Hutton - Fundamentals of Finite Element Analysis (523155), страница 44
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To illustrate the effect, recall that, for the linear, two-nodeelement the first derivative of the field variable, in this case, the temperature gradient, isconstant; that is,dTTT==dxxLeUsing the computed nodal temperatures, the element gradients areElement 1:◦dTF136.16 − 180== −43.84dx1in.Element 2:◦dTF111.02 − 136.16== −25.14dx1in.Element 3:◦dTF97.23 − 111.02== −13.79dx1in.Element 4:◦dTF90.79 − 97.23== −6.44dx1in.where the length is expressed in inches for numerical convenience. The computed gradient values show significant discontinuities at the nodal connections.
As the number ofelements is increased, the magnitude of such jump discontinuities in the gradient valuesdecrease significantly as the finite element approximation approaches the true solution.Hutton: Fundamentals ofFinite Element Analysis7. Applications in HeatTransferText© The McGraw−HillCompanies, 20047.4 Heat Transfer in Two DimensionsTable 7.1 Nodal Temperature Solutionsx (inches)Four Elements,T (◦F)Eight Elements,T (◦F)00.51.01.52.02.53.03.54.0180158.08*136.16123.59*111.02104.13*97.2394.01*90.79180155.31136.48122.19111.41103.4197.6293.6391.16To illustrate convergence as well as the effect on gradient values, an eight-element solution was obtained for this problem. Table 7.1 shows the nodal temperature solutions forboth four- and eight-element models.
Note that, in the table, values indicated by * areinterpolated, nonnodal values.7.4 HEAT TRANSFER IN TWO DIMENSIONSA case in which heat transfer can be considered to be adequately described by atwo-dimensional formulation is shown in Figure 7.5. The rectangular fin hasdimensions a × b × t , and thickness t is assumed small in comparison to a andb. One edge of the fin is subjected to a known temperature while the other threeedges and the faces of the fin are in contact with a fluid.
Heat transfer then occursfrom the core via conduction through the fin to its edges and faces, where convection takes place. The situation depicted could represent a cooling fin removing heat from some process or a heating fin moving heat from an energy sourceto a building space.To develop the governing equations, we refer to a differential element of asolid body that has a small dimension in the z direction, as in Figure 7.6, andexamine the principle of conservation of energy for the differential element. Aswe now deal with two dimensions, all derivatives are partial derivatives.
Again,on the edges x + dx and y + dy , the heat flux terms have been expanded in firstorder Taylor series. We assume that the differential element depicted is in theinterior of the body, so that convection occurs only at the surfaces of the elementand not along the edges. Applying Equation 5.53 under the assumption of steadystate conditions (i.e., U = 0 ), we obtain∂q x∂q yq x t dy + q y t dx + Qt dy dx = q x +dx t dy + q y +dy t dx∂x∂y+ 2h(T − Ta ) dy dx(7.20)235Hutton: Fundamentals ofFinite Element Analysis2367. Applications in HeatTransferCHAPTER 7Text© The McGraw−HillCompanies, 2004Applications in Heat Transferqy ⭸qydy⭸yqhdyqxbTQ, Uqx ⭸qxdx⭸xqhqhdxtaFigure 7.5 Two-dimensionalconduction fin with face andedge convection.h(T Ta)qyFigure 7.6 Differential element depictingtwo-dimensional conduction with surfaceconvection.wheret = thicknessh = the convection coefficient from the surfaces of the differential elementTa = the ambient temperature of the surrounding fluidUtilizing Fourier’s law in the coordinate directionsq x = −k x∂T∂xq y = −k y∂T∂y(7.21)then substituting and simplifying yields∂T∂T∂∂Qt dy dx =−k xt dy dx +−k yt dy dx + 2h(T − Ta ) dy dx∂x∂x∂y∂y(7.22)where k x and k y are the thermal conductivities in the x and y directions, respectively.
Equation 7.22 simplifies to∂∂T∂∂Tt kx+tky+ Qt = 2h(T − Ta )(7.23)∂x∂x∂y∂yEquation 7.23 is the governing equation for two-dimensional conduction withconvection from the surfaces of the body. Convection from the edges is alsopossible, as is subsequently discussed in terms of the boundary conditions.7.4.1 Finite Element FormulationIn developing a finite element approach to two-dimensional conduction withconvection, we take a general approach initially; that is, a specific element geometry is not used.
Instead, we assume a two-dimensional element having M nodesHutton: Fundamentals ofFinite Element Analysis7. Applications in HeatTransferText© The McGraw−HillCompanies, 20047.4 Heat Transfer in Two Dimensionssuch that the temperature distribution in the element is described byMT (x , y) =N i (x , y)Ti = [N ]{T }(7.24)i=1where N i (x , y) is the interpolation function associated with nodal temperatureTi , [N ] is the row matrix of interpolation functions, and {T } is the column matrix(vector) of nodal temperatures.Applying Galerkin’s finite element method, the residual equations corresponding to Equation 7.23 are ∂∂∂T∂TN i (x , y)t kx+tky+ Qt − 2h(T − Ta ) d A = 0∂x∂x∂y∂yAi = 1, M(7.25)where thickness t is assumed constant and the integration is over the area of theelement.
(Strictly speaking, the integration is over the volume of the element,since the volume is the domain of interest.) To develop the finite elementequations for the two-dimensional case, a bit of mathematical manipulation isrequired.Consider the first two integrals in Equation 7.25 as ∂∂∂T∂TtkxNi +kyNi d A∂x∂x∂y∂yA ∂q x∂q y= −tNi +Ni d A(7.26)∂x∂yAand note that we have used Fourier’s law per Equation 7.21. For illustration, wenow assume a rectangular element, as shown in Figure 7.7a, and examiney2 x2∂q x∂q xtNi d A = tN i dx dy(7.27)∂x∂xy1 x1Ay(x1, y2)(x2, y2)abqx(x1, y)(x1, y1)(x2, y1)(a)qx(x2, y)ab(b)Figure 7.7 Illustration of boundary heat flux in x direction.x237Hutton: Fundamentals ofFinite Element Analysis2387.
Applications in HeatTransferCHAPTER 7Text© The McGraw−HillCompanies, 2004Applications in Heat Transfer∂q xdx , we obtain, formally,∂xy2y2x2 x2∂q x∂ NitN i d A = t q x N i x dy − tqxdx dy1∂x∂xIntegrating by parts on x with u = N i and dv =y1Ay2=ty1 x1 x2q x N i x dy + tkx1y1A∂ T ∂ NidA∂x ∂x(7.28)Now let us examine the physical significance of the termy2t x2q x N i x dy = ty2[q x (x 2 , y) N i (x 2 , y) − q x (x 1 , y) N i (x 1 , y)] dy1y1(7.29)y1The integrand is the weighted value ( N i is the scalar weighting function) of theheat flux in the x direction across edges a-a and b-b in Figure 7.7b. Hence, whenwe integrate on y, we obtain the difference in the weighted heat flow rate in thex direction across b-b and a-a , respectively. Noting the obvious fact that theheat flow rate in the x direction across horizontal boundaries a-b and a -b is zero,the integral over the area of the element is equivalent to an integral around theperiphery of the element, as given bytq x N i d A = t q x N i n x dS(7.30)ASIn Equation 7.30, S is the periphery of the element and n x is the x componentof the outward unit vector normal (perpendicular) to the periphery.
In our example, using a rectangular element, we have n x = 1 along b-b , n x = 0 along b -a ,n x = −1 along a -a , and n x = 0 along a-b. Note that the use of the normal vector component ensures that the directional nature of the heat flow is accountedfor properly.
For theoretical reasons beyond the scope of this text, the integrationaround the periphery S is to be taken in the counterclockwise direction; that is,positively, per the right-hand rule.An identical argument and development will show that, for the y-directionterms in equation Equation 7.26,∂∂T∂ T ∂ NitkyN i d A = −t q y N i n y dS − k ydA(7.31)∂y∂y∂y ∂yASAThese arguments, based on the specific case of a rectangular element, areintended to show an application of a general relation known as the Green-Gausstheorem (also known as Green’s theorem in the plane) stated as follows: LetF (x , y) and G (x , y) be continuous functions defined in a region of the x-y planeHutton: Fundamentals ofFinite Element Analysis7. Applications in HeatTransferText© The McGraw−HillCompanies, 20047.4 Heat Transfer in Two Dimensions(for our purposes the region is the area of an element); then ∂F∂G+=( F n x + G n y ) dS∂x∂yAS ∂ F ∂∂ G ∂+dA−∂x ∂x∂y ∂y(7.32)AReturning to Equation 7.26, we let F = k x ∂∂Tx , G = k y ∂∂Ty , and = N i (x , y) , andapply the Green-Gauss theorem to obtain ∂∂ ∂T∂TtkxNi +Ni d A∂x∂x∂y ∂yA ∂ T ∂ Ni∂ T ∂ Ni= −t (q x n x + q y n y ) N i dS − tkx+ kydA(7.33)∂x ∂x∂y ∂ySAApplication of the Green-Gauss theorem, as in this development, is the twodimensional counterpart of integration by parts in one dimension.
The result isthat we have introduced the boundary gradient terms as indicated by the firstintegral on the right-hand side of Equation 7.33 and ensured that the conductancematrix is symmetric, per the second integral, as will be seen in the remainder ofthe development.Returning to the Galerkin residual equation represented by Equation 7.25and substituting the relations developed via the Green-Gauss theorem (beingcareful to observe arithmetic signs), Equation 7.25 becomes ∂ T ∂ Ni∂ T ∂ Nikx+ kyt d A + 2hT Ni d A∂x ∂x∂y ∂yAA=Q N i t d A + 2h TaN i d A − t (q x n x + q y n y ) N i dSi = 1, MAAS(7.34)as the system of M equations for the two-dimensional finite element formulationvia Galerkin’s method.
In analogy with the one-dimensional case of Equation 7.8, we observe that the left-hand side includes the unknown temperaturedistribution while the right-hand side is composed of forcing functions, representing internal heat generation, surface convection, and boundary heat flux.At this point, we convert to matrix notation for ease of illustration byemploying Equation 7.24 to convert Equation 7.34 to ∂N T ∂N∂N T ∂Nkx+ ky{T }t dA + 2h[N ]T [N ]{T } dA∂x∂x∂y∂yAA=Q[N ]T t dA + 2hTa[N ]T dA − qs n s [N ]T t dS(7.35)AAS239Hutton: Fundamentals ofFinite Element Analysis2407.
Applications in HeatTransferCHAPTER 7Text© The McGraw−HillCompanies, 2004Applications in Heat Transferwhich is of the form (e) (e) (e) k (e) {T } = f Q + f h + f g(7.36)as desired.Comparison of Equations 7.35 and 7.36 shows that the conductance matrix isT T (e) ∂N∂N∂N∂Nk=kx+ kyt dA∂x∂x∂y∂yA+ 2h[N ] T [N ] d A(7.37)Awhich for an element having M nodes is an M × M symmetric matrix. While weuse the term conductance matrix, the first integral term on the right of Equation 7.37 represents the conduction “stiffness,” while the second integral represents convection from the lateral surfaces of the element to the surroundings.
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