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F ..EXACT FINITE ELEMENTS FOR CONDUCTION AND CONVECTION

by

Earl A. ThorntonAssociate Professor

Pramote DechaumphaiResearch Assistant

Kama r K. TammaResearch Assistant

I Rf

Mechanical Enginecring and Mechanics DepartmentOld Dominion UniversityNorfolk, Virginia 23508

Presented at the

Second International Conference onNumerical Methods in Thermal Problems

July 7-10, 1981Venice, Italy

F- II

EXACT FINITE ELEMENTS FOR CONDUCTION AND CONVECTION

Earl A. Thornton l , Pramote Dechaumphai ii , Kumar K. Tama II

Mechanical Engineering and Mechanics DepartmentOld Dominion University, Norfolk, Virginia USA

SUMMARY

An approach for developing exact one-dimensionalconduction-convection finite elements is presented. Exactinterpolation functions are derived based on solutions to thegoverning differential equations by employing a nodelessparameter. Exact interpolation functions are presented forcombined heat transfer in several solids of different shapes,and for combined heat transfer in a flow passage. Numericalresults demonstrate the exact one dimensional elements offersignificant advantages over elements based on approximateinterpolation functions.

1. INTRODUCTION

A significant development in finite element methodologyfor forced convection analysis is the concept of upwindweighting functions [1]. For the one-dimensional convectivediffusion equation upwind weighting functions are expressedin terms of an upwind parameter which can be varied to adjustthe accuracy of the finite element solution. Optimal valuesof the upwind parameter lead to exact values o` the tempera-ture at nodal points. Other elements in one-dimensionalthermal and structural analysis also predict exact nodalvalues. For example, one dimensional solutions [2] for heatconduction with internal heat generation by linear ii-terpola-tion functions yield exact temperatures at the nodes. Theability of finite elements to predict exact nodal temperatureshas previously-been regarded as a property of a few particularequations and to be of limited generality. There is, however,

( Associate Profes:.or

II ResearchAssistant

a broad class of-one-dimensional problems governed by linearordinary differential equations for which finite elements can

be developed to yield exact solutions. The one-dimensionalconvective-diffusion equation and the one-dimensional conduc-tion equation are special cases of this class of problems. Ina previous paper [3], the authors have presented two examplesof one-dimensional thermal-structural finite elements whichyield exact solutions for linear problems.

_s

The purpose of the paper is to present an approach fordeveloping one-dimensional conduction-convection finite ele-ments which yield exact nodal temperatures and an exactvariatior, of temperature within an element for steady-statelinear analysis. The one-dimensional conduction-convectionproblems considered are first discussed. A nodeless variableapproach for deriving exact interpolation functions is thenpresented and illustrated for several cases. Next, the gener-al form of the element equations is presented and discussed.Finally, the benefits of the exact finite element formulationare illustrated for two problems by comparing numerical solu-tions from finite elements with exact and linear interpolationfunctions.

2. EXACT ELEMENT FORMULATION

2.1 Governing Equations and Boundary Conditions

The geometry and terminology for eight one-dimensionalconduction-convection cases are shown in Fig. 1. Cases 1-7are solids of various shapes where heat transfer consists ofconduction which may be combined with: (a) surface convection,(b) internal heat generation, and (c) surface heating. Case(8) is a one-dimensional flow where the heat.transfer consistsof fluid conduction and mass-transport convection which may becombined with: (a) surface convection, and (c) surface heat-ing. In the figure, Q is the volumetric heat generation rate,q is the surface heating rate, h is the convective heattransfer coefficient and T is the environmental temperaturefor the convection heat exchange. Heat transfer in Cases 1-8is described by differential equations of the form

ao (x) _7+a1(x)rx + a2 (x)T = a3 (x) (1)dx

where a i , i - 0,1,2,3, are functions which depend on thegeometry and thermal parameters of each case. Finite elementequations for eq. (1) may be formulated by the method ofweighted residuals, but due to the presence of the odd-orderderivative an unsymmetrical coefficient matrix will result.However, if eq. (1) is first cast into self-adjoint form, thedifferential operator will be symmetric. Then, finite elementmatrices can be derived by the method of weighted residuals ora variational method, and coefficient matrices will be

0 q h'T"'

xL

Case 1, ROD

q

WXhTft

h, Lq

Case 2, SLAB

TI.To

-I-

qh Tr

thel '

L

L

Q

9Case 7, SPHERICAL SHELL Case 8. FLOW PASSAGE

Figure 1ONE DIMENSIONAL CONDUCTION AND CONVECTION CASES

Case 3. HOLLOW CYLINDER Case 4, HOLLOW SPHERE

Case 5, CYLINDRICAL SHELL

° = bi

414

4 hAlmCase 6. CONICAL SHELL

- ds cosy TV -- •-

- d[PdT] h PT h PTTx_ dx kA 0

where P - exp(-mcx/kA)

A cos a kt cos a--

kA P

e

symmetrical, [2]. Eq. (1) is written in self-adjoint form bymultiplying it by the factor

P - exp[t(a 1/ao) dx]such that

az [P(x^] + Q(x)T - R (x) (2)where Q(x) - a 2 P/ao and R(x) - a3 P/ao.

The governing self-adjoint differential equations forCases 1-8 are shown in Table 1. Additional terms not previous-

• ly defined are k the thermal conductivity, p the perimeter ofthe rod or flow passage, A the cross-sectional area of the rodor flow pass.;ge, m the mass flow, rate and c the specific heat.Table 1 shows the .xefficie. cs of eq. (2) for each case. Notethat for each case a diffQ,ent differential equation may

TABLE 1r GOVERNING SELF-ADJOINT DIFFERENTIAL EQUATIONS

dx [P(x 'dx] + Q(x)T - R(x)Heat Loads

Case Conduction Convection Convection Source SurfaceFlux

(a) (a) (b) (c)

1 - dx[t] T T k2 - dx - -- -- -- .

d [ dT]3 - dr rdr -- -- kr --

4 ^[r2^] -- -- kr•2 --

d dT h h5

- ds [ds ] kt T kt T^ k kt

5 - ts Csas ] kt sT kt sT- i S ktsI 7

8e

• ^Y• V ll^i

t^ dT

i

- . • An .

result by combining conduction with each of the three heatioads or combinations of these heat loads.

The boundary conditions for &4. (2) consist of specifyingthe temperature or the temperature gradient at the endpointsof the solution domain, a

e

l

N1 (x) - f2 (x2 )f l (x) - fl(x2)f2(x)

(7b)

N (x) s fl (xl )f2 (x) - f2(xl)fl(x)(7c)2

where 0 - f1 (X1 )f2 (x2 ) - fl(x2)f2(xl).

Element interpolation functions derived in the form of eq. (2)are not generally the same as used in conventional finiteelements since they result from the differential equationsolution. Interpolation functions, in fact, may range fromsimple polynomials to higher transcendental functions depend-ing on the equation. The exact interpolation functions, eqs.(7), are derived for specified temperature boundary conditions,but they also yield exact solutions for the gradient boundaryconditions, eq. (3), when these boundary conditions are con-sistently incorporated in the finite element equations. Sincethe interpolation functions are exact, temperature gradientsand fluxes computed from these functions are also exact.

Nodeless parameters and element interpolation functionshave been derived for Cases 1-8 with heat loads (a) - (c), butfor brevity only typical results are presented herein. Typicalnodeless parameters appear in Table 2 and typical interpolationfunctions appear in Table 3.

TABLE 2

NODELESS PARAMETERS

Case To Case To

1(a) TW 1(b)^a2/k

1(b) OL2/2k 8(b) qpL/mc

3(b) Ob2/4kw 8(a) T

4(b) 0/6kwhere w - ln(b/a)

I

..

TABLE 3

ELEMENT INTERPOLATION FUNCTIONS

CaseNo

N1 N2

1(a) 1-N1-N2Binh m(L-x) sinh mx

sink mL sinh mL

1(b) N1N21 L

3(b)

2 N!-2 2

- w + - - w w In (

r) w In

(a)

4(b)(r-a)(b-rM r+a+b)

ra b_rr a

b r_ar a

l+sin a

7(b)7(b)s

log(cosa)-N2 log(cos

log

La) 1-N2

sinaL+sni allog

1•sin a J[l

8(b) L - N2 1-N2 1 - e2axZT.

8(a) 1-N -n ,1 2

eax sinh e(L-x) sinh SL

ea(x-L) sinh Bxsi—n a

where m = (hp/kA)}

w s ln(b/a)

CL = me/2kA

a - (1112 + a2 ) }

i

2.3 Element Matrices

For general interpolation functions, element matrices foreq. (2) are

x dN dN xKid = f 2 P

axdx + f

2 Q Ni N^dx (8a)

x l xl

F i f x2

R N, dxx1

where K is a typi(.al term in the symmetric conductancematrix,'And F is P. typical term in the heat load vector. Fora node on the boundary additional terms are required to repre-sent surface heating or a convective heat exchange. For theexact interpolation functions, element matrices have the-general form

KOO KO 1 K02 ToFo

K10 K11 K12 T 1 =F1

K20 K21 K22 T2 F2

since i = 0,1,2. Because To is known the first equation may

be uncoupled from the nodal unknowns in the second and thirdequations. Thus, the exact element matrices have the samesite as a conventional linear Element and can be written as

K11 K12 T 1 F1 K10- To

(9)

K21 K22T2

F2 K20

Eq. (9) can be simplified further by noting that K =0 for

self-adjoint equations. The last result can be shn$observing that N1 =1,2 are solutions of the ordinarydifferential equation, (2). Multiplying this equation by Nand integrating by parts shows that K =K, n 0. Thus, for °

self-adjoint equations, element matri12s h ged to be evaluatedonly for 1=1,2.

As emphasized previously, the development of the elementmatrices, eqs. (8), and element equations, eq. (9), are basedupon the differential equation being written in the self-adjoint form, eq. (2). As an alternate ap proach, exact finite

element matrices may also be derived from the standard form ofthe differential equation, eq. (1). However, element matrices

.

(8b)

will not be symmetrical due to the odd-order derivative, andthe exact element equations will have Koi # 0, 1-1,2.

The principal advantage of the exact one-dimensionalfinite elements is the superior accuracy in --Comparison to

elements based on approximate interpolation functions. Theprincipal disadvantage of the exact elements is the additionaleffort required to form the more complex interpolationfunctions and to evaluate the element matrices. This disad-vantage has been overcome, in part, by using a computer-basedsymbolic manipulation language MACSYMA to perform the algebraand calculus required for these derivations.

3. NUMERICAL EXAMPLES

Two numerical examples are presented to illustrate thebenefits of the exact elements; other examples for Cases 1 and:I appear in [4]. The solution of problems with exact elementstypically requires only a few elements because mesh refinementis not required in regions of large temperature gradlen,ts.The number of elements used is determined by changes ingeometry, material properties or heating variations. Afternodal temperatures are computed, temperatures are computed atseveral points within each element by using the element inter-polation functions, eq. (6).

3.1 Coffee Spoon with Conduction and Convection

Rod elements with conduction and convection (Table 1,Case 1(a)) are used to model one-dimensional heat transfer ina coffee spoon (Fig. 2). The lower one-half of the spoon isconvectively heated by the coffee at a specified temperatureof 150°F, and the upper one-half of the spoon is convectivelycooled by the atmosphere at a specified temperature of 50°F.The ends of the one-dimensional spoon model are a>sumed tohave negligible heat transfer. Element matrices for the e!oc tfinite elements appear in reference [4]. Temperatures for thtspoon model are computed from: (1) two exact finite elements,(2) two elements with linear temperature interpolation,(3) tenelements with linear temperature interpolation.

The temperature distributions (Fig. 2) show the exacttemperature distribution computed by the two exact elementswith three nodal unknown temperatures compared with predictionsfrom the elements with linear temperature interpolation. Twolinear elements predict the temperatures at the three nodeswith fair accuracy, but the linear interpolation functions aienot capable of representing the zero temperature gradientboundary conditions. Temperatures computed from ten linearelements, however, show excellent agreement with the exactfinite element solution. Solutions for similar problems havealso shown this trend; typically about five to ten elementswith linear interpolation per exact element are required for

acceptable accuracy.

^ 160 h= 0.05r Tan 50 ^`

140

h=010 L^2 x^.=150

r120

T, °F —exact fe.

1002 elements

--linear fa2 elements

--- linear fa80 10 elements

0600 0.2 0.4 0.6 0.8 1.0

x fL

Figure 2R00 ELEMENT ANALYSIS OF COFFEE SPOONWITH CONDUCTION AND CONVECTION

3.2 Heat Transfer in Merging Flows

In merging flows, significant temperature gradients mayoccur at the flow confluence; therefore, merging flows provide

i

a good test for finite element evaluations. In reference [4],the authors compared conventional (Bubnov-Galerkin) finiteelement solutions with upwind (Petrov-Galerkin) solut i ons forsteady-state and transient merging flows. Fig. 3 presentsthe geometry and terminology of a merging flow where conduction

S

is combined with mass-transport and surface convection. pa

Temperatures for the merging flow (Fig. 3) were computed 3from: (1) three exact finite elements with four nodes, ar'(2) 75 conventional elements with 76 nodes. There is goodagreement between the two solutions except in the vicinity

;.

of the flow confluence where the conventional element showsosculations indicating a need for mesh refinement in theregion of large temperature gradients. The clear superiorityof the exact elements is demonstrated. Exact nodal tempe;-a-tures are also predicted in problems of this type (withoutsurface convection) by optimum upwind elements, but a larger

s.

i'

number of nodes are required since upwind elements do notpredict exact temperatures within an element.

2.0`.T=2

° ti .^• m

1.5 -6- exact f e.3 elements.4 nodes l • 1 ^,

T(x) o conventional fe.75 elements °ip76 nodes

1.0

0

0 0.4 0.8 X/L 12 1.6 2.0

Figure 3EXACT AND CONVENTIONAL FINITE ELEMENT ANALYSISOF MERGING FLOW WITH CONDUCTION, MASS TRANSPORTCONVECTION AND SURFACE CONVECTION.

4. CONCLUDING REMARKS

An approach for developing one-dimensional, exact finiteelements for linear, steady conduction-convection analysis ispresented. The element interpolation functions employ thesolution from the governing second-order di fferenti al equati on byutilizing a nodeless parameter approach. The elements requiretwo nodes and yield exact nodal temperatures and an exactvariation of the temperature within an element. Exact nodalinterpolation functions are presented for several cases ofconduction and convection. Numerical results are presentedfor conductior. and surface convection in a rod element model,and for conduction, mass - transport com

REFERENCES

s 1. CHRISTIE, I., GRIFFITHS, 0. F., MITCHELL, A. R.,ZIENKIEWICZ, 0. C. - Finite Element Methods for SecondOrder Differential Equations with Significbn, FirstDerivatives. International Journal for NumericalMethods in Engineering, Vol 10, pp. 1339-1396, 1976.

' 2. ZIENKIEWICZ, 0. C. - The Finite Element Method, thirdedition, McGraw-Hill, 1977.

3. THORNTON, E. A., DECHAUMPHAI, P., WIETING, A. R. andTAMMA, K. K. - Integrated Transient Thermal-StructuralFinite Element Analysis. Proceedings of the AIAA/ASME/ASCE/AHS 22nd Structures, Structural Dynamics and MaterialsConference, Atlanta, Georgia, AIAA Paper No. 81-0480,Apri 1 6-8, 1981.

4. THORNTON, E. A. and DECHAUMPHAI, P. - i:onvective HeatTransport in Merging Flows, Finite Elements in Wa orResources, Ed. Wang, S. Y., Brebbia, C

A., ons

iGray, W. G. , and Pi nder, G. F., The Unversity ofMississippi, 1980.

ACKNOWLEDGEMENT

This study was supported in part by the National Aero-nautics and Space Administration under grant NSG 1321.

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