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NME 3125 pp: 1–12 (col.fig.: Nil)

ED: Selva

PROD. TYPE: COM PAGN: Babu -- SCAN:

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2011) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3125

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An artificial compressibility CBS method for modelling heat transfer and fluid flow in heterogeneous porous materials A. G. Malan1, ∗, † and R. W. Lewis2 Systems Competency, Council for Scientific and Industrial Research, P.O. Box 395, Pretoria 0001, South Africa 2 School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.

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1 Aerospace

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SUMMARY

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This work is concerned with the development of an artificial compressibility version of the characteristicbased split (CBS) method proposed by Zienkiewicz and Codina (Int. J. Numer. Meth. Fluids 1995; 20:869–885). The technique is applied to modelling both forced convection as well as heat transfer and fluid flow through heterogeneous saturated porous materials via an edge-based finite volume discretization scheme. A volume-averaged set of local thermal disequilibrium governing equations is employed to describe the general case which allows for the modelling of effects such as wall channelling and wall-bed radiative heat transfer. The resulting set of coupled non-linear partial differential equations is solved in a matrix-free manner with spatial discretization being effected with a compact vertex-centred finite volume edge-based discretization scheme. The latter was done in the interest of efficiency and accuracy. The developed scheme is validated via application to problems ranging from forced convection to natural convection in heterogeneous materials, and shown to be stable, robust and accurate. Copyright 䉷 2011 John Wiley & Sons, Ltd.

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KEY WORDS:

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1. INTRODUCTION

Modelling of heat transfer and fluid flow in saturated porous systems remains an important area of research in the industry as well as academia. This is due to both its complexity and ubiquity to systems ranging from biological tissue to electronic cooling devices and packed bed reactors [1]. For modelling to be a valuable predictive tool, the quantitative description of the involved heat and fluid flow physics to engineering accuracy is crucially important, and all dominant physics is to be properly accounted for. This remains a challenging task as a specific process or device may contain significant heterogeneity with a resulting wide spectrum of flow length scales. Further, the prevalent heat transfer phenomenon ranges from convection to radiation dominated. Mathematical models that describe the heat and fluid flow through porous materials range from overly simplistic [2] to pore-scale. The former fails to account for important physical phenomena

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edge-based; artificial compressibility CBS; matrix-free; forced convection; heterogeneous porous materials

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Received 25 February 2010; Revised 17 October 2010; Accepted 4 November 2010

∗ Correspondence

to: A. G. Malan, Aerospace Systems Competency, Council for Scientific and Industrial Research, P.O. Box 395, Pretoria 0001, South Africa. † E-mail: [email protected] Copyright 䉷 2011 John Wiley & Sons, Ltd.

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while the latter requires numerical solution, where the number of unknowns to be solved for is at present computationally prohibitively expensive. A viable alternative is however the so-called generalized mathematical description [3–11]. This approach results in a set of equations (typically non-linear coupled partial differential equations) which is written in terms of microscopic volumeaveraged hydrodynamic quantities (density, pressure, velocity, etc.). It therefore circumvents the explicit modelling of microscopic phenomena, which is accounted for in a volume-averaged sense via empirical constitutive relations. The available generalized or volume-averaged approaches may be subdivided by virtue of their treatment of fluid–solid thermal interaction as well as their ability to account for complex nonlinear effects such as material and matrix property variance due to large temperature or porosity variations. In the case of the former, the majority of models assume a state of so-called local thermal equilibrium which results in a single energy conservation governing equation [3, 7, 12–15]. The local thermal disequilibrium technique [6, 11, 16–18] is thermodynamically the more general case as it accounts for the disparity between solid and fluid phase temperatures in a volume-averaged sense. In the majority of the cited work, it is further typical to assume either constant porosity (in the porous region) or invariance of material properties with respect to temperature. An exception is however the work of Visser et al. [18] where both local thermal disequilibrium is assumed while variance of material properties are fully accounted for. The fluid density in porous flow problems are typically invariant with respect to pressure, i.e. incompressible. The two main approaches employed to solve such flow problems are the so-called pressure-based (pressure projection—PP) [19] and density-based (artificial compressibility—AC) [20] methods. Historically, the former was initially the most extensively used while artificial compressibility (AC) has only received comparable research attention in the recent years [21–25]. The main advantage of the former is that the solution of the pressure field involves the pressure Poisson equation, which is numerically naturally stable and robust for smooth fields. The AC method on the other hand allows matrix-free solution of compressible as well as incompressible flows, which is of key value to large-scale distributed memory computing. The fusion of the PP and AC methods based on the characteristic-based split (CBS) methodology [26] was recently conceived by Nithiarasu [27], in which it is shown that the advantages of the two approaches may be combined. This work was subsequently extended to porous systems with equal success [15]. In the interest of computational and programming efficiency, the chosen spatial discretization algorithm should be naturally applicable to any element type. This may be achieved by employing a purely edge-based discretization methodology, which holds the additional advantage of being computationally considerably more efficient than element-based approaches [24]. A compact vertex-centred edge-based finite volume algorithm is employed in this work as it exhibits these characteristics. Note that the compact [28, 29] variant of the scheme is employed as it avoids destabilizing odd–even decoupling while affording greater accuracy as compared with the standard vertex-centred finite volume methodology. This paper deals with the solution of a generalized set of volume-averaged partial differential equations which describes incompressible flows ranging from isothermal forced convection to natural convection in heterogeneous porous materials in which the fluid and solid phases are in a state of thermal disequilibrium. This allows for the more direct modelling of wall-related effects such as wall channelling as well as radiative heat transfer between the solid matrix and an enclosing wall. The AC CBS methodology is applied for solution purposes, where the algorithm is extended to allow for the solution of local thermal disequilibrium systems. The resulting set of non-linear coupled partial differential equations is spatially discretized via an edge-based vertexcentred finite volume method, where a compact stencil methodology is employed for the diffusive terms. This is the first instance in which the AC CBS method is employed to model heat and mass transfer in heterogeneous porous materials, as well as the said method discretized and solved via an edge-based finite volume method. The developed modelling technology is validated by application to two benchmark problems from the literature viz. forced convection in a nonporous cavity at various Reynolds numbers, and natural convection in a heterogeneous porous material. The results are compared with benchmark and analytical data proving robustness and accuracy.

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A. G. MALAN AND R. W. LEWIS

Copyright 䉷 2011 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Engng (2011) DOI: 10.1002/nme

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MODELLING HEAT AND FLUID FLOW

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3 5

2. GOVERNING EQUATIONS The volume-averaged partial differential equation set describing the heat and mass transfer through both a non-porous and a saturated rigid heterogeneous porous domain may be written in a generalized rectangular–cylindrical two-dimensional Cartesian coordinate system as [18]: *W *F j *G j *H − +r ε =S + *t *x j *x j *x j

⎞

r ε f f u 1 f

⎜ ⎜ ⎜ r ε f f u 2 f ⎜ ⎜ ⎜ W=⎜ r ε f f ⎜ ⎜ ⎜ ⎜ r ε f C f T f pf f ⎜ f ⎝

⎛

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(2)

⎞ ⎛ p f 1 j ⎟ ⎜ ⎟ ⎜ ⎜ p f ⎟ 2j⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ H=⎜ 0 ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 ⎟ ⎟ ⎜ ⎠ ⎝ 0

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⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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⎜ ⎜ ⎜ r ε2 j f ⎜ ⎜ ⎜ ⎜ 0 ⎜ j G =⎜ ⎜ ⎜ *T f f ⎜r εk f f ⎜ *x j ⎜ ⎜ ⎜ ⎝ *Ts s r kseff *x j

0

D

⎞

r ε1 j f

⎞

⎜ ⎟ ⎜ ⎟ f f f ⎜ ⎟ r ε u u 2 j f ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ f f ⎟, Fj =⎜ r ε f u j ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜r ε f C f T f u f ⎟ p f j ⎜ ⎟ f f ⎝ ⎠

r (1−ε)s s C ps s Ts s ⎛

r ε f f u 1 f u j f

OO

⎛

F

where

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(1)

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and r denotes the radius. The latter is aligned with the x 1 cartesian axis and is to be set to r = 1 in the case of a non-symmetric two-dimensional spatial domain. In the above equations, ε and , respectively, denote the porosity and the intrinsic volume-averaged density of substance . Further, u j f is the averaged velocity in the coordinate direction j , p is the static pressure and T and C p are, respectively, the averaged temperature and specific heat of substance . The intrinsic fluid and effective (superficial) solid matrix thermal conductivity are, respectively, given by k f f and kseff , while the fluid stress term i j f is defined as 2 11 = − f f 3

f

1 * *u 2 f (r u 1 f )+ r *x 1 *x 2

12 = 21 = f f

9

f

2 22 f = − f f 3


f

*u 2 f *u 1 f + *x 1 *x 2

+2 f f

*u 1 f *x 1

(3)

1 * *u 2 f (r u 1 f )+ r *x 1 *x 2

(4) +2 f f

*u 2 f *x 2

(5)


NME 3125 4

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where f f is the intrinsic fluid phase dynamic viscosity. The source term in Equation (1) is defined as ⎛ ⎞ r ε f f g1 +r B1 −(1−1 )ε33 f ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ r ε f f g2 +r B2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ S=⎜ (6) ⎟ 6 f

⎜ ⎟ f s ⎜ −r (1−ε) h eff T f −Ts ⎟ ⎜ ⎟ dp ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 6 f

−r (1−ε) h eff Ts s −T f f dp where

2 33 = − f f 3

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Vm

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31

(7)

In this work we propose the use of a vertex-centred edge-based finite volume algorithm for the purposes of spatial discretization, where a compact stencil method is employed for second-derivative terms in the interest of both stability and accuracy [29]. This is the first instance in which the aforementioned approach is applied in the context of the CBS scheme (detailed below), and was selected as the method allows natural generic mesh applicability, second-order accuracy without odd–even decoupling, and computational efficiency which is factors greater than element-based approaches. Note that the proposed edge-based approach is also particularly well suited to shared memory parallel hardware architectures. The first step in discretizing the system of governing equations is subdivision of the spatial domain V into non-overlapping volumes Vm ∈ V (see Figure 1), and application of the system of governing equations to each volume or sub-domain in weak form. After application of the Divergence theorem we write

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3. SPATIAL DISCRETIZATION

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+2 f

u 1 f r

and gi and Bi are, respectively, the gravitational acceleration and solid matrix drag in the Cartef sian coordinate direction i. Further, h eff is the effective volume-averaged heat transfer coefficient f and d p the porous particle size. Bi , h eff and kseff are to be determined from experimental data. Note that in the case of a rectangular coordinate axis = 1, while = 0 when the spatial field is axi-symmetric.

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1 * *u 2 f (r u 1 f )+ r *x 1 *x 2

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*W dV+ *t

(F −G +r Vm εVm H)n j dS = j

Sm

j

Vm

S dV

(8)

where Sm is the surface bounding Vm and n is the unit vector in the direction normal to the boundary segment dS. Further, r Vm and εVm are, respectively, the spatially integrated average radius and porosity (with respect to Vm ). All surface integrals are calculated in an edge-wise manner. For this purpose, bounding surface information is similarly stored in an edge-wise manner and termed edge-coefficients. The latter for a given internal edge mn connecting nodes m and n is defined as Cmn = nmn1 Smn1 +nmn2 Smn2 Copyright 䉷 2011 John Wiley & Sons, Ltd.

(9) Int. J. Numer. Meth. Engng (2011) DOI: 10.1002/nme

NME 3125

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Color Online, B&W in Print

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Figure 1. Schematic diagram of the construction of the median dual-mesh on hybrid grids. Here, mn depicts the edge connecting nodes m and n.

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where Smn1 is a bounding surface-segment intersecting the edge. The discrete form of the surface integral in Equation (8), computed for the volume surrounding the node m, now follows as:

j j {F j −G j +r Vm εVm H}n j dS ≈ {F −[G |tang Sm

f mn

f mn

D

mn ∩Vm j

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where all •mn quantities denote edge-averaged values, and the operator f returns the latter such that second-order accuracy of the overall scheme is ensured [30]. Further, G j |tang is calculated by employing directional derivatives and G j |norm is approximated by employing the standard finite volume first-order derivative terms.

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4. TEMPORAL DISCRETIZATION AND SOLUTION PROCEDURE

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j

|norm ]+r Vm εVm Hmn }Cmn

As noted previously, the solution procedure followed involves an AC CBS methodology due to the recent work by Nithiarasu [27]. In this paper, the scheme is applied in an edge-based finite volume method and extended in addition to model heat transfer and fluid flow in heterogeneous porous materials for which the fluid and solid phases may be at a state of local thermal disequilibrium. To developed the scheme in a manner that it is suitable for both the fluid and solid phases, we consider Equation (8) for an arbitrary volume V, from which the first incremental solution step written in semi-discrete form follows as: j (W )∗i *Fi t j j uk V=− (Fi − G i )n j dS+ n k dS+ Si V (11a) t 2 S S *x j

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+G f

where the superscript denotes the previous (existing) solution or pseudo-time-step and t = t +1 −t . Further, Wi∗ is an auxiliary variable which is used in the second step to compute the fluid static pressure in a matrix-free manner via the introduction of AC as: j (W )∗k *Hk 1 *p k W n V = − +t −r ε dS (11b) k V V c2 *t t *x j S(t) for k = [1, 2] and where the pseudo-acoustic velocity is computed as

19

c = max(cconv , cdiff ) Copyright 䉷 2011 John Wiley & Sons, Ltd.

(11c) Int. J. Numer. Meth. Engng (2011) DOI: 10.1002/nme

NME 3125 6

Here, the convective acoustic velocity cconv = max[0.5; 1.2u j u j ], while the diffusive component is determined by a relation which takes into consideration the viscous diffusion velocity as: cdiff =

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f f f f vn

+|u f |

(11d)

In the above relation, denotes the spatial discretization magnitude (effective cell/element size). Further, vn is the von Neumann number (typically set to 0.5) The final and third incremental solution step now follows as +1 wi+1 −wi (W )∗i j V= V−r V εV Hi n j dS (11e) ti t S

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for i = 1, 2, 4, 5.

5. PSEUDO-TIME-STEP CALCULATIONS

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The allowable time-step size employed in the solution scheme is calculated in the interest of stability as

k f f +|u f f | f f C p f f vn

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i=4 =

i=5 =

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s C

kseff s ps (1−ε) vn

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(12)

where the subscript i denotes the specific equation (11e) and is an approximation of the maximum eigenvalue associated with each. The latter is calculated as follows:

i=1,2 = |u f |+ u f ·u f +c2

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i

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ti =

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where the effective solid conductivity is divided by the relevant porosity function to return the intrinsic value.

6. NUMERICAL TESTS

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The developed modelling technology is evaluated in terms of both accuracy and computational cost via the solution of two benchmark problems. As the developed artificial CBS methodology is the focus of this paper, the test cases are selected to serve as a stringent test for the related modelling technology as follows:

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• Non-porous flow in a lid-driven cavity for Stokes flow and Re = 5000 (diffusion to convectiondominated flow). • Natural convective flow in a heterogeneous porous cavity. In all cases accuracy is assessed by comparison of the predicted results with analytical or experimental data. Mesh independent solutions (to within 2%) are quoted in all cases.‡ Note that all analyses were done on a Pentium 3 GHz 64 Bit CPU. ‡ The

quoted percentage refers to the deviation of the dependent variables when halving the mesh size.



NME 3125 7


2

u1

1 u2

0

p 0 u2

0 1.0 x1

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u1

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0 u2

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Y

x

Figure 2. Lid-driven cavity: schematic (left) and (right) mesh used for all flow regimes. The mesh contains 2389 points and was initially generated through an adaptive procedure for use in a semi-implicit projection type scheme for Re = 5000 [31].

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The first test case considered comprised the viscous isothermal recirculation flow in a square cavity. Flow is generated by the uniform translation of the upper surface (lid) as shown in Figure 2. The lid is driven at a uniform velocity and no fluid is allowed to escape between the lid and the cavity wall. No-slip boundary conditions are applied to all cavity walls and the pressure is prescribed at a point close to the mid-section of the bottom side. The flow regimes considered ranged from diffusion to convection-dominated viz. Stokes-flow and Reynolds number 5000 flow. The mesh used is shown in Figure 2, and was originally generated via an adaption process for Re = 5000 by Nithiarasu and Zienkiewicz [31]. This mesh contains 2389 nodes and was employed for both diffusion and convection flow regimes to demonstrate the ability of the scheme to remain robust and stable in the presence of high velocity and pressure gradients (as is present in the top corner regions). Considering first the Stokes-flow problem, the predicted pressures (shown in Figure 3 (left)) are shown to compare well with the solution of others, while the scheme converges to machine precision in circa 3 min (Figure 4). Similar accuracy was achieved for the convectiondominated flow case with the required analysis time being of similar magnitude. In conclusion, the test-case served to prove the accuracy and robustness of the developed scheme in solving incompressible fluid flow.

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6.1. Test case 1: lid-driven cavity

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6.2. Test case 2: natural convection in a heterogeneous porous material The developed numerical technology is next validated by modelling the SANA experiment at the Jülich Research Center [32]. The setup, which is shown schematically in Figure 5, consists of a cylindrical vessel with internal diameter of 1.5 m and height of 1 m. The vessel is filled with 60 mm randomly packed graphite spheres. The working fluid considered here was helium (pressurized to 1 bar). The heat input to the system is via a centrally placed electrical heater. The top and bottom of the vessel are insulated while heat is allowed to escape from the outer annular surface. The operating condition selected for the purpose of validation is a steady-state test with heating element spanning the full length of the pebble bed, and 5 kW nominal heating power input. The heat transfer processes to be modelled are the influx of heat from the inner wall to both the working fluid and solid pebbles, conduction and radiative heat transfer within the solid matrix, convective heat transfer between the solid and fluid phase, whereas within the fluid phase. The Copyright 䉷 2011 John Wiley & Sons, Ltd.


Q1

NME 3125 8


2 AC – CBS

10000 1.5 100

0.5

1

2 (Resall)

1

p

0

0.01

-0.5 0.0001 -1

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1e-06

-1.5

semi-implicit CBS [32] p

1e-08 0

0.2

0.4 0.6 x1 coordinate

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1

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0

1

2

3

4

5

CPU time (minutes)

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Figure 3. Stokesflow: predicted pressure at x2 = 0.5 (left) of the developed as well as benchmark scheme, and convergence history (right). Symbols on the left graph indicate the benchmark solution.

x1 coordinate 0.2

0.4

0.6

1

0.8

1 0.6

u1 u2

AC- CBS

0.4

0.1

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0.8

0.4

2 (Resall)

0

u2

0.2

0.6

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x2 coordinate

1

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0.001

-0.2

0.2

-0.4

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0.2 u

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-0.6

1

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1e-05 0

0.5

1

1.5

2

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3

CPUt ime (minutes)

Figure 4. Velocities at x1 = 0.5 (u 1 ) and x2 = 0.5 (u 2 ) for Reynolds number of 5000 (left) and convergence curve (right). Symbols on the left graph indicate the benchmark solution [33].

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axi-symmetric mesh employed contained 9500 nodes (Figure 5) for which the spacing normal to the wall was set to 1 mm in order to capture wall-related effects. The above was simulated via a two-dimensional axi-symmetric model to which the following boundary conditions were applied: • • • •

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No-slip boundary conditions to all bed walls. The top and bottom walls treated as adiabatic. Constant uniform heat flux applied to the left (inner) wall. A constant temperature field applied to the right (outer) wall (temperatures applied were taken from the experimental measurements).

The temperature-dependent material properties used for helium are as prescribed by the Nuclear Safety Standards Commission [34], while the material properties for graphite were taken from the International Atomic Energy Agency [35] and Niessen and Stöcker [32]. The porosity variation for an annular bed is calculated by using the exponential correlation as proposed by Hunt and Tien Copyright 䉷 2011 John Wiley & Sons, Ltd.


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x1 [m]

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Figure 5. Schematic diagram of the SANA test setup (left) with central heating element taken from [32] and the representative axi-symmetric mesh used (right) containing 9500 points.

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x2 [m]

0.6

0.8

0.6

0.5

0.4

0

0.4

0.5

0.6

0.7

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ε

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

x1 [m]

Figure 6. Porosity distribution at the horizontal (left) and vertical (right) centrelines.

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[36] (the porosity approaches the limit of 1.0 at the wall) similar to Visser et al. [18]. The resulting porosity distribution is highly heterogeneous and is shown for mid-height and width sections of the bed in Figure 6. The employed matrix-related empirical correlations were similarly also taken from the aforementioned authors. Copyright 䉷 2011 John Wiley & Sons, Ltd.


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0.35

650

600

0.09 [m] 0.50 [m] 0.91 [m]

550

exp. 0.09 [m] exp.0.50 [m] exp.0.91 [m]

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[K ]

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s

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400

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u2 f [m / s]

0.2

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0.1

0.2

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0.6

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0.7

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x1 [m]

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0.2

0.3

0.4 0.5 x1 [m]

0.6

0.7

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Figure 7. Porous test-case: vertical velocities in radial section at 0.5 m vertical elevation for 5 kW (left) and predicted temperatures compared to experimental values at three different heights (right).

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0.01

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2 (Resall)

0.1

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0.001

0.0001

1e-05

0

2

4

6

8

10

CPU time (hours)

Figure 8. Porous test-case: convergence history.

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The calculated velocities at a section through the horizontal centreline of the packed bed are shown in Figure 7 and were found to be similar to those predicted in the previous work [18]. The sharp velocity peaks seen close to the solid boundaries (wall channelling) are the result of the increased void fraction as the boundary is approached. Also shown in the figure are the predicted pebble temperature distribution at three different heights as compared to measurements [32]. These were again accurate with the resulting maximum and average normalized errors being 12 and 3%, respectively. Finally, the convergence history for the analysis is shown in Figure 8. Here, the residual is calculated as the average Euclidean norm of all five dependent variables solved for. The purely monotone drop in residual is an indication that there were no stability concerns during the solution process. The relatively long analysis time is due to the simple Jacobi iterative procedure resulting in severe pseudo-time-step limitations at the small cells near the boundaries. This will be remedied

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in the future work via the use of a matrix-free methodology [37] to solve the pressure equation implicitly.

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7. CONCLUSION

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This paper was concerned with the development of an AC version of the CBS method originally proposed by Zienkiewicz and Codina [26] for modelling both forced convection as well as heat transfer and fluid flow through heterogeneous saturated porous materials via an edge-based finite volume discretization scheme. A volume-averaged set of local thermal disequilibrium governing equations was employed to describe the general case. The resulting set of coupled non-linear partial differential equations was solved in a matrix-free manner with spatial discretization being effected with a compact vertex-centered finite volume edge-based discretization scheme. This is the first instance in which the AC-CBS methodology is employed in the context of the chosen finite volume discretization or to model flow through heterogeneous saturated porous materials in a purely matrix-free manner. The developed technology was successfully validated by application to the modelling of flows ranging from isothermal forced convection to natural convection in a saturated heterogeneous porous material. It was shown to be both accurate and robust for all cases. REFERENCES

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1. Kaviany M. Principles of Heat Transfer in Porous Media. Springer: New York, 1995. 2. Prasad V, Lauriat G, Kladias N. Non-Darcy natural convection in a vertical porous cavity. Heat and Mass Transfer in Porous Media. Elsevier: Amsterdam, 1992; 293–314. 3. Vafai K, Tien CL. Boundary and inertia effects on flow and heat transfer in porous media. International Journal for Heat and Mass Transfer 1981; 24:195–203. 4. Cheng P, Hsu CT. Fully developed, forced convection flow through an annular packed-sphere bed with wall effects. International Journal for Heat and Mass Transfer 1986; 29:1843–1853. 5. Lewis RW, Roberts PJ, Schrefler BA. Finite-element modeling of 2-phase heat and fluid-flow in deforming porous-media. Transport in Porous Media 1989; 4(4):319–334. 6. Kuipers JAM, van Duin KJ, van Beckum FPH, van Swaaij WPM. A numerical model of gas-fluidised beds. Chemical Engineering Science 1992; 47(8):1913–1924. 7. Nithiarasu P, Seetharamu KN, Sundararajan T. Natural convection heat transfer in a fluid saturated variable porosity medium. International Journal for Heat and Mass Transfer 1997; 40(16):3955–3967. 8. Becker S, Laurien E. Three-dimensional numerical simulation of flow and heat transport in high-temperature nuclear reactors. Nuclear Engineering and Design 2003; 222:189–201. 9. Massarotti N, Nithiarasu P, Carotenuto A. Microscopic and macroscopic approach for natural convection in enclosures filled with fluid saturated porous medium. International Journal of Numerical Methods for Heat and Fluid Flow 2003; 13(7):862–886. 10. Rees I, Masters I, Malan AG, Lewis RW. An edge-based finite volume scheme for saturated–unsaturated groundwater flow. Computer Methods in Applied Mechanics and Engineering 2004; 193(42–44):4741–4759. 11. du Toit CG, Rousseau PG, Greyvenstein GP, Landman WA. A systems CFD model of a packed bed high temperature gas-cooled nuclear reactor. International Journal of Thermal Sciences 2006; 45:70–85. 12. Philip JR, de Vries DA. Moisture movement in porous materials under temperature gradients. Transactions of the American Geophysics Union 1957; 38(2):222–232. 13. Luikov AV. Heat and Mass Transfer in Capillary-porous Bodies. Pergamon Press: Oxford, 1966. 14. Lebon G, Cloot A. A thermodynamical modelling of fluid flows through porous media: application to natural convection. International Journal for Heat and Mass Transfer 1986; 29(3):381–390. 15. Arpino F, Massarotti N, Mauro A, Nithiarasu P. Artificial compressibility-based CBS scheme for the solution of the generalized porous medium model. Numerical Heat Transfer Part B 2009; 55:1–33. 16. Gunn DJ. Transfer of heat or mass to particles in fixed and fluidised beds. International Journal for Heat and Mass Transfer 1978; 21:467–476. 17. Gidaspow D. Hydrodynamics of fluidization and heat transfer: supercomputer modeling. ASME Applied Mechanics Reviews 1986; 39(1):1–23. 18. Visser C, Malan AG, Meyer JP. An artificial compressibility algorithm for modelling natural convection in saturated packed pebble beds: A heterogeneous approach. International Journal for Numerical Methods in Engineering 2008; 75:1214–1237. 19. Patankar SV. Numerical Heat Transfer and Fluid Flow. McGraw-Hill: New York, 1980. 20. Chorin AJ. A numerical method for solving incompressible viscous flow problems. Journal of Computational Physics 1967; 2:12–26.

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21. Belov A, Martinelli L, Jameson A. A new implicit algorithm with multigrid for unsteady incompressible flow calculations. AIAA Paper, vol. 95-0049, 1995. 22. Manzari MT. An explicit finite element algorithm for convection heat transfer problems. International Journal of Numerical Methods for Heat and Fluid Flow 1999; 9(8):860–877. 23. Drikakis D, Iliev OP, Vassileva DP. A nonlinear multigrid method for three-dimensional incompressible Navier– Stokes equations. Journal of Computational Physics 1998; 146:301–321. 24. Zhao Y, Zhang B. A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids. International Journal of Numerical Methods in Engineering 1994; 37:3323–3341. 25. Malan AG, Lewis RW, Nithiarasu P. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part I. Theory and implementation. International Journal for Numerical Methods in Engineering 2002; 54(5):695–714. 26. Zienkiewicz OC, Codina R. A general algorithm for compressible and incompressible flow. Part I: the split characteristic based scheme. International Journal for Numerical Methods in Fluids 1995; 20:869–885. 27. Nithiarasu P. An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flow. International Journal for Numerical Methods in Engineering 2003; 56(13): 1815–1845. 28. Crumpton PI, Moinier P, Giles MB. An unstructured algorithm for high Reynolds number flows on highly stretched meshes. In Numerical Methods in Laminar and Turbulent Flow, Taylor C, Cross JT (eds), 1997; 561–572. 29. Malan AG, Lewis RW. Modeling coupled heat and mass transfer in drying non-hygroscopic capillary particulate materials. Communications in Numerical Methods in Engineering 2003; 19(9):669–677. 30. Lewis RW, Malan AG. Continuum thermodynamic modeling of drying capillary particulate materials via an edge-based algorithm. Computer Methods in Applied Mechanics and Engineering 2005; 194(18–20):2043–2057. 31. Nithiarasu P, Zienkiewicz OC. Adaptive mesh generation for fluid mechanics problems. International Journal for Numerical Methods in Engineering 2000; 47(1–3):629–662. 32. Niessen HF, Stöcker B. Data sets of SANA experiment: 1994–1996. JUEL-3409, Forschungszentrum Jülich, 1997. 33. Ghia U, Ghia KN, Shin CT. High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. Journal of Computational Physics 1982; 48:387–411. 34. KTA. Reactor core design of high-temperature gas-cooled reactors. Part 1: calculation of the material properties of helium. Nuclear Safety Standards Commission (KTA), kta3102.1 edition, 1978. 35. International Atomic Energy Agency. Heat transport and after heat removal for gas cooled reactors under accident conditions. IAEA-TECDOC-1163, 2001. 36. Hunt ML, Tien CL. Non-Darcian flow, heat and mass transfer in catalytic packed-bed reactors. Chemical Engineering Science 1990; 45:55–63. 37. Malan AG, Meyer JP, Lewis RW. Modelling non-linear heat conduction via a fast matrix-free implicit unstructuredhybrid algorithm. Computer Methods in Applied Mechanics and Engineering 2007; 196(45–48):4495–4504. 38. Zienkiewicz OC, Taylor RL. The Finite Element Method: Volume 3—Fluid Dynamics (5th edn). ButterworthHeinemann: Oxford, 2000.

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Please check the placement of Figure 4.

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Please provide the publisher details for Ref. [28].

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Please cite Ref. [38] in the text.

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