Eq. (17) lead to

sin cos t - cos sin t= Daver(/2)[aver /x +

+(sin/x) sin t+(cos/x) cos t]+ +0.5Dsin(sin/x) + 0.5Dcos(cos/x) + +Dsin(aver/x2) sin t + Dcos(aver/x2) cos t+ +daver1 (aver /x)+0.5[dsin1(sin/x)+dcos1(cos/x)] + +[daver1(sin/x)+dsin1(aver/x)] sin t + +[daver1(cos/x) + dcos1(aver/x)] cos t + +d2{(aver/x)2 + 0.5[(sin/x)2+(cos/x)2 ]+ +2(aver/x)(sin/x)sint+ 2(aver/x)(cos/x)cost }+ daver3 + dsin3 sint+ dcos3 cos t (23) with

daver3=-2(DT/)[Taver2+0.5(Tsin2+Tcos2)] dsin3=2-2(DT/)(TaverTsin) dcos3=2-2(DT/)(TaverTcos) An examination of Eq. (22) indicates that the first and second derivatives of sin and cos in order to the spatial coordinate x, in the time independent part of the equation, can be neglected because they are very small comparing with the remaining terms.

Consequently the time independent part of Eq. (23) can be written as 

Daver(/2) ( aver /x  +daver1(aver /x)+ + d2(aver/x)2+daver3=0 

(24)

aver/xaver/

(28)

After a convenient mathematical manipulation, cos and sin are discarded from Eqs. (26) and (27), being the solutions of the resulting differential equations (Yosida, 1991)

sin=1exp[(1+i)x]+2exp[(1i)x]+

+3exp[(2+i)x]+4exp[(2-i)x]+sin cos=-i1exp[(1+i)x]+i2exp[(1-i)x]+ +i3exp[(2+i)x]-i4exp[(2-i)x]-cos

(29) (30)

with

1=(0.5/Daver(/2)){{[daver1+2d2(aver/)]4+

+(4Daver(/2))2}1/4cos(0.5)-daver1-2d2(aver/)} 2=-(0.5/Daver(/2)){{[daver1+2d2(aver/)]4+ +(4Daver(/2))2}1/4cos(0.5)+daver1+2d2(aver/)} =(0.5/Daver(/2)) {[daver1+2d2(aver/)]4+ +(4Daver(/2))2}1/4sin(0.5) =arctg{4Daver(/2)/[daver1+2d2(aver/)]}2 sin=-1[dcos1(aver/)+dcos3] cos=-1[dsin1(aver/)+dsin3] The constants 1, 2, 3 and 4 can be obtained from the boundary conditions. According to Eqs. (14.1) and (14.2) the boundary conditions for the time dependent part of the solution are

sin(x=0)=0* and cos(x=0)=0 (31) sin(x=)=* cos (, ) and cos(x=)=* sin (, ) (32) Substitution of Eqs. (31) and (32) in Eqs. (29) and (30) allow us the computation of the above‐mentioned  constants.

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Introducing Eqs. (25), (29) and (30) into Eq. (18) leads to the following equation for the mass distribution in the porous material

=1x2+2x+sin sin t+cos cos t+aver,0

(33)

Equation (33) expresses the solution of Eq. (13) for periodic boundary conditions. The existence of this solution allows describing analytically the distribution of the mass concentration within the porous media.

A similar procedure can be followed to obtain the solution of Eq. (15). The solutions is T=(Taver+Tsin sin t+Tcos cos t)(x/)+ Taver,0+T0* sin (t+T,0) (34) Equation (34) allows describing the temperature distribution within the porous material.

Verification and Results The analytical solutions for periodic boundary conditions are based on few approximations (e.g., harmonics higher than first order are neglected). This implicates that the predictive accuracy of these solutions should be tested. This testing is made by comparing the data generated by the solutions against experimental (Künzel, 1992, 1995) and numerical simulations (van der Braak et al., 1997) on a 12 cm long concrete plate (D=4.7x10‐10m2/s; DT=2.3x10‐12m2/sK; kt=182) exposed to ambient conditions during 15 months. The results of this comparison are plotted in Figs. 1 and 2.

The plots indicate that both predictions made, with the analytical and the numerical solutions are within the accuracy interval of the measurements. In addition, the analytical and numerical solutions show a good agreement. Consequently, we can conclude that the approximations made are suitable and don’t affect the predictive accuracy of the solutions.

FIG. 2 Temperature distribution in a cellular concrete plate after 2 and 6 months: ____ predicted by Eq. (34), _ _ _ _ _ predicted numerically, measured after 2 (O) and 6 (x) months

Concluding Remarks One of the main interests of the analytical solutions is that they provide more physical insight into the nature of process at hand. They are also fast and easy to evaluate. In this paper, approximate analytical solutions to the mass concentration and heat transport equations in porous materials under periodic boundary conditions are presented. It is shown that these solutions are well suited for materials in which the time of response for temperature is much shorter than for the moisture content. This criterion applies to the majority of situations with practical interest.

The importance of the analytical solutions presented here, it is not only predicting mass and temperature distributions in porous materials. In‐situo or even laboratory measurements of the mass diffusivity due to mass concentration gradients and the Soret coefficient of the porous material are difficult and takes a long time. Eqs. (33)and (34) may help obtaining these coefficients, if the mass and temperature distributions are provided.

REFERENCES

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Eaton, R. R. et al., eds. Multiphase transport in porous media. FED‐122/HTD‐186, 1991.

Künzel, H. M. Simultaneous Heat and Moisture Transport in FIG. 1 Moisture distribution in a cellular concrete plate after 2 and 6 months: ____ predicted by Eq. (33), _ _ _ _ _ predicted numerically, measured after 2 (O) and 6 (x) months

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Künzel, H.M. Connection between liquid water and vapour transport in porous media and its consequences for heat and moisture transfer models. Beitrag zur Tagung des EG‐Projektes SCIENCE, London, 1992.

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Antonio F. Miguel received his Ph.D in Applied Physics from the WUR, the Netherlands, in 1998. He has been Associate Professor of Physics at University of Evora (Portugal) since 2003. His work covers a diversity of topics in Energy, Fluid Flow, and Generation of Design in Flow Systems. He has served in the advisory boards of more than 15 international conferences. He has authored or co‐ authored of over 50 refereed journal papers.

Mikhailov, M. D. “Exact solution of temperature and moisture distribution in a porous half‐space with moving evaporation front.” Int. Journal Heat Mass Transfer 19 (1976) 651‐655.

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