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Dispersion in Composite Porous Medium with Homogeneous andHeterogeneous Chemical Reactions

J. Prathap Kumar, J.C. Umavathi, and Shivakumar MadhavaraoDepartment of Mathematics, Gulbarga University, Karnataka, India

The effect of homogeneous and heterogeneous reactions on the dispersion ofa solute in a composite porous medium between two parallel plates is studied. Thesolution approach suggested by Taylor [2] is generalized for the present model. TheBrinkman model is used to define the flow through the porous medium. The fluids inboth the regions of the parallel-plate channel are incompressible and transport prop-erties are assumed to be constant. The closed-form solutions are obtained in both theregions of the channel. The results are tabulated for various values of viscosity ratio,pressure gradient, and porous parameter on the volumetric flow rate and the effectiveTaylor dispersion coefficient. It is found that for a homogeneous chemical reaction,the effective Taylor dispersion coefficient decreases as the reaction rate parameter andporous parameter increase. The validity of the results obtained for a composite porousmedium is compared with the available porous medium results for a one-fluid modeland the values tallied for the small values of porous parameter. Also a two-fluid modelin the absence of a porous matrix is compared with the available one-fluid model andgood agreement is found. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res, 40(7),608–640, 2011; Published online 26 July 2011 in Wiley Online Library (wileyonlineli-brary.com/journal/htj). DOI 10.1002/htj.20364

Key words: Taylor dispersion, immiscible fluids, horizontal channel, porousmedium

1. Introduction

The dispersion of soluble matter in an incompressible viscous fluid flowing in a circular pipehas been discussed by Taylor [2–4] wherein it is observed that the solute is dispersed with an apparentdiffusion coefficient R2Ux

2 / 48D, where R is the radius of the pipe, and Ux and D are the average

velocity and molecular diffusion coefficient, respectively. Aris [5] extended the above analysis withfewer departures from Taylor. The work was later extended to include dispersion in a non-Newtonianfluid by Fan and Hwang [6]. However, the above investigations do not take into account the chemicalreaction of a solute during the process of dispersion in a solvent. We see a wide variety of situationsin chemical engineering wherein diffusion of a solute takes place with a simultaneous chemicalreaction, namely, hydrolysis, gas absorption in an agitated tank, and ester experiments [7]. The

© 2011 Wiley Periodicals, Inc.

Heat Transfer—Asian Research, 40 (7), 2011

Contract grant sponsor: UGC-New Delhi for the financial support under UGC-Major Research Project.

608

combined first-order heterogeneous and homogeneous reactions on the dispersion of a solute havebeen studied by Walker [8] and Solomon and Hudson [9].

However, the diffusion process, combined with homogeneous and heterogeneous chemicalreactions of a solute in a porous medium, is also important in chemical engineering. The diffusioninside the tortuous void passages in the medium is pointed out by Bird et al. [7], p. 542–546). Theyalso pointed out that such study is important in a porous catalyst pellet, which is used in fixed-bed,axial-flow reactors. They described the “average” diffusion of the chemical species in terms of aneffective diffusion coefficient both for spherical porous catalyst and for non-spherical catalystparticles. Interest in dispersion in porous media has also resulted from seawater intrusions into coastalaquifers, seepage from canals and streams into and through aquifers, and the deliberate release ofherbicides into canals to kill weeds.

These problems of dispersion have been receiving considerable attention from chemical,environmental, and petroleum engineers, hydrologists, mathematicians, and soil scientists. Most ofthe works reveal common assumptions of homogeneous porous media with constant porosity, steadyseepage flow velocity, and constant dispersion coefficient. For such assumptions Ebach and White[10] studied the longitudinal dispersion problem for an input concentration that varies periodicallywith time and Ogata [11] for a constant input concentration. Hoopes and Harleman [12] studied theproblem of dispersion in radial flow from fully penetrating, homogeneous, and isotropic non-adsorb-ing confined aquifer. Bruce and Street [13] considered both longitudinal and lateral dispersion withinsemi-infinite nonadsorbing porous media in a study of unidirectional flow field for a constant inputconcentration. Marino [14] considered the input concentration varying exponentially with time.Al-Niami and Rushton [15] and Marino [16] studied the analysis of flow against dispersion in porousmedia.

Fluid flow in a system containing simultaneously a fluid reservoir and a porous mediumsaturated with fluid is of great mathematical and physical interest. More specifically the existence ofa fluid layer adjacent to a layer of fluid, saturated porous medium is a common occurrence in bothgeophysical and engineering environments. Extensive work has been performed in recent years toexamine the effects of different geometric, fluid and solid matrix parameters on the natural convectionin domains partially filled with porous materials [17–22]. Neale and Nader [23] proposed the use ofthe Brinkman extended Darcy equations to account for the macroscopic viscous stress in the porousmedium and suggested that, at the interface, the macroscopic viscous shear stress in the porousmedium is equal to the shear stress on the fluid side. Malashetty et al. [24], Umavathi et al. [25], andPrathap Kumar et al. [26] studied flow and heat transfer of fluid saturated porous medium for two-or three-fluid models. The literature on hydrodynamic dispersion in porous medium is very sparse inspite of its versatile applications in many branches of science, engineering, and technology.

Based on our review of the current literature, it is evident that very few studies are availableon the dispersion in porous medium for two-fluid models. Looking at the various applications oftwo-fluid model as mentioned above, the object of this work is to study the effect of homogeneousand heterogeneous chemical reactions on the dispersion of solute containing porous and fluid layerusing Taylor’s model.

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Nomenclature

Ci: concentration of the solute Di: molecular diffusion coefficientD: ratio of molecular diffusion coefficient (D2 / D1)dPi / dXi: pressure gradienth: distance between the platesKi: first-order reaction rate constant L: typical length along the flow directionm: viscosity ratio (µ2 / µ1)n: density ratio (ρ1 / ρ2)pi: non-dimensional pressure gradientQi: volumetric flow rateUi: velocityu_

i: non-dimensional average velocity

Greek Symbols

αi, βi: dimensionless reaction rate parametersκ: permeability of the porous mediumη: dimensionless lengthµi: dynamic viscosityρi: density of the fluidσ: porous parameter (h / √κ)

Subscripts

i = 1, 2 where 1, 2 are quantities for region-1 and region-2, respectively

2. Mathematical Formulation

The physical configuration considered in this study is shown in Fig. 1. Consider the laminarflow of two immiscible fluids between two parallel plates distant h apart, taking the X-axis along themid-section of the channel and Y-axis perpendicular to the walls. Region-1 (−h ≤ Y ≤ 0) is filled with

Fig. 1. Physical configuration.

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fluid-saturated porous medium of density ρ1, viscosity µ1, permeability κ under a uniform pressuregradient dP1 / dX, whereas Region-2 (0 ≤ Y ≤ h) is filled with another viscous fluid of density ρ2,viscosity µ2 under a uniform pressure gradient dP2 / dX. The fluids in both regions are Newtonianfluids.

It is assumed that the flow is steady, laminar, fully developed, and that fluid properties areconstant. The flow in both regions is assumed to be driven by a common constant pressure gradient.Under these assumptions, the governing equations of motion for incompressible fluids are [23]

Region-1

Region-2

where Ui is the X-component of fluid velocity and Pi is the pressure. The subscripts 1 and 2 denotethe values for Region-1 and Region-2, respectively.

The boundary conditions on velocity are no-slip conditions requiring that the velocity mustvanish at the walls. In addition, continuity of velocity and shear stress at the interface is assumed.With these assumptions, the boundary and interface conditions on velocity become

Using the non-dimensional parameters

Equations (1) to (3) become

(3)

(4)

(2)

(5)

(6)

(1)

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where p1 = dp1∗

/ dx, p2 = dp2∗

/ dx, m = µ2 / µ1, and n = ρ1 / ρ2.

Solutions of Eqs. (5) and (6) are

where a1, a2, a3, and a4 are integrating constants that are evaluated using boundary and interfaceconditions as given in Eq. (7).

From Eqs. (8) and (9) the average velocities become

Case 1: Diffusion with a homogeneous first-order chemical reaction

We assume that a solute diffuses and simultaneously undergoes a first-order irreversiblechemical reaction in the liquid under isothermal conditions. The equation for the concentration C1 ofthe solute for Region-1 satisfies

Similarly, the equation for the concentration C2 of the solute for Region-2 satisfies

in which D1 and D2 are the molecular diffusion coefficients (assumed constant) for Region-1 andRegion-2, respectively, and K1 and K2 are the first-order reaction rate constants. In deriving Eqs. (12)and (13), it is assumed that the solute is present in a small concentration, the last term −K1C1/molcm–3s–1 and −K2C2/mol cm–3s–1 represents the volume rate of disappearance of the solute due tochemical reaction. We now assume that

If we now consider convection across a plane moving with the mean speed of the flow, thenrelative to this plane the fluid velocities are given by

(8)

(7)

(9)

(10)

(11)

(12)

(13)

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Region-1

Region-2

where u_

is the sum of average velocities of Region-1 and Region-2.

Introducing the dimensionless quantities

and using Eq. (16), Eqs. (12) and (13) become

Region-1

Region-2

where L is the typical length along the flow direction. Following Taylor [2], we now assume thatpartial equilibrium is established in any cross-section of the channel so that the variations of C1 andC2 with η are calculated from Eqs. (17) and (18) as

Region-1

Region-2

where α1 = h√K1 / D1 and α2 = h√K2 / D2 .

To solve these equations we use the following two types of boundary conditions.

The first one is connected with an insulated type of boundary conditions, namely

which expresses the fact that the walls of the channel are impermeable. However, in many biologicalproblems the condition at the upper wall is conducting and the lower is insulating. In other words

(14)

(15)

(16)

(17)

(18)

(20)

(19)

(21)

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where the former represents the impermeable and the latter the permeable.

To find the exact solutions of Eqs. (19) and (20), we require two more interface conditionsalong with boundary conditions (21) and (22) for two different cases, which are given by

Case 1a: Concentration distribution with impermeable wall conditions

Equations (19) and (20) are solved exactly for C1 and C2 which are given by

Region-1

Region-2

where b1, b2, b3, and b4 are the integrating constants which are evaluated using boundary and interfaceconditions as defined in Eqs. (21) and (23). The expressions for C1 and C2 can also be written as

where

The volumetric flow rates at which the solute is transported across a section of the channel ofunit breadth Q1 (Region-1) and Q2 (Region-2) using Eqs. (14), (15) and (24), (25) are

Region-1

where

(26)

(22)

(23)

(24)

(25)

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Region-2

where

Following Taylor [2], we assume that the variations of C1 and C2 with η are small comparedwith those in the longitudinal direction, and if Cm1 and Cm2 are the mean concentration over a section,∂C1 / ∂ξ1 and ∂C2 / ∂ξ2 are indistinguishable from ∂Cm1 / ∂ξ1 and ∂Cm2 / ∂ξ2, respectively, so that Eqs.(26) and (27) may be written as

Region-1

Region-2

The fact that no material is lost in the process is expressed by the continuity equation for Cm1 andCm2, namely

Region-1

Region-2

Equations (28) and (29) using Eqs. (26) and (27) become

Region-1

Region-2

which are the equations governing the longitudinal dispersion,

(28)

(29)

(30)

(31)

(27)

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where

Values of Fii are computed for different values of dimensionless parameters such as viscosity ratiom, pressure gradients p1, p2, and por ous parameter σ for var iations of α1 and α2 are shown inTable 1. Volumetric flow rate is also computed for variations of viscosity ratio, pressure gradients,and porous parameter and is displayed in Fig. 2.

Case 1b: Concentration distribution with lower wall impermeable (insulating) and upperwall permeable (conducting) wall conditions

Equations (19) and (20) are solved exactly for C1 and C2 which are given by

Region-1

Region-2

where b1, b2, b3, and b4 are the integrating constants obtained using boundary and interface conditionsas defined in Eqs. (22) and (23).

The volumetric flow rates at which the solute is transported across a section of the channel ofunit breadth Q1 (Region-1) and Q2 (Region-2) and the evaluation of effective dispersion coefficientsFii are evaluated as explained in the above case and are shown in Table 3. Volumetric flow rate is alsocomputed for variations of viscosity ratio, pressure gradients, and porous parameter and is displayedin Fig. 3.

Case 1c: Diffusion with combined homogeneous and heterogeneous first-order chemical reac-tion

We now discuss the problem of diffusion in a channel with a first-order irreversible chemicalreaction taking place both in the bulk of the fluid as well as at the walls which are assumed to be

(32)

(33)

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catalytic. In this case the diffusion equations remain the same as defined in Eqs. (19) and (20) subjectto the dimensionless boundary and interface conditions as

where β1 = f1h and β2 = f2h are the heterogeneous reaction rate parameters corresponding to catalyticreaction at the walls.

The solutions of Eqs. (19) and (20) are

Region-1

Region-2

where b1, b2, b3, and b4 are the integrating constants obtained using boundary and interface conditionsas defined in Eq. (34).

The volumetric rates at which the solute is transported across a section of the channel of unitbreadth Q1 (Region-1) and Q2 (Region-2) and the evaluation of effective dispersion coefficients Fii

are evaluated as explained in Case 1a and tabulated in Table 5. Volumetric flow rate is also computedfor variations of viscosity ratio, pressure gradients, and porous parameter and is displayed in Fig. 3.

Case 2: The channel filled with porous matrix (one-fluid model)

To validate the results obtained for composite porous medium (two-fluid model), the problemis solved considering the channel filled with only a porous medium (one-fluid model) which wasstudied by Rudraiah and Ng [29].

The following are the solutions of one-fluid model obtained by Rudraiah and Ng [29].

The non-dimensional equation of motion is

along with boundary conditions

(34)

(35)

(36)

(38)

(37)

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The solution of Eq. (37) is

The average velocity is given by

The concentration equation for one-fluid model using Taylor [2] becomes

where

The solution of Eq. (39) is

where the constants b1 and b2 are evaluated using impermeable boundary conditions ∂C / ∂η = 0 atη = ±1.

The volumetric flow rate at which the solute is transported across a section of the channel ofunit breadth is given by

Following Taylor [2], we assume that the variations of C with η are small compared with thosein the longitudinal direction, and if Cm is the mean concentration over a section, ∂C / ∂ξ is indistin-guishable from ∂Cm / ∂ξ so that Eq. (41) may be written as

where

(40)

(41)

(42)

(43)

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and F(α, σ, p) represents the terms in square brackets of Eq. (43).

The fact that no material is lost in the process is expressed by the continuity equation for Cm,namely,

Equation (44) using Eq. (42) becomes

which is the equation governing the longitudinal dispersion. The values for velocity, concentration,mass flow rate, and the effective diffusion coefficient F(α, σ, p) are computed and are shown in Table7 for the two-fluid and one-fluid models considering σ = 0.01, p1 = p2 = 1, m = 1, and n = 1.

Case 3a: Effect of homogeneous and heterogeneous reactions on the dispersion of solute inthe absence of porous matrix (two-fluid model)

Since the results for a composite porous medium (two-fluid model) tally with Rudraiah andNg [29] (one-fluid model) only for small values of the porous parameter σ, we further justify ourresults by comparing them with the results obtained by Gupta and Gupta [1] for a one-fluid model inthe absence of porous parameter.

The solutions of Eqs. (5) and (6) using boundary and interface conditions (7) in the absenceof porous parameter σ become

The average velocities as defined in Eqs. (10) and (11) in the absence of a porous parameter become

The solutions of Eqs. (19) and (20) in the absence of a porous parameter σ yield

(45)

(44)

(48)

(49)

(46)

(47)

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where b1, b2, b3, and b4 are integrating constants to be evaluated using boundary and interfaceconditions as defined in Eqs. (21) and (23).

The volumetric rates at which the solute is transported across a section of the channel of unitbreadth Q1 (Region-1) and Q2 (Region-2) and the evaluation of effective dispersion coefficients Fii

are evaluated as explained in Case 1a in the absence of a porous parameter. The values ofFii(α1, α2, p1, p2, m, n) are computed for different values of the dimensionless reaction rate parametersαi, pi, and m and are shown in Table 8.

Case 3b: The channel filled with only viscous fluid (one-fluid model)

The following are the solutions of a one-fluid model obtained by Gupta and Gupta [1].

The non-dimensional equation of motion is

along with boundary conditions

The solution of Eq. (50) is

The average velocity is given by

The concentration equation for one-fluid model using Taylor [2] becomes

where ux = pη2

2 −

p6

.

The solution of Eq. (52) using boundary conditions ∂C / ∂η = 0 at η = ±1 is

The volumetric flow rate in which the solute is transported across a section of the channel ofunit breadth is

Comparing Eq. (54) with Fick’s law of diffusion, we find that the solute is dispersed relativeto a plane moving with the mean speed of the flow with an effective dispersion coefficient D∗ givenby D∗ = [(h2p2)/D]F(α)

(50)

(51)

(52)

(53)

(54)

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Values of F(α) are computed for different values of the dimensionless reaction rate parameterα and are shown in Table 8. When α → 0, Eq. (55) gives

so that the value for D*can be written as (h2p2 / D) (2 / 945) which agrees with the results of Wooding

[30] where p is the non-dimensional pressure gradient.

The solution for a heterogeneous chemical reaction is also found for the two-fluid and one-fluid models and the results are shown in Table 9.

All the constants appearing above are defined in the Appendix.

3. Results and Discussion

The problem concerned is with the longitudinal dispersion of a solute subject to moleculardiffusion when it is introduced into a channel for composite porous medium following Taylor’sdispersion model with a homogeneous and heterogeneous first-order chemical reaction.

In order to find out average velocity in both regions, no-slip conditions at the boundaries andcontinuity of velocity and shear stress is assumed at the interface. The closed form solutions areobtained for the concentrations in Region-1 and Region-2. The volumetric flow rates in both regionsof the channel are also found. The effective dispersion coefficient in each region is also evaluated andthe values are tabulated for variations of governing parameters. The homogeneous first-order chemicalreaction is analyzed for two types of boundary conditions. The first one is connected with the insulatedtype of boundary conditions and another condition is that with lower wall insulating and the upperwall conducting. The effective dispersion coefficient is also found with a heterogeneous first-orderchemical reaction. The physical parameters such as viscosity ratio and pressure gradients are fixed asone whereas the porous parameter is fixed as 0.01 except the varying parameters in all the tabulatedvalues.

Case 1: Diffusion with a homogeneous first-order chemical reaction

Case 1a: Concentration distribution with impermeable wall conditions

The effects of viscosity ratio, pressure gradients, and porous parameter on volumetric flowrate is shown in Fig. 2. It is seen that as the viscosity ratio and porous parameter σ increase volumetricflow rate increases in magnitude for small values of m, and σ remains almost constant for values ofviscosity ratio m greater than 1 and porous parameter σ greater than 8. Volumetric flow rate issymmetric for negative and positive values of pressure gradients and the optimum flow rate is attainedin the absence of pressure gradients.

(55)

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The effective dispersion coefficients F1(α1, α2) (Region-1) and F2(α1, α2) (Region-2) forvariations of viscosity ratio m, pressure gradient p, and porous parameter σ are shown in Table 1. Asthe reaction rate parameter α(= α1 = α2) increases, F1(α1, α2) and F2(α1, α2) decrease in both theregions for any value of viscosity ratio m, pressure gradient p, and porous parameter σ. The decreasein F(= F1 + F2) with an increase in α is natural based on physical grounds since an increase in α leadsto an increase in the number of molecules of solute undergoing chemical reaction, resulting in adecrease in Taylor’s dispersion coefficient. As m increases, the total effective dispersion coefficientdecreases for m ≤ 1 and increases for m ≥ 1. The total effective dispersion coefficient F = (F1 + F2)decreases as p increases for values of p < 0 and increases as p increases for values of p > 0. In theabsence of pressure gradients p the values of F are very small and are of order 10−5. It is observedthat the effective dispersion coefficient F decreases very rapidly with increasing porous parameter σin both regions, which is similar to the result obtained by Rudraiah and Ng [29]. This is due to thefact that as σ grows, the velocity profiles continue to flatten out and tend to plug flow as σ → ∞. Thisflattening effect is the cause for the decrease in F with an increase in σ.

To understand the nature of distribution of concentration for the real field, physical numbersfor porous parameter and diffusivity coefficients are chosen and shown in Table 2. The experimentalvalues of diffusion coefficients in gases at one atmosphere are chosen from Cussler [27]. It isinteresting to note that for any combination of gases, as the porous parameter σ increases, concentra-tion decreases in magnitude in both regions near the boundaries and increases at the interface. Thisis due to the fact that an increase in the grain size increases the permeability κ and hence porousparameter σ decreases, which in turn reduces the value of the diffusion coefficient (this behavior isin conformity with the experimental results of Harleman et al. [28] which results in the reduction ofconcentration. Two different fluids are chosen in each region, such as carbon dioxide and hydrogen,with both regions filled with air. Another combination of nitrogen and helium is chosen with both

Fig. 2. Volumetric flow rate Q versus viscosity m, porous parameter σ, and pressure gradient p.

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Table 1. Values of Effective Dispersion Coefficient for Variations of Viscosity Ratio, PressureGradient, and Porous Parameter for Homogeneous Impermeable Wall Conditions

623

regions filled with water. It is seen that the concentration is high in magnitude for the channel filledwith air compared to the channel filled with water in both regions for variations of diffusivitycoefficient.

Case 1b: Concentration distribution with lower wall impermeable (insulating) and upperwall permeable (conducting) wall conditions

The effect of viscosity ratio m, porous parameter σ, and pressure gradient p on the volumetricflow rate is depicted in Fig. 3. It is seen that as the viscosity ratio increases volumetric flow rateincreases in magnitude for small values of m. It remains almost constant for values of viscosity ratiom greater than 2. Volumetric flow rate is symmetric for negative and positive values of pressuregradients and the optimum flow rate is attained in the absence of pressure gradients. Volumetric flowrate decreases as porous parameter increases for small values of σ and remains almost constant forvalues of porous parameter σ greater than 10.

The effects of viscosity ratio m, porous parameter σ, and pressure gradient p on the effectivedispersion coefficient with variation of homogeneous chemical reaction rate parameter α are shownin Table 3. As the reaction rate parameter α increases, the effective dispersion coefficient decreasesin both regions for any values of m, p, and σ in both regions as seen in Table 3. Further as viscosityratio m increases the total effective dispersion coefficient F decreases in magnitude for m ≤ 1 andincreases for m ≥ 1. A similar effect is observed for variations of pressure gradients p on F. As theporous parameter σ increases the effective dispersion coefficient F increases as σ increases for smallvalues of σ whereas F decreases for large values of σ. It is clear that the effect of the upper wall being

Table 2. Values of Concentration for Impermeable Wall Conditions

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permeable is to increase the Taylor dispersion coefficient and hence to increase the transport of fluidin the viscous fluid region.

The effect of the porous parameter on the concentration for soil, silica grains, and wire crampsusing the values of diffusivity coefficients from experimental values is shown in Table 4. The effectof σ on the concentration is not effective when compared to impermeable wall conditions. The valuesof concentration are less for water when compared to air which is in contradiction with the resultobtained for impermeable wall conditions.

Case 1c: Diffusion with combined homogeneous and heterogeneous first-order chemical reac-tion

The effect of viscosity ratio, pressure gradient, and porous parameter on volumetric flow rateis shown in Fig. 4. It is seen that as the viscosity ratio and porous parameter σ increase volumetricflow rate increases in magnitude for small values of m, σ, and remains almost constant for values ofviscosity ratio m greater than 1 and porous parameter σ greater than 8. Volumetric flow rate issymmetric for negative and positive values of pressure gradients and the optimum flow rate is attainedin the absence of pressure gradients.

The effects of heterogeneous reaction rate parameter β for a fixed value of homogeneousreaction rate parameter β are tabulated in Table 5. From this it is clear that as in the case of the

Fig. 3. Volumetric flow rate Q versus viscosity ratio m, porous parameter σ, and pressure gradientp.

625

Table 3. Values of Effective Dispersion Coefficient for Variations of Viscosity Ratio, PressureGradients, and Porous Parameter for Homogeneous Insulating and Conducting Wall Conditions

626

Fig. 4. Volumetric flow rate Q versus viscosity ratio m, porous parameter σ, and pressure gradientp.

Table 4. Values of Concentration With Lower Wall Impermeable and Upper Wall Permeable WallConditions

627

homogeneous reaction, the increase in the wall catalytic parameter causes a decrease in the effectivedispersion coefficient for all values of viscosity ratio m, pressure gradient p, and porous parameter σ.Further as the viscosity ratio increases the total effective dispersion coefficient F decreases, whereasas the pressure gradient p increases F decreases for p ≤ 0 and increases for p ≥ 0. The effectivedispersion coefficient F decreases very rapidly as porous parameter σ increases, which is a similarresult as in the case of a homogeneous reaction. This result is similar to the result obtained by Rudraiahand Ng [29] for heterogeneous reactions.

Table 5. Values of Effective Diffusion Coefficient for Variations of Viscosity Ratio, Pressure Gradients, and Porous Parameter for Heterogeneous Wall Conditions

628

Case 2: The channel filled with a porous matrix (one-fluid model)

The problem of dispersion in porous media with and without chemical reaction was analyzedby Rudraiah and Ng [29]. In this problem the channel is filled with only a porous matrix. The velocitydistribution is evaluated for this model and shown in Table 7a. It is seen that for small values of porousparameter, the velocity distribution is the same in the porous region for the two-fluid and one-fluidmodels. As the porous parameter σ increases the values of velocity remain the same near the lowerwall but very near the interface and in the viscous fluid region. It is also observed that the velocitydecreases as the porous parameter σ increases. For a large porous parameter, the frictional dragresistance against the flow in the porous region is very large and, as a result, the velocity is very small

Table 6. Values of Concentration of Heterogeneous First-Order Chemical Reactions

Table 7a. Values of Velocity for Different Porous Parameters

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in the porous region. The velocity in the clear fluid region also decreases with an increase in the valueof the porous parameter. This is due to the coupling effect.

Table 7b displays the values of concentration, volumetric flow rate, and effective diffusioncoefficient for the two-fluid and one-fluid models. It is observed that the values of concentration,volumetric flow rate, and effective dispersion coefficient tally very well for the two-fluid and one-fluidmodels for σ = 0.01.

Case 3: Effect of homogeneous and heterogeneous reactions on the dispersion of solute in theabsence of porous matrix

The problem is solved considering viscous fluids in both regions (two-fluid model) andcompares the results with the one-fluid model [1]. It is seen from Table 8a that the values of velocity,concentration, volumetric flow rate, and effective dispersion coefficient agree very well (two-fluidmodel) with the results of Gupta and Gupta [1] for the one-fluid model which justify the two-fluidmodel for both homogeneous and heterogeneous chemical reactions.

The effect of viscosity ratio m and pressure gradient p for variations of homogeneous reactionrate parameter on the effective dispersion coefficient is tabulated in Table 8b. It is seen that as thehomogeneous reaction rate parameter α increases, the effective dispersion coefficient decreases forall values of viscosity ratio m and pressure gradient p. Further as viscosity ratio m increases the totaleffective dispersion coefficient F decreases. As the pressure gradient p increases F decreases for valuesof p ≤ 0 and increases for values of p ≥ 0 which is a similar result observed for composite porousmedium.

Table 7b. Values of Concentration, Volumetric Flow Rate (Q), and Effective Diffusion Coefficientfor Homogeneous Impermeable Wall Conditions

630

Table 8b. Values of Effective Dispersion Coefficient for Variations of Viscosity Ratio and Pressure Gradients for Homogeneous Insulating and Conducting Wall Conditions

Table 8a. Value of Concentration, Volumetric Flow Rate (Q), and Effective Dispersion Coefficientfor Homogeneous Impermeable Wall Conditions

631

The effective dispersion coefficient is evaluated for variations of homogeneous and hetero-geneous reaction rate parameters for two- and one-fluid models and is shown in Table 9a. The valuesof effective dispersion coefficient tally with the results of Gupta and Gupta [1].

Table 9a. Values of Effective Dispersion Coefficient for Heterogeneous Impermeable Wall Conditions

Table 9b. Values of Effective Dispersion Coefficient for Variations of Viscosity Ratio and Pressure Gradients for Heterogeneous Impermeable Wall Conditions

632

The effect of heterogeneous reaction rate parameter β for a fixed value of homogeneous rateparameter α is tabulated in Table 9b. As the wall catalytic parameter β increases, the effectivedispersion coefficient decreases in both regions for variations of viscosity ratio m and pressuregradient p. Further as m increases, F decreases for values of p ≤ 0 and increases for p ≥ 0 which aresimilar results observed for homogeneous chemical reaction. The values of effective dispersioncoefficient tally with the results of Gupta and Gupta [1].

4. Conclusions

The problem of solute dispersion of a solute for a composite porous medium between twoparallel plates was studied using Taylor’s dispersion model in the presence of first-order homogeneousand heterogeneous chemical reactions. Exact solutions were obtained for two different types ofboundary conditions such as insulating and insulating-impermeable wall conditions for homogeneouschemical reaction.

The results obtained for insulating homogeneous first-order chemical reactions were:

1. The velocity decreases as the porous parameter σ increases in both regions.

2. The values of velocity, concentration, volumetric flow rate, and effective dispersioncoefficient tally very well for composite porous medium (present model) with the one-fluid model[29] for small values of porous parameter σ.

3. As the homogeneous reaction rate parameter increases the effective dispersion coefficientdecreases for variations of viscosity ratio, pressure gradients, and porous parameter.

4. The effective dispersion coefficient decreases for values of viscosity ratio and pressuregradients less than one and increases for values of viscosity ratio and pressure gradients greater thanone. The effective dispersion coefficient decreases very rapidly with increasing porous parameterwhich is the similar result obtained by Rudraiah and Ng [29].

5. The effect of the porous parameter for soil, silica grains, and wire cramps was to reduce theconcentration which is a similar result obtained experimentally by Harleman et al. [28]. Further it wasalso observed that concentration distribution was high for air when compared to water.

The results obtained for homogeneous insulating-permeable wall boundary conditions were:

1. As the homogeneous reaction rate parameter increases the effective dispersion coefficientdecreases for any values of viscosity ratio, pressure gradients, and porous parameter.

2. The effect of viscosity ratio and pressure gradient was to reduce the effective dispersioncoefficient for values less than one and increases for values greater than one. The effective dispersioncoefficient increases for small values of porous parameter and decreases for large values of porousparameter.

633

3. The effect of porous parameter on the concentration for soil, silica grains, and wire crampswere not effective when compared to insulating wall boundary conditions. The distribution ofconcentration for water is less when compared to air.

The results obtained for heterogeneous first-order chemical reactions were:

1. As the wall catalytic parameter increases the effective dispersion coefficient decreases forvariations of viscosity ratio, pressure gradient, and porous parameter.

2. The effect of viscosity ratio, pressure gradient, and porous parameter on the effectivedispersion coefficient was similar to the results observed for homogeneous chemical reactions withinsulated wall boundary conditions.

3. The effect of the porous parameter on the concentration for soil, silica grains, and wirecramps for air and water showed a similar nature to those obtained for homogeneous insulating wallconditions.

The volumetric flow rate increases in magnitude as the viscosity ratio and porous parameterincreases for small values and then remained invariant for large values for both homogeneous andheterogeneous wall boundary conditions.

The results obtained for composite porous media agreed very well for small values of porousparameter with the results of Rudraiah and Ng [29].

To justify the results, the problem was solved in the absence of the porous parameter and theresults were in good agreement with results of Gupta and Gupta [1].

Acknowledgments

The authors thank UGC-New Delhi for the financial support under UGC-Major ResearchProject.

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Appendix

Case 1: Diffusion with a homogeneous first-order chemical reaction

Case 1a: Concentration distribution with impermeable wall conditions

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Case 1b: Concentration distribution with lower wall impermeable (insulating) and upperwall permeable (conducting) wall conditions

Case 1c: Diffusion with combined homogeneous and heterogeneous first-order chemical reac-tion

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