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Universität Stuttgart - Institut für Wasser- und Umweltsystemmodellierung Lehrstuhl für Hydromechanik und Hydrosystemmodellierung Prof. Dr.-Ing. Rainer Helmig Master’s Thesis Forchheimer Porous-media Flow Models - Numerical Investigation and Comparison with Experimental Data Submitted by Vishal A. Jambhekar Matrikelnummer 2550192 Stuttgart, 26. November 2011 Examiner: Prof. Dr.-Ing Rainer Helmig Supervisors: Dipl.-Ing. Philipp Nuske and Dipl.-Ing. Katherina Baber

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Page 1: Forchheimer Porous-media Flow Models - Numerical Investigation

Universität Stuttgart - Institut für Wasser- undUmweltsystemmodellierung

Lehrstuhl für Hydromechanik und HydrosystemmodellierungProf. Dr.-Ing. Rainer Helmig

Master’s Thesis

Forchheimer Porous-media Flow Models -Numerical Investigation and Comparison with

Experimental Data

Submitted by

Vishal A. JambhekarMatrikelnummer 2550192

Stuttgart, 26. November 2011

Examiner: Prof. Dr.-Ing Rainer HelmigSupervisors: Dipl.-Ing. Philipp Nuske and Dipl.-Ing. Katherina Baber

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To my parents

To my parents

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To my parents

I hereby acknowledge that I have prepared this master’s thesis independently, and that onlythose sources, aids and advisors that are duly noted herein have been used and / orconsulted.

Signature:

Date:

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Contents

1 Introduction 11.1 Applications and motivation . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Fundamentals of Porous Media Flow 42.1 Scales - the continuum approach . . . . . . . . . . . . . . . . . . . . . . . 42.2 Local equilibrium in porous media . . . . . . . . . . . . . . . . . . . . . . 62.3 Effective parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Porosity (φ): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Saturation (S): . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.3 Intrinsic permeability(K): . . . . . . . . . . . . . . . . . . . . . . 72.3.4 Capillary pressure (Pc): . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.4.1 Micro-scale capillarity (pc) . . . . . . . . . . . . . . . . . 82.3.4.2 Macro-scale capillarity (pc) . . . . . . . . . . . . . . . . 9

2.3.5 Relative permeability (kr,α): . . . . . . . . . . . . . . . . . . . . . 112.4 Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Momentum balance . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2.1 Darcy law . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2.2 Forchheimer law . . . . . . . . . . . . . . . . . . . . . . 15

2.4.3 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3.1 Local thermal equilibrium . . . . . . . . . . . . . . . . . 172.4.3.2 Local thermal non-equilibrium . . . . . . . . . . . . . . 17

2.5 Multiphase non-Darcy flow . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5.1 Modified Ergun equation . . . . . . . . . . . . . . . . . . . . . . 202.5.2 Barree-Conway equation . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2.1 Barree-Conway model for single and multiphase flows . . 24

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2.5.2.2 Barree-Conway approach for relative permeability-saturationrelationship . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 DuMuX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 ITLR Experiment 293.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Isothermal experiment . . . . . . . . . . . . . . . . . . . . . . . . 313.1.2 Non-isothermal experiment . . . . . . . . . . . . . . . . . . . . . . 323.1.3 Motivation for the current work . . . . . . . . . . . . . . . . . . . 32

4 Results and Discussion 344.1 Intrinsic permeability and Forchheimer coefficient . . . . . . . . . . . . . 34

4.1.1 Linear regression analysis for intrinsic permeability K and nonlin-ear regression analysis for Forchheimer coefficient β . . . . . . . . 354.1.1.1 Linear regression analysis for intrinsic permeability K . 354.1.1.2 Nonlinear regression for Forchheimer coefficient β . . . . 37

4.1.2 Nonlinear regression analysis for both intrinsic permeabilityK andForchheimer coefficient β . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.3 Apparent permeability . . . . . . . . . . . . . . . . . . . . . . . . 414.1.3.1 Constant Forchheimer coefficient β for complete range of

experimental data . . . . . . . . . . . . . . . . . . . . . 444.1.3.2 Forchheimer coefficient β for limited Re ranges . . . . . 444.1.3.3 Linear Forchheimer coefficient β(Re) . . . . . . . . . . . 47

4.2 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.1 Incompressible isothermal case . . . . . . . . . . . . . . . . . . . . 504.2.2 Non-isothermal case . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2.1 With local thermal equilibrium . . . . . . . . . . . . . . 564.2.2.2 With local thermal non-equilibrium . . . . . . . . . . . . 65

5 Conclusion 68

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List of Figures

1.1 Applications of porous media : (a) Catalytic converter for automobileexhaust system. (b) Porous heat exchanger for air cooled condensers. (c)Cooling pores in a gas turbine blade. . . . . . . . . . . . . . . . . . . . . 1

2.1 Micro-scale to macro-scale transition [30] . . . . . . . . . . . . . . . . . . 52.2 Definition of the REV [21] . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Capillary forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Capillary pressure-saturation relationship [21] . . . . . . . . . . . . . . . 112.5 Deviation of experimental data from Forchheimer linear equation [23] . . 232.6 Relative permeability-saturation relationship[3] . . . . . . . . . . . . . . 262.7 Corrected relative permeability-saturation relationship [3] . . . . . . . . . 26

3.1 Photograph of ITLR experimental setup [30] . . . . . . . . . . . . . . . . 303.2 Porous structure used in ITLR experiments [29] . . . . . . . . . . . . . . 303.3 Schematic representation of ITLR experimental setup [30] . . . . . . . . 31

4.1 Linear regression with Darcy law . . . . . . . . . . . . . . . . . . . . . . 374.2 Linear Darcy regression for intrinsic permeability K and nonlinear Forch-

heimer regression for Forchheimer coefficient β . . . . . . . . . . . . . . . 384.3 Nonlinear regression with Forchheimer law . . . . . . . . . . . . . . . . . 414.4 Apparent permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5 Constant Forchheimer coefficient β for the complete range of experimental

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.6 Forchheimer coefficient β for limited Re ranges . . . . . . . . . . . . . . 454.7 Apparent permeability Kapp for limited Re ranges . . . . . . . . . . . . . 454.8 Forchheimer coefficient β as a function of Reynolds number (Re) . . . . 474.9 Apparent permeability β as a function of velocity . . . . . . . . . . . . . 484.10 Incompressible isothermal model domain . . . . . . . . . . . . . . . . . . 51

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4.11 Pressure distribution across porous domain for an incompressible isother-mal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.12 Velocity distribution across porous domain for an incompressible isother-mal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.13 Friction coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.14 Model domain non-isothermal case . . . . . . . . . . . . . . . . . . . . . 564.15 Unphisical heating along edges . . . . . . . . . . . . . . . . . . . . . . . . 584.16 Pressure distribution across porous domain for a compressible non-isothermal

flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.17 Temperature distribution across porous domain for a compressible non-

isothermal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.18 Velocity and density distribution across porous domain for a compressible

non-isothermal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.19 Evolution of pressure (non-isothermal model with local thermal equilibrium) 614.20 Evolution of temperature (non-isothermal model with local thermal equi-

librium) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.21 Evolution of velocity (non-isothermal model with local thermal equilibrium) 634.22 Evolution of temperature (non-isothermal model with local thermal non-

equilibrium) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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List of Tables

2.1 Exponents for relative permeability kr and relative passability ηr for theliquid phase (l) and the gas phase (g) [36, 37] . . . . . . . . . . . . . . . 22

4.1 Experimental data for isothermal case (ITLR) . . . . . . . . . . . . . . . 364.2 Forchheimer coefficient β for different subsets of the experimental data . 384.3 Percentage error for the calculated pressure gradients using different Forch-

heimer coefficients given in Table 4.2 . . . . . . . . . . . . . . . . . . . . 394.4 Forchheimer coefficient β for different subsets of the experimental data . 404.5 Percentage error of the calculated pressure gradients using different in-

trinsic permeabilities and Forchheimer coefficients given in Table 4.4 . . . 424.6 Forchheimer coefficient β for limited Re ranges . . . . . . . . . . . . . . 464.7 Comparison of percentage errors of calculated pressure gradients using

different approaches for Forchheimer coefficient β . . . . . . . . . . . . . 494.8 Ergun coefficient CE for different Forchheimer coefficient β approaches . 504.9 Experimental and numerical velocity data . . . . . . . . . . . . . . . . . 534.10 Experimental and numerical friction coefficients . . . . . . . . . . . . . . 544.11 Boundary conditions non-isothermal case . . . . . . . . . . . . . . . . . 574.12 Material parameters and input data . . . . . . . . . . . . . . . . . . . . . 574.13 Experimental and numerical wall temperatures (non-isothermal model

with local thermal equilibrium) . . . . . . . . . . . . . . . . . . . . . . . 644.14 Experimental and numerical wall temperatures (non-isothermal model

with local thermal non-equilibrium) . . . . . . . . . . . . . . . . . . . . . 67

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Nomenclature

λ lumped thermal condictivity [W/mK]

T transition constant [1/m]

β Forchheimer coefficient [1/m]

ηr,α relative passability of phase α [-]

η intrinsic passability [m]

α thermal diffusivity of the fluid phase [m2/s]

h convective heat transfer coefficient [W/m2K]

λf thermal condictivity of the fluid phase [W/mK]

λs thermal condictivity of the porous matrix [W/mK]

g gravity vector [m/s2]

K intrinsic permeability tensor [m2]

Kf hydraulic conductivity tensor [m/s]

v seepage velocity vector [m/s]

vf Darcy or Forchheimer velocity vector [m/s]

µ dynamic viscosity [kg/ms]

ν kinematic viscosity [m2/s]

φ solid matrix porosity [-]

σ surface tension [N/m]

% density [kg/m3]

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CE Ergun coefficient [-]

cpf specific heat capacity of the fluid phase [J/kgK]

d capillary diameter [m]

dp averaged particle diameter [m]

g gravity [m/s2]

h piezometric head [m]

hf specific enthalpy of fluid [J/kg]

K intrinsic permeability [m2]

Kf hydraulic conductivity [m/s]

Kapp apparent passability [m2]

kr,α relative permeability for phase α [-]

L characteristic length [m]

pc capillary pressure [Pa]

pn pressure of the non-wetting phase [Pa]

pw pressure of the wetting phase [Pa]

qfs exchange energy between the solid matrix and the fluid phase [W/m3]

sv specific interfacial area [1/m]

Sα saturation of the fluid phase α [-]

T temperature [K]

uf specific internal energy of fluid [J/kg]

vf Darcy or Forchheimer velocity [m/s]

z elevation head [m]

Nu Nusselt number [-]

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Pr Prandtl number [-]

Re Reynolds number [-]

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1 Introduction

1.1 Applications and motivation

Fluid flow and transport processes through porous structures is a topic of great interestin various scientific and technical fields. In particular, in engineering applications such ascatalytic converters (used for reducing toxicity of exhaust emissions from automobiles engines,see Figure 1.1a), condensers (used as heat exchanger for cooling condensers, see Figure1.1b) and gas turbines (used for cooling gas turbine blades, see Figure 1.1c), fluid flowthrough porous media in the high-velocity regime becomes relevant. In heating and coolingapplications, in addition to high-velocity flow, developing a deep understanding about non-isothermal flow and related heat-transfer processes becomes crucial.

Motivated by real world engineering applications and scientific interest, many scientists havechanneled research efforts towards developing a detailed understanding about these flowand transport processes by means of experimentation and numerical analysis. Authors like[22, 36, 4, 32, 37] used the Forchheimer law (see Section 2.4.2.2) [19] to describe highvelocity flow. They supported their choice by stating that the Forchheimer law accounts forhigh velocity inertial effects.

(a) (b) (c)

Figure 1.1: Applications of porous media : (a) Catalytic converter for automobile ex-haust system. (b) Porous heat exchanger for air cooled condensers. (c)Cooling pores in a gas turbine blade.

1

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Mayer et al. [30] observed that metallic porous media offer an effective solution in manyheating and cooling applications by the virtue of their inherent properties such as largesurface area-to-volume ratio, high permeability and thermal dispersion of the fluid due tothe complex flow within pore channels. Pressure loss and convective heat transfer in thesemetallic structures have been investigated for the above mentioned engineering applications.

At the Institute of Aerospace Thermodynamics / Institut für Thermodynamik der Luft- undRaumfahrt (ITLR), University of Stuttgart, experiments are carried out to analyze and opti-mize a uniform metallic porous structure for cooling applications. The pressure drop acrossthe structure and temperatures at different locations are measured to validate the numericalmodels for both isothermal and non-isothermal flow systems.

In the scope of this master’s thesis, the existing numerical model for a single-phase isothermalDarcy (creeping) flow is modified for high velocity non-Darcy flow by using the Forchheimerequation. The Forchheimer flow model is also extended to allow the description of non-isothermal flow with local thermal equilibrium and with local thermal non-equilibrium.

The ultimate goal of the current work is to develop satisfactory isothermal and non-isothermalForchheimer flow models. We achieve this by means of linking experimental data withnumerical studies. Determination of accurate effective parameters from the experimental datais an important component of the current work. Thus, we review the Darcy and Forchheimerlaws in the context of determination of the intrinsic permeability K and the Forchheimercoefficient β and validate our numerical models against experimental measurements.

1.2 Objectives

Given below are the objectives of the current work

• Detailed analysis of the experimental data and determination of the intrinsic perme-ability K and the Forchheimer coefficient β.

• Implementation of the Forchheimer model for a single-phase isothermal flow.

• Implementation of the Forchheimer model for a single-phase non-isothermal flow as-suming local thermal equilibrium.

• Extension of the single-phase non-isothermal Forchheimer model for local thermal non-equilibrium.

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• Setup of relevant numerical examples with appropriate boundary conditions.

• Attempt to validate above mentioned isothermal and non-isothermal models by com-paring numerical simulations with experimental results.

• Detailed literature review for multiphase Forchheimer flow.

1.3 Structure of the thesis

• Chapter 2: In this chapter, the theoretical background about the different scales, effec-tive parameters, equilibrium criteria and balance equations necessary for the descriptionof a complex porous media flow system is explained. In addition, this chapter also pro-vides a brief introduction to the multiphase Forchheimer flow (see Section 2.5) andDuMuX, an open source simulation software for flow and transport processes in porousmedia (see Section 2.6).

• Chapter 3: The experimental setup and procedure are discussed in detail in this chapter.

• Chapter 4: Analytical determination of the intrinsic permeability K and the Forch-heimer coefficient β are discussed in detail in this chapter. The determined intrinsicpermeability K and the Forchheimer coefficient β are used for numerical simulationsand the results are compared with corresponding experimental data.

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2 Fundamentals of Porous MediaFlow

2.1 Scales - the continuum approach

According to [32], the treatment of the problem of flow through a porous structure is largelydependent on the scale considered. At a small scale, one looks at just a few small pores(micro-scale or pore-scale). It is therefore convenient to use conventional fluid mechanicsapproach to describe the flow phenomenon in the fluid-filled spaces. However, when the scaleis large (macro-scale), the field of vision includes a large number of pore spaces. In such acase, the complicated flow paths and the need to describe complex spatial resolution of theporous structure rule out the possibility of using the conventional fluid mechanics approach.Hence, a volume averaging (continuum) approach is used [32].

The finite scale an engineer would look at is the molecular scale. The continuum mechanicsbased approach is used for transition from the molecular scale to the micro-scale [1]. Theconsideration of a continuum corresponds to replacing the molecular properties by averagedproperties over a large number of molecules. According to [21, 13], the consideration of acontinuum at the macro-scale is a fundamental concept of fluid mechanics. For example, airis used as the fluid phase at ITLR for experiments and numerical simulations. Here, ratherthan looking at the movement of every single molecule, the overall air-flow is observed usingaveraged fluid properties such as density % and viscosity µ. In the context of the currentwork, the term “phase” is used to differentiate between physical continua separated by asharp interface (e.g., solid and fluid phase).

Initially, the continuum approach is used to transfer the molecular properties to the micro-scale or pore-scale in order to resolve the flow in pore spaces. In the context of the currentwork, micro-scale can be represented by the dimension of a unit cell forming the uniformporous structure (see Figure 2.1). At ITLR, numerical simulations are performed at the pore-

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scale using conventional fluid mechanics approach. However, for reasons discussed previously,for a large scale problem, the possibility of using the conventional approach is ruled out. Thus,further volume averaging is needed to describe the flow properties on the macro-scale.

Figure 2.1: Micro-scale to macro-scale transition [30]

Figure 2.2: Definition of the REV [21]

The Figure 2.1 shows the different scales involved in the averaging process. The micro-scaleproperties are averaged over a representative elementary volume (REV) in order to obtain amacro-scale description of the system with effective parameters such as porosity φ, saturationS and intrinsic permeability K (see Section 2.3). The macro-scale is also referred to as the

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REV-scale. It can be seen from Figure 2.1 that at the REV-scale, detailed spatial resolutionof solid matrix and fluid phase is lost, and effective volume averaged parameters (effectiveparameters) are available.

For the process of volume averaging, proper selection of REV size is very crucial, as thevolume-averaged quantities need to be independent of the REV-size. Figure 2.2 shows thatthe selected REV should not only be smaller than the flow domain, but it also should belarger than a single pore in the porous medium. A very small REV leads to oscillations dueto existence of inhomogeneities at the micro-scale. On the other hand, a very large REVleads to fluctuations caused by macroscopic heterogeneities of the medium [21].

2.2 Local equilibrium in porous media

The local thermodynamic equilibrium in porous media mainly consists of thermal, chemicaland mechanical equilibria as follows:

Thermal equilibrium:

A system is said to be in local thermal equilibrium, if at any given point of the system all thephases exist at the same temperature

T = Ts = Tf [K].

Here, Ts and Tf are the temperatures of the solid matrix and fluid phase respectively.

Chemical equilibrium:

A system is said to be in chemical equilibrium, if the potential for exchange of chemicalcomponents across different phases or within a phase is zero. In other words, there is noexchange of components within a phase or between different phases.

Mechanical equilibrium:

When multiple fluid phases are present in the system, mechanical equilibrium refers to theexistence of equal pressure on either side of the phase boundary (e.g., a lake surface).However, in the context of the porous media flow, one must account for the pressure jump

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at the fluid phase boundaries due to capillarity i.e., the capillary pressure (see Section 2.3.4)[21].

2.3 Effective parameters

2.3.1 Porosity (φ):

As discussed earlier, porous media consists of interconnected voids and a solid matrix. Poros-ity is defined as the ratio of the volume of pores in REV to the volume of REV:

φ =Volume of pores in REV

Volume of REV[−]. (2.1)

Here, 1 − φ is the volume fraction of the solid matrix. For the current study porosity isassumed to be constant i.e. the solid matrix is assumed to be a rigid structure. Porosity isa dimensionless parameter.

2.3.2 Saturation (S):

With the macro-scale approach and adaptation of REV for multiphase flow problems, anew effective parameter called saturation is introduced at the macro-scale. This parameteraccounts for the existence of different fluid phases in a given REV. In other words, it accountsfor the fractions of the pore space occupied by different fluid phases. The saturation of afluid phase α is the ratio of the volume of fluid phase α in REV to the volume of pores inREV and is given as follows:

Sα =Volume of fluid phase α in REV

Volume of pores in REV[−]. (2.2)

Saturation, like porosity φ, is also a dimensionless parameter.

2.3.3 Intrinsic permeability(K):

The intrinsic permeability K of a porous medium represents its ability to allow the fluid toflow through. It is a macro-scale property. The intrinsic permeability tensor K is a part ofthe definition of the hydraulic conductivity tensor Kf . “Hydraulic conductivity is a macro-scale parameter which accounts for the influence of viscosity and adhesion at the soil grain

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surfaces” [21]. The hydraulic conductivity tensor Kf for a single-phase flow is given as:

Kf = K%g

µ

[m

s

], (2.3)

where K is the intrinsic permeability tensor, % and µ are the density and viscosity of the fluidrespectively and g is the gravitational acceleration. The intrinsic permeability tensor is asecond order tensor with nine components indicating permeabilities in different directions:

K =

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

[m2]. (2.4)

Intrinsic permeability K is only dependent on the porous structure and can be same or differ-ent in different directions, depending on whether the porous matrix is isotropic or anisotropic.

2.3.4 Capillary pressure (Pc):

2.3.4.1 Micro-scale capillarity (pc)

For a multiphase flow through porous media, a certain force acts at the interfacial areabetween different fluid phases. This force is called surface tension and is strongly influencedby the solid and fluid properties. The surface tension caused by the interaction of the solidand the different fluid phases leads to capillary pressure (pc) [21]. Capillary pressure is definedas difference between the pressure of the non-wetting (pn) and wetting (pw) phases at theinterface as shown in Figure 2.3:

pc = pn − pw [Pa]. (2.5)

Here, wetting and non-wetting phases are relative terms. For a two-phase flow system, thewetting phase is the phase which has higher affinity with the solid phase. The Laplaceequation for the capillary pressure is as follows:

pc =4σ cos θ

d= pn − pw [Pa], (2.6)

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Figure 2.3: Capillary forces

where σ is the surface tension for the fluid phase, θ is the angle of contact between the fluidphase and the solid phase and d is the pore diameter.

2.3.4.2 Macro-scale capillarity (pc)

From the detailed study of capillary effects, it is observed that the fluid saturations have astrong influence on the capillary pressure [21]. The macro-scale capillary effects are taken intoaccount based on the fundamental correlation between the capillary pressure and saturationof the wetting phase and the non-wetting phase.

Typical examples of multiphase flow system are: Imbibition - injection of a wetting phase intoa porous medium to displace a non-wetting phase and draining - injection of a non-wettingphase into a porous medium to displace a wetting phase.

For example, during a draining process, as the saturation of the wetting phase decreases,the wetting phase retreats into the smaller pores in the porous medium. It is noticed that ahigher capillary pressure is needed for further displacement of the wetting phase. In this way,the required capillary pressure keeps increasing as the wetting phase moves into finer pores ofthe porous matrix. The capillary pressure required for the displacement of the wetting phasecan be expressed as a function of its saturation (see Equation 2.7). Detailed discussion onthe macro-scale capillary pressure-saturation relationship can be studied in [21].

pc = pc(Sw). (2.7)

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Many scientists came up with ideas to describe capillary pressure-saturation relationship.According to [21], the best known capillary pressure-saturation relationships for air-watersystems could be found in the literature from Leverett (1941) [25], Brooks and Corey (1964)[10] and Van Genuchten (1980) [20].

• Brooks and Corey:

The capillary pressure-saturation relation proposed by Brooks and Corey is discussed below:

Se(pc) =Sw − Swr1− Swr

=

(pdpc

)λfor pc ≥ pd, (2.8)

where λ is the Brooks Corey parameter, pc is the capillary pressure, pd is the entry pressure,Se is the effective saturation, Sw is the wetting phase saturation and Swr is the residualsaturation of the wetting phase (w). Here, Swr refers to the saturation of the detached partof a phase which is held back within the porous medium. Detailed description of residualsaturation can be found in [21]. λ indicates the material grain size distribution and it rangesbetween 0.2 and 0.3. The Brooks-Corey relation is valid only for capillary pressure greaterthan entry pressure pd (see Figure 2.4). Here, entry pressure pd is the minimum pressureneeded for the non-wetting phase to enter the porous medium.

According to Brooks and Corey [10], the capillary pressure as a function of wetting phasesaturation Sw is given as:

pc(Sw) = pdS− 1λ

e ; for pc ≥ pd. (2.9)

• Van Genuchten:

The capillary pressure-saturation relation proposed by Van Genuchten is as follows:

Se(pc) =Sw − Swr1− Swr

= [1 + (α · pc)n]m for pc > 0, (2.10)

where m, n and α are Van Genuchten parameters, such that m = 1− (1/n). m and n andare dimensionless parameters whereas, α has a dimension of [1/Pa]. The Van Genuchtenrelation is valid for all capillary pressures greater than zero.

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According to Van Genuchten [20], the capillary pressure as a function of wetting phasesaturation Sw is given as:

pc(Sw) =1

α(S− 1m

e − 1)1n ; for pc > 0. (2.11)

Figure 2.4: Capillary pressure-saturation relationship [21]

Figure 2.4 shows the capillary pressure-saturation relationship for an air-water flow systemby Van Genuchten [20] and Brooks-Corey [10].

2.3.5 Relative permeability (kr,α):

Hydraulic conductivity of a porous medium is already discussed in Section 2.3.3. However,for a multiphase system, the hydraulic conductivity is defined as follows:

Kf = Kkr,α%αg

µα

[m

s

], (2.12)

where K is the intrinsic permeability tensor, %α and µα are the density and viscosity of thefluid phase α respectively and g is the gravitational acceleration. The relative permeabilitykr,α is a dimensionless parameter and it depends on the fluid phase saturation Sα [21].The relative permeability kr,α accounts for the dependence of effective permeability K kr,α

on saturation Sα through the relative permeability-saturation relationship. The best known

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relative permeability-saturation relationships are proposed by Brooks and Corey (1964) [10]and Van Genuchten (1980) [20].

• Brooks and Corey:

The relative permeability-saturation relationship proposed by Brooks and Corey [21] for atwo phase air-water system is as follows:

kr,w = S2+3λλ

e and (2.13)

kr,n = (1− Se)2(1− S2+λλ

e ), (2.14)

where, as discussed in Section 2.3.4.2, λ and Se are the Brooks-Corey parameter and effectivesaturation respectively (see Equation 2.8). Here, kr,w and kr,n are the relative permeabilitiesof the wetting phase (water) and the non-wetting phase (air) respectively.

• Van Genuchten:

The relative permeability-saturation relationship proposed by Van Genuchten [21] for a twophase air-water system is as follows:

kr,w = Sεe[1− (1− S1me )m]2 and (2.15)

kr,n = (1− Se)γ[1− S1me ]2m, (2.16)

where, as discussed in Section 2.3.4.2, m and Se are the Van Genuchten parameter andeffective saturation respectively (see Equation 2.10). The parameters ε and γ describe theconnectivity of pores. Generally, ε = 1

2and γ = 1

3[21].

2.4 Balance equations

The choice of the continuum approach for the current work necessitates the definition of thelaws for conservation of mass, momentum and energy.

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2.4.1 Mass balance

The continuity equation or the mass balance equation ensures that the overall change ofmass within a continuum is zero. The equation for conservation of mass for a free flowsystem is as follows:

∂%f∂t

+∇ · (%fv)− q = 0, (2.17)

where %f is the density of the fluid, t is time, v is the velocity vector of the fluid and q isthe external source or sink.

Equation 2.17 indicates that “the rate of change of mass per unit control volume, fluxesacross the faces of the control volume and the potential sources and sinks must balance” [1].In the context of the current work, porosity is included in Equation 2.17, as the continuity isonly considered for the fluid flow through porous matrix:

φ∂%f∂t

+∇ · (%fφv)− q = 0. (2.18)

Neglecting the source or sink term q and using the relation between the Darcy / Forchheimervelocity vector vf and seepage velocity vector v given by:

v =vfφ, (2.19)

equation 2.18 can be rewritten as follows:

φ∂%f∂t

+∇ · (%fvf ) = 0. (2.20)

The velocity vector vf is referred to as Darcy velocity vector in Section 2.4.2.1 and Forch-heimer velocity vector in Section 2.4.2.2.

Assuming an incompressibile fluid, a rigid porous medium and no source or sink (q = 0),Equation 2.20 is reduced to the following mass balance equation:

∇ · (vf ) = 0. (2.21)

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On the other hand, an assumption of a compressible fluid, a rigid porous medium and nosource or sink (q = 0) leads to the following mass balance equation:

φ∂%f∂t

+∇ · (%fvf ) = 0. (2.22)

2.4.2 Momentum balance

This section discusses the macro-scale momentum balance equations for the fluid flowthrough porous media. According to [32], the Darcy law (see Section 2.4.2.1) is used todescribe slow or creeping flows and the Forchheimer law (see Section 2.4.2.2) is used for thedescription of high velocity flows. Both the Darcy law and the Forchheimer law are obtainedexperimentally in order to describe the flow through a porous medium and have become themacro-scale momentum equation of choice in literature [21, 32]. According to [13], theseequations for momentum description allow the decoupling of the continuity and momentumbalance.

2.4.2.1 Darcy law

A French scientist, Henry Darcy in his work on the investigation of hydrological systems forwater supply in the city of Dijon performed steady-state unidirectional flow experiments fora uniform sand column [14]. From his experimental observations, he proposed the Darcy lawas follows:

vf = −Kf · ∇h, (2.23)

where vf is the Darcy velocity vector, Kf is the hydraulic conductivity tensor as explainedin Section 2.3.3 and ∇h is the gradient of the piezometric head h given by:

h =p

%fg+ z [m], (2.24)

where p%fg

is the pressure head and z is the elevation head. From Equation 2.24 and Equation2.3, the Darcy law can be rewritten as follows:

vf = −K

µ· (∇p+ %fg∇z), (2.25)

where µ is the dynamic viscosity of the fluid , ∇p is the applied pressure gradient and K isthe intrinsic permeability tensor of the porous matrix. For the current study, gravitational

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effects are neglected and as the porous matrix is isotropic, the intrinsic permeability is treatedas a scalar. Thus, the Darcy law boils down to:

vf = −Kµ∇p. (2.26)

According to [21], Bear and Bachmat (1986) [6] derived the Darcy law. In the derivation,they have neglected inertial or time dependent effects. Thus, the Darcy law is only valid forslow (creeping) flows (Re << 1). Here, the Reynolds number (Re) is defined as the ratio ofthe inertia force to the viscous force:

Re =Inertia ForceViscous Force

=%fvfL

µ[−], (2.27)

where %f , vf and µ are the density, velocity, and dynamic viscosity of the fluid respectively.For the current work, the characteristic length (L= 1/901 [m]) is obtained from the specificinterfacial area (sv= 901

[1m

]) [30]. The specific interfacial area sv is defined as the area of

contact between the solid and fluid phase per unit volume.

2.4.2.2 Forchheimer law

An Austrian scientist Phillip Forchheimer (1901) [19] in his work “Wasserbewegung durchBoden”, investigated fluid flow through porous media in the high velocity regime. During thisstudy, he observed that as the flow velocity increases, the inertial effects start dominating theflow. In order to account for these high velocity inertial effects, he suggested the inclusion ofan inertial term representing the kinetic energy of the fluid to the Darcy equation [38]. TheForchheimer equation is given as follows:

∇p = − µKvf − β%fv2

f . (2.28)

Here, the parameter β is called the Forchheimer coefficient and vf stands for the Forchheimervelocity. The Forchheimer equation in the vector form is given below [22, 38, 32]:

∇p = − µK

vf − β%f |vf |vf , (2.29)

where vf is the Forchheimer velocity vector.

Theoretical evaluation of the Forchheimer coefficient β is cumbersome. Thus, for mostpractical applications, this parameter is obtained from the best fit to the experimental data.

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Ergun and Orning (1949) [16] worked for the investigation of fluid flow through packedcolumns and fluidized beds. Based on this work, Ergun (1952) [15] proposed an expressionfor the Forchheimer coefficient β:

β =CE√K

[1

m

], (2.30)

where CE is called Ergun constant and it accounts for inertial (kinetic) effects. K is theintrinsic permeability (see Section 2.3.3). From Equation 2.30 and Equation 2.29, we getthe Ergun equation. This equation is also referred to as Forchheimer equation with Ergunexpression for Forchheimer coefficient [22, 32]:

∇p = − µK

vf −CE√K%f |vf |vf . (2.31)

The Ergun coefficient CE is strongly dependent on the flow regime. For slow flows, CE isvery small. Thus, the second term on the right hand side of Equation 2.31 is very small andcan be neglected. This reduces the Forchheimer equation to the Darcy equation.

As the flow velocity increases, inertial effects also increase and the flow adapts to the Forch-heimer flow regime [34]. These inertial effects are accounted for by the Ergun coefficient CEand the kinetic energy of the fluid %f |vf |vf [38]. However, according to [31, 4, 2], a constantErgun coefficient CE is valid as long as the fluid flow is laminar. Thus, in the high velocityflow regime, the Ergun coefficient CE needs to be adapted to reflect the experimental inertialeffects.

2.4.3 Energy balance

As discussed in Section 1.1, in the scope of the current work, numerical models are imple-mented for both isothermal and non-isothermal flows. The non-isothermal model is imple-mented for two possible thermodynamic scenarios - namely, with local thermal equilibriumand with local thermal non-equilibrium. For the first scenario, the system is not necessarilyin thermal equilibrium globally. However, thermal equilibrium exists locally. This means thatat any given point in the system, different phases exists at same temperature. In such a case,only one energy equation is required for the description of the temperature of each phase atany given point in the system. For the other scenario, there exists no thermal equilibrium.That is, at any given point in the system, different phases exist at different temperatures.

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Thus, different energy equations are required for the description of the temperature of eachphase. In what follows, these two scenarios are discussed in detail.

2.4.3.1 Local thermal equilibrium

The behavior of non-isothermal flow system is strongly dependent on the flow velocity. Fora slow flow system, as different phases (e.g., solid phase and fluid phase) are in contact fora sufficient period of time, the possibility for the system to exchange energy locally and toestablish local thermal equilibrium always exists. In such a case, only one energy equationis sufficient for the description of temperature of all phases at any given location within thesystem. For a single-phase flow through porous matrix, the energy balance equation is givenas follows:

∂t(φ%fuf ) +

∂t((1− φ)%scpsT ) +∇ · (%fhfvf )−∇ · (λ∇T )− q = 0. (2.32)

Here, uf , hf , cps and T stands for the internal energy of the fluid, the enthalpy of the fluid,specific heat capacity of the solid matrix and temperature respectively. q is the externalsource or sink. λ is averaged thermal conductivity of the solid matrix and the fluid phaseand is given by Equation 2.33 [11, 9, 8] :

λ = (1− φ)λs + φλf

[W

mK

], (2.33)

where λs and λf are the thermal conductivities of the solid matrix and the fluid phaserespectively.

2.4.3.2 Local thermal non-equilibrium

For high velocity flow, the interaction between different phases is rapid. Thus, differentphases cannot exchange sufficient amount of energy to establish local thermal equilibrium.Therefore, at any given location in the system, different phases exist at different temperatures.In such a case, one needs different energy equations for the description of the temperatureof each phase. In the context of the current work, the energy equations for the solid matrixand the fluid phase are given by Equation 2.34 and Equation 2.35 respectively:

∂t((1− φ)%scpsTs)−∇ · ((1− φ)λs∇Ts)− qs = 0 and (2.34)

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∂t(φ%fuf ) +∇ · (%fhfvf )−∇ · (φλf∇Tf )− qf = 0, (2.35)

where qs and qf represent the exchange energy with other phases or external source/sink. Inthe context of the current work, qs = −qf = qfs is the exchange energy between the solidmatrix and the fluid phase.

The idea now, is to obtain an expression for exchange of energy between different phases.For this purpose, we present some fundamentals of heat transfer.

In the context of the current work, transfer of thermal energy in the solid matrix and thefluid phase takes place by means of two different processes: Firstly, by means of conduction- involving the transfer of thermal energy between different parts of a system due to temper-ature difference. Secondly, by means of convection - involving the transfer of thermal energydue to the motion of molecules. Convection occurs only in fluids, while conduction takesplace in solids as well as fluids.

In order to determine the dominance of conductive or convective heat transfer, a new dimen-sionless number called Nusselt number is used. Nusselt number is defined as follows:

Nu =Convective Heat Transfer CoefficientConductive Heat Transfer Coefficient

=hL

λf[−]. (2.36)

Here, h is the convective heat transfer coefficient and λfL

is the conductive heat transfercoefficient.

The Nusselt number can also be described in terms of other dimensionless numbers, namely- Reynolds number and Prandtl number:

Nu = Nu(Re,Pr). (2.37)

For the definition of Reynolds number see Equation 2.27. Prandtl number is defined as theratio of viscous diffusion rate to the thermal diffusivity:

Pr =Viscous Diffusion RateThermal Diffusivity

α[−]. (2.38)

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Here, ν stands for the kinematic viscosity and the thermal diffusivity α is the ratio of thermalconductivity λf to the volumetric heat capacity %fcpf :

α =λf%fcpf

[m2

s

], (2.39)

where cpf is the specific heat capacity of fluid. For a single-phase flow system, the exchangeof energy between the solid matrix and the fluid phase is given by following equation:

qfs = hsv(Ts − Tf ).[W

m3

], (2.40)

where qfs is the heat exchange between the solid and the fluid phase, Ts is temperatureof the solid, Tf is temperature of the fluid, h is convective heat transfer coefficient and svis specific interfacial area. From the definition of Nusselt number (see Equation 2.36), theconvective heat transfer coefficient h can be expressed as:

h =λfL

Nu(Re,Pr)[W

m2K

]. (2.41)

Various problem-specific Nusselt correlations are available in literature. The following Nusseltnumber correlation used in the current work is taken from [27]:

Nu (Re, Pr)=0.023 Re0.8Pr0.33. (2.42)

2.5 Multiphase non-Darcy flow

This section provides a detailed overview of the different approaches available for modelingmultiphase non-Darcy flow. These approaches would be useful for extension of the currentwork to high velocity multiphase flow in the future. The Forchheimer flow (see Section2.4.2.2) is also referred to as non-Darcy flow in literature [12, 17, 4, 5, 3]. Realizing theimportance of the non-Darcy multiphase flow in industrial applications, many authors [7, 17,36, 3] have come up with different modeling approaches.

Schäfer and Lohnert (2006) [35] mentioned that most of the multiphase dryout models fornuclear research are based on the Ergun equation (see Section 2.5.1). Bennethum and Giorgi(1997) [7] derived the generalized Forchheimer equation for the description of single phase andmultiphase flows. Ewing et al. (1999) [17] developed a numerical model for the description

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of the non-Darcy multiphase flow through isotropic porous media. Barree and Conway (2007)[3] proposed a new Forchheimer type equation for the description of a multiphase non-Darcyflow (see Section 2.5.2). Wu et al. (2011) [39] supported and discussed the Barree-Conwayapproach for modeling multiphase non-Darcy flow through porous media.

The classical approaches for capillary pressure-saturation relationship discussed in Section2.3.4 and the relative permeability-saturation relationship discussed in Section 2.3.5 are validfor the Darcy flow regime (Re << 1). For a multiphase non-Darcy flow, the capillary forcesare neglected for the sake of simplicity and only relative permeability kr,α is taken into account[35, 37]. The relative permeability-saturation relationships for a multiphase non-Darcy floware discussed in Section 2.5.1 and Section 2.5.2.2.

2.5.1 Modified Ergun equation

According to [36, 35], the modified Ergun equation is based on the Ergun equation (seeEquation 2.31) and is used to describe multiphase non-Darcy flows. The modified Ergunequation for each phase α in a multiphase non-Darcy flow is given by:

∇pα = −%αg −µα

kr,αKvfα −

%αηr,αη

|vfα|vfα, (2.43)

where ∇pα is the pressure gradient, %α is the density, µα is the dynamic viscosity, K isthe intrinsic permeability, kr,α is the relative permeability, vfα is the Forchheimer velocityvector, η is the intrinsic passability and ηr,α is the relative passability for phase α. ComparingEquation 2.43 with Equation 2.31 one can define intrinsic passability η as the ratio of thesquare root of intrinsic permeability to the Ergun coefficient CE:

η =

√K

CE=

1

β[m]. (2.44)

Modified Ergun equation for the representation of the multiphase non-Darcy flow differsfrom the Ergun equation (see Equation 2.31) in terms of interpretation of the intrinsicpermeability K and intrinsic passability η. In the case of the modified Ergun equation, theintrinsic permeability and passability are interpreted in terms of spatial parameters as follows[28, 36, 35, 37, 26]:

K =d2pφ

3

A(1− φ)2and (2.45)

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η =dpφ

3

B(1− φ), (2.46)

where φ stands for porosity and dp is the averaged diameter of particles in the porous medium[35]. According to the definition of the modified Ergun equation, A = 150 and B = 1.75

[36, 37].

During multiphase flow through a porous medium, the flow area for one phase is reduceddue to the existence of other phases. “Relative permeability kr,α and relative passability ηr,αare the parameters which quantify the effect of the reduced flow area for each phase” [37].We have already discussed relative permeability kr,α in Section 2.3.5. Relative passabilityηr,α for a certain phase α is a part of the definition of effective passability ηα = η ηr,α. Fora gas-liquid non-Darcy flow system, relations between effective and intrinsic permeabilitiesand effective and intrinsic passabilities are given as follows:

For the gas phase,Kg = Kkr,g(Sg) and (2.47)

ηg = ηηr,g(Sg). (2.48)

For the liquid phase,Kl = Kkr,l(1− Sg) and (2.49)

ηl = ηηr,l(1− Sg). (2.50)

Here, Sg is saturation of the gas phase. The relative permeability and passability for a gas-liquid flow system are given as kr,g = Sng , kr,l = (1 − Sg)

n,ηr,g = Smg , ηr,l = (1 − Sg)m

[36, 37]. Inserting the expression for the relative permeability and passability in Equation2.43, we get the modified Ergun equation to estimate the pressure drop caused by each phase

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in a non-Darcy gas-liquid flow system as follows:

∇pg = −%gg −µgSngK

vfg −%gSmg η|vfg|vfg and (2.51)

∇pl = −%lg −µl

(1− Sg)nKvfl −

%l(1− Sg)mη

|vfl|vfl, (2.52)

where ∇pg is the pressure gradient for the gas phase and ∇pl is the pressure gradient for theliquid phase. Here, vfg and vfl are the Forchheimer velocity vectors for the gas and liquidphases respectively.

Table 2.1: Exponents for relative permeability kr and relative passability ηr for the liquidphase (l) and the gas phase (g) [36, 37]

kr,l kr,g ηr,l ηr,gExponent n n m m

Lipinski 3 3 3 3Reed 3 3 5 5Theofanous 3 3 6 6

According to [36], commonly used values of the exponent n and m in Equation 2.51 andEquation 2.52 for relative permeability kr,α and relative passability ηr,α are proposed byLipinski, Reed and Theofanous and are given in Table 2.1. Detailed discussion on the selectionof exponents for relative permeability kr,α and relative passability ηr,α functions can be foundin [36, 37] .

According to [35], Tung and Dhir (1988) accounted the interfacial drag between differentflow phases and added a new term to the Ergun equation as given below:

∇pg = −%gg −µgSngK

vfg −%gSmg η|vfg|vfg +

FiSg

and (2.53)

∇pl = −%lg −µl

(1− Sg)nKvfl −

%l(1− Sg)mη

|vfl|vfl −Fi

(1− Sg), (2.54)

where Fi[Nm3

]is the volumetric interfacial drag force [35].

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Alternative to the modified Ergun equation, Barree and Conway proposed a new Forchheimertype formulation for the description of Darcy and Forchheimer flow regime. Their approachis discussed below.

2.5.2 Barree-Conway equation

Barree and Conway (2004) performed an experimental analysis of the non-Darcy flow throughporous media [4]. They represented the Forchheimer equation in a form similar to the Darcyequation as given below:

∇p = − µvfKapp

, (2.55)

where vf is the Forchheimer velocity vector and Kapp is called apparent permeability and isdefined as below:

1

Kapp

=1

K

(1 + β

K%f |vf |µ

). (2.56)

Thus, the Forchheimer equation can be rewritten as follows:

∇p = −µvfK

(1 + β

K%f |vf |µ

). (2.57)

Figure 2.5: Deviation of experimental data from Forchheimer linear equation [23]

As shown in Figure 2.5, the experimental data of Barree and Conway (thick blue line) did notfollow the linear apparent permeability Kapp (thin blue and red lines) given by Equation 2.56for a constant Forchheimer coefficient β (slope). Thus, Barree and Conway (2004) argued

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that the Forchheimer coefficient β and thus, the apparent permeability Kapp must vary withthe flow rate [4]. Barree and Conway also stated that a general model for a non-Darcy flowcan be obtained by giving up on the expectation for a constant Forchheimer coefficient β [4].The literature from Kaviany (1991) [22] and Nield & Bejan (2006) [32] discussed in Section4.1.3 also supports this argument.

From Equation 2.56, Barree and Conway suggested that the apparent permeability Kapp canalso be given as follows:

kapp =K

1 + Re, (2.58)

where the Reynolds number Re (see Equation 2.27) is evaluated based on the characteristiclength βK [m]. There exists no direct relationship for the interpretation of the Forchheimercoefficient β and it has to be determined from the experimental data.

2.5.2.1 Barree-Conway model for single and multiphase flows

Barree and Conway developed a new Forchheimer type equation for the representation ofsingle and multiphase non-Darcy flows. They proposed an alternative relationship for theapparent permeability based on the Log-Dose equation [4] as follows:

Kapp,α = kr,α +K − kr,α

(1 + ReFα )E, (2.59)

where, Kapp,α is the apparent permeability, kr,α is the relative permeability and Reα is theReynolds number for phase α. The exponents F and E are selected such that Equation2.59 follows the experimental data. Most of the literature based on the Barree-Conwayapproach uses F = 1 [4, 3, 39]. According to [3], the Barree-Conway equation for apparentpermeability is given as follows:

Kapp,α = K

(kmr,α +

(1− kmr,α)

(1 + %α|vfα|/µαT )E

), (2.60)

where T[

1m

]is called the transition constant and the minimum permeability ratio kmr,α is

the ratio of the relative permeability kr,α to the intrinsic permeability K [39]. SubstitutingEquation 2.60 in Equation 2.55, we get the general Barree-Conway relationship for single

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and multiphase non-Darcy flows:

∇pα = − µvfα

K

(kmr,α + (1−kmr,α)

(1+%α|vfα|/µαT)E

) , (2.61)

where ∇pα is the pressure gradient for phase α. Substituting E = 1 and the minimumpermeability ratio kmr,α = 0 in Equation 2.61, we get the Forchheimer equation for a singlephase flow. Here, the transition constant T is related to the Forchheimer coefficient β asT = 1

βK

[1m

].

Barree and Conway (2007) [3] stated that their model takes into account the Reynoldsnumber for each phase dependent on its intrinsic velocity (vfα) and the possibility of definingphase Reynolds number distinguishes their work from the previous ones. Wu et al. (2011)[39] presented a mathematical and numerical model to implement the Barree-Conway modelfor multiphase non-Darcy flows and also compared the numerical results with experiments.

2.5.2.2 Barree-Conway approach for relative permeability-saturationrelationship

The relative permeability-saturation relationship for a nitrogen-water non-Darcy porous mediaflow is predicted by many authors [33, 12, 3]. Barree and Conway (2007) tested nitrogen-water non-Darcy flow through various poppants (porous matrices) at different pressure gra-dients [3]. Given below is the brief summary of their experimental work.

Barree and Conway experimentally evaluated the relative permeability-saturation relationshipfor a multiphase non-Darcy flow system. The long solid matrix, initially fully saturated withwater was drained using nitrogen gas at different inlet pressures. Every single time prior togas injection, it was carefully ensured that the porous matrix is completely saturated withwater.

A typical relative permeability-saturation relationship for a non-Darcy gas-water flow systemis shown in Figure 2.6. Barree and Conway stated that the measured data cannot be used tocalculate the relative permeability as long as the injected gas breaks through the other endof the porous medium. The break through point was observed to be at around 38% of thegas saturation. Figure 2.6 shows the calculated gas and water relative permeability only afterthe injected gas breaks through the other end. “Directly measured data for both gas and

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Figure 2.6: Relative permeability-saturation relationship[3]

Figure 2.7: Corrected relative permeability-saturation relationship [3]

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water curves is affected by the non-Darcy flow, proportional to the Reynolds number (Re).Using individual phase saturation and velocity, the relative permeability curve of each phaseis corrected for non-Darcy effects” [3]. Figure 2.6 also shows corrected curves for non-Darcyrelative permeability-saturation relationship of gas (magenta) and water (green) phases [3].For details on non-Darcy corrections, please refer to [3, 24].

Barree and Conway performed five tests at different injection pressures and observed that afterapplication of non-Darcy corrections, the relative permeability data for a phase collapses to aconsistent data set (see Figure 2.7). This data set is extrapolated beyond the measured valuesin order to predict relative permeabilities for both lower and higher saturations. For additionaldata and further details regarding the experiments performed by Barree and Conway, pleaserefer to [3].

The modified Ergun equation discussed in Section 2.5.1 is a commonly used model in nuclearresearch for multiphase dryouts [36, 35, 37]. The acceptance of modified Ergun equation forthe description of a multiphase flow through a well structured porous matrix is convenientas the intrinsic permeability (see Equation 2.45) and passability (see Equation 2.46) can becalculated in terms of spatial parameters. However, for a complex porous matrix, the intrinsicpermeability K and passability η need to be determined using the experimental data.

On the other hand, according to [3, 39], the Barree-Conway model is valid for both Darcyand Forchheimer flow regimes. The Barree-Conway equation is a new approach mainly usedin the petroleum industry. For this approach, even though there is no need to calculatethe Forchheimer coefficient β, mathematical regression of the experimental data is anywayrequired to determine the transition constant T and the exponent E [4, 3].

The current work is restricted to single phase flow through porous medium and uses theForchheimer equation with Ergun interpretation of the Forchheimer coefficient (see Equation2.31).

2.6 DuMuX

DuMuX (DUNE for multi-{phase, component, scale, physics, . . . } flow and transport inporous media) is an open-source software for simulating the flow and transport processesin porous media [18]. DuMuX is built on top of DUNE (Distributed and Unified Numerics

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Environment). The main purpose of the DuMuX software is to provide a sustainable andconsistent framework for the implementation and application of model concepts, constitutiverelations, discretizations, and solvers for porous media applications. DuMuX can also bereferred to as an additional DUNE module as it inherits its functionalities from the DUNEcore modules.

DuMuX mainly consists of two different sorts of module implementations, fully-coupledmodules and decoupled modules. Fully-coupled modules describe the flow system usinga strongly coupled system of equations, which can be mass balance equation, energy balanceequation and balance equations for different phases. However, decoupled modules consistof a pressure equation which is iteratively coupled to a saturation equation, energy balanceequations etc.

As discussed in Chapter 1, Forchheimer models for single-phase isothermal and non-isothermalflow through porous media are implemented for numerical simulations in DuMuX for eachof the following thermodynamic assumptions:

• Isothermal single-phase Forchheimer flow.

• Non-isothermal single-phase Forchheimer flow with local thermal equilibrium.

• Non-isothermal single-phase Forchheimer flow with local thermal non-equilibrium.

Balance equations for the conservation of mass, momentum and energy for the above men-tioned models are given in Section 2.4. Relevant numerical examples with appropriate bound-ary conditions are set up for each model and numerical simulations are performed for thesame. Detailed description of numerical examples and comparison of numerical simulationswith the experimental results are given in Chapter 4.

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3 ITLR Experiment

As discussed in Section 1.1, a metallic porous medium offers an effective solution in manyheating and cooling engineering applications. Motivated by this, an experimental analysiswas performed in order to understand the convective cooling behavior of a well-structuredhomogeneous porous material at ITLR. The objective of this analysis is to develop a goodunderstanding about the convective heat transfer process and to use this knowledge toenhance the efficiency in cooling applications.

3.1 Experimental setup

Mayer et al. [30] conducted the experimental analysis for the current work at ITLR. In orderto determine the pressure loss and the overall heat transfer, an experiment was set up asshown in Figure 3.1. The porous matrix for this work is a uniform honeycomb like cylindricalstructure as shown in Figure 3.2. This cylindrical porous medium is held horizontally withthe help of a metallic structure. On its surface, the porous cylinder is wound with a heatingcoil along its complete length and the heating coil in turn is covered with a thick layerof insulating material in order to minimize the heat loss. During experiments, the porousstructure is exposed to a forced convective flow.

The schematic representation of the experimental setup is shown in Figure 3.3. The experi-mental setup consists of three major parts as follows.

Air Supply: Air at high pressure is supplied with the help of a compressor. A valve is usedto regulate the air pressure and flow rate. The mass flow is measured using a Venturi nozzle.

Test Section: The test section is a circular pipe filled with the porous medium. The porouscylinder has a diameter, D = 30 mm and a total length, L = 295 mm. The porous cylinderis made up of Ni-alloyed steel. Thermocouples with a diameter of 0.25 mm are embedded

29

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Figure 3.1: Photograph of ITLR experimental setup [30]

Figure 3.2: Porous structure used in ITLR experiments [29]

30

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Figure 3.3: Schematic representation of ITLR experimental setup [30]

into the tube wall at the inlet and the outlet of the porous cylinder. The thermocouples arealso distributed at regular intervals along the tube length in order to measure the surfacetemperature of the porous structure.

Data Acquisition System: The energy flow into the specimen is monitored and recordedby a single-phase energy meter. Pressure measurement modules with differential pressuretransducers are used to determine the pressure differences between the inlet and the outletof the specimen. Measurements of fluid and wall temperatures were taken using a dataacquisition unit. In the measurement procedure, the wall heat flux and the mass flow rateare held constant.

The inlet of the horizontally placed cylindrical assembly is connected to the air supply unitwhich continuously supplies air at the desired inlet pressure. The outlet is positioned suchthat the air can escape directly into the atmosphere after traveling through the porousmatrix. The pressure and the temperature are monitored by the data acquisition system.Two different experiments were conducted at ITLR for the current work and are discussedbelow.

3.1.1 Isothermal experiment

In this experiment, the complete system is assumed to be isothermal. That is, the tempera-ture at different locations in the porous domain is assumed to be the same and equal to theatmospheric temperature. During the experiment, compressed air at high pressure is injectedthrough the inlet and is released into the atmosphere at the outlet. As the experiment is

31

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isothermal, no heat flux is supplied to the porous structure. The velocity and pressure lossacross the cylindrical porous medium are measured.

Mayer et al. performed pore-scale computational fluid dynamics (CFD) simulations to eval-uate a CFD model against the experimental data [30]. In the current work, the isothermalexperimental data is used for the determination of intrinsic permeability K and Forchheimercoefficient β and to validate the REV-scale isothermal Forchheimer numerical model (seeSection 4.2.1).

3.1.2 Non-isothermal experiment

During this experiment, compressed air at high pressure is injected through the inlet and isreleased into the atmosphere at the outlet. Once the flow is established through the porousmedium, a constant heat flux is applied at the surface of the cylindrical porous structure.Upon reaching the steady-state, velocity and pressure are measured across the cylindricalporous matrix. Using thermocouples, surface temperature is measured at different locationsalong the length of the porous cylinder.

Similar to the isothermal case, Mayer et al. also performed pore-scale simulations to evaluatea CFD model against the non-isothermal experimental data [30]. In the current work, theexperimental data is used to validate the REV-scale non-isothermal Forchheimer numericalmodel (see Section 4.2.2).

3.1.3 Motivation for the current work

At ITLR, experiments are performed to analyze the heat-transfer properties of a uniformporous medium. In addition to the experimental analysis, pore-scale CFD simulations arealso performed for the reduced domain for both isothermal and non-isothermal cases [30].

Even with the reduced domain, the complexity of flow paths and the porous structure makeit very difficult to perform a detailed pore-scale numerical investigation [30]. Moreover,computational cost and time for the pore-scale CFD simulation has always been an issue.Thus, for a large scale application, with the need to describe a complex porous structure andlimited computational resources, it would be practically impossible to perform a pore-scaleCFD simulation. This motivates the use of the volume averaged (REV-scale) approach forthe current work.

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In the scope of the current work, as discussed in Section 2.6, the isothermal and non-isothermal models are implemented as a part of DuMuX. Numerical simulations are per-formed and an attempt is made to validate the implemented DuMuX models against theexperimental data from ITLR (see Section 4.2).

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4 Results and Discussion

As mentioned in Chapter 1, the ultimate goal of the current work is to develop thermody-namic models as discussed in Section 2.6 and validate them against experimental data. Forthis purpose, several numerical simulations have been carried out and are discussed in thischapter. Firstly, we look at and compare various approaches for the determination of intrinsicpermeability K and Forchheimer coefficient β in Section 4.1. Secondly, in Section 4.2, thenumerical results for an isothermal case are compared with the isothermal experimental data(see Section 4.2.1) and the numerical results for a non-isothermal case are compared withthe non-isothermal experimental data (see Section 4.2.2).

4.1 Intrinsic permeability and Forchheimer coefficient

As discussed in Section 2.4.2.2, for high velocity flow through porous media, the momentumis described by the Forchheimer equation. In order to perform numerical simulations with theForchheimer model determining accurate values of intrinsic permeability K and the Forch-heimer coefficient β for the flow system becomes very crucial. The intrinsic permeabilityK and the Forchheimer coefficient β for a flow system are either determined analyticallyby fitting the experimental data with the Forchheimer equation or by using some standardrelationships in terms of spatial parameters (see Section 2.5.1). For the current work, bothintrinsic permeability K and Forchheimer coefficient β are determined by fitting the experi-mental data for the isothermal case (see Table 4.1) with the Forchheimer equation given byEquation 2.28.

Initial attempts to fit both intrinsic permeability K and Forchheimer coefficient β over thecomplete range of the isothermal experimental data (see Table 4.1) using nonlinear regressionanalysis with the Forchheimer equation did not lead to a physically meaningful intrinsicpermeability K. In order to overcome this issue, it was decided to use following threeapproaches.

1. Determine the intrinsic permeability K by performing linear regression analysis of the

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experimental data with the Darcy equation and use this intrinsic permeability K forthe nonlinear regression analysis with the Forchheimer equation to calculate the Forch-heimer coefficient β (see Section 4.1.1).

2. Determine both intrinsic permeability K and Forchheimer coefficient β by performingnonlinear regression analysis for subsets of the experimental data (Re < 180) with theForchheimer equation (see Section 4.1.2).

3. Use intrinsic permeability K from the above approach and adapt the Forchheimercoefficient β in order to account for high velocity (Re > 180) inertial effects [22, 32](see Section 4.1.3).

4.1.1 Linear regression analysis for intrinsic permeability K and

nonlinear regression analysis for Forchheimer coefficient β

For this approach, the idea is to first calculate the intrinsic permeability K by performinglinear regression analysis for different subsets of the experimental data given in Table 4.1with the Darcy equation (see Equation 2.26). The calculated intrinsic permeability K is usedfor the nonlinear regression analysis of different subsets of the experimental data with theForchheimer equation (see Equation 2.28) in order to determine the Forchheimer coefficientβ.

4.1.1.1 Linear regression analysis for intrinsic permeability K

For the current work, the available experimental data is well beyond the Darcy range, i.e,(Re >> 1). Thus, in order to determine the intrinsic permeability K, linear regressionanalysis is performed only for the set of first three experimental data values in Table 4.1.

The Darcy system of equations for the linear regression is given below:

[−∇pi] =1

K[µivfi] for i = {1, 2, 3}, (4.1)

where the subscript i indicates the experiment number given in Table 4.1. The intrinsicpermeability K calculated by linear regression is given below:

K = 2.8× 10−8 [m2]. (4.2)

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Table4.1:

Exp

erim

entald

ataforisotherm

alcase

(ITLR

)Exp

t.no

Pressuregrad

ient

(∇p E

xpt)

Density

(%)

Viscosity

(µ)

Velocity

(vf)

Reyno

ldsno

.(R

e)(i)

[Pam

][ kg m

3

][ kg m·s

][ m s

][−

]

118

1.66

01.13

1846

076

1.80

074×

10−

50.24

0404

334

16.552

265

1.76

01.13

3291

891

1.80

101×

10−

50.53

3709

961

36.794

321

20.25

1.13

8238

830

1.80

131×

10−

51.00

5893

441

69.650

449

93.44

1.14

8195

312

1.80

158×

10−

51.57

8759

265

110.27

580

04.28

1.15

8573

870

1.80

191×

10−

52.03

7208

190

143.58

612

263.25

1.17

3422

397

1.80

218×

10−

52.49

9192

423

178.40

719

448.21

1.19

8658

910

1.80

241×

10−

53.08

8510

077

225.20

824

294.88

1.21

5673

561

1.80

257×

10−

53.40

5504

908

251.84

930

284.81

1.23

6787

096

1.80

266×

10−

53.75

2667

697

282.34

1040

987.76

1.27

4604

158

1.80

273×

10−

54.31

9910

287

334.95

1149

150.42

1.30

3478

797

1.80

274×

10−

54.62

8402

860

367.00

1258

746.00

1.33

7432

461

1.80

274×

10−

54.96

1807

053

403.69

1368

519.69

1.37

2085

828

1.80

267×

10−

55.35

3980

962

446.88

1478

375.66

1.40

7044

240

1.80

261×

10−

55.52

9258

966

473.27

1588

688.64

1.44

3628

417

1.80

252×

10−

55.86

0607

217

514.67

1610

0100

.93

1.48

4166

729

1.80

238×

10−

56.06

0703

046

547.19

1711

6934

.34

1.54

3826

676

1.80

232×

10−

56.37

2508

522

598.47

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100 200 300 400 500 600 700 800 900 1000−12

−10

−8

−6

−4

−2

0x 10

4

Re [−]

∇ p

[P

a/m

]

ITLR Exp

Linear Regression

Figure 4.1: Linear regression with Darcy law

The intrinsic permeability given by Equation 4.2 and the experimental data for velocity andviscosity (see Table 4.1) are used with Equation 4.1 for the back calculation of the corre-sponding pressure gradients. The calculated pressure gradients ∇pCalc (black plus marks)and the experimental pressure gradients (blue stars) are plotted against the flow Reynoldsnumber (Re) as shown in Figure 4.1. The Reynolds number for each experimental data isdetermined using Equation 2.27 and is given in Table 4.1.

From Figure 4.1, one can clearly observe that the calculated Darcy pressure gradients (blackplus marks) immediately start diverging from the experimental pressure gradients (blue stars)and follow an inclined line. Moreover, the experimental pressure gradient drops nonlinearlywith increasing Reynolds number (Re). From [19, 15, 22, 5, 32], one can say that, thisnonlinear drop in the pressure gradient is caused by the high velocity inertial effects. Asdiscussed in Section 2.4.2.2, these inertial effects are accounted for by the kinetic energyterm in the Forchheimer equation.

4.1.1.2 Nonlinear regression for Forchheimer coefficient β

In this section, the Forchheimer coefficient β is determined by performing nonlinear regressionanalysis of the experimental data with the Forchheimer equation (see Equation 2.28). Here,the intrinsic permeability K is given by Equation 4.2. The system of equations for nonlinearForchheimer regression is as follows:

[−∇pi] =[µivfi %iv

2fi

] [1/K β

]Tfor i = {1, 2, 3, . ., 17}, (4.3)

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Table 4.2: Forchheimer coefficient β for different subsets of the experimental dataExpt. dataset i Forch. coeff. β [1/m] Ergun coeff.CE{1, 2, 3} β1= 1608.66 CE1= 0.2499{1, 2, . ., 6} β2= 1573.28 CE2= 0.2444{1, 2, . ., 9} β3= 1757.76 CE3= 0.2731{1, 2, . ., 17} β4= 1902.16 CE4= 0.2955

100 200 300 400 500 600 700 800 900 1000−14

−12

−10

−8

−6

−4

−2

0x 10

4

Re [−]

∇ p

[P

a/m

]

ITLR ExpForch coeff β

1

Forch coeff β2

Forch coeff β3

Forch coeff β4

Figure 4.2: Linear Darcy regression for intrinsic permeability K and nonlinear Forch-heimer regression for Forchheimer coefficient β

where the subscript i indicates the experiment number given in Table 4.1. The nonlinearregression analysis is performed for different subsets of the experimental data (i.e., i =

{1, 2, 3}, i = {1, 2, 3, . ., 6}, i = {1, 2, 3, . ., 9} and i = {1, 2, 3, . ., 17}).

The Forchheimer coefficients obtained for different subsets of the experimental data are givenin Table 4.2. These Forchheimer coefficients β are used along with intrinsic permeabilityK given by Equation 4.2, the experimental data for velocity and viscosity in Table 4.1 andEquation 4.3 to back calculate the corresponding pressure gradients. The calculated pressuregradients ∇pCalc are plotted against the Reynolds number as shown in Figure 4.2. It appearsfrom Figure 4.2, that the calculated pressure gradients match well with experimental ones,especially for lower Reynolds numbers (Re). However, it is clear from Table 4.3 that withthe current approach, the pressure gradients at lower Reynolds numbers are predicted poorly.

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Table 4.3: Percentage error for the calculated pressure gradients using different Forch-heimer coefficients given in Table 4.2

∇pExpt[Pam ] % Error∇p(β1) % Error∇p(β2) % Error∇p(β3) % Error∇p(β4)

181.660 46.27 45.00 51.64 56.84651.760 34.27 32.52 41.66 48.812120.25 18.87 16.95 26.97 34.814993.44 13.00 10.97 21.54 29.828004.28 13.23 11.11 22.19 30.8612263.25 9.260 7.151 18.17 26.8019448.21 4.580 2.500 13.35 21.8424294.88 2.060 3.010 10.71 19.0930284.81 3.080 1.940 8.660 16.9640987.76 7.390 2.450 8.250 16.6349150.42 8.140 5.150 5.320 13.5258746.00 5.060 7.040 3.290 11.3868519.69 3.350 5.380 5.200 13.4878375.66 7.910 9.860 0.260 8.19088688.64 6.600 8.580 1.730 9.800100100.93 9.300 11.23 1.180 6.670116934.34 11.08 12.98 3.090 4.640

Here, the percentage error for calculated pressure gradient ∇pCalc is defined as follows:

%Error(∇p): = 100 ·(∇pExpt −∇pCalc

∇pExpt

), (4.4)

where ∇pCalc and ∇pExpt stand for the calculated and experimental pressure gradients re-spectively.

In Section 4.1.3, the apparent permeability Kapp for the current approach is determined usingthe intrinsic permeability K given by Equation 4.2 and the Forchheimer coefficient β4 andcompared with the apparent permeability Kapp of the approach discussed in Section 4.1.2.

4.1.2 Nonlinear regression analysis for both intrinsic permeability

K and Forchheimer coefficient β

As discussed earlier in this chapter, determination of both intrinsic permeability K and Forch-heimer coefficient β using the quadratic Forchheimer regression analysis for the complete setof the experimental data given in Table 4.1 did not lead to a physically meaningful intrinsic

39

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permeability K. Thus, nonlinear regression analysis is repeated for different subsets of theexperimental data. It is observed that the quadratic regression analysis of the experimentaldata with Reynolds number Re < 180 leads to a reasonable and physically meaningful valueof the intrinsic permeability K and the Forchheimer coefficient β. The system of equationsfor the nonlinear Forchheimer regression is as follows:

[−∇pi] =[µivi %iv

2i

] [1/K β

]Tfor i = {1, 2, 3, 4, 5, 6}, (4.5)

where the subscript i indicates the experiment number for Re < 180 given in Table 4.1.Here, the nonlinear regression analysis is performed for different subsets of the experimentaldata (i.e., i = {1, 2, 3}, i = {1, 2, 3, 4}, i = {1, 2, 3, 4, 5} and i = {1, 2, 3, 4, 5, 6}).

Table 4.4: Forchheimer coefficient β for different subsets of the experimental dataExpt. dataset i Intr. perm. K [m2] Forch. coeff. β [1/m] Ergun coeff.CE{1, 2, 3} K5 = 6.0687× 10−8 β5= 1608.66 CE5= 0.3962{1, 2, 3, 4} K6 = 5.7589× 10−8 β6= 1593.02 CE6= 0.3822{1, 2, 3, 4, 5} K7 = 4.1427× 10−8 β7= 1510.57 CE7= 0.3074{1, 2, 3, 4, 5, 6} K8 = 5.7299× 10−8 β8= 1573.28 CE8= 0.3766

The Intrinsic permeability K and the Forchheimer coefficient β calculated from the nonlinearForchheimer regression of the above mentioned experimental datasets are given in Table 4.4.The Ergun coefficient CE is calculated for each Forchheimer coefficient β using Equation2.30. It is observed that the intrinsic permeability K for all the datasets mentioned in Table4.4 has the same order of magnitude as obtained by Mayer et al. [30].

The intrinsic permeability K and the Forchheimer coefficient β for each dataset given inTable 4.4 are used along with Equation 4.5 and the experimental data for density, velocityand viscosity given in Table 4.1 for the back calculation of pressure gradients ∇pCalc. Thecalculated pressure gradients are plotted against Reynolds number (Re) as shown in Figure4.3. The dotted line connecting the calculated pressure gradients indicate the quadraticbehavior of the Forchheimer equation for the corresponding intrinsic permeability K andForchheimer coefficient β pair in Table 4.4.

Table 4.5 shows the percentage error for calculated pressure gradients ∇pCalc. Here, thepercentage error is determined using Equation 4.4. Comparing percentage errors in Table

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100 200 300 400 500 600 700 800 900 1000−12

−10

−8

−6

−4

−2

0x 10

4

Re [−]

∇ p

[P

a/m

]

ITLR Exp

Forch coeff β5

Forch coeff β6

Forch coeff β7

Forch coeff β8

Figure 4.3: Nonlinear regression with Forchheimer law

4.3 and Table 4.5, it is clear that the current approach fits better over the complete range ofthe experimental data, especially for low Reynolds numbers (Re < 180). Thus, from Table4.3 and Table 4.5, the best fitted intrinsic permeability K and the Ergun coefficient CE arechosen for numerical simulations as follows:

K = K8 = 5.73× 10−8m2 and (4.6)

CE = CE8 = 0.3766. (4.7)

The numerical results are compared with the experimental data in Section 4.2.

From Figure 4.3 and Table 4.5 it is also clear that at higher Reynolds numbers (Re > 180)the experimental data starts diverging from the current approach. Similar behavior is alsoobserved by [22, 5, 32] and is discussed in Section 4.1.3.

4.1.3 Apparent permeability

As discussed in Section 2.5.2, Barree and Conway [4] in their work stated that if one gives upon the expectation of a constant Forchheimer coefficient β, any porous media flow regime canbe described using the Forchheimer equation. They represented the Forchheimer equationby a Darcy-like equation (see Equation 2.55) , where the apparent permeability given by

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Table 4.5: Percentage error of the calculated pressure gradients using different intrinsicpermeabilities and Forchheimer coefficients given in Table 4.4

∇pExpt[Pam ] % Error∇p(β5) % Error∇p(β6) % Error∇p(β7) % Error∇p(β8)

181.660 6.9189 5.5958 5.8862 5.1192651.760 1.4013 1.7924 6.6393 0.93052120.25 0.0870 0.2641 0.4089 1.26954993.44 0.4754 0.0233 1.2985 1.06388004.28 3.2408 2.6558 0.4181 1.505212263.25 1.3640 0.7096 2.0713 0.442219448.21 1.4409 2.1465 5.3562 3.285824294.88 3.1797 3.9012 7.2606 5.028230284.81 4.4719 5.2096 8.7130 6.328840987.76 4.1592 4.9333 8.6960 6.065649150.42 6.4367 7.2086 11.000 8.318458746.00 7.9362 8.7108 12.551 9.807068519.69 5.9489 6.7547 10.783 7.878678375.66 10.200 10.977 14.883 12.05288688.64 8.6869 9.4873 13.531 10.584100100.93 11.163 11.949 15.935 11.017116934.34 12.692 13.473 17.458 12.526

Equation 2.56 can be rewritten as follows:

1

Kapp

=1

K+ β

%|vf |µ

. (4.8)

The apparent permeabilities Kapp are determined using the intrinsic permeability K and theForchheimer coefficient β for approaches discussed in Section 4.1.1 and Section 4.1.2. Theapparent permeabilities Kapp for experiments are calculated using Equation 2.55 and theexperimental data given in Table 4.1. Figure 4.4 shows a plot of the apparent permeabilityfor the approaches discussed in Section 4.1.1 and Section 4.1.2.

In Figure 4.4, slope of each data-line represents the Forchheimer coefficient β for the corre-sponding approach. It is very clear from Figure 4.4, that the approach discussed in Section4.1.1 under-predicts the intrinsic permeability K and thus the apparent permeability Kapp

over the complete range of the experimental data.

It can also be seen from Figure 4.4, that for higher flow velocities, the apparent permeabilityKapp for experiments (blue stars) starts behaving nonlinearly and diverges from the approach

42

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0 1 2 3 4 5 6

x 105

0

2

4

6

8

10

12x 10

8

ρ.v/µ [m−1

]

1/K

ap

p [

m−

2]

ITLR Exp Data

Forch Regression (K & β) (1−6)

Darcy Regession (K) & Forch Regression (β)

Figure 4.4: Apparent permeability

discussed in Section 4.1.2. Barree and Conway (2005) [5] explained such behavior by statingthat, the Forchheimer coefficient β should vary with the flow rate in order to account forinertial nonlinearities. Kaviany (1991) and Nield & Bejan (2006) [22, 32] stated that, thequadratic Forchheimer equation with a constant Forchheimer coefficient β is only valid upto a specific Reynolds number (for the current work Re ≈ 180), and for higher Reynoldsnumbers (Re > 180), the Forchheimer coefficient β should be adjusted in order to accountfor high velocity inertial effects. In the scope of the current work, an attempt is made toaccount for the inertial effects using one of the following approaches:

1. A constant Forchheimer coefficient β is defined for the complete range of the experi-mental data (see Section 4.1.3.1).

2. The complete set of the experimental data in Table 4.1 is divided into different rangeson the basis of the Reynolds number (Re) and a constant Forchheimer coefficient β isdetermined for each range (see Section 4.1.3.2).

3. For the high velocity flow regime (Re > 180), based on the experimental data, theForchheimer coefficient β is described as a linear function of the Reynolds number (seeSection 4.1.3.3).

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4.1.3.1 Constant Forchheimer coefficient β for complete range of experimentaldata

For this approach, the intrinsic permeability K is set to a constant value given by Equation4.6. Using calculated apparent permeabilities Kapp for experiments, intrinsic permeability Kand the experimental data along with Equation 4.8, a constant Forchheimer coefficient β(slope) is determined for the complete range of the experimental data as shown (green line) inFigure 4.5. The red line and the blue stars in Figure 4.5 describe the apparent permeabilitiesKapp for the nonlinear regression approach (see Section 4.1.2) and experiments respectively.

0 1 2 3 4 5 6

x 105

0

2

4

6

8

10

12x 10

8

ρ.v/µ [m−1

]

1/K

ap

p [

m−

2]

ITLR Exp Data

Forch Regression (1−6)

Forch Coeff β Comp Range

Figure 4.5: Constant Forchheimer coefficient β for the complete range of experimentaldata

Intrinsic permeability K given by Equation 4.6, Ergun coefficient CE = 0.4904 determinedusing the Forchheimer coefficient β for the current approach and Equation 2.30 are used fornumerical simulations. Pressure gradients ∇pCalc are back calculated using Equation 2.29and the experimental data from Table 4.1. The percentage errors for the calculated pressuregradients are determined using Equation 4.4 and compared with other approaches in Table4.7.

4.1.3.2 Forchheimer coefficient β for limited Re ranges

For this approach, the intrinsic permeability K is set to a constant value given by Equation4.6. Using calculated apparent permeabilities Kapp for the experiments, intrinsic permeability

44

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0 100 200 300 400 500 600 7001000

1500

2000

2500

Re [−]

β [

1/m

]

ITLR Exp

Forch Coeff Nonlinear Regression

Forch Coeff (Re<180)

Forch Coeff (180<Re<340)

Forch Coeff (340<Re<475)

Forch Coeff (Re>475)

Figure 4.6: Forchheimer coefficient β for limited Re ranges

0 1 2 3 4 5 6

x 105

0

2

4

6

8

10

12x 10

8

ρ.v/µ [m−1

]

1/K

ap

p [

m−

2]

ITLR Exp

Forch Regression

App Perm (Re<180)

App Perm (180<Re<340)

App Perm (340<Re<475)

App Perm (Re>475)

Figure 4.7: Apparent permeability Kapp for limited Re ranges

45

Page 58: Forchheimer Porous-media Flow Models - Numerical Investigation

K and the experimental data along with Equation 4.8, corresponding Forchheimer coefficientsβ are determined and are shown (blue stars) in Figure 4.6.

The calculated Forchheimer coefficients β are divided into different ranges on the basis ofReynolds number (Re) and an average Forchheimer coefficient β is determined for eachReynolds number range and is shown (horizontal magenta lines) in Figure 4.6. Using theaverage Forchheimer coefficients β, intrinsic permeability K and the experimental data alongwith Equation 4.8, apparent permeabilities Kapp are back calculated for each Reynolds num-ber range and are shown (inclined magenta lines) in Figure 4.7.

Figure 4.6 and Figure 4.7 show a plot of the Forchheimer coefficient β against the Reynoldsnumber (Re) and a plot of the corresponding apparent permeability Kapp equation (seeEquation 4.8) respectively. It can be seen from Figure 4.6 and Figure 4.7 that the Forchheimercoefficient β for nonlinear regression approach (red line) discussed in Section 4.1.2 is constantfor all flow velocities. Whereas, for the current approach, the Forchheimer coefficient β andtherefore the apparent permeabilityKapp are adjusted in order to account for nonlinear inertialeffects at higher Reynolds numbers (Re > 180).

Table 4.6: Forchheimer coefficient β for limited Re rangesReynolds number (Re) Forch. coeff. β Ergun coeff. CERe < 180 β9= 1573.28 CE9= 0.3766180 < Re < 340 β10= 1615.25 CE10= 0.3866340 < Re < 475 β11= 1709.87 CE11= 0.4093Re > 475 β12= 1772.92 CE12= 0.4244

The Forchheimer coefficient β obtained for the different ranges of Reynolds numbers are givenin Table 4.6. For each Reynolds number range, the Ergun coefficient CE is calculated usingEquation 2.30 and the corresponding Forchheimer coefficient β. The intrinsic permeabilityK given by Equation 4.6 and the Ergun coefficients CE for different Reynolds number rangesfrom Table 4.6 are used for numerical simulations. Pressure gradients ∇pCalc for the currentapproach are back calculated using Equation 2.29 and the experimental data from Table 4.1.The percentage errors for the calculated pressure gradients are determined using Equation4.4 and compared with other approaches in Table 4.7.

46

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0 100 200 300 400 500 600 7001000

1500

2000

2500

Re [−]

β [

1/m

]

ITLR Exp Data

Forch Coeff Nonlinear Regression

Forch Coeff (Re<180)

Forch Coeff Fit (180<Re<340)

Forch Coeff Fit (340<Re<475)

Forch Coeff Fit (Re>475)

Forch Coeff f(Re)

Figure 4.8: Forchheimer coefficient β as a function of Reynolds number (Re)

4.1.3.3 Linear Forchheimer coefficient β(Re)

Similar to Section 4.1.3.1 and Section 4.1.3.2, for this approach, the intrinsic permeability Kis set to a constant value given by Equation 4.6. As discussed in Section 4.1.3.2, Forchheimercoefficients β for experiments (blue stars), nonlinear Forchheimer regression approach (redline) and Forchheimer coefficient β for limited Re ranges approach (magenta line) are shownin Figure 4.6. It is clear from Figure 4.6, that for Re > 180, the Forchheimer coefficient βfor experiments varies linearly with the flow rate. Thus, as suggested by [22, 32], for the highvelocity flow regime (Re > 180), a linear relationship between the Forchheimer coefficient βand the flow Reynolds number (Re) is determined (see Figure 4.8) as follows:

β(Re) = 0.6364Re + 1560.3. (4.9)

It can be seen from Figure 4.8, that the linear description of the Forchheimer coefficientβ (cyan line) accounts for nonlinear inertial effects at higher flow velocities (Re > 180).Equation 4.9 for the Forchheimer coefficient β allows the cubic behavior of the Forchheimerequation and follows the experimental data for Re > 180. A cubic form of the Forchheimerequation is given below:

∇p =µvfK

(1 + β(Re(|vf |))

K%|vf |µ

). (4.10)

47

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0 1 2 3 4 5 6

x 105

0

2

4

6

8

10

12x 10

8

ρ.v/µ [m−1

]

1/K

ap

p [

m−

2]

ITLR Exp Data

Forch Coeff Nonlinear Regression

Forch Coeff (Re<180)

Forch Coeff (180<Re<340)

Forch Coeff (340<Re<475)

Forch Coeff (Re>475)

Forch Coeff f(Re)

Figure 4.9: Apparent permeability β as a function of velocity

Using Equation 4.8, the intrinsic permeability K given by Equation 4.6, the Forchheimercoefficient β given by Equation 4.9 and the experimental data (see Table 4.1), the appar-ent permeabilities Kapp are calculated for the high velocity flow regime (Re > 180). Thecalculated apparent permeabilities (cyan line) are plotted as shown in Figure 4.9.

Pressure gradients ∇pCalc for the current approach are back calculated using Equation 2.29and the experimental data from Table 4.1. The percentage errors for the calculated pressuregradients are determined using Equation 4.4 and compared with other approaches in Table4.7. From Figure 4.9 and Table 4.7, it is very clear that the current approach describes theexperimental pressure gradients better than any other approach discussed in Section 4.1.1,Section 4.1.2 and Section 4.1.3.

The Ergun coefficient CE is determined from Equation 2.30 and Equation 4.9 as follows:

CE =√K(0.6364Re + 1560.3). (4.11)

For the current approach, a constant Ergun coefficient CE = 0.3766 (see Equation 4.7) forRe < 180 and a linear Ergun coefficient CE (see Equation 4.11) for Re > 180 are used alongwith intrinsic permeability K given by Equation 4.6 for numerical simulations.

48

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Table4.7:

Com

parisonof

percentage

errors

ofcalculated

pressure

grad

ientsusingdiffe

rent

approa

ches

forFo

rchh

eimer

coef-

ficientβ

∇p E

xpt[Pam

]Non

linear

regression

∇p(βcomp)∇p(βranges

)∇p(β

(Re)

)∇p(β

5)∇p(β

6)

∇p(β

7)∇p(β

8)

181.66

06.91

895.59

585.88

625.11

9210

.830

5.11

925.11

9265

1.76

01.40

131.79

246.63

930.93

052.42

860.93

050.93

0521

20.25

0.08

700.26

410.40

891.26

953.75

691.26

951.26

9549

93.44

0.47

540.02

331.29

851.06

385.72

051.06

381.06

3880

04.28

3.24

082.65

580.41

811.50

529.23

721.50

521.50

5212

263.25

1.36

400.70

962.07

130.44

227.65

450.44

220.21

0519

448.21

1.44

092.14

655.35

623.28

585.02

460.82

020.88

6724

294.88

3.17

973.90

127.26

065.02

823.31

412.59

631.67

3230

284.81

4.47

195.20

968.71

306.32

882.06

553.91

871.88

4340

987.76

4.15

924.93

338.69

606.06

562.57

063.63

370.36

8649

150.42

6.43

677.20

8611

.000

8.31

840.21

460.56

150.85

7258

746.00

7.93

628.71

0812

.551

9.80

701.31

542.15

421.13

3168

519.69

5.94

896.75

4710

.783

7.87

860.88

740.04

182.58

7378

375.66

10.200

10.977

14.883

12.052

3.63

173.55

851.12

0588

688.64

8.68

699.48

7313

.531

10.584

1.95

830.57

562.02

9710

0100

.93

11.163

11.949

15.935

11.017

4.58

112.14

690.40

0211

6934

.34

12.692

13.473

17.458

12.526

6.17

733.82

480.43

79

49

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4.2 Test cases

After analyzing the experimental data for the determination of the intrinsic permeability Kand the Forchheimer coefficient β, the immediate objective is the validation of implementedDuMuX models by setting up a relevant numerical example for each model. Thus, a test casewith appropriate boundary conditions is setup for an isothermal model (see Section 4.2.1)and a non-isothermal model (see Section 4.2.2).

4.2.1 Incompressible isothermal case

During incompressible isothermal experiments at ITLR, different pressure gradients are ap-plied across the porous cylinder and the corresponding flow velocities are recorded (see Table4.1). These pressure gradients are used as boundary conditions for numerical simulationswith the corresponding DuMuX model and numerical results are compared with the experi-mental data. Nitrogen is used as a working fluid for the numerical simulations in the currentwork. The incompressible behavior of the fluid is ensured by fixing the density and viscosityof the fluid phase as given in Table 4.1.

A set of simulations is performed using the intrinsic permeability K given by Equation 4.6and the Ergun coefficients CE for different Forchheimer coefficient β approaches discussedin Section 4.1.2 and Section 4.1.3 (see Table 4.8).

Table 4.8: Ergun coefficient CE for different Forchheimer coefficient β approachesApproach for Forchheimer coefficient β Ergun Coefficient CENonlinear Forchheimer regression 0.3766Forchheimer coefficient β for complete data range 0.4904

Forchheimer coefficient β for limited Re ranges

0.3766 (Re < 180)0.3866 (180 < Re < 340)0.4093 (340 < Re < 475)0.4244(Re > 475)

Forchheimer coefficient β = β(Re) 0.3766 (Re < 180)√K(0.6364Re + 1560.3) (Re > 180)

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Model domain:

Figure 4.10: Incompressible isothermal model domain

For simulating incompressible isothermal flow, due to the symmetric shape of the porousdomain and the unidirectional flow velocity, only one quarter of the cylinder is modeled. Forthis domain, a uniform mesh with hexahedral elements is created using IcemCFD 12.1 as themeshing tool. The dimensions of the mesh are the same as that of the porous domain, i.e.,length (L) = 295 mm and diameter (d) = 30 mm. The model domain is shown in Figure4.10. Boundary conditions, input data and results for the incompressible isothermal case arediscussed below.

Boundary conditions:

For the incompressible isothermal case, the flow is only governed by the pressure differenceapplied across the cylinder. Thus, a Dirichlet boundary condition for pressure is applied at theinlet and the outlet of the domain. The Dirichlet boundary condition at the inlet is calculatedfrom the pressure gradient data provided by ITLR. Whereas, the Dirichlet boundary conditionat the outlet is set to the atmospheric pressure (100000Pa). The other boundaries are setto no flow (Neumann) boundary condition. As the problem is isothermal, the temperaturethroughout the domain is set to a constant atmospheric temperature, i.e., 300K.

Input data:

• The intrinsic permeability K for the solid matrix is obtained earlier in this chapter asK = 5.73× 10−8m2 (see Equation 4.6).

51

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• The porosity of the solid matrix is provided by ITLR, i.e., φ= 0.558.

• The Ergun coefficients CE belonging to different approaches are given in Table 4.8.The selection of the Ergun coefficient CE depends on the approach one wants to useand the flow Reynolds number (Re).

Results and discussion:

(a) Qualitative pressure distribution (b) Quantitative pressure distribution

Figure 4.11: Pressure distribution across porous domain for an incompressible isothermalflow

(a) Qualitative velocity distribution (b) Quantitative velocity distribution

Figure 4.12: Velocity distribution across porous domain for an incompressible isothermalflow

Simulations are performed for each of the experimental observations using Ergun coefficientsCE belonging to different approaches mentioned in Table 4.8. In order to exhibit the distri-bution of the pressure and velocity fields across the porous cylinder, one of the simulationsrun is presented (see Figure 4.11 and Figure 4.12). It can be observed from Figure 4.11b and

52

Page 65: Forchheimer Porous-media Flow Models - Numerical Investigation

Table 4.9: Experimental and numerical velocity data∇pExpt [Pa

m] vfExpt.

[ms

]vf[ms

](β8) vf

[ms

](βcomp) vf

[ms

](βranges) vf

[ms

](β(Re))

181.660 0.2404 0.2423 0.2571 0.2423 0.2571651.760 0.5337 0.5283 0.5269 0.5283 0.54322120.25 1.0059 1.0073 0.9869 1.0072 1.03144993.44 1.5788 1.5785 1.5694 1.5785 1.60538004.28 2.0372 2.0096 1.9477 2.0096 2.031512263.25 2.4992 2.4899 2.4687 2.4899 2.500919448.21 3.0885 3.1228 3.0135 3.1016 3.109524294.88 3.4055 3.4751 3.3507 3.4515 3.443430284.81 3.7527 3.8559 3.7152 3.7239 3.799640987.76 4.3199 4.4318 4.2662 4.2798 4.325649150.42 4.6284 4.8068 4.6248 4.6416 4.664758746.00 4.9618 5.1954 4.9965 5.0167 5.009468519.69 5.3539 5.5486 5.3319 5.3551 5.307378375.66 5.5293 5.8629 5.6348 5.6009 5.584988688.64 5.8606 6.1622 5.9210 5.8437 5.8287100100.93 6.0607 6.4615 6.2071 6.1272 6.0783116934.34 6.3725 6.8534 6.5819 6.4987 6.3922

Figure 4.12b, that for an incompressible isothermal flow, the pressure is linearly distributedand the flow velocity is constant along the axis of the porous cylinder.

The numerical data for velocity is collected for the different sets of simulations performed us-ing Ergun coefficients CE for different approaches mentioned in Table 4.8. The experimentaldata for the pressure gradient and the corresponding experimental and numerical data forvelocity are given in Table 4.9.

The experimental and the numerical velocity data presented in Table 4.9 is used for theevaluation of the friction coefficient f . The friction coefficient of a porous medium is definedin terms of porosity φ and specific interfacial area sv as follows [30]:

f =∇pφ3

%f |vf |2sv. (4.12)

Table 4.10 shows calculated friction coefficients and corresponding percentage errors belong-ing to the experimental and numerical velocities given in Table 4.9. Here, the percentage

53

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Table4.10

:Exp

erim

entala

ndnu

merical

friction

coeffi

cients

∇p E

xpt

[Pam

]f E

xpt

f Num(β

Regr.)

f Num(β

comp)

f Num(β

ranges)

f Num(β

(Re))

f Num

%Error

f Num

%Error

f Num

%Error

f Num

%Error

181.66

00.53

550.52

711.57

160.46

7912

.609

0.52

711.57

160.46

791.57

1665

1.76

00.38

930.39

732.03

650.39

942.58

570.39

722.03

650.37

582.03

6521

20.25

0.35

500.35

410.27

110.36

873.87

960.35

400.27

110.33

760.27

1149

93.44

0.33

640.33

650.03

710.34

041.18

840.33

650.03

710.32

540.03

7280

04.28

0.32

100.32

982.76

240.35

119.40

120.32

982.76

240.32

282.76

2412

263.25

0.32

260.32

500.74

130.33

062.47

930.32

500.74

130.32

210.14

2919

448.21

0.32

790.32

082.18

220.34

455.03

460.32

520.84

030.32

351.34

6824

294.88

0.33

220.31

913.96

360.34

323.29

310.32

342.64

580.32

502.19

1230

284.81

0.33

530.31

755.28

430.34

202.02

520.34

041.54

720.32

702.45

4640

987.76

0.33

220.31

574.98

900.34

062.52

990.33

851.88

180.33

130.26

7449

150.42

0.33

940.31

477.28

340.33

990.15

280.33

740.56

910.33

411.55

3658

746.00

0.34

400.31

378.79

080.33

921.38

740.33

652.17

720.33

751.89

0768

519.69

0.33

590.31

276.89

070.33

870.82

920.33

570.04

250.34

181.76

6578

375.66

0.35

130.31

2411

.058

0.33

823.71

470.34

242.54

190.34

431.98

5188

688.64

0.34

490.31

199.54

860.33

792.03

090.34

690.57

980.34

871.09

8210

0100

.93

0.35

410.31

1512

.024

0.33

754.66

330.34

642.16

170.35

200.57

6811

6934

.34

0.35

970.31

0913

.542

0.33

716.26

230.34

583.84

650.35

740.61

51

54

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10 100 1000 100000.1

1

Re [−]

f [−

]

ITLR Exp

Forch Coeff Nonlinear Regression

Forch Coeff Comp Expt Data

Forch Coeff Re Ranges

Forch Coeff Funct of Re

ITLR Num

Figure 4.13: Friction coefficient

error of the friction coefficient f is defined as follows:

% Error (f) := 100 ·(fExpt − fNum

fExpt

), (4.13)

where fExpt and fNum are the experimental and numerical friction coefficients respectively.

Figure 4.13 shows a plot of the friction coefficient f against the Reynolds number (Re).From Figure 4.13, Table 4.9 and Table 4.10, it is clear that, all of the approaches, viz.,Quadratic Forchheimer regression (see Section 4.1.2), Forchheimer coefficient β for thecomplete range of experimental data (see Section 4.1.3.1), Forchheimer coefficient β forlimited Re ranges (see Section 4.1.3.2) and Forchheimer coefficient as a linear function ofReynolds number (β = β(Re)) (see Section 4.1.3.3) fit with the experimental data withinan acceptable tolerance. However, the approach discussed in Section 4.1.3.3 gives the bestfit with a maximum deviation of 2− 3% from the experimental data.

It is natural to assume that after using properly fitted intrinsic permeability K and Forch-heimer parameter β, the numerical results should go well with experiments. However, oneneeds to take into account that the determination of a proper intrinsic permeability K andForchheimer parameter β was one of the most challenging tasks in the current work. TheForchheimer coefficient β is a function of both the porous media structure and the flowregime, which makes it problem-specific [32]. Thus, it was very important to cross check the

55

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selected intrinsic permeability K and Forchheimer coefficient β by validating the numericalresults against the experimental data.

It was observed that the DuMuX simulations for an incompressible isothermal flow througha porous medium takes considerably less time compared to the pore-scale CFD simulationsperformed at ITLR. Moreover, the pore-scale CFD simulations are performed for a smallsection of the experimental porous domain [30]. DuMuX takes approximately 150 sec ofCPU time for a simulation of an isothermal case using the macro-scale approach. Whereas,the pore-scale CFD simulation performed by Mayer et al. (2010) [30] requires a comparativelylarger computation time. Thus, one can say that, in addition to better numerical results (seeFigure 4.13), the macro-scale approach for an isothermal flow system is very advantageousin terms of computational cost and time, especially for large scale problems.

4.2.2 Non-isothermal case

4.2.2.1 With local thermal equilibrium

For heating and/or cooling applications it is very important to analyze the porous structure forits heat-transfer properties. Experimental analysis of heat transfer properties of a cylindricalporous structure is performed at ITLR. The experimental data from ITLR is used as thebase, and an attempt is made to validate a non-isothermal DuMuX model with local thermalequilibrium against this experimental data. Nitrogen is used as a working fluid for simulationsin the current work. The model domain, boundary conditions and results for the non-isothermal case are discussed below.

Model domain:

Figure 4.14: Model domain non-isothermal case

56

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Unlike in the isothermal case, the complete cylindrical domain is modeled for the non-isothermal case. The model domain was extended to a length L = 400 mm whereas, thediameter is maintained the same as in the isothermal case i.e d = 30 mm. The extendeddomain is used to simulate the atmosphere, i.e., the model domain consists of 295 mm ofporous matrix and 105 mm of atmosphere (atmosphere domain) as shown in Figure 4.14 bywhite and red colors respectively. The reasons for modeling the complete cylinder and usingan extended domain are explained below in “challenges”. A uniform mesh with hexahedralelements is created using IcemCFD 12.1 as a meshing tool for this cylindrical model domain.

Boundary conditions:

Table 4.11: Boundary conditions non-isothermal caseBoundary condition ValueInlet pressure 101131.443 PaOutlet pressure 100000.000 PaSurface heat flux 22059.088 W/m2

Inlet temperature 299.749 KOutlet temperature 299.749 K

For the non-isothermal case, a Dirichlet boundary condition is applied for pressure and tem-perature at the inlet and the outlet of the domain (see Table 4.11). A constant heat flux(Neumann) boundary condition is applied only at the surface of the porous matrix. Here,the outlet boundary condition refers to the outlet of the model domain and not the porousdomain. Only one set of the experimental data as shown in Table 4.11 is available for thenon-isothermal model validation.

Table 4.12: Material parameters and input dataParameters Value Source

λa 0.028 W/mK Expert’s opinionλs 15 W/mK ITLRcps 510 J/kg K Expert’s opinion% 7900 kg/m3 ITLRφ 0.558 ITLRK 5.73× 10−8m2 Current workβ 0.3866 Current work

57

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Input data:

Apart from boundary conditions, other additional input data and material parameters areneeded for the simulations as shown in Table 4.12. This data was either provided by ordiscussed with an expert from ITLR.

Some challenges:

Extension of the model domain and addition of the atmospheric part to the model domainis a modeling trick. This trick is used in order to be able to simulate the flow at the outletof the porous cylinder. Before using the extended domain, an attempt was also made to usethe existing outflow boundary condition available in DuMuX. However, it was found thatthis boundary condition fails for energy closure with the Forchheimer approach.

Use of the extended domain for the atmospheric part also produced a need to describeparameters such as intrinsic permeability K and porosity φ for this domain. The intrinsicpermeability and the porosity values of the extended domain are K = 1 × 10−5m2 andφ = 1 respectively. With the usage of permeabilities orders of magnitude different for theporous medium and the atmospheric part, an unphysical jump in the flow properties viz.,velocity, pressure and temperature was observed at the permeability junction during DuMuX

simulations. This jump existed due to an averaging problem while writing the output at thepermeability junction. This problem was countered by using the Piola transformation forevaluation of velocity fluxes while writing the output.

Figure 4.15: Unphisical heating along edges

Initially, only one quarter of the cylinder was modeled for the non-isothermal case (see Figure4.15). However it was observed that, when the heat flux is applied at the surface of the

58

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porous matrix, the long surface edges of the domain heat up more than the rest of theporous medium, which is unphysical. Hence, it was decided to model the entire cylindricaldomain.

One of the major problems with the non-isothermal model was its slow convergence. It wasobserved that the DuMuX model converges very slowly with the given boundary conditions.Thus, once a linear pressure field is developed across the porous medium, the heat fluxboundary condition is switched on, i.e., the heat flux is switched on at time t = 100 s. Thishelped to achieve faster convergence.

Results and discussion:

Typical distributions of the pressure and the temperature across the porous cylinder for a non-isothermal case is as shown in Figure 4.16 and Figure 4.17 respectively. It can be observedfrom Figure 4.16b, that unlike in the incompressible isothermal case, in the non-isothermalcase, the pressure distribution is nonlinear over the solid matrix. This nonlinearity in thepressure distribution is due to the compressible behavior of the fluid (i.e Nitrogen) and thetemperature-dependency of the fluid properties.

It can also be seen from Figure 4.16b, that the pressure-drop across the atmospheric part ofthe model domain is very small. This low value of the pressure drop across the atmosphericpart of the model domain is due to very high permeability and porosity in this part. In orderto compensate for this pressure drop across the atmospheric part, the Dirichlet boundarycondition for the inlet pressure is adjusted in such a way that the pressure drop across theporous cylinder matches with that of the experiment.

Figure 4.17b shows a plot of the distribution of temperature along the length of the modeldomain. From Figure 4.17b, it is observed that the slope of the bulk temperature (brownline) and the slope of the surface temperature (black line) are equal. It is very importantto know that this behavior of the slopes of the bulk and the surface temperatures is alsoobserved for the experimental data at ITLR.

Figure 4.18 shows the distribution of the velocity and density over the length of the porousmedium. From Figure 4.18 and Figure 4.17b, it is observed that the velocity of the fluidwithin the porous matrix increases with increase in temperature and the resulting decreasein fluid density.

59

Page 72: Forchheimer Porous-media Flow Models - Numerical Investigation

(a) Qualitative pressure distribution (b) Quantitative pressure distribution

Figure 4.16: Pressure distribution across porous domain for a compressible non-isothermal flow

(a) Qualitative temperature distribution (b) Quantitative temperature distribution

Figure 4.17: Temperature distribution across porous domain for a compressible non-isothermal flow

(a) Velocity distribution (b) Density distribution

Figure 4.18: Velocity and density distribution across porous domain for a compressiblenon-isothermal flow

60

Page 73: Forchheimer Porous-media Flow Models - Numerical Investigation

(a) 0.0 s (b) 100.0 s

(c) 200.0 s (d) 300.0 s

(e) 400.0 s (f) 500.0 s

(g) 600.0 s (h) 700.0 s

Figure 4.19: Evolution of pressure (non-isothermal model with local thermal equilib-rium)

61

Page 74: Forchheimer Porous-media Flow Models - Numerical Investigation

(a) 0.0 s (b) 100.0 s

(c) 200.0 s (d) 300.0 s

(e) 400.0 s (f) 500.0 s

(g) 600.0 s (h) 700.0 s

Figure 4.20: Evolution of temperature (non-isothermal model with local thermal equi-librium)

62

Page 75: Forchheimer Porous-media Flow Models - Numerical Investigation

(a) 0.0 s (b) 100.0 s

(c) 200.0 s (d) 300.0 s

(e) 400.0 s (f) 500.0 s

(g) 600.0 s (h) 700.0 s

Figure 4.21: Evolution of velocity (non-isothermal model with local thermal equilibrium)

63

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Table 4.13: Experimental and numerical wall temperatures (non-isothermal model withlocal thermal equilibrium)

Dist. from inlet [mm] Expt. temp TExp [K] Num. temp TNum [K]

0 299.75 299.7543 439.24 452.0568 454.72 526.9684 464.63 573.83105 477.63 634.01203 538.32 892.92235 558.15 968.39295 595.29 1061.49

We have already discussed that for a non-isothermal model, the heat flux boundary conditionfor the surface of the cylinder is switched on at time t = 100 s, i.e., once a linear pressurefield is developed across the porous medium. In Figure 4.19, Figure 4.20 and Figure 4.21,this can be observed. It can also be inferred, that the fluid behaves incompressibly as longas the heat flux boundary condition is switched off. This incompressible behavior might alsoexist due to the low pressure difference applied across the porous cylinder which might notbe the case for higher pressure differences. However, in the current work, no conclusion canbe made in this regard due to limited experimental data.

For the non-isothermal model with local thermal equilibrium, comparison of numerical andexperimental wall temperatures along the length of the porous cylinder is given in Table4.13. We have already discussed from Figure 4.17b that for the current problem, the slopeof the bulk temperature is equal to the slope of the surface temperature, as expected fromthe experimental observations at ITLR. However, from Table 4.13, it is very clear that thetemperatures at the surface of the porous cylinder are considerably over-predicted by thecurrent model.

There could be various reasons for this over-prediction of the surface temperature. Someof the possible reasons are: unavailability of exact thermodynamic properties of the porousmaterial and the fluid (see Table 4.12) and selection of nitrogen as fluid phase for numericalsimulations instead of air (used for experiments at ITLR). However, the most probable reasonis the selection of an improper thermodynamic model. As discussed in Section 2.4.3.1, theassumption of local thermal equilibrium is very well suited for the Darcy flow regime. As theflow velocity for the ITLR experiments is beyond the Darcy flow regime (see Table 4.1), it

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is most likely that the phases are not in contact for a sufficient period of time and no localthermal equilibrium exists within the system (see Section 2.4.3.2).

Thus, it is concluded that the assumption of local thermal equilibrium is not appropriate tonumerically simulate the non-isothermal experiments performed at ITLR. In order to simulatethese experiments, local thermal non-equilibrium is assumed (see Section 4.2.2.2).

4.2.2.2 With local thermal non-equilibrium

As discussed in Section 4.2.2.1, the possibility of existence of local thermal non-equilibriummotivates the use of a non-isothermal model with local thermal non-equilibrium for the cur-rent work. The implementation of non-isothermal model with local thermal non-equilibriumin DuMuX is discussed in Section 2.6. Similar to the model discussed in Section 4.2.2.1, anattempt is made to validate the current model against the experimental data. The modeldomain, boundary conditions and input data for the current numerical problem are same asthat of the problem for non-isothermal model with local thermal equilibrium. The numericalresults for this model are discussed below.

Results and discussion:

Figure 4.22 shows the evolution of the temperature field for the fluid (black) and the solid(magenta) phases. The plots for the evolution of the pressure and velocity fields for thiscase are not presented here as they are similar to that of the non-isothermal case with localthermal equilibrium (see Figure 4.19 and Figure 4.21). It can be clearly observed from Figure4.22, that the wall heat flux boundary condition for the porous domain is switched on aftertime t = 100 s, (i.e., once a linear pressure field is developed across the porous medium).

For the non-isothermal model with local thermal non-equilibrium, comparison of experimentaland numerical temperatures of the fluid phase, along the length of the model domain surfaceis given in Table 4.14. Comparing Table 4.13 and Table 4.14, it is very clear that theassumption of local thermal non-equilibrium has definitely produced better numerical results.However, it can also be observed from Table 4.14, that even with the assumption of localthermal non-equilibrium, the numerical fluid temperatures deviate from the experiments inthe range of 8− 20%.

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(a) 0.0 s (b) 100.0 s

(c) 200.0 s (d) 300.0 s

(e) 400.0 s (f) 500.0 s

(g) 600.0 s (h) 700.0 s

Figure 4.22: Evolution of temperature (non-isothermal model with local thermal non-equilibrium)

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Table 4.14: Experimental and numerical wall temperatures (non-isothermal model withlocal thermal non-equilibrium)

Dist. from inlet [mm] Expt. temp TExp [K] Num. temp TNum [K]

0 299.75 299.7543 439.24 372.3568 454.72 406.1284 464.63 424.62105 477.63 461.33203 538.32 597.22235 558.15 643.84295 595.29 709.35

Thus, in order to achieve better numerical results, in the future work, some modificationsare recommended viz., radial mesh grading (with finer mesh at the surface of the cylindricaldomain), modification of the outflow boundary condition in DuMuX for the high velocityForchheimer flow, implementation of air as the fluid phase in DuMuX and determination ofa Nusselt number (Nu) relationship which precisely describes the exchange of energy betweendifferent phases [27].

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5 Conclusion

In this work, the intrinsic permeability K and the Forchheimer coefficients β are determinedby performing nonlinear regression analysis of the experimental data from ITLR in Section4.1.1, Section 4.1.2 and Section 4.1.3. From the numerical results discussed in Section 4.2.1,it is concluded that an appropriate intrinsic permeability K (see Equation 4.6) is selected anddepending on the desired precision, different approaches can be used for the determinationof the Forchheimer coefficient β.

From the Forchheimer equation (see Equation 2.28), we know that the inertial effects aredirectly proportional to the flow kinetic energy. From the detailed analysis of the experimentaldata, it is observed that, in order to imitate the inertial effects, one has to given up on theexpectation of a constant Forchheimer coefficient β. Various approaches to determine theForchheimer coefficient β for different flow regimes are explained in Section 4.1.3.

As discussed in Section 4.2.1, fitting the Forchheimer coefficient β to the experimental dataproduced better numerical results. However, before proceeding to the non-isothermal models,it was extremely important to get the momentum balance right. From Section 4.2.1, it isvery clear that the Forchheimer coefficient β approach discussed in Section 4.1.3.3 preciselyaccounts for nonlinear inertial effects, especially in the high velocity flow regime (Re > 180)and provides the best fit to the experimental data with a maximum deviation of 2− 3%.

Thus, it is concluded that the REV-scale approach provides an effective, efficient and eco-nomical solution for the numerical modeling of fast isothermal single-phase flow throughporous media, especially for large scale applications.

From the comparison of non-isothermal numerical results with the corresponding experimentaldata, it is concluded that the assumption of local thermal equilibrium is not appropriate as itfails to accurately describe the non-isothermal experimental data (see Section 4.2.2.1). Theassumption of local thermal non-equilibrium definitely describes the experimental results moreaccurately (see Section 4.2.2.2). Also, with this assumption, the numerical results deviate

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from the experiments in a reasonable range of 8− 20%. However, for further improvementof results, radial mesh grading (with finer mesh at the surface of the cylinder), modificationof the outflow boundary condition in DuMuX, use of air as the fluid phase and a preciseNusselt number (Nu) correlation for the exchange of energy between different phases arerecommended.

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