1D Heat Propagation Problems

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Robert Berry
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1 Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2 with the simpe case of no heat generation, for which the equation reads T t = λ 2 T cρ x 2 1.2) that in the stationary case reduces to 2 T x 2 =. 1.3) The stationary case of heat conduction in a one dimension domain, ike the one represented in Figure??, is particuary simpe to be soved. Ta Tb x Figure 1.1: The temperature within a soid media with prescribed Dirichet boundary conditions. 1

2 1. 1D Heat Propagation Probems 2 In fact the genera soution of the equation reads in this case T x) = Ax + B, 1.4) with A and B coefficients to be determined by imposing the boundary conditions. Assigned temperatures at both eft and right surfaces T x= = T a, T x= = T b, 1.5) we have T a = B, and T b = A + T a, or In concusion T b T a = A. 1.6) T x) = T b T a x + T a. 1.7) Assigned temperature at eft surface and assigned heat fux entering the body at the right surface T x= = T a, q x= = q b, or λ dt dx = q b, as a consequence, being dt dx Finay T x= = B, λ dt dx = q b. T x) = q b λ x + T a. = A, we have 1.8) 1.9) 1.1) Considering that by imposing on one side the entering heat fux the derivative of temperature, that is the scope of the straight ine representing the temperature distribution, is determined, it is not possibe to find a stationary soution by imposing an heat fux condition on both sides. In fact if q a and q b are not equa the temperature response is not stationary since in presence of a variation of heat content the temperature of the body shoud way with time. If q a = q b the sope of the ine is determined ony the temperature difference can be obtained by no assumption on the temperature vaue on either surface can be done. The case is anaog to the mechanica response of an eastic spring free in the space at the end of which two forces with opposite sign are appied. The absoute vaue of dispacement cannot be obtained since the rigid body motion of the spring has not been constrained.

3 1. 1D Heat Propagation Probems 3 Let us now consider a transient probem in which the temperature at x = is equa to T a, the temperature at x = is equa to zero and the initia condition is set as T = T i x) The governing equations read as foows T = λ 2 T t cρ x 2, T x= = T a T x= = T t= = T i x k = λ cρ 1.11) The soution is found by separation of variabes, as we assume that the temperature can be expressed by the product of a function of position ony fx) and a function of time gt) T x, t) = fx)gt), 1.12) by substituting in the governing equation we obtain fx) dgt) dt that can aso be written as = kgt) d2 fx) dx 2, 1.13) 1 dgt) = 1 d 2 fx) kgt) dt fx) dx 2, 1.14) since the first member does not depend upon x, the second one does not depend upon t and the two members are equa, they can be set equa to a constant. By setting the constant equa to s 2 we obtain two separate equations and dgt) dt + ks 2 gt) =, 1.15) d 2 fx) dx 2 + s 2 fx) =. 1.16) The genera soution of the first equation can be easiy obtained by searching soution of the kind gt) = e αt and by finding the characteristic equation α + ks 2 =, 1.17) that eads to the genera soution gt) = c 1 for s 2 =, gt) = c 2 e ks2 t for s ) The genera soution of the second equation can be sought in the same way or set directy as fx) = c 3 x + c 4 for s 2 =, 1.19) fx) = c 5 sin sx + c 6 cos sx for s 2.

4 1. 1D Heat Propagation Probems 4 In concusion the genera soution of the origina equation, that shoud be vaid for arbitrary vaues of s can be written as with T x, t) = e ks2t A sin sx + B cos sx) + Cx + D, 1.2) A = c 2 c 5, B = c 2 c 6, C = c 1 c 3, D = c 1 c ) We now impose the satisfaction of boundary conditions by substituting then in the atter expression: T, t) = Be ks2t + D = T a, 1.22) T, t) = e ks2t A sin s + B cos s) + C + D =. In order to satisfy the first equation for every vaue of t since e ks2t is never equa to zero it is necessary that B =, D = T a. 1.23) For the second equation we have now Ae ks2t sin s + C + T a = 1.24) that can ony be satisfied for sin s, and C = T a. 1.25) The vaues of k for which sin s = are s n = nπ n = 1, 2, ) they are the eigenvaues of our probem. The temperature can then be expanded in an infinite series of form T x, t) = T a 1 x ) + A n e ks2 n t sin s n x, 1.27) when A n are unknown coefficients sti to be determined. We now impose the initia condition T x, ) = T i x) by substitution in the preceding equation we obtain T i x) T a a x ) = A n sin s n x. 1.28) In order to determine the coefficients A n corresponding to initia condition we can profit of the properties of the sinusoida function. In fact by mutipying both sides of the equation

5 1. 1D Heat Propagation Probems 5 by sin s m x and integrating it from to that is by finding the vaue of the scaar product between the terms at the first and the second member of the equation and the generic sinusoida function sin s m x) we obtain: sin s n x sin s m xdx = for m n 1.29) sin 2 s n x = 2 for m = n And, for the coefficients A n A n = 2 [T i x) T a 1 x )] sin s n xdx. 1.3) In concusion the expression of the temperature can be written as T x, t) = T a 1 x ) + 2 { [T i x) T a 1 x )] sin s n xdx} sin s n xe ks2nt. 1.31) From the form of the soution it is cear that the first term represents the stationary response that wi eventuay reached after a transient. The second term is on the contrary a timedependent one, with the tendency of decaying to zero as time increases. In the case that different boundary conditions are imposed, say on both sides x =, imposed heat fux by convection, as expressed beow T a b q or b q λ x Figure 1.2: The temperature profie within a wa with prescribed Dirichet and Neumann boundary conditions. q a = h a T T a ), 1.32) q b = h b T T b),

6 1. 1D Heat Propagation Probems 6 with q a ) = k T x, q b) = k T x, 1.33) and, consequenty k T x = h at T a ) onx =, 1.34) k T x = h bt T b) onx =. It is possibe to use the same expression of the soution obtained before: T x, t) = e ks2t A sin kx + B cos kx) + Cx + D. 1.35) For obtaining the soution in this case simiar steps can be foowed to the ones used for the previous set of boundary conditions. The soution in this case wi resut as foows h b h a x + λ) T x, t) = T a + T b T a ) λh a + h b ) + h a h b + λ 2 s 2 n + h 2 ) b ha sin s n x + λs n cos s n x) +2 λ 2 s 2 n + h 2 a) λ 2 s 2 n + h 2 ) nt b + k ha + h b ) λ 2 s 2 n + h a h b ) e ks2 [ {T i x) T a + T b T a ) h b h a x + λ) λh a + h b ) + h a h b with s n being any positive root of the transcendenta equation ] } h a sin s n x + λs n cos s n x) dx, 1.36) tan s = λs h a + h b ) λ 2 s 2 h a h b. 1.37) The stationary part of soution h b h a x + λ) T x, t ) = T a + T b T a ) λh a + h b ) + h a h b can be further examined. In fact the boundary condition in x = reads 1.38) q a = λ T n = h a T T a ) 1.39) if the conductivity of the materia is ow or the convention therma coefficient is high, the temperature on the wa reaches the temperature of the fuid T a. In fact, with ow λ and high h c we have k λ 1 and k T λ x = T T a) 1.4) with the first term of the equation equa to zero and T = T a. This resuts can aso obtained from the above expression of the stationary part of the soution in presence of convective conditions on both sides, that can be written as h b h a x + λ) T x) = T a + T b T a ) λh a + h b ) + h a h b = ) h b x + λ h a = T a + T b T a ) [λh a + h b ) + h a h b ] 1.41)

7 1. 1D Heat Propagation Probems 7 and, considering the same assumption k λ 1 we finay have T x) = T a + T b T a ) x. 1.42) That is the temperature of the media at both sides of the soid can be directy assumed as the wa temperature. It is aso interesting to note that,aso with entering fuxes at both sides q a and q b, the temperature of the body T in the stationary response cannot have temperatures higher than the highest of the media by which the body is surrounded in perfect agreement with the physics of heat transfer. It is aso obvious that, aso in the presence of an entering fux, the body cannot increase indefinitey its temperature that cannot increase higher than the one from which the fux is generated. T a q a qb n a n b T b x Figure 1.3: The temperature profie within a soid media which separates two semi-infinite fuid media. T a and T b correspond, respectivey, to the temperature of the externa media for x < and for x >. The 1D therma behavior just described can be appied to the case of an indefinite pate of thickness with the side at isotherma condition where either the temperature, aso variabe with the time t, or the heat fux can be appied. The same resuts are aso vaid for a monodimensiona soid ike a bar or a cabe) with uniform section, provided that the atera surface of the bar is isoated, that is no heat fux is present through it.

1D heat conduction problems

Chapter 1D heat conduction problems.1 1D heat conduction equation When we consider one-dimensional heat conduction problems of a homogeneous isotropic solid, the Fourier equation simplifies to the form: