1D Heat Transfer. Background. Local Equations. The temperature assumption. Balance law. Lecture 7

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1 1D Heat Transfer Backgroun Consier a true 3D boy, where it is reasonable to assume that the heat transfer occurs only in one single irection. The heat conuctivity λ [J/sC m] an the internal heat generation per unit length Q(x) [J/sm] are given constants. Assume bounary conitions are that the temperature T (x = ) = g an that the heat flux q(x = l) = h per unit surface [J/sm 2 ] in the right en are given. Isolate bounary A(x), λ S g S h T() = g Q(x) q(l) = h x x =. x = L The temperature assumption The temperature is assume to have a constant value at every position x. This means that heat flux q only occurs along the x-irection. That is, from a mathematical point of view this problem is one-imensional! Local Equations Balance law Q(x) x H(x) H(x+ x) x Heat flux balance requires H(x) + Q(x) x = H(x + x) (1) 1

2 where Taylor s formula gives H(x + x) H(x) + H(x) x (2) an the total heat flux H(x) can be expresse in the heat flux per unit surface q(x) as follows H(x) = A(x)q(x). (3) These two equations (??) an (??) efines the Balance Law as (A(x)q(x)) Q(x) = (4) Constitutive relation Assume that Fourier s law hol T (x) q(x) = λ (5) Compatibility relation This is a scalar-value problem an no compatibility relation is require. Strong Form After elimination of the heat flux q(x) an introuction of bounary conitions, we immeiately receive the following Bounary-Value problem. Box: S Strong form of 1D Linear Heat Transfer Given Q(x), h an g. Fin T (x) such that ( ) T (x) A(x)λ + Q(x) = x ], L[ T () = g T (L) q(l) = h = λ on S g on S h This is ientically the same formulation as the 1D bar problem if we change T (x) u(x), Q(x) b(x) an λ E 2

3 Weak Form A Strong form S can always be turne over into an equivalent Weak form W by: multiplication by an arbitrary Weight function w(x) an integration over the omain. ( ) w (AλT ) + Q = Integration by parts of the first term gives [ waλ T ] L w AλT + w (AλT u ) + wq = (6) wq = (7) The first term in the equation above can be rewritten as the natural bounary conition h can be ientifie form box S as [ waλ T ] L T (L) T () = w(l)a(l) λ w()a()λ (8) }{{ } = h A Weak form W of a 1D Bar problem can be summarize as follows. Box: W Weak form of 1D Linear Static Elasticity Given Q(x), h an g. Fin T (x) such that w(x) (x) A(x)λT = T () w()a()λ w(x)q(x) w(l)a(l)h T () = g on S g for all choices of weight functions w(x) The partial integration step is performe because that opens a possibility to later on en up in a symmetric system of linear algebraic equations that is more efficiently solve in the computer compare to a non-symmetric system. The natural bounary conition is implicit containe in the integral equation. S W 3

4 Galerkin an Matrix Formulation Now move over to a iscrete approximative Galerkin formulation T (x) T h (x) = N 1 (x)a 1 +N 2 (x)a N n (x)a n + N 1 (x)g = n N i a i + N 1 (x)g (9) i=1 or shorter in a matrix notation T (x) = N(x)a + N 1 (x)g (1) In a Galerkin formulation the arbitrariness is limite to w(x) = N(x)c w(x) = c T N T (x) (11) where c is an arbitrary vector N i C The set of all continuous functions! N i (x j ) = δ ij linear inepenency The unknown vector a contains noe temperatures Putting in the approximation T (x) an the weight function w(x) into the weak form W the following iscrete Galerkin formulation will be achieve. (ct N T )Aλ (Na + N 1 g) = c T N T Q c T N T (L)A(L)h The vector c T can be brought out from left in all terms as follows { c T N T Aλ } (Na + N 1 g) N T Q + N T (L)A(L)h = an a matrix B(x) can be efine as B(x) = N(x) [ N1 (x) N = 2 (x) N n (x) which then can be inserte into the equation above an the following is receive ] (12) { c T B T AλB a } {{ } =K ( N T Q N T (L)A(L)h B T Aλ )} N 1 g =. }{{} =f 4

5 where K is the Global conuctivity matrix an f is the Global loa vector. A iscrete Galerkin formulation for this 1D problem now reas Box: G Galerkin form of 1D Linear Static Elasticity Fin a such that c T (Ka f) = c T r = for all choices of the vector c (the weight function) The vector c must be possible to select completely arbitrary. That implies r must be equal to a zero vector. A matrix problem consisting of n linear algebraic equations can now be ientifie. Box: M Matrix form of 1D Linear Static Elasticity Fin a such that Ka = f where K an f are known quantities After solving of M the temperatures in the vector a are use for calculation of the previously eliminate heat flux q(x). This is one in an element-wise fashion. q(x) = λ (N(x)a + N 1 (x)g) = λ(b(x)a + N 1 (x) g) (13) Both the matrix K an vector f are completely efine by the choice of number of shape functions N i (elements) an the behavior of these shape functions (the element type) From a mathematical point of view this iscussion can be summarize as L S W G M an sources for errors in this mathematical moel of reality are eviations from reality in the constitutive an the compatibility relations, eviations in the selecte bounary conitions an numerical errors ue to use of a limite number of finite elements with a specific behavior in each element. One can show that the solution to the matrix problem M always exists an has a unique solution if the global conuctivity matrix K is non-singular. If there exist at least one essential bounary conition which fixes the temperature scale the conuctivity matrix will be non-singular. That is the global stiffness matrix K is positive efinite an the following hols a T Ka > a et(k) > 5