The central thermodynamical function of the model is the Helmholtz function, especially in the bulk, introduced by Eq. (1). It is calculated from the equation of state of the mixture considered, which must be van der Waals or related (Redlich-Kwong, ...) so that, for temperatures less than the critical, regions of unstable fluid insure the existence of interphase regions; association terms must be included for the polar molecules of the mixture (see Ref. [16] and references given there). It is shown in Ref. [16] that, independently of the equation of state that is chosen, the Helmholtz function contains the sum of the Helmholtz functions of the components, each considered as a gas where molecular interactions are neglected (ideal). Each ideal gas Helmholtz function contains a function of *T* that drops out under differentiation when *T* is assumed constant, but is important for the thermal model in accounting for the energy stored in the internal degrees of freedom of the molecules. The Helmholtz function of one mole of an ideal gas of component *α* is

The transport coefficients, *k* and *η*, of Eqs. (97) and (98), are needed as

equation, Eq. (75), in terms of temperature: see Section 2.3 of Ref. [3].

functions of *T* and of the *Rα*. See Ref. [3] and references given there.

variables. See Ref. [3] for a detailed presentation.

(see Refs. [2, 16] and references given there):

**20**

third of expressions (93) one gets the diffuse mass current of component *α*:

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

*I α k* ¼ �Φ

ℓ*αβ* ¼ �*a RαRβ* � �2

model one then sets

**5.3 Closing details**

a matrix in the case *ν* ¼ 3, see Refs. [2, 3].

that such upgrading is required.

$$f^{\text{id.gas},a} = R\_G T \ln \frac{\mathcal{V}^a(T)}{ev^a},\tag{104}$$

where *vα* is the molar volume, *RG* is the gas constant, *e* ¼ exp 1ð Þ, and V*α*ð Þ *T* is a function of temperature with the dimension of a volume per mole. V*α* depends on the atomic masses, the principal moments of inertia of the molecule, .... It can be obtained from statistical physics, as shown in Ref. [16].

It is briefly shown below that V*α* can also be obtained from an expression of *f* id*:*gas,*α* in terms of the heat capacity of the component. Assuming *Pvα* ¼ *RGT*, one easily finds that, with *T* and *vα* as independent variables, the molar internal energy *uα* (the "id.gas" superscript is dropped for simplicity) is independent of *vα*, and that one can write

$$\mathbf{d}u^{a} = c\_{V}^{a}(T)\mathbf{d}T,\tag{105}$$

where *cα V* is the molar heat capacity of component *α*. The molar entropy, *s α*, obeying *T*d*s α* ¼ d*uα* þ *P*d*vα*, one obtains

$$\mathbf{ds}^a = \mathbf{c}\_V^a(T)\mathbf{d}T/T + \mathbf{R}\_G\mathbf{d}\boldsymbol{\nu}^a/\boldsymbol{\nu}^a. \tag{106}$$

Integrating the last two displayed equations, one obtains *f* id*:*gas,*α* as *uα* � *Tsα*:

$$f^{\text{id.gas},a} = u\_0^a - Ts\_0^a - R\_G T \ln \frac{\nu^a}{v\_0^a} + \int\_{T\_0}^{T} \left(1 - \frac{T}{T'}\right) c\_V^a(T') \text{d}T',\tag{107}$$

where *uα* 0 and *s α* 0 are the internal energy and the entropy at a reference state where volume, temperature, and pressure are *vα* 0, *T*0, and *P*0 ¼ *RGT*0*=vα* 0. Equating the right-hand sides of Eqs. (104) and (107), one obtains an expression for *T* ln V*α* in terms of the experimentally measurable function *cα V*. As stated in Ref. [16], it is in fact the derivative with respect to *T* of *T* ln V*α* that is needed in the differential equations of the thermal model. Assuming *s α* 0 ¼ 0, one obtains

$$\frac{d}{dT}[R\_G T \ln \mathcal{V}^a(T)] = R\_G \ln \frac{eR\_G T\_0}{P\_0} - \int\_{T\_0}^{T} \frac{c\_V^a(T')}{T'} \mathbf{d}T'.\tag{108}$$

Concerning examples of numerical solutions of the equations of the minimal model, see Ref. [2] for phase segregation, and for coning at uniform temperature; see Ref. [3] for an injection-production situation at variable temperature.

#### **A. Appendices**

#### **A.1 Appendix 1: Assumptions**

**A0:** None of the chemical species completely wets the rock

**A1:** There are no sources or sinks

**A2:** There is no loss of energy by radiation

**A3:** There are no chemical reactions between the chemical components

**A4:** The solid is perfectly rigid

**A5:** The external force per unit volume *fi* is approximately constant in the averaging volume

**A6:** At each point, the difference between the solid and fluid temperatures is negligible

**A7:** The Λ*α* are constant numbers inside the porous medium

**A8:** The porosity Φ is uniform in space

#### **A.2 Appendix 2: Derivatives of distibutions – Formulas**

The following is a set of formulas for the space derivatives of distributions having discontinuities across a given surface Σ. For proofs, see [12, 13].

Consider a function *f*ð Þ **x**, *t* symbolising a physical quantity, continuous in time but dicontinuous in space across a surface Σ that divides space in two regions, called A-side and B-side; *f* ¼ *f B* on the B-side, *f* ¼ *f A* on the A-side. Σ has a normal vector **n** at each point, pointing towards A:

$$f(\mathbf{x},t) = \begin{cases} f^{\mathcal{A}}(\mathbf{x},t) & \mathbf{x} \quad \text{on the A-side of } \Sigma\\ f^{\mathcal{B}}(\mathbf{x},t) & \mathbf{x} \quad \text{on the B-side of } \Sigma, \end{cases} \tag{109}$$

One now defines the following regular distributions:

$$\{\partial\_{\mathbf{k}}f(\mathbf{x},t)\} = \begin{cases} \partial\_{\mathbf{k}}f^{\mathcal{A}}(\mathbf{x},t) & \mathbf{x} \quad \text{on the A-side of } \Sigma\\ \partial\_{\mathbf{k}}f^{\mathcal{B}}(\mathbf{x},t) & \mathbf{x} \quad \text{on the B-side of } \Sigma, \end{cases} \tag{110}$$

$$\{\partial\_{\varnothing}f(\mathbf{x},t)\} = \begin{cases} \partial\_{t}f^{A}(\mathbf{x},t) & \mathbf{x} \quad \text{on the A-side of } \Sigma\\ \partial\_{t}f^{B}(\mathbf{x},t) & \mathbf{x} \quad \text{on the B-side of } \Sigma. \end{cases} \tag{111}$$

Then [12, 13].

$$
\partial\_{\varnothing}f = \{\partial\_{\varnothing}f\},
\tag{112}
$$

If *Zij* is a tensor, then *Z* is a scalar, *Z*^ is a symmetric traceless tensor, and *Z*~*k* is a pseudo-vector. A most useful property is as follows: if *Uij* and *Vij* are two tensors

*UijVij* ¼ 3*UV* þ *U*^ *ijV*^ *ij* þ 2*U*~ *iV*~ *i:* (116)

then

*Porous Flow with Diffuse Interfaces*

*DOI: http://dx.doi.org/10.5772/intechopen.95474*

**Author details**

Paul Papatzacos

**23**

University of Stavanger, Stavanger, Norway

provided the original work is properly cited.

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

$$
\partial\_{\mathbb{k}}f = \{\partial\_{\mathbb{k}}f\} + \left[ \left( \left. f^{\mathbb{A}} \right|^{\Sigma} - \left( \left. f^{\mathbb{B}} \right|^{\Sigma} \right) \right] n\_{\mathbb{k}} \delta\_{\Sigma}, \tag{113}
$$

where *δ*Σ is a surface-Dirac distribution, its action h i *δ*Σ, *φ* on a so-called test function *φ* [13] being defined as:

$$
\langle \delta\_{\Sigma}, \rho \rangle = \int\_{\Sigma} \rho \, \mathbf{d} \Sigma. \tag{114}
$$

#### **A.3 Appendix 3: Formulas concerning tensors**

Let *Zij* be an arbitrary second order tensor. It can be shown (see Ref. [15]) that the following expression has general validity:

$$Z\_{i\bar{j}} = Z\delta\_{i\bar{j}} + \hat{Z}\_{i\bar{j}} + \varepsilon\_{i\bar{j}k}\tilde{Z}\_k, \text{where} \begin{cases} Z = \frac{1}{3}Z\_{kk}, \\\\ \hat{Z}\_{i\bar{j}} = \frac{1}{2}\left(Z\_{i\bar{j}} + Z\_{j\bar{i}}\right) - \frac{1}{3}Z\_{kk}\delta\_{i\bar{j}}, \\\\ \hat{Z}\_k = \frac{1}{2}\varepsilon\_{klm}Z\_{lm}. \end{cases} \tag{115}$$

*Porous Flow with Diffuse Interfaces DOI: http://dx.doi.org/10.5772/intechopen.95474*

**A6:** At each point, the difference between the solid and fluid temperatures is

The following is a set of formulas for the space derivatives of distributions

Consider a function *f*ð Þ **x**, *t* symbolising a physical quantity, continuous in time but dicontinuous in space across a surface Σ that divides space in two regions, called

*A*ð Þ **x**, *t* **x** on the A‐side of Σ

ð Þ **x**, *t* **x** on the B‐side of Σ,

*A*ð Þ **x**, *t* **x** on the A‐side of Σ

*B*ð Þ **x**, *t* **x** on the B‐side of Σ,

*A*ð Þ **x**, *t* **x** on the A‐side of Σ

*B*ð Þ **x**, *t* **x** on the B‐side of Σ*:*

� *f B* � �Σ � �

*A* � �Σ

where *δ*Σ is a surface-Dirac distribution, its action h i *δ*Σ, *φ* on a so-called test

ð Σ

Let *Zij* be an arbitrary second order tensor. It can be shown (see Ref. [15]) that

*Z* ¼ 1

8 >>>>>>><

>>>>>>>:

*Z*^*ij* ¼ 1

*Z*~*k* ¼ 1

3 *Zkk*,

2 *Zij* þ *Zji* � � � 1

2 *εklmZlm:*

h i *δ*Σ, *φ* ¼

*∂tf* ¼ f g *∂tf* , (112)

*nk δ*Σ, (113)

*φ* dΣ*:* (114)

3 *Zkkδij*,

*A* on the A-side. Σ has a normal vector

(109)

(110)

(111)

(115)

**A7:** The Λ*α* are constant numbers inside the porous medium

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

**A.2 Appendix 2: Derivatives of distibutions – Formulas**

having discontinuities across a given surface Σ. For proofs, see [12, 13].

*B* on the B-side, *f* ¼ *f*

**A8:** The porosity Φ is uniform in space

negligible

A-side and B-side; *f* ¼ *f*

Then [12, 13].

**22**

**n** at each point, pointing towards A:

f g *∂kf*ð Þ **x**, *t* ¼ *∂k f*

f g *∂tf*ð Þ **x**, *t* ¼ *∂t f*

function *φ* [13] being defined as:

*f*ð Þ¼ **x**, *t f*

(

(

*f B*

One now defines the following regular distributions:

*∂k f*

*∂t f*

**A.3 Appendix 3: Formulas concerning tensors**

the following expression has general validity:

*Zij* ¼ *Zδij* þ *Z*^*ij* þ *εijkZ*~*k*, where

*∂kf* ¼ f g *∂kf* þ *f*

(

If *Zij* is a tensor, then *Z* is a scalar, *Z*^ is a symmetric traceless tensor, and *Z*~*k* is a pseudo-vector. A most useful property is as follows: if *Uij* and *Vij* are two tensors then

$$U\_{\vec{\eta}}V\_{\vec{\eta}} = \Im UV + \hat{U}\_{\vec{\eta}}\hat{V}\_{\vec{\eta}} + 2\tilde{U}\_i\tilde{V}\_i. \tag{116}$$

#### **Author details**

Paul Papatzacos University of Stavanger, Stavanger, Norway

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Papatzacos P: Macroscopic two-phase flow in porous media assuming the diffuse-interface model at pore level. Transport in Porous Media 2002;49: 139–174. DOI: 10.1023/A:<PHONE_NUMBER>189

[2] Papatzacos P: A model for multiphase and multicomponent flow in porous media, built on the diffuse-interface assumption. Transport in Porous Media 2010;82:443–462. DOI: 10.1007/ s11242-009-9405-2

[3] Papatzacos P: A model for multiphase, multicomponent, and thermal flow in neutrally wetting porous media, built on the diffuseinterface assumption. Journal of Petroleum Science and Engineering 2016;143:141–157. DOI: 10.1016/j. petrol.2016.02.027

[4] Papatzacos P, Skjæveland SM: Diffuse-interface modelling of twophase flow for a one-component fluid in a porous medium. Transport in Porous Media 2006;65:213–236. DOI: 10.1007/ s11242-005-6081-8

[5] van der Waals JD: The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, Journal of Statistical Physics 1979;20:200–244. DOI: 10.1007/BF01011514

[6] Korteweg DJ: Sur la forme que prennent les equations etc, Archives Neerlandaises des Sciences Exactes et Naturelles, 1901;II:6–20. DOI: Not known

[7] Anderson DM, McFadden GB, Wheeler AA: Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics 1998;30:139–165. DOI: 10.1146/annurev.fluid.30.1.139

[8] Ganesan V, Brenner H: A diffuseinterface model of two-phase flow in porous media. Proceedings of the Royal Society London A 2000;456:731–803. DOI: 10.1098/rspa.2000.0537

**Chapter 2**

Multiscale Modeling of

Porous Media

*Kieu Hiep Le*

**1. Introduction**

**25**

**Abstract**

Non-Isothermal Fluid Transport

To preserve the product quality as well as to reduce the logistics and storage cost, drying process is widely applied in the processing of porous material. In consideration of transport phenomena that involve a porous medium during drying, the complex morphology of the medium, and its influences on the distribution, flow, displacement of multiphase fluids are encountered. In this chapter, the recent advanced mass and energy transport models of drying processes are summarized. These models which were developed based on both pore- and continuum-scales, may provide a better fundamental understanding of non-isothermal liquid–vapor transport at both the continuum scale and the pore scale, and to pave the way for designing, operating, and optimizing drying and relevant industrial processes.

**Keywords:** drying model, multiscale – modeling, porous media, pore-network

In both natural systems (i.e. clays, aquifers, oil and gas reservoirs, plants, and biological tissues) and industrial systems (i.e. fuel cells, concrete, textiles, polymer composites, capillary heat pipe, and paper, etc.), porous media are often encountered. From a morphological point of view, porous media is composed of a persistent solid matrix and an interconnected void space that can be occupied by fluid phases. In several porous systems, the void space is initially filled by liquid water. To maintain the product quality, prolong the storage time as well as reduce the logistics cost, the liquid water is needed to be removed by using various drying techniques. During the drying process, the complex morphology of the media and its influences on the capillary flow, liquid – vapor phase change under thermal effects are faced. These processes are complex by themselves [1, 2]. Additionally, it should be noted that the industrial dryers consume approximately 12% of the total energy used in manufacturing processes [3]. For several industrial sectors (i.e. tissue, food, and agriculture) the energy consumption ratio of drying may reach 33%. As a complicated, important, and high energy consumption process, the dryer designing, and operation should not be done by trial and error. A better fundamental understanding of this drying process at the pore- and continuous scales (i.e. understanding the influence of porous structure on the drying behavior, drying

model, continuum model, diffusion model, upscaling strategy

Involved in Drying Process of

[9] Adler PM, Brenner H: Multiphase phase flow in porous media. Annual Review of Fluid Mechanics 1988;20:35– 59. DOI: 10.1146/annurev. fl.20.010188.000343

[10] Papatzacos P, Skjæveland SM: Relative permeability from thermodynamics. Society of Petroleum Engineers Journal 2004;9:213–236. DOI: 10.2118/87674-PA

[11] Hirschfelder JO, Curtiss CF, Bird RB: Molecular theory of gases and liquids, John Wiley and Sons, Inc. New York, Chapman and Hall, Limited, London, 1954. DOI: 10.1002/ pol.1955.120178311

[12] Marle CM: On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. International Journal of Engineering Science 1982;20:643–662. DOI: 10.1016/0020-7225(82)90118-5

[13] Appel W: Mathematics for physics and physisists, Princeton University Press 2007. ISBN-13: 978–0–691-13102-3

[14] Weinstock R: Calculus of Variations, Dover 1974. ISBN-13: 0– 486–63069-2

[15] Reichl LE: A Modern Course in Statistical Physics, 2nd Edition, John Wiley and Sons 1998. ISBN: 0–471– 59520-9

[16] Papatzacos P. The Helmholtz free energy of pure fluid substances and fluid mixtures. In: Proceedings of the 8th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, (HEFAT2011); 11–13 July 2011; Mauritius.

#### **Chapter 2**

**References**

1016091821189

s11242-009-9405-2

petrol.2016.02.027

s11242-005-6081-8

[5] van der Waals JD: The

DOI: 10.1007/BF01011514

known

**24**

thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, Journal of Statistical Physics 1979;20:200–244.

[6] Korteweg DJ: Sur la forme que prennent les equations etc, Archives Neerlandaises des Sciences Exactes et Naturelles, 1901;II:6–20. DOI: Not

[7] Anderson DM, McFadden GB, Wheeler AA: Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics 1998;30:139–165. DOI:

10.1146/annurev.fluid.30.1.139

[8] Ganesan V, Brenner H: A diffuseinterface model of two-phase flow in

[3] Papatzacos P: A model for multiphase, multicomponent, and thermal flow in neutrally wetting porous media, built on the diffuseinterface assumption. Journal of Petroleum Science and Engineering 2016;143:141–157. DOI: 10.1016/j.

[4] Papatzacos P, Skjæveland SM: Diffuse-interface modelling of twophase flow for a one-component fluid in a porous medium. Transport in Porous Media 2006;65:213–236. DOI: 10.1007/

139–174. DOI: 10.1023/A:

[1] Papatzacos P: Macroscopic two-phase flow in porous media assuming the diffuse-interface model at pore level. Transport in Porous Media 2002;49:

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

porous media. Proceedings of the Royal Society London A 2000;456:731–803.

[9] Adler PM, Brenner H: Multiphase phase flow in porous media. Annual Review of Fluid Mechanics 1988;20:35–

[10] Papatzacos P, Skjæveland SM: Relative permeability from

thermodynamics. Society of Petroleum Engineers Journal 2004;9:213–236. DOI:

[11] Hirschfelder JO, Curtiss CF, Bird RB: Molecular theory of gases and liquids, John Wiley and Sons, Inc. New York, Chapman and Hall, Limited, London, 1954. DOI: 10.1002/

[12] Marle CM: On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. International Journal of Engineering Science 1982;20:643–662. DOI: 10.1016/0020-7225(82)90118-5

[13] Appel W: Mathematics for physics and physisists, Princeton University Press 2007. ISBN-13: 978–0–691-13102-3

[14] Weinstock R: Calculus of Variations, Dover 1974. ISBN-13: 0–

[15] Reichl LE: A Modern Course in Statistical Physics, 2nd Edition, John Wiley and Sons 1998. ISBN: 0–471–

[16] Papatzacos P. The Helmholtz free energy of pure fluid substances and fluid mixtures. In: Proceedings of the 8th International Conference on Heat Transfer, Fluid Mechanics and

Thermodynamics, (HEFAT2011); 11–13

July 2011; Mauritius.

486–63069-2

59520-9

DOI: 10.1098/rspa.2000.0537

59. DOI: 10.1146/annurev. fl.20.010188.000343

10.2118/87674-PA

pol.1955.120178311

[2] Papatzacos P: A model for multiphase and multicomponent flow in porous media, built on the diffuse-interface assumption. Transport in Porous Media 2010;82:443–462. DOI: 10.1007/

## Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of Porous Media

*Kieu Hiep Le*

### **Abstract**

To preserve the product quality as well as to reduce the logistics and storage cost, drying process is widely applied in the processing of porous material. In consideration of transport phenomena that involve a porous medium during drying, the complex morphology of the medium, and its influences on the distribution, flow, displacement of multiphase fluids are encountered. In this chapter, the recent advanced mass and energy transport models of drying processes are summarized. These models which were developed based on both pore- and continuum-scales, may provide a better fundamental understanding of non-isothermal liquid–vapor transport at both the continuum scale and the pore scale, and to pave the way for designing, operating, and optimizing drying and relevant industrial processes.

**Keywords:** drying model, multiscale – modeling, porous media, pore-network model, continuum model, diffusion model, upscaling strategy

#### **1. Introduction**

In both natural systems (i.e. clays, aquifers, oil and gas reservoirs, plants, and biological tissues) and industrial systems (i.e. fuel cells, concrete, textiles, polymer composites, capillary heat pipe, and paper, etc.), porous media are often encountered. From a morphological point of view, porous media is composed of a persistent solid matrix and an interconnected void space that can be occupied by fluid phases. In several porous systems, the void space is initially filled by liquid water. To maintain the product quality, prolong the storage time as well as reduce the logistics cost, the liquid water is needed to be removed by using various drying techniques. During the drying process, the complex morphology of the media and its influences on the capillary flow, liquid – vapor phase change under thermal effects are faced. These processes are complex by themselves [1, 2]. Additionally, it should be noted that the industrial dryers consume approximately 12% of the total energy used in manufacturing processes [3]. For several industrial sectors (i.e. tissue, food, and agriculture) the energy consumption ratio of drying may reach 33%. As a complicated, important, and high energy consumption process, the dryer designing, and operation should not be done by trial and error. A better fundamental understanding of this drying process at the pore- and continuous scales (i.e. understanding the influence of porous structure on the drying behavior, drying

time determination, and energy consumption prediction) may help to design better products and processes.

In this chapter, a review of the recent advanced model in a selection of drying phenomena involved in porous media is illustrated. We discussed the state-of-theart numerical methods as complementary ways to get more insight. The future challenges and the hint at solutions to accommodate for them are also given.

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

In 1950s, a new modeling approach to describe the transport process in the porous medium was proposed by Fatt. In his model, the void space of the porous media was modeled by a network of cylindrical tubes. In individual tubes, the mass transfer equation has been written and the discrete mass conservation equations are written for each pore node. This model has been named as pore network model. Afterward, several pore-scale models, such as the direct numerical approach, the volume of fluid approach, and method, have been presented in literature based on the idea of pore network model. Since the transport phenomena are considered directly at the pore-scale, the better fundamental understanding on the interaction between solid structure and fluid flow can be provided. As discussed in Section 1, the mentioned interaction can be upscaled to determine the macroscopic transport coefficients of the continuous models such as the relative permeabilities of the

In this section, we focus on the non – isothermal drying pore network models

Based on the invasion percolation algorithm, several non-isothermal pore network models were developed to simulate the hot air drying process [2, 7–9]. These models accounted capillary, gravity effects, viscous effects, and the transport of vapor by diffusion in the gas phase under the thermal effect. Recently, the pore network model has been applied to describe the non-isothermal mixture vapor liquid water transport during the superheated steam drying process [10]. The features of this pore network model is the fully treatment of condensed liquid by introducing the newly formulated liquid invasion rules in a two-dimensional domain. In this chapter, this two-dimensional pore network model is extended to three-dimensional model in this work to simulate the transport processes inside a capillary porous medium undergoing superheated steam drying with two different heating modes: (i) the convective heating at the top surface (convective- heating mode) and (ii) the simultaneous convective and conductive heating at the top and

Due to the absence of diffusion in the gas phase, the water vapor is transported by convective mechanism only and the Hagen-Poiseuille law is applied to calculate

> *ij* 8*υvL pi* � *p j*

where the vapor mass flow rate in the cylindrical throat connecting pores i and j is denoted by *M*\_ *v*,*ij*, pi and pj present the pressure of the vapor phase at pores I and j, respectively. The length and radius of throat are respectively indicated by *L* and *r*. The kinematic viscosity of the vapor phase is denoted by *υv*. Under the quasi-steady assumption, the mass conservation of water vapor in a empty/partially pore is

(1)

and their application in estimating the macroscopic transport coefficients.

**2.1 Pore network model for superheated steam drying**

bottom surfaces (contact-heating mode) (c.f. **Figure 2**).

the vapor mass flow rate between two adjacent pores [10].

written as

**27**

*M*\_ *v*,*ij* ¼ *πr*4

**2. Pore-scale models**

liquid–gas mixture, the liquid diffusivity.

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

The drying process of porous media may be considered at different scales. At different scales, the approaches used to describe the transport processes may be quite different [4]. For instance, in a drying plant, thousands or millions of drying products can be dried simultaneously and the state of the drying agent evolves over time and space. Thus, the intra-sample distribution of process parameters should be lumped to avoid the extremely expensive computational cost. As a result, the lumped model is the suitable model that should be implemented in drying plant simulation. Coming to the sample drying process, the internal transport process is taken into account. The heat, mass, and momentum conservation equations are developed based on the first physics principles. In a control volume, the process properties are assumed to be homogenous and isotropic and the impact of microheterogeneity on the transport process is omitted. The macro-scale models often result in a system of partial differential equations. Both globe and local drying behavior of the sample can be well predicted by using the continuous models if the size of the domain is large enough. For a thin layer of the porous medium, where micro-heterogeneity plays a role, the continuous models are not valid. Due to the interaction between the solid skeleton and the fluid phases, the transport equations should be derived for individual pores. Since the transport phenomena are directly investigated at the pore-scale, the impact of changes at the pore or microstructure such as wettability, pore shape, pore size distribution, and pore structure on drying behavior has been effectively interpreted by pore-scale models [5, 6]. One should notes that these models are not standing individually. The results of pore scale model can be upscaled to generate the effective parameters of macro scale models which can be reduced to the lumped models. An example multiscale modeling approach for wood particle drying process is presented in **Figure 1**.

#### **Figure 1.**

*Multiscale modeling approach for porous particle drying process.*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

In this chapter, a review of the recent advanced model in a selection of drying phenomena involved in porous media is illustrated. We discussed the state-of-theart numerical methods as complementary ways to get more insight. The future challenges and the hint at solutions to accommodate for them are also given.

#### **2. Pore-scale models**

time determination, and energy consumption prediction) may help to design better

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

The drying process of porous media may be considered at different scales. At different scales, the approaches used to describe the transport processes may be quite different [4]. For instance, in a drying plant, thousands or millions of drying products can be dried simultaneously and the state of the drying agent evolves over time and space. Thus, the intra-sample distribution of process parameters should be lumped to avoid the extremely expensive computational cost. As a result, the lumped model is the suitable model that should be implemented in drying plant simulation. Coming to the sample drying process, the internal transport process is taken into account. The heat, mass, and momentum conservation equations are developed based on the first physics principles. In a control volume, the process properties are assumed to be homogenous and isotropic and the impact of microheterogeneity on the transport process is omitted. The macro-scale models often result in a system of partial differential equations. Both globe and local drying behavior of the sample can be well predicted by using the continuous models if the size of the domain is large enough. For a thin layer of the porous medium, where micro-heterogeneity plays a role, the continuous models are not valid. Due to the interaction between the solid skeleton and the fluid phases, the transport equations should be derived for individual pores. Since the transport phenomena are directly investigated at the pore-scale, the impact of changes at the pore or microstructure such as wettability, pore shape, pore size distribution, and pore structure on drying behavior has been effectively interpreted by pore-scale models [5, 6]. One should notes that these models are not standing individually. The results of pore scale model can be upscaled to generate the effective parameters of macro scale models which can be reduced to the lumped models. An example multiscale modeling

approach for wood particle drying process is presented in **Figure 1**.

products and processes.

**Figure 1.**

**26**

*Multiscale modeling approach for porous particle drying process.*

In 1950s, a new modeling approach to describe the transport process in the porous medium was proposed by Fatt. In his model, the void space of the porous media was modeled by a network of cylindrical tubes. In individual tubes, the mass transfer equation has been written and the discrete mass conservation equations are written for each pore node. This model has been named as pore network model. Afterward, several pore-scale models, such as the direct numerical approach, the volume of fluid approach, and method, have been presented in literature based on the idea of pore network model. Since the transport phenomena are considered directly at the pore-scale, the better fundamental understanding on the interaction between solid structure and fluid flow can be provided. As discussed in Section 1, the mentioned interaction can be upscaled to determine the macroscopic transport coefficients of the continuous models such as the relative permeabilities of the liquid–gas mixture, the liquid diffusivity.

In this section, we focus on the non – isothermal drying pore network models and their application in estimating the macroscopic transport coefficients.

#### **2.1 Pore network model for superheated steam drying**

Based on the invasion percolation algorithm, several non-isothermal pore network models were developed to simulate the hot air drying process [2, 7–9]. These models accounted capillary, gravity effects, viscous effects, and the transport of vapor by diffusion in the gas phase under the thermal effect. Recently, the pore network model has been applied to describe the non-isothermal mixture vapor liquid water transport during the superheated steam drying process [10]. The features of this pore network model is the fully treatment of condensed liquid by introducing the newly formulated liquid invasion rules in a two-dimensional domain. In this chapter, this two-dimensional pore network model is extended to three-dimensional model in this work to simulate the transport processes inside a capillary porous medium undergoing superheated steam drying with two different heating modes: (i) the convective heating at the top surface (convective- heating mode) and (ii) the simultaneous convective and conductive heating at the top and bottom surfaces (contact-heating mode) (c.f. **Figure 2**).

Due to the absence of diffusion in the gas phase, the water vapor is transported by convective mechanism only and the Hagen-Poiseuille law is applied to calculate the vapor mass flow rate between two adjacent pores [10].

$$
\dot{M}\_{v,\dot{\imath}} = \frac{\pi r\_{\dot{\imath}}^4}{8\nu\_v L} \left(p\_i - p\_j\right) \tag{1}
$$

where the vapor mass flow rate in the cylindrical throat connecting pores i and j is denoted by *M*\_ *v*,*ij*, pi and pj present the pressure of the vapor phase at pores I and j, respectively. The length and radius of throat are respectively indicated by *L* and *r*. The kinematic viscosity of the vapor phase is denoted by *υv*. Under the quasi-steady assumption, the mass conservation of water vapor in a empty/partially pore is written as

**Figure 2.** *Two different heating modes used in the pore network simulations.*

$$
\sum\_{j} \dot{M}\_{v, \vec{\mu}} - \sum\_{j} \pi r\_{t, \vec{\mu}} \, ^2 \dot{m}\_{\text{evp}/\text{con}, \vec{\mu}} - \pi r\_{p, i} \, ^2 \dot{m}\_{\text{evp}/\text{con}, i} = \mathbf{0} \tag{2}
$$

Due to the non-uniform pore size distribution, the liquid within the network is transported under capillary action. Thus, the large pores/throats are preferentially emptied or filled to maintain the lowest liquid pressure according to Eq. 4. As a result, the liquid phase disintegrates into several liquid clusters. Since the liquid mass balance must be satisfied in the entire network as well as for each liquid cluster, the morphological information of liquid cluster is required. The Hoshen– Kopelman algorithm is applied for labeling the liquid clusters.

$$p\_l = p\_v - \frac{2\sigma(T)\cos\theta}{r\_{p/t}}\tag{3}$$

balance equation written for the control volume around pore i in time step Δt is

*Heat balance at a vapor pore. The arrows indicate the direction of the heat flux towards each face of the control*

*The 2D sketch of liquid invasion rules: Liquid invasion from single liquid throat to its neighboring pore (a), liquid invasion from fully filled liquid throat in a liquid cluster to its neighboring empty pore (b), liquid*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

*redistribution from fully filled pore in a liquid cluster to its neighboring empty throat (c).*

*L* ¼ *ρeff*,*iceff*,*iVi*

The convective heat transfer boundary condition presented in Eq. 8 is applied for both heating modes. At the bottom of the network, perfect thermal insulation (Eq. 9) is imposed for the convective heating mode, whereas uniform constant

> � � *y*¼*Ly*

*∂T ∂y* � � � � *y*¼0 Δ*t* ð Þ *Ti*,*t*þΔ*t* � *Ti*

•*n* ¼ *α Tsteam* � *T*j

2 *m*\_ *evp=con*,*i*

*y*¼*Ly* � � (8)

¼ 0 (9)

(7)

*m*\_ *evp=con*,*ji* þ Δ*Hevpπrp*,*i*

*Ti*,*t*þΔ*t* � *T j*,*t*þΔ*t*

þΔ*Hevp*X

temperature (Eq. 10) is set for the contact-heating mode.

" #

*j πrt*,*ij* 2

þ Δ*Hevpρlvl*

discretized as

**Figure 4**

*volume.*

**29**

**Figure 3.**

X *j*

� *Acv*,*ijλeff*,*ij*

*λeff ∂T ∂z* � � � � *y*¼*Ly*

To fully model the phase transition including both evaporation and condensation, in addition the emptying and refilling events, the liquid invasion represented in **Figure 3** is introduced in the present model. The time steps for the emptying and refilling events of meniscus pores or throats are computed by

$$\Delta t\_{\text{empty},cn/st,p/t} = \frac{V\_{p/t} \rho\_l(T) \mathbf{S}\_{p/t}}{\sum\_{c \le \text{tr}} \dot{\mathbf{M}}\_{\text{exp}/con}} \text{ and } \Delta t\_{\text{refill},cn/st,p/t} = -\frac{V\_{p/t} \rho\_l(T) \left(\mathbf{1} - \mathbf{S}\_{p/t}\right)}{\sum\_{c \le \text{tr}} \dot{\mathbf{M}}\_{\text{exp}/con}} \tag{4}$$

The time step for the first two liquid invasion events is calculated by

$$
\Delta t\_{\rm in,cn/st} = -\frac{V\_{p/t,p}\rho\_l(T)}{\sum\_{cn/st} \dot{M}\_{evp/con}} \tag{5}
$$

The pore/throat saturations after the capillary re-equilibration event are computed by

$$\mathbf{S}\_{t,ih} = \mathbf{1} \text{ and } \mathbf{S}\_{p,h} = \frac{V\_{p,h} - V\_{t,ih}}{V\_{p,h}} \tag{6}$$

The thermal energy supplied to a control volume (c.f. **Figure 4**) lead to a change of the enthalpy in the control volume: the control volume temperature increases, and the phase transition occurs. Based on a fully implicit scheme, the energy

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

#### **Figure 3.**

X *j*

**Figure 2.**

*Δtempty*,*cn=st*,*p=t* ¼ *Vp=tρl*ð Þ *T Sp=t* P

computed by

**28**

*M*\_ *v*,*ij* �X *j πrt*,*ij* 2

*Two different heating modes used in the pore network simulations.*

Kopelman algorithm is applied for labeling the liquid clusters.

refilling events of meniscus pores or throats are computed by

*cn=stM*\_ *evp=con*

*m*\_ evp*=*con,*ji* � *πrp*,*i*

Due to the non-uniform pore size distribution, the liquid within the network is transported under capillary action. Thus, the large pores/throats are preferentially emptied or filled to maintain the lowest liquid pressure according to Eq. 4. As a result, the liquid phase disintegrates into several liquid clusters. Since the liquid mass balance must be satisfied in the entire network as well as for each liquid cluster, the morphological information of liquid cluster is required. The Hoshen–

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*pl* ¼ *pv* � 2*σ*ð Þ *T* cos *θ*

To fully model the phase transition including both evaporation and condensation, in addition the emptying and refilling events, the liquid invasion represented in **Figure 3** is introduced in the present model. The time steps for the emptying and

The time step for the first two liquid invasion events is calculated by

*Δtin*,*cn=st* ¼ � *Vp=t*,*ρl*ð Þ *T* P

The pore/throat saturations after the capillary re-equilibration event are

*St*,*ih* ¼ 1 and*Sp*,*h* ¼ *Vp*,*h* � *Vt*,*ih*

The thermal energy supplied to a control volume (c.f. **Figure 4**) lead to a change of the enthalpy in the control volume: the control volume temperature increases, and the phase transition occurs. Based on a fully implicit scheme, the energy

*rp=t*

and *Δt*refill,*cn=st*,*p=t* ¼ � *Vp=tρl*ð Þ *T* 1 � *Sp=t*

*cn=stM*\_ *evp=con*

*Vp*,*h*

P

2

*m*\_ *evp=con*,*i* ¼ 0 (2)

� �

*cn=stM*\_ *evp=con*

(3)

(4)

(5)

(6)

*The 2D sketch of liquid invasion rules: Liquid invasion from single liquid throat to its neighboring pore (a), liquid invasion from fully filled liquid throat in a liquid cluster to its neighboring empty pore (b), liquid redistribution from fully filled pore in a liquid cluster to its neighboring empty throat (c).*

#### **Figure 4**

*Heat balance at a vapor pore. The arrows indicate the direction of the heat flux towards each face of the control volume.*

balance equation written for the control volume around pore i in time step Δt is discretized as

$$\begin{split} \sum\_{j} -A\_{\text{ev}, \text{ji}} \lambda\_{\text{eff}, \text{ji}} \frac{T\_{i, t + \Delta t} - T\_{j, t + \Delta t}}{L} &= \frac{\rho\_{\text{eff}, i} c\_{\text{eff}, i} V\_i}{\Delta t} (T\_{i, t + \Delta t} - T\_i) \\ + \Delta H\_{\text{ev}} \sum\_{j} \pi r\_{t, \text{ji}} \,^2 \dot{m}\_{\text{ev}/\text{con}, \text{ji}} + \Delta H\_{\text{ev}} \pi r\_{p, i} \,^2 \dot{m}\_{\text{ev}/\text{con}, i} \end{split} \tag{7}$$

The convective heat transfer boundary condition presented in Eq. 8 is applied for both heating modes. At the bottom of the network, perfect thermal insulation (Eq. 9) is imposed for the convective heating mode, whereas uniform constant temperature (Eq. 10) is set for the contact-heating mode.

$$\left[\lambda\_{\rm eff} \frac{\partial T}{\partial \mathbf{z}}\Big|\_{\mathbf{y}=L\_{\gamma}} + \Delta H\_{\rm evp} \rho\_{\rm I} v\_{\rm l}\Big|\_{\mathbf{y}=L\_{\gamma}}\right] \bullet \mathbf{n} = a \left(T\_{\rm steam} - \left.T\right|\_{\mathbf{y}=L\_{\gamma}}\right) \tag{8}$$

$$\left.\frac{\partial T}{\partial \mathbf{y}}\right|\_{\mathbf{y}=\mathbf{0}} = \mathbf{0} \tag{9}$$

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$$\left.T\right|\_{\mathcal{Y}=L\_{\mathcal{Y}}} = T\_{\text{bottom}}\tag{10}$$

**2.2 Effective parameters of continuous model assessed from the pore network**

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

As mentioned in the previous section, the heat and mass transfer inside the porous domain can be simulated by using pore network models. However, since the pore network model considers the non-isothermal fluid flow in the pore individually, the number of the discrete equations of the model increases with the size of the domain. Thus, the pore network model cannot be used to simulate the macroscale drying process of porous media which is consists of billions of pores. To describe the drying process of macroscopic porous material, the continuous models developed based on the macroscopic effective transport parameters should be used. These effective parameters are often correlated from a serial of experimental data. To provide a fundamental understanding of the link between the transport coefficient and the fluid transport mechanisms, the transport coefficients are revisited by using the pore network simulation results. For example, the influence of network saturation on the non-local equilibrium effect, the effective moisture transport is investigated in the works of Kharaghani et al. [11–13] for the isothermal drying process. The incorporation of these effective parameters in the highly non-isothermal drying process is still an open issue. In the future, the impact of non-isothermal pore scale

fluid transport on the effective parameter should be taken into account.

derived based on the first physical principles using effective parameters.

In continuum-scale models, the porous media is assumed isotropic and homogeneous. Both solid structure and fluid properties are averaged in a representative elementary volume (REV) based on the volume averaging technique. The size of REV should not large enough to avoid the fluctuations of the morphological properties due to micro heterogeneity [14, 15]. The energy and mass conservation equations are

In diffusion model, the mixture of liquid water and vapor water flow is considered as single phase named moisture and the gradient of moisture content is solely

**modeling**

*Impact of heating mode on the drying kinetic curve.*

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 6.**

**3. Continuum-scale models**

**3.1 Diffusion model**

**31**

The simulations are carried out for a 10 � 10 � 10 pore network with throat radius distribution 100 � 10 μm, pore radius distribution 250 � 10 μm and uniform distance between two adjacent pores L = 1000 μm. Because of the high computational demand required for a full simulation of drying, only one realization of network was considered here. The solid skeleton of the network is made of typical glass with thermal conductivity λs = 1 W/mK. The steam velocity at the network surface is 5 m/s. For the contact-heating mode, a uniform temperature boundary condition Tbottom = 105°C is imposed at the network bottom (c.f. Eq. 10), whereas the impermeable heat transfer boundary condition is set for the convective heating mode (c.f. Eq. 9). To depict the influence of heating mode on drying characteristics, the evolution of drying rate and of the local network saturation over network saturation obtained for these two heating modes are shown in **Figures 5** and **6**.

Since a large amount of thermal energy was supplied to the network with the contact-heating mode, the drying rate remains high and drying time is thus short compared to the network with the convective-heating mode. Also, since the surface temperature of the network with the contact-heating mode reaches boiling temperature fast, the surface condensation period is short and thus the amount of condensed liquid at the network surface is less, compared to the convective-heating mode. Moreover, it can be seen that the semi-constant drying rate period obtained from the network with the contact-heating mode is long compared to the convective heating model. This tendency can be explained by the temperature dependency of the surface tension; a lower surface tension obtained at high temperature leads to a lower capillary pressure and a higher liquid pressure at the liquid–vapor-solid interface. The pores/throats near the bottom of the network with the contactheating mode are preferentially emptied. As a result, the vapor fingers appear markedly, and the bottom of the network is dried completely when the middle and top zones are still partly wet. The impact of vapor fingering on the phase distribution is shown in **Figure 5**. More number of liquid clusters is obtained with the contact-heating mode compared to the convective-heating mode. On the other hand, the network with the convective-heating mode is dried evenly from the top to the bottom zone. Since the top and middle zones remain saturated for long time, an extended quasi-constant drying rate period is obtained for the network with the contact-heating mode (c.f. **Figure 6**).

**Figure 5.** *Impact of heating mode on the local saturation evolution over network saturation.*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 6.** *Impact of heating mode on the drying kinetic curve.*

*T*j

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contact-heating mode (c.f. **Figure 6**).

*Impact of heating mode on the local saturation evolution over network saturation.*

**Figure 5.**

**30**

The simulations are carried out for a 10 � 10 � 10 pore network with throat radius distribution 100 � 10 μm, pore radius distribution 250 � 10 μm and uniform distance between two adjacent pores L = 1000 μm. Because of the high computational demand required for a full simulation of drying, only one realization of network was considered here. The solid skeleton of the network is made of typical glass with thermal conductivity λs = 1 W/mK. The steam velocity at the network surface is 5 m/s. For the contact-heating mode, a uniform temperature boundary condition Tbottom = 105°C is imposed at the network bottom (c.f. Eq. 10), whereas the impermeable heat transfer boundary condition is set for the convective heating mode (c.f. Eq. 9). To depict the influence of heating mode on drying characteristics, the evolution of drying rate and of the local network saturation over network saturation obtained for these two heating modes are shown in **Figures 5** and **6**. Since a large amount of thermal energy was supplied to the network with the contact-heating mode, the drying rate remains high and drying time is thus short compared to the network with the convective-heating mode. Also, since the surface temperature of the network with the contact-heating mode reaches boiling temperature fast, the surface condensation period is short and thus the amount of condensed liquid at the network surface is less, compared to the convective-heating mode. Moreover, it can be seen that the semi-constant drying rate period obtained from the network with the contact-heating mode is long compared to the convective heating model. This tendency can be explained by the temperature dependency of the surface tension; a lower surface tension obtained at high temperature leads to a lower capillary pressure and a higher liquid pressure at the liquid–vapor-solid interface. The pores/throats near the bottom of the network with the contactheating mode are preferentially emptied. As a result, the vapor fingers appear markedly, and the bottom of the network is dried completely when the middle and top zones are still partly wet. The impact of vapor fingering on the phase distribution is shown in **Figure 5**. More number of liquid clusters is obtained with the contact-heating mode compared to the convective-heating mode. On the other hand, the network with the convective-heating mode is dried evenly from the top to the bottom zone. Since the top and middle zones remain saturated for long time, an extended quasi-constant drying rate period is obtained for the network with the

*y*¼*Ly* ¼ *Tbottom* (10)

#### **2.2 Effective parameters of continuous model assessed from the pore network modeling**

As mentioned in the previous section, the heat and mass transfer inside the porous domain can be simulated by using pore network models. However, since the pore network model considers the non-isothermal fluid flow in the pore individually, the number of the discrete equations of the model increases with the size of the domain. Thus, the pore network model cannot be used to simulate the macroscale drying process of porous media which is consists of billions of pores. To describe the drying process of macroscopic porous material, the continuous models developed based on the macroscopic effective transport parameters should be used. These effective parameters are often correlated from a serial of experimental data. To provide a fundamental understanding of the link between the transport coefficient and the fluid transport mechanisms, the transport coefficients are revisited by using the pore network simulation results. For example, the influence of network saturation on the non-local equilibrium effect, the effective moisture transport is investigated in the works of Kharaghani et al. [11–13] for the isothermal drying process. The incorporation of these effective parameters in the highly non-isothermal drying process is still an open issue. In the future, the impact of non-isothermal pore scale fluid transport on the effective parameter should be taken into account.

#### **3. Continuum-scale models**

In continuum-scale models, the porous media is assumed isotropic and homogeneous. Both solid structure and fluid properties are averaged in a representative elementary volume (REV) based on the volume averaging technique. The size of REV should not large enough to avoid the fluctuations of the morphological properties due to micro heterogeneity [14, 15]. The energy and mass conservation equations are derived based on the first physical principles using effective parameters.

#### **3.1 Diffusion model**

In diffusion model, the mixture of liquid water and vapor water flow is considered as single phase named moisture and the gradient of moisture content is solely driving force of moisture transport. Since the water diffusion flow leads to the evolution of water concentration in the control volume, the mass conservation of water is derived as

$$\frac{\partial \rho\_0 X}{\partial t} + \frac{\partial}{\partial \mathbf{x}} \cdot \left[ -D\_{\text{eff}} \frac{\partial}{\partial \mathbf{x}} (\rho\_0 X) \right] = \mathbf{0}.\tag{11}$$

The validated diffusion model of wood drying is implemented in a CFD model is

*The evolution of drying history during optimization process for Dref determination (a) and experimental and*

(L = 900 mm, D = 200 mm, and H = 25 mm) is presented in **Figure 8**. The air with a temperature of 140°C and moisture content of 7 g ram vapor/kg dry air flows into the dryer with a velocity of 0.1 m/s. The initial moisture and temperature of wood are 0.61 kg water/kg dried solid and 20°C. The 2D velocity profile of drying agent is presented in **Figure 8** and the evolution of moisture content of plate number (1), (2), (3), and (4) are plotted together with the mean moisture content of all plates in **Figure 9**. The results indicate a noticeable moisture content maldistribution of wood plates which should be remedied by wood plate disturbing during the drying

In the 1970s, the simultaneous heat, mass, and momentum transfer for nonisothermal drying process is proposed by Whitaker based on the volume averaging technique. In this model, all pore-level mechanisms for heat and mass transfer are considered. The liquid flow due to capillary forces, vapor and gas flow due to convection and diffusion are taken into account in this model. Using the volume averaging technique, the properties of fluid and solid phases such as velocity, density, pressure are averaged in REV. Afterward, the macroscopic differential equations were defined in terms of average field quantities. The detail descriptions of Whitaker's model can be found in Vu et al. [20, 21]. In this section, the mass and

used to investigate the drying behavior of a pilot dryer. The sketch of dryer (L = 1000 mm, D = 3000 mm, and H = 500 mm) where 30 plates of wood

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

process.

**Figure 8.**

**33**

**Figure 7.**

**3.2 Whitaker's continuum model**

*numerical mean moisture content evolutions over time (b).*

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

energy conservation equations are briefly recalled.

*Pilot dryer model and velocity profile inside the dryer.*

In Eq. 11, the apparent density of the dry porous medium where the void volume is taken into account is computed as *ρ*0 ¼ *Ms V* (kg dry solid/m3 ), *X* ¼ *Ml Ms* (kg water/kg dry solid) is the dry-based moisture content of the wet solid. *Deff* (m2 /s) is the effective diffusivity of moisture in the porous medium. Similarly, the energy conservation equation is written as the change of energy density caused by the enthalpy flow which consists of contributions from diffusive heat flow and heat conduction, namely,

$$\frac{\partial}{\partial t} \left[ (\rho\_0 c\_{p,t} + \rho\_0 c\_{p,l} X) T \right] - \nabla \cdot \left[ D\_{\rm eff} c\_{p,l} T \nabla (\rho\_0 X) \right] - \nabla \cdot \left[ \lambda\_{\rm eff} \nabla (T) \right] = \mathbf{0}. \tag{12}$$

In Eq. 12, *cp*,*s* and *cp*,*l* (J/kg.K) denote the specific heat capacity of solid and liquid water, respectively; *λeff* (W/m.K) denotes the effective thermal conductivity of the porous medium.

To perform the diffusional model simulation, the effective thermal conductivity and effective moisture diffusivity needed to be known. Several methods such as fitch method, laser flash, hot-disk method and hot-wire method, can be used to measure the effective thermal conductivity directly whereas measurement of effective moisture diffusivity is rather more complex. Generally, the effective diffusivity of wetted porous materials often decreases during the water dehydration process since the liquid water is strongly captured in the small pores under a higher capillary action compared to large pores [16, 17]. Recently, Khan et al. [18] reported that the moisture diffusivity for isotropic porous materials, therein food products, can be described as a function of moisture content as

$$D\_{\rm eff} = D\_{\rm ref} \left( \frac{\mathbf{1} + \mathbf{X}\_0}{\mathbf{1} + \mathbf{X}} \frac{\rho\_l (\mathbf{1} + \mathbf{X}) + \rho\_0 \mathbf{X}}{\rho\_l (\mathbf{1} + \mathbf{X}\_0) + \rho\_0 \mathbf{X}\_0} \right)^2,\tag{13}$$

where *D*ref (m2 /s) is the reference diffusivity, *X0* is the initial moisture content. The question raised here is how to determine the reference effective diffusivity *D*ref. Instead of measurement directly, it can be computed numerically by an optimization routine using invert method. The objective optimizing function *g Deff* � � is the sum of square of difference between numerical moisture content *Xi*,*sim* and experimental *Xi*, exp as

$$\lg\left(D\_{\text{eff}}\right) = \sum\_{i=1:nnd} \left(X\_{\text{exp}}(t\_i) - X\_{sim}(t\_i)\right)^2,\tag{14}$$

where *t*i denotes the time interval *i* of the data sampling. By minimizing the value of objective, the numerical moisture evolution is fitted to the experimental observation and the value of effective diffusivity is obtained. For an example, this method is applied for single wood particle drying process in superheated steam environment. The detail description of the geometrical information, initial and boundary conditions is presented in Le et al. [19]. The effective diffusivity of 6.633 � 10�9 m2 /s determined using the experimental data observed at a drying temperature of 140°C can be used reliable for a large range of drying temperature as shown in **Figure 7**. It implies that the moisture diffusivity model presented in Eq. 13 can be used resonable for wooden porous media.

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 7.**

driving force of moisture transport. Since the water diffusion flow leads to the evolution of water concentration in the control volume, the mass conservation of

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*∂x* � �*Deff*

diffusivity of moisture in the porous medium. Similarly, the energy conservation equation is written as the change of energy density caused by the enthalpy flow which consists of contributions from diffusive heat flow and heat conduction, namely,

*∂ ∂x* ð Þ *ρ*0*X* � �

In Eq. 11, the apparent density of the dry porous medium where the void volume is

*∂t ρ*0*cp*,*s* þ *ρ*0*cp*,*lX* � �*T* � � � ∇ � *Deffcp*,*lT*∇ð Þ *ρ*0*X* � � � ∇ � *λeff* ∇ð Þ *T* � � ¼ 0*:* (12)

In Eq. 12, *cp*,*s* and *cp*,*l* (J/kg.K) denote the specific heat capacity of solid and liquid water, respectively; *λeff* (W/m.K) denotes the effective thermal conductivity of the

To perform the diffusional model simulation, the effective thermal conductivity and effective moisture diffusivity needed to be known. Several methods such as fitch method, laser flash, hot-disk method and hot-wire method, can be used to measure the effective thermal conductivity directly whereas measurement of effective moisture diffusivity is rather more complex. Generally, the effective diffusivity of wetted porous materials often decreases during the water dehydration process since the liquid water is strongly captured in the small pores under a higher capillary action compared to large pores [16, 17]. Recently, Khan et al. [18] reported that the moisture diffusivity for isotropic porous materials, therein food products, can be

> *ρl*ð Þþ 1 þ *X ρ*0*X ρl*ð Þþ 1 þ *X*0 *ρ*0*X*0

/s) is the reference diffusivity, *X0* is the initial moisture content. The

*X*expð Þ� *ti Xsim*ð Þ*ti* � �2

/s determined using the experimental data observed at a drying

� �2

*V* (kg dry solid/m3

¼ 0*:* (11)

*Ms* (kg water/kg dry

/s) is the effective

, (13)

� � is the sum of square of

, (14)

), *X* ¼ *Ml*

*∂ρ*0*X ∂t* þ *∂*

solid) is the dry-based moisture content of the wet solid. *Deff* (m2

taken into account is computed as *ρ*0 ¼ *Ms*

described as a function of moisture content as

*Deff* ¼ *Dref*

*g Deff*

can be used resonable for wooden porous media.

using invert method. The objective optimizing function *g Deff*

� � ¼ X

1 þ *X*0 1 þ *X*

question raised here is how to determine the reference effective diffusivity *D*ref. Instead of measurement directly, it can be computed numerically by an optimization routine

where *t*i denotes the time interval *i* of the data sampling. By minimizing the value of objective, the numerical moisture evolution is fitted to the experimental observation and the value of effective diffusivity is obtained. For an example, this method is applied for single wood particle drying process in superheated steam environment. The detail description of the geometrical information, initial and boundary conditions is presented in Le et al. [19]. The effective diffusivity of

temperature of 140°C can be used reliable for a large range of drying temperature as shown in **Figure 7**. It implies that the moisture diffusivity model presented in Eq. 13

difference between numerical moisture content *Xi*,*sim* and experimental *Xi*, exp as

*i*¼1:*end*

water is derived as

*∂*

porous medium.

where *D*ref (m2

6.633 � 10�9 m2

**32**

*The evolution of drying history during optimization process for Dref determination (a) and experimental and numerical mean moisture content evolutions over time (b).*

The validated diffusion model of wood drying is implemented in a CFD model is used to investigate the drying behavior of a pilot dryer. The sketch of dryer (L = 1000 mm, D = 3000 mm, and H = 500 mm) where 30 plates of wood (L = 900 mm, D = 200 mm, and H = 25 mm) is presented in **Figure 8**. The air with a temperature of 140°C and moisture content of 7 g ram vapor/kg dry air flows into the dryer with a velocity of 0.1 m/s. The initial moisture and temperature of wood are 0.61 kg water/kg dried solid and 20°C. The 2D velocity profile of drying agent is presented in **Figure 8** and the evolution of moisture content of plate number (1), (2), (3), and (4) are plotted together with the mean moisture content of all plates in **Figure 9**. The results indicate a noticeable moisture content maldistribution of wood plates which should be remedied by wood plate disturbing during the drying process.

#### **3.2 Whitaker's continuum model**

In the 1970s, the simultaneous heat, mass, and momentum transfer for nonisothermal drying process is proposed by Whitaker based on the volume averaging technique. In this model, all pore-level mechanisms for heat and mass transfer are considered. The liquid flow due to capillary forces, vapor and gas flow due to convection and diffusion are taken into account in this model. Using the volume averaging technique, the properties of fluid and solid phases such as velocity, density, pressure are averaged in REV. Afterward, the macroscopic differential equations were defined in terms of average field quantities. The detail descriptions of Whitaker's model can be found in Vu et al. [20, 21]. In this section, the mass and energy conservation equations are briefly recalled.

**Figure 8.** *Pilot dryer model and velocity profile inside the dryer.*

**Figure 9.** *Moisture content evolution over time of wood.*

Mass conservation equation of liquid water

$$\frac{\partial}{\partial t}(\rho\_l \varepsilon\_l) + \nabla \cdot (\rho\_l v\_l) + \dot{m}\_v = 0 \tag{15}$$

calculated as *Kr*,*l* ¼ *εl*

bound and free water:

Tref = 0°C.

**Figure 10.**

**35**

*with different drying temperature (d).*

*ψ* 2

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 10**, indicates the validity of the model.

The mass conservation equation of liquid can be rewrite as

*∂t εl*,*f ρl* þ *ρb*

*∂*

and *Kr*,*g* ¼ 1 � *εl*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

*ψ* 2

The Whitaker model's is suitable for the porous media made by impermeable solid phase. This model is applied to describe the drying process of wood material [23]. A good agreement between experimental and numerical data, shown in

For porous media structured by permeable solid, the liquid water is not only transported in the void space, but it also diffuses in the solid phase. In these hygroscopic porous materials, the liquid water is available in the form of both

where *Ml*, *Ml*,*f* and *Mf*,*b* denote the mass of total, free and bound liquid water, respectively. As a result, the total moisture content is the sum of the free and bound moisture content *X* ¼ *Xl*,*f* þ *Xf*,*b*. It should be noted that the bound water is mainly accumulated inside the cells of cellular porous products. A small amount of liquid water is located in small pores inside the solid matrix. In Le et al. [24], the bound water is assumed to be accumulated in the small pores of the macro-solid matrix. Thus, the bound water is removed by both local evaporation and liquid diffusion.

*The distribution of moisture content (a), temperature (b), vapor pressure (c) over drying time for wood particle drying at 120 °C and the comparison between experimental and numerical mean moisture content obtained*

(J/kg) are the specific enthalpy of solid, liquid, vapor and air, respectively. Using the assumption of constant specific heat capacity, the specific enthalpy of these components can be calculated respectively by *hs* ¼ *cp*,*s T* � *Tref* , *hl* ¼ *cp*,*l T* � *Tref* , *ha* ¼ *cp*,*a T* � *Tref* and *hv* ¼ *cp*,*v T* � *Tref* þ *Δhevp Tref* at reference temperature

[21, 22]. In Eq. 19, *hs*, *hl*, *hv* and *ha*

*Ml* ¼ *Ml*,*f* þ *Mf*,*b* (21)

þ ∇ � ð Þþ *ρlv*l,f þ *ρbvb m*\_ *v* ¼ 0 (22)

Mass conservation equation of water vapor

$$\frac{\partial}{\partial t} \left( \rho\_v \varepsilon\_{\rm g} \right) + \nabla \cdot \left( \rho\_v v\_{\rm g} \right) - \nabla \cdot \left( \rho\_{\rm g} D\_{\rm eff} \nabla \left( \frac{\rho\_v}{\rho\_{\rm g}} \right) \right) - \dot{m}\_v = 0 \tag{16}$$

Mass conservation of water in both vapor and liquid phases (sum of Eqs. 15 and 16)

$$\frac{\partial}{\partial t} \left( \rho\_l \mathbf{e}\_l + \rho\_v \mathbf{e}\_\mathbf{g} \right) + \nabla \cdot \left( \rho\_l \mathbf{v}\_l + \rho\_v \mathbf{v}\_\mathbf{g} \right) - \nabla \cdot \left( \rho\_\mathbf{g} D\_{\text{eff}} \nabla \left( \frac{\rho\_v}{\rho\_\mathbf{g}} \right) \right) = \mathbf{0} \tag{17}$$

Mass conservation of air in the gas phase

$$\frac{\partial}{\partial t} \left( \rho\_a \varepsilon\_\mathfrak{g} \right) + \nabla \cdot \left( \rho\_a v\_\mathfrak{g} \right) - \nabla \cdot \left( \rho\_\mathfrak{g} D\_{\mathfrak{eff}} \nabla \left( \frac{\rho\_a}{\rho\_\mathfrak{g}} \right) \right) = \mathbf{0} \tag{18}$$

Energy conservation equation

$$\begin{split} \frac{\partial}{\partial t} \left( \rho\_s \varepsilon\_b h\_t + \rho\_l \varepsilon\_l h\_l + \rho\_v \varepsilon\_g h\_v + \rho\_a \varepsilon\_g h\_a \right) + \nabla \cdot \left[ \rho\_l h\_l \upsilon\_l + (\rho\_v h\_v + \rho\_a h\_a) \upsilon\_g \right] \\ - \nabla \cdot \left[ h\_a \rho\_g D\_{\text{eff}} \nabla \left( \frac{\rho\_a}{\rho\_\mathcal{g}} \right) \right] - \nabla \cdot \left[ h\_v \rho\_g D\_{\text{eff}} \nabla \left( \frac{\rho\_v}{\rho\_\mathcal{g}} \right) \right] - \nabla \cdot \left[ \lambda\_{\text{eff}} \nabla (T) \right] = \mathbf{0} \end{split} \tag{19}$$

In these equations, *ρl*, *ρv* and *ρs* (kg/m3) are the mass density of liquid, vapor and air, respectively. *εl* ¼ *Vl V*, *εg* ¼ *Vg V* are the volume fraction of the liquid and gas phases. The porosity of the medium is computed as *ψ* ¼ *εl* þ *εg* and *εs* ¼ 1 � *ψ* denotes the volume fraction of the solid phase. The superficial velocity of the liquid and gas phased *vl* and *vg* (m/s) are calculated as

$$v\_l = -\frac{KK\_{r,l}}{\mu\_l} \nabla p\_l \text{ and } v\_\mathbf{g} = -\frac{KK\_{r,\mathbf{g}}}{\mu\_\mathbf{g}} \nabla p\_\mathbf{g} \tag{20}$$

where *K* (m2 ) is the absolute permeability of the porous medium. *Kr*,*l* and *Kr*,*v* are the relative permeabilities of the liquid and gas phases, which are respectively *Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

calculated as *Kr*,*l* ¼ *εl ψ* 2 and *Kr*,*g* ¼ 1 � *εl ψ* 2 [21, 22]. In Eq. 19, *hs*, *hl*, *hv* and *ha* (J/kg) are the specific enthalpy of solid, liquid, vapor and air, respectively. Using the assumption of constant specific heat capacity, the specific enthalpy of these components can be calculated respectively by *hs* ¼ *cp*,*s T* � *Tref* , *hl* ¼ *cp*,*l T* � *Tref* , *ha* ¼ *cp*,*a T* � *Tref* and *hv* ¼ *cp*,*v T* � *Tref* þ *Δhevp Tref* at reference temperature Tref = 0°C.

The Whitaker model's is suitable for the porous media made by impermeable solid phase. This model is applied to describe the drying process of wood material [23]. A good agreement between experimental and numerical data, shown in **Figure 10**, indicates the validity of the model.

For porous media structured by permeable solid, the liquid water is not only transported in the void space, but it also diffuses in the solid phase. In these hygroscopic porous materials, the liquid water is available in the form of both bound and free water:

$$\mathbf{M}\_l = \mathbf{M}\_{l\underline{f}} + \mathbf{M}\_{\underline{f},b} \tag{21}$$

where *Ml*, *Ml*,*f* and *Mf*,*b* denote the mass of total, free and bound liquid water, respectively. As a result, the total moisture content is the sum of the free and bound moisture content *X* ¼ *Xl*,*f* þ *Xf*,*b*. It should be noted that the bound water is mainly accumulated inside the cells of cellular porous products. A small amount of liquid water is located in small pores inside the solid matrix. In Le et al. [24], the bound water is assumed to be accumulated in the small pores of the macro-solid matrix. Thus, the bound water is removed by both local evaporation and liquid diffusion. The mass conservation equation of liquid can be rewrite as

$$\frac{\partial}{\partial t} \left( \varepsilon\_{l\,f} \rho\_l + \overline{\rho}\_b \right) + \nabla \cdot \left( \rho\_l v\_{l,\rm f} + \overline{\rho\_b v\_b} \right) + \dot{m}\_v = \mathbf{0} \tag{22}$$

#### **Figure 10.**

*The distribution of moisture content (a), temperature (b), vapor pressure (c) over drying time for wood particle drying at 120 °C and the comparison between experimental and numerical mean moisture content obtained with different drying temperature (d).*

Mass conservation equation of liquid water

Mass conservation equation of water vapor

� � þ ∇ � *ρvvg*

� � þ ∇ � *ρlvl* þ *ρvvg*

� � þ ∇ � *ρavg*

Mass conservation of air in the gas phase

*∂ ∂t ρaεg*

Energy conservation equation

" # !

phased *vl* and *vg* (m/s) are calculated as

�∇ � *haρgDeff*∇ *ρa*

air, respectively. *εl* ¼ *Vl*

where *K* (m2

**34**

*∂t ρsεshs* þ *ρlεlhl* þ *ρvεghv* þ *ρaεgha*

*ρg*

*V*, *εg* ¼ *Vg*

*vl* ¼ � *KKr*,*l μl*

*∂ ∂t ρvεg*

*Moisture content evolution over time of wood.*

*∂t ρlεl* þ *ρvεg*

*∂*

**Figure 9.**

*∂*

*∂*

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

*∂t ρl* ð Þþ *εl* ∇ � ð Þþ *ρlvl m*\_ *v* ¼ 0 (15)

*ρg*

! !

*ρg*

� �

*V* are the volume fraction of the liquid and gas phases.

*μg*

� ∇ � *λeff*∇ð Þ *T* � � ¼ 0

∇*pg* (20)

! !

*ρg*

� *m*\_ *v* ¼ 0 (16)

¼ 0 (17)

¼ 0 (18)

(19)

! !

� � � ∇ � *ρgDeff* ∇ *ρv*

Mass conservation of water in both vapor and liquid phases (sum of Eqs. 15 and 16)

� � � ∇ � *ρgDeff* ∇ *ρv*

� � � ∇ � *ρgDeff* ∇ *ρa*

*ρg*

� � þ ∇ � *ρlhlvl* þ ð Þ *ρvhv* þ *ρaha vg*

� ∇ � *hvρgDeff*∇ *ρv*

" # !

In these equations, *ρl*, *ρv* and *ρs* (kg/m3) are the mass density of liquid, vapor and

∇*pl* and *vg* ¼ � *KKr*,*g*

) is the absolute permeability of the porous medium. *Kr*,*l* and *Kr*,*v*

The porosity of the medium is computed as *ψ* ¼ *εl* þ *εg* and *εs* ¼ 1 � *ψ* denotes the volume fraction of the solid phase. The superficial velocity of the liquid and gas

are the relative permeabilities of the liquid and gas phases, which are respectively

For the superheated steam drying process of rice seed, the bound water diffusivity can be calculated as

$$
\overline{D}\_b = \overline{D}\_{b, \text{ref}} \left( \frac{\rho\_w + \frac{X}{1+X} \rho\_s}{\rho\_w + \frac{X\_0}{1+X\_0} \rho\_s} \right)^2 \text{with} \, \overline{D}\_{b, \text{ref}} = 0.71 \times 10^{-9} \,\text{m}^2/\text{s} \tag{23}
$$

The internal structure of rice seed obtained by μ-CT and the comparison between the experimental observations and numerical results are presented in **Figure 11**. It indicates the validity of the proposed model.

Beside the porous material made by permeable solid phase, we consider the drying process of cellular plant product (i.e. fruits and vegetables) which is comprised of several types of cells such as parenchyma, collenchyma, and solute-conducting cells shown in **Figure 12** [25, 26]. In these materials, the water is mainly accumulated in the cell space. For several cellular product; i.e. eggplant, cucumber; the amount of intracellular water make up more than 95% of total water [27]. During the drying process, both intracellular and extracellular water is removed. The extracellular water is transported to the medium surface due to both capillary flow and internal evaporation. The changing of extracellular water content leads to the different of water potential between the cell space and intercellular void space which is the driving force of the intracellular water transport across the cell membrane as shown in **Figure 12**. A advance heat and mass transfer model for superheated steam drying process of cellular porous material was developed in Le et al. [28]. The conservation equations of extracellular and intracellular water are recalled as

$$
\rho\_l \frac{\partial \varepsilon\_{l,\text{ex}}}{\partial t} + \rho\_l \nabla . (v\_{l,\text{ex}}) = -\dot{m}\_v + j\_{w,1} A\_v \frac{\varepsilon\_{l,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} \tag{24}
$$

In Eqs. 24–26, *εl*,*ex* ¼ *Vl*,*ex*

**Figure 12.**

tion flux is denoted by *m*\_ *v*(kg/m3

*removal (reprinted from [28] with the permission).*

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

in a unit volume. The term *Av εl*,*ex*

diffusive water flux through the cell walls *j*

*∂ ρv* ð Þ *εv*,*ex ∂t*

*ρl ∂εl*,*ex ∂t*

*ρl ∂εl*,in *∂t* þ *j w*,1*Av*

the cell-matrix reads

where *εl*,*in* ¼ *Vl*,*in*

membrane, i.e. *j*

**37**

control volume reads [26, 28].

*ρcp ∂T ∂t*

*w*,1 and *j*

(Pa) and water conductivity of the cell membrane as

�*j w*,1*Av*

the water in the intercellular void space can be written as

þ *ρl*∇*:*ð Þþ *vl*,*ex*

*εl*,*ex εl*,*ex* þ *εv*,*ex*

> *εl*,*ex εl*,*ex* þ *εv*,*ex*

*V* denotes the volume fraction of cell water. Using the local thermal equilibrium assumption, the heat balance equation of the

þ *cp*,*lρlvl*,*ex* þ *cp*,*vρvvv*,*ex* ∇ � *T*

In conservation equation system, the diffusive water fluxes through the cell

These diffusive fluxes are determined by using the concept of the water potential *ϕ*

¼ ∇ � *λeff* ∇*T* � Δ*hevp m*\_ *v* þ *j*

*V* , *εv*,*ex* ¼ *Vv*,*ex*

water vapor in the intercellular void space, respectively. The volumetric evapora-

the medium, which is the exchange area between cells and intercellular void space

liquid, where *ψ* ¼ *εl*,*ex* þ *εv*,*ex* denotes the intercellular porosity of the medium. The

mass conservation equation for the extracellular water vapor can thus be written as

After rearranging of Eqs. 24 and 25, the overall mass conservation equation for

� *j w*,2*Av*

We assumed that the water diffusion across the cell membrane is the mechanism of cell water removal, the mass conservation equation for intracellular water inside

> þ *j w*,2*Av*

*∂ ρv* ð Þ *εv*,*ex ∂t*

þ ∇*:*ð Þ¼ *ρvvv*,*ex m*\_ *v* þ *j*

/m3

*w*,2 (kg/m2

*w*,2*Av*

s). *Av* (m2

*Pictorial representation of extracelluar liquid water (in blue) and intracellular liquid water (in green)*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

*V* are the volume fractions of liquid water and

*εv*,*ex εl*,*ex* þ *εv*,*ex*

þ ∇*:*ð Þ *ρvvv*,*ex*

*εv*,*ex εl*,*ex* þ *εv*,*ex*

*εv*,*ex εl*,*ex* þ *εv*,*ex* (30)

*w*,2*Av*

*w*,2 (red and violet arrows in **Figure 12**), need to be known.

¼ 0

*εv*,*ex εl*,*ex* þ *εv*,*ex*

*εl*,*ex*þ*εv*,*ex*refers to the cell surface area wetted by the

) is the volumetric specific area of

s) serve as vapor sources. The

(27)

(28)

¼ 0 (29)

$$\frac{\partial(\rho\_v \varepsilon\_{v,\text{ex}})}{\partial t} + \nabla.(\rho\_v v\_{v,\text{ex}}) = \dot{m}\_v + j\_{w,2} A\_v \frac{\varepsilon\_{v,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} \tag{25}$$

$$\begin{split} \rho\_l \frac{\partial \varepsilon\_{l,\infty}}{\partial t} + \rho\_l \nabla . (v\_{l,\infty}) + \frac{\partial (\rho\_v \varepsilon\_{v,\infty})}{\partial t} + \nabla . (\rho\_v v\_{v,\infty}) \\ -j\_{w,1} A\_v \frac{\varepsilon\_{l,\infty}}{\varepsilon\_{l,\infty} + \varepsilon\_{v,\infty}} - j\_{w,2} A\_v \frac{\varepsilon\_{v,\infty}}{\varepsilon\_{l,\infty} + \varepsilon\_{v,\infty}} = 0 \end{split} \tag{26}$$

**Figure 11.**

*The morphology of rice seed (a) and the numerical temperature and moisture content evolution over time obtained from diffusion simulations (b).*

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Figure 12.**

For the superheated steam drying process of rice seed, the bound water diffu-

The internal structure of rice seed obtained by μ-CT and the comparison between the experimental observations and numerical results are presented in

þ *ρl*∇*:*ð Þ¼� *vl*,*ex m*\_ *v* þ *j*

þ ∇*:*ð Þ¼ *ρvvv*,*ex m*\_ *v* þ *j*

*∂ ρv* ð Þ *εv*,*ex ∂t*

� *j w*,2*Av*

*The morphology of rice seed (a) and the numerical temperature and moisture content evolution over time*

þ *ρl*∇*:*ð Þþ *vl*,*ex*

*εl*,*ex εl*,*ex* þ *εv*,*ex* *w*,1*Av*

*w*,2*Av*

þ ∇*:*ð Þ *ρvvv*,*ex*

*εv*,*ex εl*,*ex* þ *εv*,*ex*

*εl*,*ex εl*,*ex* þ *εv*,*ex*

*εv*,*ex εl*,*ex* þ *εv*,*ex*

¼ 0

Beside the porous material made by permeable solid phase, we consider the drying process of cellular plant product (i.e. fruits and vegetables) which is comprised of several types of cells such as parenchyma, collenchyma, and solute-conducting cells shown in **Figure 12** [25, 26]. In these materials, the water is mainly accumulated in the cell space. For several cellular product; i.e. eggplant, cucumber; the amount of intracellular water make up more than 95% of total water [27]. During the drying process, both intracellular and extracellular water is removed. The extracellular water is transported to the medium surface due to both capillary flow and internal evaporation. The changing of extracellular water content leads to the different of water potential between the cell space and intercellular void space which is the driving force of the intracellular water transport across the cell membrane as shown in **Figure 12**. A advance heat and mass transfer model for superheated steam drying process of cellular porous material was developed in Le et al. [28]. The conservation equations of

with*Db*,*ref* ¼ 0*:*71 � 10�9m2

*=*s (23)

(24)

(25)

(26)

sivity can be calculated as

*Db* ¼ *Db*,*ref*

*ρw* þ *X* 1þ*X ρs*

!2

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

*ρw* þ *X*0 1þ*X*0 *ρs*

**Figure 11**. It indicates the validity of the proposed model.

extracellular and intracellular water are recalled as

*ρl ∂εl*,*ex ∂t*

*ρl ∂εl*,*ex ∂t*

**Figure 11.**

**36**

*obtained from diffusion simulations (b).*

*∂ ρv* ð Þ *εv*,*ex ∂t*

> �*j w*,1*Av*

*Pictorial representation of extracelluar liquid water (in blue) and intracellular liquid water (in green) removal (reprinted from [28] with the permission).*

In Eqs. 24–26, *εl*,*ex* ¼ *Vl*,*ex V* , *εv*,*ex* ¼ *Vv*,*ex V* are the volume fractions of liquid water and water vapor in the intercellular void space, respectively. The volumetric evaporation flux is denoted by *m*\_ *v*(kg/m3 s). *Av* (m2 /m3 ) is the volumetric specific area of the medium, which is the exchange area between cells and intercellular void space in a unit volume. The term *Av εl*,*ex εl*,*ex*þ*εv*,*ex*refers to the cell surface area wetted by the liquid, where *ψ* ¼ *εl*,*ex* þ *εv*,*ex* denotes the intercellular porosity of the medium. The diffusive water flux through the cell walls *j w*,2 (kg/m2 s) serve as vapor sources. The mass conservation equation for the extracellular water vapor can thus be written as

$$\frac{\partial(\rho\_v \varepsilon\_{v,\text{ex}})}{\partial t} + \nabla.(\rho\_v v\_{v,\text{ex}}) = \dot{m}\_v + j\_{w,2} A\_v \frac{\varepsilon\_{v,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} \tag{27}$$

After rearranging of Eqs. 24 and 25, the overall mass conservation equation for the water in the intercellular void space can be written as

$$\begin{split} \rho\_l \frac{\partial \varepsilon\_{l, \text{ex}}}{\partial t} + \rho\_l \nabla . (v\_{l, \text{ex}}) + \frac{\partial (\rho\_v \varepsilon\_{v, \text{ex}})}{\partial t} + \nabla . (\rho\_v v\_{v, \text{ex}}) \\ -j\_{w, 1} A\_v \frac{\varepsilon\_{l, \text{ex}}}{\varepsilon\_{l, \text{ex}} + \varepsilon\_{v, \text{ex}}} - j\_{w, 2} A\_v \frac{\varepsilon\_{v, \text{ex}}}{\varepsilon\_{l, \text{ex}} + \varepsilon\_{v, \text{ex}}} = \mathbf{0} \end{split} \tag{28}$$

We assumed that the water diffusion across the cell membrane is the mechanism of cell water removal, the mass conservation equation for intracellular water inside the cell-matrix reads

$$\rho\_l \frac{\partial \varepsilon\_{l,\text{in}}}{\partial t} + j\_{w,1} A\_v \frac{\varepsilon\_{l,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} + j\_{w,2} A\_v \frac{\varepsilon\_{v,\text{ex}}}{\varepsilon\_{l,\text{ex}} + \varepsilon\_{v,\text{ex}}} = \mathbf{0} \tag{29}$$

where *εl*,*in* ¼ *Vl*,*in V* denotes the volume fraction of cell water.

Using the local thermal equilibrium assumption, the heat balance equation of the control volume reads [26, 28].

$$\begin{split} \left< \rho \varepsilon\_{p} \right> & \frac{\partial T}{\partial t} + \left( c\_{p,l} \rho\_{l} v\_{l,\infty} + c\_{p,v} \rho\_{v} v\_{v,\infty} \right) \nabla \cdot T \\ &= \nabla \cdot \left( \lambda\_{\text{eff}} \nabla T \right) - \Delta h\_{\text{exp}} \left( \dot{m}\_{v} + j\_{w,\mathfrak{L}} A\_{v} \frac{\varepsilon\_{v,\infty}}{\varepsilon\_{l,\infty} + \varepsilon\_{v,\infty}} \right) \end{split} \tag{30}$$

In conservation equation system, the diffusive water fluxes through the cell membrane, i.e. *j w*,1 and *j w*,2 (red and violet arrows in **Figure 12**), need to be known. These diffusive fluxes are determined by using the concept of the water potential *ϕ* (Pa) and water conductivity of the cell membrane as

**Figure 13.**

*Simulated and experimental evolution of the normalized total, intracellular and extracellular moisture content over time for potato drying in superheated steam environment at 180°C (replotted from [28] with the permission).*

$$j\_{w,1} = -\rho\_l k\_p \left(\phi\_{w,in} - \phi\_{l, \text{ex}}\right) \tag{31}$$

scales shall be carried out. For example, several effective macroscopic parameters such as the capillary pressure curve or the relative permeabilities may be obtained from a post-processing of the Monte-Carlo simulation results of pore-scale modeling. The moisture diffusivity under the thermal effect also should be revisted and compared with the insothermal drying process. This upscaling statergy can help on

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

This work is funded by the VIETNAMESE MINISTRY OF EDUCATION AND

School of Heat Engineering and Refrigeration, Hanoi University of Science and

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to<EMAIL_ADDRESS>
both the product and system design, operation, and optimization.

TRAINING (MOET) under research project B2021-BKA-12.

**Acknowledgements**

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

**Author details**

Technology, Hanoi, Vietnam

provided the original work is properly cited.

Kieu Hiep Le

**39**

$$j\_{w,2} = -\rho\_l k\_p \left(\phi\_{w,in} - \phi\_{v,\infty}\right) \tag{32}$$

The water potential *ϕ* is computed as the difference of the total energy of one kilogram water molecules compared to liquid water molecule energy at the free liquid – water interface at 1 bar [25]. For the intracellular water, the water potential *ϕw*,*in*, which was empirically determined in [29] as a function of the moisture content *ϕw*,*in* ¼ *f X*ð Þ *in* . The potential of liquid water in the intercellular void space *ϕl*,*ex* is calculated as the capillary pressure, i.e. *ϕl*,*ex* ¼ �*pc* . The water vapor potential *ϕv*,*ex* is computed from the gas theory based on the Gibbs free energy [30].

$$\phi\_{v,\text{ex}} = \frac{\tilde{R}T\rho\_l}{\tilde{M}\_w} \ln \frac{p\_{v,\text{ex}}}{p\_{v,\text{sat}}} \tag{33}$$

This model has been used to describe the superheated steam drying process of potato. As can be seen in **Figure 13**, the experimental observations can be reflected fairly by numerical results. It implies the predictive potential of the proposed model. Furthermore, the extracellular moisture is removed very soon when the dry process commences. The drying process is mainly driven by the dehydration of intracellular water by the diffusion mechanism. Although the Whitaker's continuum approach is effectively used to predict the drying behavior, the parameters of the continuum model is seemingly difficult to be measured. Thus, in the future, with the help of pore-scale model, these effective parameters should be theoretically extracted from the pore level simulations.

#### **4. Conclusions**

In this chapter, some recent advance models on heat and mass transfer during non-isothermal drying process of porous media have been reviewed. It indicates that the drying process at pore-scale and macro-scale has been thoroughly investigated. However, a system study on the bridges between the pore-, macro- and plant *Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

scales shall be carried out. For example, several effective macroscopic parameters such as the capillary pressure curve or the relative permeabilities may be obtained from a post-processing of the Monte-Carlo simulation results of pore-scale modeling. The moisture diffusivity under the thermal effect also should be revisted and compared with the insothermal drying process. This upscaling statergy can help on both the product and system design, operation, and optimization.

### **Acknowledgements**

*j*

*j*

*ϕl*,*ex* is calculated as the capillary pressure, i.e. *ϕl*,*ex* ¼ �*pc*

extracted from the pore level simulations.

**4. Conclusions**

**38**

**Figure 13.**

*permission).*

*w*,1 ¼ �*ρlkp ϕw*,*in* � *ϕl*,*ex*

*Simulated and experimental evolution of the normalized total, intracellular and extracellular moisture content over time for potato drying in superheated steam environment at 180°C (replotted from [28] with the*

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

*w*,2 ¼ �*ρlkp ϕw*,*in* � *ϕv*,*ex*

The water potential *ϕ* is computed as the difference of the total energy of one kilogram water molecules compared to liquid water molecule energy at the free liquid – water interface at 1 bar [25]. For the intracellular water, the water potential *ϕw*,*in*, which was empirically determined in [29] as a function of the moisture content *ϕw*,*in* ¼ *f X*ð Þ *in* . The potential of liquid water in the intercellular void space

*ϕv*,*ex* is computed from the gas theory based on the Gibbs free energy [30].

*ϕv*,*ex* ¼ *RT*~ *ρl M*~ *w*

This model has been used to describe the superheated steam drying process of potato. As can be seen in **Figure 13**, the experimental observations can be reflected fairly by numerical results. It implies the predictive potential of the proposed model. Furthermore, the extracellular moisture is removed very soon when the dry process commences. The drying process is mainly driven by the dehydration of intracellular water by the diffusion mechanism. Although the Whitaker's continuum approach is effectively used to predict the drying behavior, the parameters of the continuum model is seemingly difficult to be measured. Thus, in the future, with the help of pore-scale model, these effective parameters should be theoretically

In this chapter, some recent advance models on heat and mass transfer during non-isothermal drying process of porous media have been reviewed. It indicates that the drying process at pore-scale and macro-scale has been thoroughly investigated. However, a system study on the bridges between the pore-, macro- and plant

ln *pv*,*ex pv*,*sat*

(31)

(32)

. The water vapor potential

(33)

This work is funded by the VIETNAMESE MINISTRY OF EDUCATION AND TRAINING (MOET) under research project B2021-BKA-12.

### **Author details**

Kieu Hiep Le School of Heat Engineering and Refrigeration, Hanoi University of Science and Technology, Hanoi, Vietnam

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] M. Prat, Recent advances in porescale models for drying of porous media, Chemical Engineering Journal 86 (2002) 153–164.

[2] V.K. Surasani, T. Metzger, E. Tsotsas, Consideration of heat transfer in pore network modelling of convective drying, International journal of heat and mass transfer 51 (2008) 2506–2518.

[3] A.S. Mujumdar, Handbook of Industrial Drying, CRC Press, 2014.

[4] T. Defraeye, Advanced computational modelling for drying processes – A review, Applied Energy 131 (2014) 323–344.

[5] M. Sahimi, Flow and transport in porous media and fractured rock: From classical methods to modern approaches, 2nd rev. and enlarged ed. ed., Wiley-Vch Verlag GmbH & Co. KGaA, Weinheim, op. 2011.

[6] T. Sochi, Pore-Scale Modeling of Non-Newtonian Flow in Porous Media, PhD dissertation, London, 2007.

[7] S. Taslimi Taleghani, M. Dadvar, Two dimensional pore network modelling and simulation of nonisothermal drying by the inclusion of viscous effects, International Journal of Multiphase Flow 62 (2014) 37–44.

[8] V.K. Surasani, T. Metzger, E. Tsotsas, Drying Simulations of Various 3D Pore Structures by a Nonisothermal Pore Network Model, Drying Technology 28 (2010) 615–623.

[9] V.K. Surasani, T. Metzger, E. Tsotsas, Influence of heating mode on drying behavior of capillary porous media: Pore scale modeling, Chemical engineering science 63 (2008) 5218–5228.

[10] K.H. Le, A. Kharaghani, C. Kirsch, E. Tsotsas, Discrete pore network modeling

of superheated steam drying, Drying Technology 35 (2017) 1584–1601.

appropriate effective diffusivity for different food materials, Drying Technology 35 (2017) 335–346.

*DOI: http://dx.doi.org/10.5772/intechopen.97317*

Isothermal Drying of Plant Cellular Materials Based on the Pore Network Approach, Drying Technology 32

[27] A. Halder, A.K. Datta, R.M. Spanswick, Water transport in cellular tissues during thermal processing, AIChE J. 57 (2011) 2574–2588.

Continuum-scale modeling of superheated steam drying of cellular plant porous media, International journal of heat and mass transfer 124

A moisture transfer model for isothermal drying of plant cellular materials based on the pore network approach, Drying Technology 32 (2014)

[28] K.H. Le, E. Tsotsas, A. Kharaghani,

[29] B. Xiao, J. Chang, X. Huang, X. Liu,

[30] P.S. Nobel, Physicochemical and environmental plant physiology, 4th ed., Academic Press, Amsterdam,

(2014) 1071–1081.

*Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of…*

(2018) 1033–1044.

1071–1081.

Boston, 2009.

[19] K.H. Le, T.T.H. Tran, E. Tsotsas, A. Kharaghani, Superheated steam drying of single wood particles: Modeling and comparative study with hot air drying, Chem. Eng. Technol. 44 (2021) 114–123.

[20] H.T. Vu, E. Tsotsas, Mass and Heat Transport Models for Analysis of the Drying Process in Porous Media: A Review and Numerical Implementation, International Journal of Chemical Engineering 2018 (2018) 1–13.

[21] H.T. Vu, E. Tsotsas, A Framework and Numerical Solution of the Drying Process in Porous Media by Using a Continuous Model, International Journal of Chemical Engineering 2019

[22] K.H. Le, N. Hampel, A. Kharaghani, A. Bück, E. Tsotsas, Superheated steam drying of single wood particles: A characteristic drying curve model deduced from continuum model simulations and assessed by

experiments, Drying Technology 36

[23] A. Kharaghani, K.H. Le, T.T.H. Tran, E. Tsotsas, Reaction engineering approach for modeling single wood particle drying at elevated air temperature, Chemical engineering science 199 (2019) 602–612.

[24] N. Hampel, K.H. Le, A. Kharaghani, E. Tsotsas, Continuous modeling of superheated steam drying of single rice grains, Drying Technology 37 (2019)

[25] P.S. Nobel, Physicochemical and environmental plant physiology, 4th ed.

[26] B. Xiao, J. Chang, X. Huang, X. Liu,

ed., Academic Press, [Place of publication not identified], 2009.

A Moisture Transfer Model for

(2019) 1–16.

(2018) 1866–1881.

1583–1596.

**41**

[11] A. Attari Moghaddam, A. Kharaghani, E. Tsotsas, M. Prat, Kinematics in a slowly drying porous medium: Reconciliation of pore network simulations and continuum modeling, Physics of Fluids 29 (2017) 22102.

[12] X. Lu, A. Kharaghani, E. Tsotsas, Transport parameters of macroscopic continuum model determined from discrete pore network simulations of drying porous media: Throat-node vs. throat-pore configurations, Chemical engineering science 223 (2020) 115723.

[13] X. Lu, E. Tsotsas, A. Kharaghani, Insights into evaporation from the surface of capillary porous media gained by discrete pore network simulations, International journal of heat and mass transfer 168 (2021) 120877.

[14] P. Perré, The proper use of mass diffusion equations in drying modeling: Introducing the drying intensity number, Drying Technology 33 (2015) 1949–1962.

[15] S. Whitaker, Simultaneous heat, mass, and momentum transfer in porous media: A theory of drying, in: Advances in Heat Transfer Volume 13, Elsevier, 1977, pp. 119–203.

[16] E. Tsotsas, A.S. Mujumdar (Eds.), Modern Drying Technology, Wiley-Vch Verlag GmbH & Co. KGaA, Weinheim, Germany, 2007.

[17] P. Suvarnakuta, S. Devahastin, A.S. Mujumdar, A mathematical model for low-pressure superheated steam drying of a biomaterial, Chemical Engineering and Processing: Process Intensification 46 (2007) 675–683.

[18] M.I.H. Khan, C. Kumar, M.U.H. Joardder, M.A. Karim, Determination of *Multiscale Modeling of Non-Isothermal Fluid Transport Involved in Drying Process of… DOI: http://dx.doi.org/10.5772/intechopen.97317*

appropriate effective diffusivity for different food materials, Drying Technology 35 (2017) 335–346.

**References**

153–164.

[1] M. Prat, Recent advances in porescale models for drying of porous media, Chemical Engineering Journal 86 (2002)

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

of superheated steam drying, Drying Technology 35 (2017) 1584–1601.

[12] X. Lu, A. Kharaghani, E. Tsotsas, Transport parameters of macroscopic continuum model determined from discrete pore network simulations of drying porous media: Throat-node vs. throat-pore configurations, Chemical engineering science 223 (2020) 115723.

[13] X. Lu, E. Tsotsas, A. Kharaghani, Insights into evaporation from the surface of capillary porous media gained by discrete pore network simulations, International journal of heat and mass

[14] P. Perré, The proper use of mass diffusion equations in drying modeling: Introducing the drying intensity number, Drying Technology 33 (2015)

[15] S. Whitaker, Simultaneous heat, mass, and momentum transfer in porous media: A theory of drying, in: Advances in Heat Transfer Volume 13,

[16] E. Tsotsas, A.S. Mujumdar (Eds.), Modern Drying Technology, Wiley-Vch Verlag GmbH & Co. KGaA, Weinheim,

[17] P. Suvarnakuta, S. Devahastin, A.S. Mujumdar, A mathematical model for low-pressure superheated steam drying of a biomaterial, Chemical Engineering and Processing: Process Intensification

[18] M.I.H. Khan, C. Kumar, M.U.H. Joardder, M.A. Karim, Determination of

Elsevier, 1977, pp. 119–203.

Germany, 2007.

46 (2007) 675–683.

transfer 168 (2021) 120877.

1949–1962.

[11] A. Attari Moghaddam, A. Kharaghani, E. Tsotsas, M. Prat, Kinematics in a slowly drying porous medium: Reconciliation of pore network simulations and continuum modeling, Physics of Fluids 29 (2017) 22102.

[2] V.K. Surasani, T. Metzger, E. Tsotsas, Consideration of heat transfer in pore network modelling of convective drying, International journal of heat and mass transfer 51 (2008) 2506–2518.

[3] A.S. Mujumdar, Handbook of Industrial Drying, CRC Press, 2014.

computational modelling for drying processes – A review, Applied Energy

[5] M. Sahimi, Flow and transport in porous media and fractured rock: From

approaches, 2nd rev. and enlarged ed. ed., Wiley-Vch Verlag GmbH & Co.

[6] T. Sochi, Pore-Scale Modeling of Non-Newtonian Flow in Porous Media, PhD dissertation, London, 2007.

[7] S. Taslimi Taleghani, M. Dadvar, Two dimensional pore network modelling and simulation of nonisothermal drying by the inclusion of viscous effects, International Journal of Multiphase Flow 62 (2014) 37–44.

[8] V.K. Surasani, T. Metzger, E. Tsotsas, Drying Simulations of Various 3D Pore Structures by a Nonisothermal Pore Network Model, Drying Technology 28

[9] V.K. Surasani, T. Metzger, E. Tsotsas, Influence of heating mode on drying behavior of capillary porous media: Pore scale modeling, Chemical engineering

[10] K.H. Le, A. Kharaghani, C. Kirsch, E. Tsotsas, Discrete pore network modeling

science 63 (2008) 5218–5228.

(2010) 615–623.

**40**

[4] T. Defraeye, Advanced

classical methods to modern

KGaA, Weinheim, op. 2011.

131 (2014) 323–344.

[19] K.H. Le, T.T.H. Tran, E. Tsotsas, A. Kharaghani, Superheated steam drying of single wood particles: Modeling and comparative study with hot air drying, Chem. Eng. Technol. 44 (2021) 114–123.

[20] H.T. Vu, E. Tsotsas, Mass and Heat Transport Models for Analysis of the Drying Process in Porous Media: A Review and Numerical Implementation, International Journal of Chemical Engineering 2018 (2018) 1–13.

[21] H.T. Vu, E. Tsotsas, A Framework and Numerical Solution of the Drying Process in Porous Media by Using a Continuous Model, International Journal of Chemical Engineering 2019 (2019) 1–16.

[22] K.H. Le, N. Hampel, A. Kharaghani, A. Bück, E. Tsotsas, Superheated steam drying of single wood particles: A characteristic drying curve model deduced from continuum model simulations and assessed by experiments, Drying Technology 36 (2018) 1866–1881.

[23] A. Kharaghani, K.H. Le, T.T.H. Tran, E. Tsotsas, Reaction engineering approach for modeling single wood particle drying at elevated air temperature, Chemical engineering science 199 (2019) 602–612.

[24] N. Hampel, K.H. Le, A. Kharaghani, E. Tsotsas, Continuous modeling of superheated steam drying of single rice grains, Drying Technology 37 (2019) 1583–1596.

[25] P.S. Nobel, Physicochemical and environmental plant physiology, 4th ed. ed., Academic Press, [Place of publication not identified], 2009.

[26] B. Xiao, J. Chang, X. Huang, X. Liu, A Moisture Transfer Model for

Isothermal Drying of Plant Cellular Materials Based on the Pore Network Approach, Drying Technology 32 (2014) 1071–1081.

[27] A. Halder, A.K. Datta, R.M. Spanswick, Water transport in cellular tissues during thermal processing, AIChE J. 57 (2011) 2574–2588.

[28] K.H. Le, E. Tsotsas, A. Kharaghani, Continuum-scale modeling of superheated steam drying of cellular plant porous media, International journal of heat and mass transfer 124 (2018) 1033–1044.

[29] B. Xiao, J. Chang, X. Huang, X. Liu, A moisture transfer model for isothermal drying of plant cellular materials based on the pore network approach, Drying Technology 32 (2014) 1071–1081.

[30] P.S. Nobel, Physicochemical and environmental plant physiology, 4th ed., Academic Press, Amsterdam, Boston, 2009.

Section 2

Mathematical Models in

Porous Media and Solutions

**43**

Section 2

## Mathematical Models in Porous Media and Solutions

**Chapter 3**

**Abstract**

*Twinkle R. Singh*

differential equations

**1. Introduction**

**45**

Study on Approximate Analytical

This chapter is about the, Variational iteration method (VIM); Adomian decomposition method and its modification has been applied to solve nonlinear partial differential equation of imbibition phenomenon in oil recovery process. The important condition of counter-current imbibition phenomenon as *vi* ¼ �*vn*, has been considered here main aim, here is to determine the saturation of injected fluid *Si*ð Þ *x*, *t* during oil recovery process which is a function of distance *ξ* and time *θ*, therefore saturation *Si* is chosen as a dependent variable while *x and t* are chosen as independent variable. The solution of the phenomenon has been found by VIM, ADM and Laplace Adomian decomposition method (LADM). The effectiveness of

**Keywords:** Variational Iteration method (VIM), Adomian decomposition method (ADM), Laplace Adomian decomposition method (LADM), nonlinear partial

First, the variational iteration method was proposed by He [1] in 1998 and was successfully applied to autonomous ordinary differential equation, to nonlinear partial differential equations with variable coefficients. In recent times a good deal of attention has been devoted to the study of the method. The reliability of the method and the reduction in the size of the computational domain give this method a wide applicability. The VIM based on the use of restricted variations and correction functional which has found a wide application for the solution of nonlinear ordinary and partial differential equations, e.g., [2–10]. This method does not require the presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives and whereas the Adomian decomposition method was before the Nineteen Eighties, it was developed by Adomian [11, 12] for solving linear or nonlinear ordinary, partial and Delay differential equations. A large type of issues in mathematics, physics, engineering, biology, chemistry and other sciences have been solved using the ADM, as reported by many authors [13]. The Adomian decomposition method (ADM) [11–28] is well set systematic method for practical solution of linear or nonlinear and deterministic or stochastic operator equations, including ordinary differential equations (ODEs), partial differential equations

Method with Its Application

Arising in Fluid Flow

our method is illustrated by different numerical.

**Chapter 3**

## Study on Approximate Analytical Method with Its Application Arising in Fluid Flow

*Twinkle R. Singh*

#### **Abstract**

This chapter is about the, Variational iteration method (VIM); Adomian decomposition method and its modification has been applied to solve nonlinear partial differential equation of imbibition phenomenon in oil recovery process. The important condition of counter-current imbibition phenomenon as *vi* ¼ �*vn*, has been considered here main aim, here is to determine the saturation of injected fluid *Si*ð Þ *x*, *t* during oil recovery process which is a function of distance *ξ* and time *θ*, therefore saturation *Si* is chosen as a dependent variable while *x and t* are chosen as independent variable. The solution of the phenomenon has been found by VIM, ADM and Laplace Adomian decomposition method (LADM). The effectiveness of our method is illustrated by different numerical.

**Keywords:** Variational Iteration method (VIM), Adomian decomposition method (ADM), Laplace Adomian decomposition method (LADM), nonlinear partial differential equations

#### **1. Introduction**

First, the variational iteration method was proposed by He [1] in 1998 and was successfully applied to autonomous ordinary differential equation, to nonlinear partial differential equations with variable coefficients. In recent times a good deal of attention has been devoted to the study of the method. The reliability of the method and the reduction in the size of the computational domain give this method a wide applicability. The VIM based on the use of restricted variations and correction functional which has found a wide application for the solution of nonlinear ordinary and partial differential equations, e.g., [2–10]. This method does not require the presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives and whereas the Adomian decomposition method was before the Nineteen Eighties, it was developed by Adomian [11, 12] for solving linear or nonlinear ordinary, partial and Delay differential equations. A large type of issues in mathematics, physics, engineering, biology, chemistry and other sciences have been solved using the ADM, as reported by many authors [13]. The Adomian decomposition method (ADM) [11–28] is well set systematic method for practical solution of linear or nonlinear and deterministic or stochastic operator equations, including ordinary differential equations (ODEs), partial differential equations

**44**

(PDEs), integral equations, integro-differential equations, etc. The ADM is considered as a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering. It allows us to solve both nonlinear initial value problems (IVPs) and boundary value problems (BVPs) [17, 29–46] without unphysical restrictive assumptions such as required by linearization, perturbation, ad hoc assumptions, guessing the initial term or a set of basic functions, and so forth. The accuracy of the analytic approximate solutions obtained can be verified by direct substitution. More advantages of the ADM over the variational iteration method is mentioned in Wazwaz [22, 28]. A key notion is the Adomian polynomials, which are tailored to the particular nonlinearity to solve nonlinear operator equations. A key concept of the Adomian decomposition series is that it is computationally advantageous rearrangement of the Banach-space analog of the Taylor expansion series about the initial solution component function, which permits solution by recursion. The selection behind choice of decomposition is nonunique, which provides a valuable advantage to the analyst, permitting the freedom to design modified recursion schemes for ease of computation in realistic systems.

The mathematical condition for imbibition phenomenon is given by Scheidegger

Where *vi* & *vn* are the seepage velocities of injected & native liquids respectively.

*vn* ¼ �*vi*

*ki* ¼ *S*3 *i kn* ¼ 1 � *αSn*, α ¼ 1*:*11

Where *ki* & *kn* denotes fictitious relative permeability. *Si* & *Sn* denotes satura-

According to the Darcy's law, the basic equations of the phenomenon as; [78]

*δn K*

> þ *∂vi*

þ *∂vn*

Combing equations (1)-(5) and using the relation for capillary pressure as,

*KD S*ð Þ*i β ∂Si*

*δnki*þ*δikn* and β being small capillary pressure coefficient.

*φL*2 , 0≤ *x*≤

*∂x*

*LSio*

Where v*i* and v*n* are the seepage velocities, k*i* and kare the relative permeabilities *δi* and *δn* are the kinematic viscosities (which are constants), p*i* and p*n* are pressure of the injected and native liquid respectively, *φ* and *K* are the porosity and the permeability of the homogeneous porous medium; S*i* is the saturation of the injected liquid; p*c* is the capillary pressure and t is the time. The co-ordinate x is measured along the axis of the cylindrical medium, the origin being located at the

*∂pi ∂x*

*∂pn ∂x*

*vi* ¼ �*vn* (3) *pc* ¼ *pn* � *pi* (4)

*∂x* ¼ 0 (5)

*∂x* ¼ 0 (6)

¼ 0 (7)

*B :* (8)

(1)

(2)

*vi* ¼ � *ki δi K*

*vn* ¼ � *kn*

*φ ∂Si ∂t*

*φ ∂Sn ∂t*

þ *∂ ∂x*

*L* , *θ* ¼ *Lt*

It is assumed is that an average value of D(S*i*) =*D* (S*i*)

*ξ* ¼ *x*

The relation between relative permeability and phase-saturation,

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow*

tions of injected and native liquids respectively.

*DOI: http://dx.doi.org/10.5772/intechopen.97548*

**3. Mathematical structure of the model**

imbibition face x=0.

p*c*=*β Si* [70], we get,

Where D(S*i*) = *kikn*

Eq. (7), becomes;

**47**

Using the transformation,

*φ ∂Si ∂t*

[78]); viz,

Same way Laplace Adomian's Decomposition Method (LADM) was first introduced by Khuri [47, 48]. The Laplace Adomian Decomposition Method (LADM) is formed with combination of the Adomian Decomposition Method (ADM) Adomian [29, 49] and Laplace transforms. LADM is a promising method and has been applied in solving various nonlinear systems of differential equations [36, 50–56]. In a variety of applied sciences, systems of partial differential equations have attracted much attention e.g. [50, 57–75]. The general ideas and the essentiality of these systems are of wide applicability. Agadjanov [56] solved Duffing equation with the help of LDM. Elgazery [51, 76] had applied Laplace decomposition method for the solution of Falkner-Skan equation.

In the solution procedure of VIM; many repeated computations and computations of the unneeded forms, which take more time and effort beyond it, so a modification has been shown to reduce these unneeded forms.

On the other hand, few researchers have been discussed imbibition phenomenon in homogenous porous media with different point of view for example, researchers taking different perspectives for this phenomenon; [77, 78] and some others have analyzed it for homogeneous porous medium.

In this Present investigated model, Imbibition takes place over a small part of a large oil formatted region taken as a cylindrical piece of homogeneous porous medium. In this model, we have considered the important condition of countercurrent imbibition phenomenon as *vi* ¼ �*vn*, Our purpose is to determine the saturation of injected fluid *Si*ð Þ *x*, *t* during oil recovery process which is a function of distance *ξ* and time *θ*, therefore saturation *Si* has been chosen as a dependent variable while *x* and *t* are chosen as independent variable.

#### **2. Imbibition phenomenon**

It is the process by which a wetting fluid displaces a non-wetting fluid the initially saturates a porous sample, by capillary forces alone. Suppose a sample is completely saturated with a non-wetting fluid, and same wetting fluid is introduced on its surface. There will be spontaneous flow of wetting fluid into the medium, causing displacement of the non-wetting fluid. This is called imbibition phenomenon. The rate of imbibition is greater if the wettability of the porous medium, by the imbibed fluid, is higher.

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow DOI: http://dx.doi.org/10.5772/intechopen.97548*

The mathematical condition for imbibition phenomenon is given by Scheidegger [78]); viz,

$$v\_n = -v\_i$$

Where *vi* & *vn* are the seepage velocities of injected & native liquids respectively. The relation between relative permeability and phase-saturation,

$$k\_i = \mathbb{S}\_i^3$$

$$k\_n = \mathbf{1} - a\mathbf{S}\_n, \alpha = \mathbf{1}.\mathbf{1}\mathbf{1}$$

Where *ki* & *kn* denotes fictitious relative permeability. *Si* & *Sn* denotes saturations of injected and native liquids respectively.

#### **3. Mathematical structure of the model**

(PDEs), integral equations, integro-differential equations, etc. The ADM is considered as a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering. It allows us to solve both nonlinear initial value problems (IVPs) and boundary value problems (BVPs) [17, 29–46] without unphysical restrictive assumptions such as required by linearization, perturbation, ad hoc assumptions, guessing the initial term or a set of basic functions, and so forth. The accuracy of the analytic approximate solutions obtained can be verified by direct substitution. More advantages of the ADM over the variational iteration method is mentioned in Wazwaz [22, 28]. A key notion is the Adomian polynomials, which are tailored to the particular nonlinearity to solve nonlinear operator

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

equations. A key concept of the Adomian decomposition series is that it is computationally advantageous rearrangement of the Banach-space analog of the Taylor expansion series about the initial solution component function, which permits solution by recursion. The selection behind choice of decomposition is nonunique, which provides a valuable advantage to the analyst, permitting the freedom to design modified recursion schemes for ease of computation in realistic

Same way Laplace Adomian's Decomposition Method (LADM) was first introduced by Khuri [47, 48]. The Laplace Adomian Decomposition Method (LADM) is formed with combination of the Adomian Decomposition Method (ADM) Adomian [29, 49] and Laplace transforms. LADM is a promising method and has been applied in solving various nonlinear systems of differential equations [36, 50–56]. In a variety of applied sciences, systems of partial differential equations have attracted much attention e.g. [50, 57–75]. The general ideas and the essentiality of these systems are of wide applicability. Agadjanov [56] solved Duffing equation with the help of LDM. Elgazery [51, 76] had applied Laplace decomposition method for the

In the solution procedure of VIM; many repeated computations and computa-

On the other hand, few researchers have been discussed imbibition phenomenon in homogenous porous media with different point of view for example, researchers taking different perspectives for this phenomenon; [77, 78] and some others have

In this Present investigated model, Imbibition takes place over a small part of a

tions of the unneeded forms, which take more time and effort beyond it, so a

large oil formatted region taken as a cylindrical piece of homogeneous porous medium. In this model, we have considered the important condition of countercurrent imbibition phenomenon as *vi* ¼ �*vn*, Our purpose is to determine the saturation of injected fluid *Si*ð Þ *x*, *t* during oil recovery process which is a function of distance *ξ* and time *θ*, therefore saturation *Si* has been chosen as a dependent

It is the process by which a wetting fluid displaces a non-wetting fluid the initially saturates a porous sample, by capillary forces alone. Suppose a sample is completely saturated with a non-wetting fluid, and same wetting fluid is introduced on its surface. There will be spontaneous flow of wetting fluid into the medium, causing displacement of the non-wetting fluid. This is called imbibition phenomenon. The rate of imbibition is greater if the wettability of the porous medium, by the

modification has been shown to reduce these unneeded forms.

variable while *x* and *t* are chosen as independent variable.

systems.

solution of Falkner-Skan equation.

**2. Imbibition phenomenon**

imbibed fluid, is higher.

**46**

analyzed it for homogeneous porous medium.

According to the Darcy's law, the basic equations of the phenomenon as; [78]

$$v\_i = -\left(\frac{k\_i}{\delta\_i}\right) K \frac{\partial p\_i}{\partial \mathbf{x}} \tag{1}$$

$$v\_n = -\left(\frac{k\_n}{\delta\_n}\right) K \left.\frac{\partial p\_n}{\partial \mathbf{x}}\right|\_{\mathbf{x}}\tag{2}$$

$$
v\_i = -v\_n\tag{3}$$

$$p\_c = p\_n - p\_i \tag{4}$$

$$
\rho \left( \frac{\partial \mathbf{S}\_i}{\partial t} \right) + \frac{\partial v\_i}{\partial \mathbf{x}} = \mathbf{0} \tag{5}
$$

$$
\rho \left( \frac{\partial \mathbf{S}\_n}{\partial t} \right) + \frac{\partial v\_n}{\partial \mathbf{x}} = \mathbf{0} \tag{6}
$$

Where v*i* and v*n* are the seepage velocities, k*i* and kare the relative permeabilities *δi* and *δn* are the kinematic viscosities (which are constants), p*i* and p*n* are pressure of the injected and native liquid respectively, *φ* and *K* are the porosity and the permeability of the homogeneous porous medium; S*i* is the saturation of the injected liquid; p*c* is the capillary pressure and t is the time. The co-ordinate x is measured along the axis of the cylindrical medium, the origin being located at the imbibition face x=0.

Combing equations (1)-(5) and using the relation for capillary pressure as, p*c*=*β Si* [70], we get,

$$\log\left(\frac{\partial \mathbf{S}\_i}{\partial t}\right) + \frac{\partial}{\partial \mathbf{x}} \left[ \mathbf{K} \mathbf{D}(\mathbf{S}\_i) \beta \left(\frac{\partial \mathbf{S}\_i}{\partial \mathbf{x}}\right) \right] = \mathbf{0} \tag{7}$$

Where D(S*i*) = *kikn δnki*þ*δikn* and β being small capillary pressure coefficient. It is assumed is that an average value of D(S*i*) =*D* (S*i*) Using the transformation,

$$\xi = \frac{\varkappa}{L} \quad , \theta = \frac{Lt}{\varrho L^2} , \quad 0 \le \varkappa \le \frac{LS\_{io}}{B} . \tag{8}$$

Eq. (7), becomes;

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

$$\left(\frac{\partial \mathbf{S}\_i}{\partial \theta}\right) + \beta \overline{D}(\mathbf{S}\_i) \frac{\partial^2 \mathbf{S}\_i}{\partial \xi^2} = \mathbf{0}$$

*∂Si ∂θ* ¼ �*βD S*ð Þ*i ∂Si* 2 *∂ξ*2

$$\frac{\partial \mathbf{S}\_i}{\partial \theta} = \varepsilon \frac{\partial \mathbf{S}\_i^{\;2}}{\partial \xi^2} \text{ Where } \varepsilon = -\beta \overline{D}(\mathbf{S}\_i) \tag{9}$$

By the Hopf-Cole transformation [79, 80] equation (9) reduces to the Burger's equation.

$$\mathbf{S}\_{i\_{\theta}}^{\*} + \mathbf{S}\_{i}^{\*} \mathbf{S}\_{i\_{\xi}}^{\*} = \varepsilon \mathbf{S}\_{i}^{\*} \mathbf{g}\_{\xi} \tag{10}$$

To solve equation (10) by VIM, substituting in equation (14) by

*Study on Approximate Analytical Method with Its Application Arising in Fluid Flow*

And can obtain the following variational iteration formula:

*in* � ð *θ*

0

*S* ∗

*S* ∗ *inτ* þ *S* ∗

Using (14), the approximate solutions *Un*ð Þ *x*, *t* are obtained by substituting;

*i* ð Þ¼ *ξ*, 0 *S* ∗

1 *θ*; *where β*0

1 *θ* þ *β*1 1 *θ*2

*S* ∗

*DOI: http://dx.doi.org/10.5772/intechopen.97548*

Approximate solutions are given below;

*S* ∗ *i*3 ¼ *S* ∗ *i*0 *e ξ* � *β*1 2 *θ*2 2! þ *β*2 2 *θ*3 3!

1.VIM can contain a series solution not exactly like ADM.

2.VIM needs many modifications to overcome the wasted time in the repeated

*θ S* ∗ *i S* ∗ *iξ* � *εS* ∗

*Si*ð Þ¼ *ξ*, 0 *e*

*i*1 ð Þ¼ *ξ*, *θ β*0

12 *θ*3 3 � *εβ*1 2 *θ*2 2

*ξ*

1 *θ*

To overcome these problems, following ADM and LADM is suggested.

*i* ð Þ¼ *ξ*, *θ L*�1

*S* ∗

*i*2 ð Þ¼ *ξ*, *θ β*0

*S* ∗

From the analysis we can observed is this:

calculations and unneeded terms.

And recursive relation is:

Then:

**49**

Now applying ADM to equation (10); we get

*S* ∗

*S* ∗ *i*1 ¼ *S* ∗ *i*0 *e ξ* � *β*0