3D contaminant transport by GFEM with hexahedral elements

International Communications in Heat and Mass Transfer 42 (2013) 43–50

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International Communications in Heat and Mass Transfer

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3D contaminant transport by GFEM with hexahedral elements☆

Estaner Claro Romão a,⁎, Luiz Felipe Mendes de Moura b

a Federal University of Itajubá, Advanced Campus of Itabira, Itabira-MG, Brazilb Thermal and Fluids Engineering Department, Mechanical Engineering Faculty, State University of Campinas, Campinas, SP, Brazil

☆ Communicated by W.J. Minkowycz.⁎ Corresponding author.

E-mail addresses: [email protected], estanerr(E.C. Romão), [email protected] (L.F.M. de Moura)

0735-1933/$ – see front matter © 2012 Elsevier Ltd. Allhttp://dx.doi.org/10.1016/j.icheatmasstransfer.2012.10.0

a b s t r a c t
a r t i c l e i n f o
Available online 8 January 2013

Keywords:GFEMHexahedral elementsError normsConvective problems

Thiswork aims to apply the GFEM (Galerkin Finite ElementMethod),with hexahedral elements in concentrationtransport problems in a three-dimensional domain. Since these problems are highly convective in nature, a studyof the GFEMwith 8 and 27-node hexahedron is performed. For such, an applicationwith an analytical solution isperformed by examining the norms L2 and L∞ from the error in the numerical solution; especially considering thevariation of the velocity components, and the geometry of the proposal. Finally, the application involved thetransport of an ammonia concentration through a tubulation containing holes caused by ruptures, whichallows for inward air flow; thus, in this study, the influence of the variation of hole quantities, the maximumdrainage velocity, and the geometry employed are analyzed.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The need to study techniques and/or methods able to analyze masstransportation issues with extreme precision, especially those related tothe transport of pollutants, becomes increasingly paramount, especiallynowadays where sustainability became a common theme. Experimentalmethods are essential for the analysis of situations involving masstransportation; however, because of the high costs of research, opera-tional difficulties, and the length of time for such studies, more effortin the numerical simulations of such problems is encouraged.

Over the past decades, several authors have been investing a lot ofeffort in these studies; among whom [1] can be noted, where theauthors present an analytical solution for the transport equation bylinear one dimensional convection–diffusion, permanent and transient,through the classical version of the generalized integral transform tech-nique. Regardless of the excellent results, it remains very different fromthe actual applications, because of its one-dimensional nature. A similarsituation occurs with the work of [2], where the authors solve threeone-dimensional problems using the Laplace transforms technique.Meanwhile in [3], the finite difference and finite element methods areapplied in a case involving the transport of contaminants in a transienttwo-dimensional view. Using porous environments, the authors dem-onstrate that in this application the standard finite element method(The Galerkin Weighted Residual Method) provides better results.

A very interesting work on the transport of contaminants, althoughin one-dimensional transient domain, is presented in [4], using themesh-free point collocation method. The authors distinguish themselves

[email protected].

rights reserved.16

by the applications presented in thiswork, in particular by the problemoftransporting contaminants in an aquifer in area of groundwater nearVadodara-India. Also regarding the transport of contaminants in ground-water, the work of [5] is worthy of mention, with the caveat that this is atransient one-dimensional domain, and the authors succeed with theapplication of the Taylor–Galerkin method. There are many other relatedstudies, among which are the works of [6–8], which, generally, employthe finite element method.

To analyze the transport of contaminants, this paper proposes anumerical solution by the Galerkin Finite Element Method (GFEM)with 27-node hexahedron of the concentration transport equationgiven in the form:

u∂CA

∂x þ v∂CA

∂y þw∂CA

∂z ¼

DAB∂2CA

∂x2þ ∂2CA

∂y2þ ∂2CA

∂z2

! ð1Þ

where u, v and w are the velocity components in the directions x, yand z respectively, DAB is the binary diffusion coefficient, and CA isthe concentration. A similar equation can be found in [9].

2. Method of weighted residuals

The goal here is to use the Method of Weighted Residuals to obtainan approximate solution to the differential Eq. (1). By introducingeach element, we present the following set of functions in the form,

CA≈C eA ¼

XNnodes

i¼1

NiCeAi ð2Þ

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Nomenclature

Latin charactersCA concentrationDAB binary diffusion coefficientH Hilbert spaceN interpolation functionNNodes number of nodes in the elementu velocity in the direction xv velocity in the direction yw velocity in the direction z

Greek charactersΩ three-dimensional domainΩe three-dimensional domain in elementΓ contour of the domainΓe contour of the element

44 E.C. Romão, L.F.M. de Moura / International Communications in Heat and Mass Transfer 42 (2013) 43–50

in which C eAi are the values of the functions in the element nodes, and

Ni are the interpolation functions. Since Eq. (2) is an approximatesolution, by substituting Eq. (2) in Eq. (1), it does not completely satisfythe governing differential equation. Thus we define residual R as

R≅DAB∂2C A∂x2 þ DAB

∂2C A∂y2 þ DAB

∂2C A∂z2 −

u∂C A

∂x −v∂C A

∂y −w∂C A

∂z :

ð3Þ

We followed this by a set of weight functions vi (i=1, 2,…, Nnodes)and define an inner product (R,vi). Determining that this inner productis equal to zero,

R; við Þ ¼ 0 ð4Þ

is the equivalent of forcing the approximation error of the differentialequation, on the average, to be equal to zero.

3. Formulation by the Galerkin method

For the introduction of this approach, it is necessary to define thevariational formulation of problem (1), as follows: we must findC eA∈ Ve with Ve∈H1(Ω) for,

∫ΩeR vei dΩ ¼ 0; ∀vei∈Ve; i ¼ 1;2;…;Nnodes: ð5Þ

in which Ω⊂R3 is a limited and closed domain.In the Galerkin Finite Element Method, the weight function is the

same as the interpolation function, i.e., vje=Nj, j=1, 2,…, Nnodes. By

Table 1Numerical results for the solution of CAB, DAB=1 and u=v=w=1.

h Nelem* 8-nodes 27-nodes

Nnost* L2 L∞ Nnost L2 L∞

1/8 512 729 8.16E−05 2.14E−04 4913 4.84E−07 2.13E−061/10 1000 1331 5.35E−05 1.39E−04 9261 3.64E−07 2.06E−051/12 1728 2197 3.78E−05 9.67E−05 15,625 5.05E−07 2.15E−051/16 4096 4913 2.17E−05 5.42E−05 35,937 9.80E−08 5.45E−071/20 8000 9261 1.41E−05 3.45E−05 68,921 1.99E−07 1.18E−06

⁎Nelem: number of elements in the mesh.⁎Nelem: number of nodes in the mesh.

substituting Eq. (3) in Eq. (5) and taking the weight function as theinterpolation function, it shows that,

∫Ωe

"DAB

∂2C A

∂x2þDAB

∂2C A

∂y2þDAB

∂2C A

∂z2−

: u∂C A

∂x −v∂C A

∂y −w∂C A

∂z

#NjdΩ ¼ 0:

ð6Þ

For the section integration (the “^”will be omitted for this time only),

∫Ωe DAB∂2CA

∂x2þ DAB

∂2CA

∂y2þ DAB

∂2CA

∂z2

" #:NjdΩ: ð7Þ

Integration by parts is used as defined on page 153 of [10]; thus,the section described in Eq. (7) can be written as,

∫Ωe DAB∂2CA

∂x2þ ∂2CA

∂y2þ ∂2CA

∂z2

!" #:NjdΩ ¼

−∫Ωe DAB∂Ni

∂x∂CA

∂x þ ∂Ni

∂y∂CA

∂y þ ∂Ni

∂z∂CA

∂z

� �� dΩþ

þ∫ΓqNiDAB∂CA

∂x ldΓq þ ∫ΓqNiDAB∂CA

∂y mdΓq þ :

∫ΓqNiDAB∂CA

∂z ndΓq:

ð8Þ

In this work, the boundary conditions are considered as being ofthe first and the second kind, which mathematically can be writtenas follows,

CA ¼ CAbin Γb ð9Þ

and

DAB∂CA

∂x lþ ∂CA

∂y mþ ∂CA

∂z n� �

þ _NA þ

h CA−CA∞� � ¼ 0 in Γq:

ð10Þ

where Γb∪Γq=Γ and Γb∩Γq=0, Γ represents the boundary; l, m and nrepresent the cosine directors; h is the concentration transfer coefficient,CA∞ is the free chain concentration, andNA is the flux concentration at theboundary. Now, by applying Eq. (10) to Eq. (8), we will have,

∫Ωe DAB∂2CA

∂x2þ ∂2CA

∂y2þ ∂2CA

∂z2

!" #:NjdΩ ¼

−∫Ωe DAB∂Ni

∂x∂CA

∂x þ ∂Ni

∂y∂CA

∂y þ ∂Ni

∂z∂CA

∂z

� �� dΩþ

−∫ΓqNj NA þ h CA−CA∞� ��

dΓq:

ð11Þ

By applying Eq. (11) to Eq. (6), we have,

∫Ωe

"DAB

∂Ni

∂x∂C A

∂x þ∂Ni

∂y∂C A

∂y þ∂Ni

∂z∂C A

∂z

!þ

u∂C A

∂x þv∂C A

∂y þw∂C A

∂z

#:NjdΩ ¼

−∫ΓqNj NA þ h CA−C A∞

� � dΓq:

ð12Þ

Table 2Numerical results for the solution of ∂CAB

∂x ≅ ∂CAB∂y ≅ ∂CAB

∂z , DAB=1 and u=v=w=1.

h GFEM/FEM GFEM/FDM

L2 L∞ L2 L∞

8-nodes 27-nodes 8-nodes 27-nodes 8-nodes 27-nodes 8-nodes 27-nodes

1/8 6.37E−02 1.06E-03 9.48E−02 1.97E−03 4.45E−04 7.49E−05 7.19E−04 1.91E−041/10 5.11E−02 6.80E-04 7.65E−02 1.27E−03 2.44E−04 4.97E−05 5.10E−04 1.68E−031/12 4.27E−02 4.73E-04 6.41E−02 1.80E−03 1.55E−04 7.09E−05 3.90E−04 1.90E−031/16 3.21E−02 2.66E-04 4.84E−02 5.03E−04 8.18E−05 2.12E−05 2.45E−04 9.39E−051/20 2.57E−02 1.70E-04 3.88E−02 3.26E−04 5.16E−05 5.08E−05 1.67E−04 2.41E−04

45E.C. Romão, L.F.M. de Moura / International Communications in Heat and Mass Transfer 42 (2013) 43–50

Now, the function approximation C A is performed by the functionC eA according to Eq. (2) in Eq. (12), thus we have,

∫Ωe

"DAB

∂Ni

∂x∂Nj

∂x þ ∂Ni

∂y∂Nj

∂y þ ∂Ni

∂z∂Nj

∂z

!þ

u∂Ni

∂x Nj þ v∂Ni

∂y Nj þw∂Ni

∂z Nj

#dΩ:Ce

A i ¼

−∫ΓqNj NA þ h CA−C Aa

� � dΓq;

ð13Þ

with i, j=1, …, NNodes.Eq. (13) generates the following linear system,

K½ � CAei

� ¼ Ff g; ð14Þ

in which

Kij ¼ ∫ΩeDAB∂Ni

∂x∂Nj

∂x dΩþ ∫ΩeDAB∂Ni

∂y∂Nj

∂y dΩþ

∫ΩeDAB∂Ni

∂z∂Nj

∂z dΩþ ∫Ωe u∂Ni

∂x NjdΩþ

þ∫Ωe v∂Ni

∂y NjdΩþ ∫Ωew∂Ni

∂z NjdΩ;

ð15Þ

Fi ¼ −∫ΓqNj NA þ h CA−C Aa

� � dΓq: ð16Þ

4. Numerical applications

The matrix coefficients are obtained by numerical integration usingGauss [11] and mapping the real elements in the master element in thelocal coordinates ξ, η and ζ (−1≤ξ,η,ζ≤1). The interpolation functionsand their derivatives for the hexahedral element with 8- and 27-nodescan be found in [12].

The system of algebraic equations represented by Eq. (14) wassolved by the Gauss–Seidel method, and the criteria of stop withmaximum error Emax≤10−10. The computational code was developedin Fortran language. The meshes were refined until reaching the limitof the computer memory capacity.

Table 3Numerical results for solution of CAB and its first derivatives, DAB=1, u=v=w=1 varying

h Min⁎ Max⁎ CAB

L2 L∞

1/8 0 1 4.84E−07 2.13E−062/8 −2.72 1 1.62E−05 8.77E−053/8 −10.10 1 1.87E−04 1.14E−034/8 −30.19 1 1.41E−03 9.27E−035/8 −84.79 1 8.48E−03 5.81E−02

Min⁎: minimum value of CAB throughout the proposed domain.Max⁎: maximum value of CAB throughout the proposed domain.

The L2 norm of the error was defined as in [13]:

‖e‖ ¼XNnosti¼1

e2i

!=Nnost

" #1=2. In this equation, Nnost is the total num-

ber of nodes in the mesh, and ei ¼ CA numð Þi−CA anð Þi�� , where CA(num) is

the result from the numerical solution, and CA(an) is the result formthe analytical solution, respectively.

The next segment will present two applications; the first being ananalysis of the error based on the comparison with an analytical solu-tion, and the second is a study of the internal flow of the ammonia ina tubulation containing structural holes; thus allowing for air intake.

4.1. Application 1 (error analysis)

To perform the error analysis in the numerical solution of Eq. (1)the following analytical solution is proposed,

CA x; y; zð Þ ¼ euxDAB−eu

1−euþ e

vyDAB−ev

1−evþ e

wzDAB−ew

1−ew

with DAB, u, v and w not void of real constants.Primarily, it is important to note that, in this application, h=Δx=

Δy=Δz will be considered in all the refinements. Tables 1 and 2show the numeric results for a situation where the binary diffusioncoefficient, and the three velocity components are equal to 1 (one)and the domain is a unitary cube. Table 1 shows that the GFEMprovidesexcellent results in the concentration numerical solution. However,for the less refined meshes, with the use of 27-node hexahedrons ina mesh with h=1/16 for h=1/20, the results worsen, demonstratingthat under these circumstances, the refinement acquires the greatestprecision, around h=1/16; thus, further refinement becomesdetrimental. Conversely, Table 2 shows numerical results from theconcentration's first three derivatives. Since they have very similarresults, they were considered as approximately equal. Two techniqueswere used for the calculation of the derivatives: the first using thesame principle of Eq. (2) only for the derivatives, in other words,

∂CA

∂x ≈∂C eA

∂x ¼XNnodes

i¼1

∂Ni

∂x CeAi

the edges of the cube.

∂CAB∂x ≅ ∂CAB

∂y ≅ ∂CAB∂z

GFEM/FEM GFEM/FDM

L2 L∞ L2 L∞

1.06E−03 1.97E−03 7.49E−05 1.91E−048.61E−03 2.05E−02 1.22E−03 3.68E−031.19E−01 4.31E−02 9.09E−03 3.00E−021.80E−01 5.54E−01 4.92E−02 1.73E−016.81E−01 2.24E−00 2.25E−01 8.29E−01

Table 4Numerical results for solution of CAB and its first derivatives, u=v=w and DAB=1.

u Min Max CAB∂CAB∂x ≅ ∂CAB

∂y ≅ ∂CAB∂z

L2 L∞ GFEM/FEM GFEM/FDM

L2 L∞ L2 L∞

1 0 1 4.84E−07 2.13E−06 1.06E−03 1.97E−03 7.49E−05 1.91E−042 −0.73 1 4.36E−06 2.35E−05 4.63E−03 1.10E−02 6.60E−04 1.97E−035 −0.99 1 9.89E−05 6.78E−04 3.97E−02 1.31E−01 1.31E−02 4.83E−0210 −1 1 1.06E−03 8.79E−03 2.13E−01 8.36E−01 1.16E−01 4.78E−0120 −1 1 9.03E−03 7.16E−02 1.03E−00 4.63E−00 8.07E−01 3.63E−00


which is analogously performed for the derivatives in the y and zdirections. However, the author's have used this technique in previousworks, as in [14,15], with less than exceeding results, especially forthe 8-node hexahedrons. To present a better technique, this paperproposes that for such a calculation, the third-order finite differenceshould be used as follows,

∂CAi

∂x ≅−11CAi þ 18CAiþ1−9CAiþ2 þ 2CAiþ3

6Δx:

with this expression being known as the third order forward finite dif-ference [16]. Since the mesh proposed in this paper consists of regularhexahedron elements, there are points in the mesh, where this expres-sion cannot be applied; in these cases, the third order backward finitedifferences were used as follows,

∂CAi

∂x ≅11CAi−18CAi−1 þ 9CAi−2−2CAi−3

6Δx:

The other derivatives in the last two expressions adhere to the analogreasoning. Table 2 shows that the use of FDM for the derivative

Fig. 1. Concentration in the XZ plane with y=0.5, U=0.01 m/s, mesh: Δx=Δy=1/30 andholes 1 and 2.

calculation, produced on average, one order of precision more than theFEM calculation.

Two situations were then proposed: to vary the cube size and tovary the velocity components to a unitary cube domain.

In Table 3, the cube with edges 1, 2, 3, 4 and 5 using 512 elementsand 4913 nodes in the mesh for 27-node hexahedrons was analyzed.It is noted that for the first row of Table 3, h=1/8, with the standarderror around 10−6. While for h=5/8 (cube with 125 times largervolume) the precision drops to around 10−2, which under thesecircumstances can be considered a good result, especially because thedomain becomes 125 times larger with the same 512 elements fromthe first case (h=1/8). Another important point to note is that in thecase of h=1/8, the error is around 10−6 for concentration valuesranging between 0 and 1. Whereas in the case of m=5/8 theconcentration varies from −84.79 to 1 (it is noteworthy that thesenegative values occur due to the adopted analytical solution, notnecessarily representing a real engineering application).

Meanwhile, in Table 4, the value of the velocity components variedin the 1, 2, 5, 10, and 20 values, and this value was equally applied tothe three components. Once again using 27-node hexahedrons in aunitary cube with h=1/8, Table 4 shows that the numerical resultspresent a significant decrease in the order of accuracy, e.g., whencomparing u=1 and u=20, where approximately four orders of

Δz=1/80, 8-nodes, Nnost=77,841, Nelem=72,000. Left: with hole 1, and right: with

Fig. 2. Concentration in the YZ plane with x=0.5, mesh: Δx=Δy=1/30 and Δz=1/80, 8-nodes, Nnost=77,841, Nelem=72,000, with hole 1.

Fig. 3. Concentration in the YZ plane with x=0.5, mesh: Δx=Δy=1/30 and Δz=1/80, 8-nodes, Nnost=77,841, Nelem=72,000, with holes 1 and 2.

Fig. 4. Concentration in the XZ plane with y=0.5, U=0.01 m/s, mesh: Δx=Δy=1/30 and Δz=1/80, 8-nodes, Nnost=77,841, Nelem=72,000. Left: with holes 1, 2 and 3, and right:with holes 1, 2, 3, and 4.


image of Fig.�2

image of Fig.�3

image of Fig.�4

Fig. 5. Concentration in the YZ plane with x=0.5, mesh: Δx=Δy=1/30 and Δz=1/80, 8-nodes, Nnost=77,841, Nelem=72,000, with holes 1, 2 and 3.

Fig. 6. Concentration in the YZ plane with x=0.5, mesh: Δx=Δy=1/30 and Δz=1/80, 8-nodes, Nnost=77,841, Nelem=72,000, with holes 1, 2, 3 and 4.


accuracy are lost. Importantly, in Tables 3 and 4, the GFEM/FDMtechnique again presented on average, one order of accuracy more thanthe GFEM/FEM.

4.2. Application 2 (internal flow)

In this application we assume a situation where there is anammonia flow into a rectangular tubulation with the followingdimensions Lx×Ly×Lz. It is accepted that the ammonia enters atthis stretch of the tubulation, in a position of fully developedflow, and the adopted velocity profile will be of the form,

Az ¼16ULx⋅Ly

−x2

Lxþ x

!−y2

Lyþ y

!;

where U is the maximum velocity in the cross section.

Fig. 7. Schematic diagram for the calculation of the

To study the transport of concentration, it is supposed that theinner surface this tubulation, in the section XZ with y=0, rupturescausing the formation of holes, whose positions are set out as follows:

Hole 1— x; y; zð Þ ¼ 0:05; 0; 0:05ð Þ

Hole 2— x; y; zð Þ ¼ 0:05; 0; 0:10ð Þ

Hole 3— x; y; zð Þ ¼ 0:05; 0; 0:15ð Þ

Hole 4— x; y; zð Þ ¼ 0:05; 0; 0:20ð Þ:

Thus, the following boundary conditions for this situation will beadopted:

CA=1 in z=0;CA=0 in the holes(air intake);

average concentration in the output section.

image of Fig.�5

image of Fig.�6

image of Fig.�7

Table 5Average concentration in the output section with a mesh with Δx=Δy=1/30, number of holes and Δz variable.

CAm

Δz Nnost Nelem Hole 1 Holes 1 and 2 Holes 1, 2 and 3 Holes 1, 2, 3 and 4 Dif med⁎

1/80 77,841 72,000 0.99621 0.99302 0.98986 0.98671 0.003321/90 87,451 81,000 0.99638 0.99337 0.99039 0.98743 0.003141/100 97,061 90,000 0.99650 0.99365 0.99081 0.98800 0.003001/110 106,671 99,000 0.99660 0.99386 0.99115 0.98844 0.002891/120 116,281 108,000 0.99666 0.99404 0.99141 0.98881 0.002801/130 125,891 117,000 0.99672 0.99417 0.99163 0.98910 0.002731/140 135,501 126,000 0.99675 0.99428 0.99180 0.98934 0.00267Difference (CAm(Δz=1/130)−CAm(Δz=1/140)) −3.00E−05 −1.10E−04 −1.70E−04 −2.40E−04

⁎Dif med=(1−CAm (holes 1, 2, 3, and 4))/4: defines the average decrease in the concentration exerted by each hole in the tubulation.

Fig. 8. Concentration in the XZ plane with y=0.5 and with holes 1, 2, 3 and 4. Left: U=0.015 m/s, mesh: Δx=Δy=1/30 and Δz=1/240, 8-nodes, Nnost=231,601, Nelem=216,000. Right: U=0.02 m/s, mesh: Δx=Δy=1/30 and Δz=1/160, 8-nodes, Nnost=154,721, Nelem=144,000.


Ammonia–air (25 °C): DAB=0.28×10−4 m2/s (Table 8 of [9]).

Before presenting the results, it is important to mention that thissituation will have an extremely dominant convective case, becausewe have a U/DAB ratio in box 3.57×102 for U=0.01 m/s, when

Fig. 9. Concentration the YZ planewith x=0.5,U=0.015 m/s, mesh:Δx=Δy=1/30 andΔz=

compared to Table 4. In relation to U/DAB in box 2×101, where thenumerical results were already in the order of 10−2, we would need avery accurate refinement for this application. In application 1 thereare no test caseswhereU/DABwas at box 3.57×102 because the adoptedanalytical solution induced the situations as eU=DAB ¼ e3:57�102→∞.

1/240, 8-nodes,Nnost=231,601,Nelem=216,000, CAm=0.99236,with holes 1, 2, 3 and 4.

image of Fig.�8

image of Fig.�9

Fig. 10. Concentration the YZ plane with x=0.5, U=002 m/s, mesh: Δx=Δy=1/30 and Δz=1/160, 8-nodes, Nnost=154,721, Nelem=144,000, CAm=0.99319, with holes 1, 2, 3and 4.


Owing to the computational limitations and the need for further refine-ment of the mesh, in this application the study will be conducted with8-node hexahedrons.

To study this application, a low maximum velocity will be used inthe section, U=0.01 m/s, emphasizing the concentration's behavior,by inserting one, two, three or four holes in the domain. For such asituation, we adopted Lx=Ly=0.1 m and Lz=0.5 m.

Figs. 1–6 show that the number of holes (1, 2, 3, or 4) has littleeffect on the concentration's transport. It should be interesting toanalyze the average concentration in the output section, XY inz=0.5. The calculation used the following reasoning,

CAm ¼

XNNosXYi¼1

TiAreai

NNosXY⋅AreaT

where NNosXY is the number of nodes in the output section, Ti isthe temperature of each node of the output section; Areai is the areaelement referring to Ti, according to the diagram shown in Fig. 7,and AreaT is the area of the output section.

There is a 0.00267 decrease in the mean concentration of the out-put section for each hole inserted in the tubulation, i.e., for eachadded hole, considering the proposed geometry and maximum Uvelocity, there was an average decrease in the output of 0.267% in theconcentration (see Table 5, Dif med with Δz=1/140). Importantly,the refinement in the z direction shows that the error in the averageconcentration drops to a maximum of 0.00024 (0.024%), which canalready be considered an excellent result.

Figs. 8–10, show the numerical results for the velocities U=0015 m/sand U=0.02 m/s.

5. Conclusions

The GFEM with hexahedral elements proved to be very efficient insolving concentration transport problems; especially, concerning theanalysis of application 1, where the use of 27-node hexahedronsproved to be extremely effective. However, because the transfer of theconcentration problem refers to dominant convective situations, andthe computational capacity being limited, application 2 used only8-node hexahedrons instead. Nevertheless, the low velocity situationspresented good results. The results for the velocities of 0.01, and 0.020015 m/s (maximum velocity in cross section), show that the higherthe velocity, the greater the length of tubulation required (z direction)for the holes to influence the average concentration of the XY sectionoutput. Therefore, longer lengths and higher velocities would requirefurther refinement than what was applied in this work, refer to Fig. 9,in which case 216,000 elements were used. It is important to

emphasize that the difficulties in solving this problem (application 2),are not only related to the high convectivity and dimensions ofthe domain, but also deal with the issue that for each hole in thetubulation, in its subregion, there is a singularity, i.e., a region wherethe concentration transport numerical value is around 1, and suddenly,because of the hole's presence, its concentration becomes equal to zero.

Acknowledgments

The National Council of Scientific Development and Technology,CNPq, Brazil (Proc. 500382/2011-5) supported the present work.

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image of Fig.�10

3D contaminant transport by GFEM with hexahedral elements

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