Pressure diffusion and chemical viscosity in the filtration

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IOP Conf. Series: Journal of Physics: Conf. Ser1i2e3s415162788(9200‘1’8“)”012036 doi:10.1088/1742-6596/1128/1/012036

Pressure diffusion and chemical viscosity in the filtration models with state equation in differential form

A G Knyazeva
Tomsk Polytechnic University, 634050, Tomsk, Lenin Av. Institute of Sthrength Physics and Material Scinece SB RAS, 634055, Tomsk, pr. Academicheskii, 2/4
E-mail: [email protected]
Abstract. Some options of coupling filtration models are suggested using irreversible state equations in differential form. State equations include explicitly the coefficient of compressibility, coefficients of concentration expansion and other physical properties affecting rheological properties and composition. To construct the models, the improvement of thermodynamic relations is used. New physical factors are introduced with the help of new thermodynamic variables. The chemical viscosity, pressure diffusion and concentration expansion phenomena are taken into account. The simplest particular problems illustrating the role of new effects are distinguished for stationary filtration regime. The revealed nonlinear effects can be important when considering biology liquid flows in porous biomaterials where deviations from classical laws are possible.

1. Introduction Many biology objects are the composites with complex structure of porous space. Artificial media synthesized for catalysis and chemical technology should be similar to natural objects [1]. This assumes that simulation of flows in porous natural and man-made biology objects should be based on the filtration laws. However interrelation between various physical effects can lead to the deviation on habitual conceptions.
Classical filtration Darсy’s law (1856) connects filtration velocity w of fluid with pressure field p

w   k p , (1) 

where k is the permeability, and  is the friction coefficient of fluid with pore walls or viscosity.
There are various generalizations of this law for high rate flows, irreversible conditions, heterogeneous fluids, two-phase flows, media with double porosity, and etc. [2-4]. Thermodynamically based generalization of law (1) for multicomponent and multi-phase flows was made in [5,6]. General form of motion equation includes (1) as particular case:

 v  vv  p  F  F  ,

(2)

t

12

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IOP Conf. Series: Journal of Physics: Conf. Ser1i2e3s415162788(9200‘1’8“)”012036 doi:10.1088/1742-6596/1128/1/012036

where v  w a ,a  S p S , and S p is the area occupied by pores in the section S ; F1 is gravity force,
and F2 is the force of internal friction depending on filtration velocity. If fluid is multicomponent, the following equations for species are necessary [2,3]

 

m

C

k

  wC

 







J



,

(3)

 t

k k k

where Ck are mass concentrations, J k are diffusion fluxes, k are sources of species due to chemical
reactions and mass exchange with solid skeleton; m –is porosity. We should add the continuity equation and state equations to the previous one. First of them has usual form

 m    w   , (4) t
where σ is possible mass source from porous skeleton. State equations define the connection between the pressure and other variables, and can be found based on experiment or irreversible thermodynamics.
Naturally, in multicomponent system, the balance of masses and diffusion fluxes take place This work demonstrates the step-by-step complication of filtration models by taking into account the coupling effects between various physical phenomena, when constitutive equations are based on irreversible thermodynamics, and rheological features are associated with physical effects.

2. Constitutive equations in thermodynamics Thermodynamics allows ascertaining two groups of constitutive equations [7,8], and assumes that the pressure consists of two parts p  pe  pV . First term increment depends linearly with the change in
volume (strains), concentrations, and other thermodynamical variables. Second term depends on the rates of change of thermodynamical variables. First group of constitutive equations follows from Gibbs equation written in suitable form. These are

n

dpe  T 1d  pk dCk ;

(5)

k 1

n

dgk



pk d







k i

d

Ci

,

(6)

i1

where pk  kT1 , k is the concentration expansion coefficient, T is isothermal compressibility. We can call the coefficients pk as partial pressures. The coefficients lk depend on the mixture under study. Here the coupling or cross effect consists of the interrelation between mechanical and chemical processes. The pressure can change due to the liquid composition variation, and the chemical potentials of the liquid components change when the volume or pressure varies.
Linear Onsager theory gives the second group of constitutive equations. For example, when only one chemical reaction takes place with the rate φ, we can write

pV  V   v  V A ;

(7)

  V   v  kchA ,

(8)

where V is the volume viscosity, kch is the reaction rate constant; A is the chemical affinity,
A  mR gR  mP gP ; then k=R,P in (3),(5) and (6); coefficient V describes the viscosity connecting
with chemical reaction and the dependence of reaction rate on fluid mobility.

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IOP Conf. Series: Journal of Physics: Conf. Ser1i2e3s415162788(9200‘1’8“)”012036 doi:10.1088/1742-6596/1128/1/012036

More complex generalizations are possible in modern thermodynamics [9,10] for multi-phase flows in porous media and for the media with double porosity.

3. Examples for stationary filtration regimes Naturally, all classical filtration models follow from above presented equations. Here we consider
only the simplest models to illustrate the influence of “new” factors appearing due to interrelation between phenomena of various physical natures. Note that obtained effects are similar to those which appear when the dependencies of permeability, porosity, viscosity and etc. on pressure are taken into consideration [11,12].

3.1. Imperfect no viscous gas For imperfect one-component gas, the state equation for “elastic” part of the pressure

dpe    d  1 d or pe  1 

(9)

T

T 

T 

follows from (5). “Viscous” part of the pressure equals to zero, and p  pe . For stationary flow, two equations remain

 v  0 and v   k p . (10) a

If the properties do not change with the pressure, the equation for p takes the form

 p  T p2  0 .

(11)

This leads to the simple problem for plane layer

d 2 p  dp 2

dx2

 T    dx 

0; x0:

p  pA ; x  L :

p  pB .

(12)

It is convenient to write exact analytical solution for this problem in dimensionless variables

p  p p , u V V ,   x L; V  k pA .

A

*

* a L

Consequently we obtain

  p  1 ln  C C , u  1 exp  T 1  .

(13)

  T

12

1   exp  T 1 

where T  T pA ,   pB pA are parameters, C1, C2 are integration constants depending on T , (not presented here). The tendency of the solution change at compressibility decrease is shown from
Figure 1. For T  0.5 the result does not distinguish from the solution for incompressible liquid. More strong effects appear when coupling phenomena are taken into account.

3.2. Compressible viscous liquid For imperfect single component compressible liquid, the state equation stays the same. However the pressure consists of two parts and Darcy law turns to

  v   k p  k pe  pV .

(14)

a

a

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IOP Conf. Series: Journal of Physics: Conf. Ser1i2e3s415162788(9200‘1’8“)”012036 doi:10.1088/1742-6596/1128/1/012036

Using (9) for first part of the pressure (“elastic”) and relation

pV  V   v

(15)

for “viscous” part, we find the equation for the velocity

T vV   v   v  aT k 1vv  0 .

(16)

p 1,0

u

0,60

0,9

12

3 0,45

0,8 0,30

1
2 3

0,7
0,0 0,2 0,4 0,6 0,8 

0,15
0,0 0,2 0,4 0,6 0,8 

Figure 1. Pressure and velocity in plane layer for the case of filtration of imperfect gas;
T 1. 5; 2. 2; 3. 0,5 , points correspond to incompressible liquid

For example, for plane layer, this equation is reduced to

d 2V dV a 2

T VV dx2  dx  k TV  0 .

(17)

It is not difficult using (14) to obtain the equation for total pressure for this case

k  dp  d3 p d 2 p  dp 2

T V

 a  dx 

dx3



dx2

 T    dx 

0.

(18)

Additionally we have the relation V  const from continuity equation.

3.3. Binary non-viscous imperfect mixture For imperfect two-component compressible mixture without viscosity ( p  pe ) and without chemical
reaction, the state equation will contain new parameter, namely – concentration expansion coefficient C :

1  d



dp  T

 



 C dC . 

(19)

This parameter C has thermodynamical definition: C  2  1 , k   CT  ; the
coefficients 2 ,1 can be calculated based on atomic volumes. Neglecting the pressure diffusion, we come for stationary filtration regime to the coupling equation system including (10) and

1  



p



T

 



 CC  , 

(20)

avC  DC .

(21)

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IOP Conf. Series: Journal of Physics: Conf. Ser1i2e3s415162788(9200‘1’8“)”012036 doi:10.1088/1742-6596/1128/1/012036

Here usual Fick law and condition D  const are assumed correct.

From these equations one can find the equation for the pressure

p  T p  CCp  0 ,

(22)

coupled with the concentration. Than the problem for plane layer with given pressure drop will include the coupled equations:

d 2 p  dp

dC  dp

dx2

 T 

dx  C

 dx  dx

0;

(23)

 k dp dC  D d 2C . (24)  dx dx dx2

The boundary conditions for pressure are similar to (12), and for concentration are

x  0 : C  C1; p  pA and x  L : C  C2; p  pB .

(25)

The problem will be a more complex, if diffusion coefficient depends on the pressure and concentration.

3.4. Filtration together with the pressure diffusion Here we use the state equation (6) for chemical potentials and take into account that diffusion fluxes for species are proportional to gradient of their chemical potentials. For binary imperfect mixture and T=const, one can write the relations

 J  

L



gi

 

(26)

k

i ki  T 

for mass fluxes of species, where chemical potentials follow from (6) or similar equation is written
based on Gibbs equation for Gibbs energy g   gkCk . Usual diffusion theory gives
k

J  Df CC  DCm Cpe .

(27)

RT

where m is the molar mass of diffused species, R – is universal gas constant. The equations for parts of the pressure and for velocity stay the same. The diffusion equation gains additional term:

avC  Df CC BCpe ,

(28)

where B  DmC is pressure diffusion coefficient. This is not independent value and is calculated RT
from other properties.

3.5. Filtration together with chemical reactions The most complex model appears when chemical reaction can accompany the flows in porous body. Different situation are possible here. Firstly, when part of fluid is absorbed by solid skeleton, in (3)
k  Ck  0 . In this case, diffusion equations (21), (28) do not change. Secondly, when reactions happen in volume of pores,   0 , and

k  mk kii ,

(29)

i

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IOP Conf. Series: Journal of Physics: Conf. Ser1i2e3s415162788(9200‘1’8“)”012036 doi:10.1088/1742-6596/1128/1/012036

where i is the rate of reaction i. For one reaction with the rate (8), it is necessary to calculate chemical affinity. By definition, the chemical affinity equals to the sum of chemical potentials of species multiplied by corresponding molar mass and stoichiometric coefficients. So, for reaction
1R1  2R2  3 Pr oduct , we can write

A  1m1g1  2m2g2  3m3g3 .

(30)

Hence, A  Ape ,C1,C2 ,C3 , it is not difficult, using the Gibbs-Duhem equation and differential
relations of type (6), to obtain

dA   f p Ck dpe  f1Ck dC1  f3Ck dC3

with functions f p , f1, f3 depending on composition and physical properties. This speaks that “viscous” pressure (7) and “elastic” pressure (5) will be interrelated with each other. Hence, chemical viscosity will include primary (connected with composition change) and secondary (connected with viscosity change in chemically reacted mixture) effects. Note the bother effects can appear in incompressible liquid, when
pV  f1  f p p1  p2 C1  f3  f p p3  p2  C3

These effects demand a special investigation.

4. Conclusion So, the resources of irreversible thermodynamics are demonstrated to reveal the features of filtration flows leading to the deviation from Darcy law. Some limiting situations are studied to show the nonlinearities associated with coupling effects between various phenomena. Pressure diffusion and chemical viscosity can be important for biological applications where mass transfer is insured by diffusion together with filtration.

Acknowledgments The work was conducted in the frameworks of the program of fundamental scientific investigations on State Academies of Science for Years 2018-2020, project No 23.2.5.

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