ChE 120B Lumped Parameter Models for Heat Transfer and the Blot Number


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1 ChE 0B Lumed Parameter Models for Heat Transfer and the Blot Number Imagine a slab that has one dimension, of thickness d, that is much smaller than the other two dimensions; we also assume that the slab is homogeneous with temerature and ositionindeendent hysical arameters. At time t = 0, the slab is laced into contact with a liquid of constant temerature, T c. Heat transfer from the slab to the fluid is governed by Newton's law of cooling: s c QhA T T () in which Q is the heat flux. A is the surface area for heat transfer, T s is the surface temerature of the secimen, hwcm K is the heat transfer coefficient. Within the secimen itself, heat transfer is by conduction and T Q ka x () k W cm K is the thermal conductivity of the secimen. At the boundary of the samle, the heat flux given by the two exressions must be equal: Ts ht s Tck () x T s x is the temerature gradient at the secimen surface. Equation is used as a one boundary condition in solving for the transient temerature distribution within the secimen,, T T s ; k t x C T x t : g cm C J g K the heat caacity of the secimen. is known as the thermal diffusivity and has units of cm sec . If the samle is cooled simultaneously from both sides, as is tyically the case, the symmetry of the roblem gives a second boundary condition: is the density and (4) T x 0, t 0 (5) The secimen is initially assumed to be at a uniform temerature T 0 :,0 T0 T x (6) To simlify the roblems and to hel in identifying imortant arameters, we define the following dimensionless variables: 8
2 ChE 0B x T Tc t ; ; t d T T d 0 c which reduces the roblem and the initial and boundary conditions to the following: (7),0, Initial Condition (7a) 0, 0,, hd,, k Boundary x 0 (7b) Boundary x 0 (7c) The dimensionless grou hd k is known as the Biot modulus, Bi. This modulus is a measure of the relative rates of convective to conductive heat transfer. With this choice of boundary conditions, the roblem can be solved by a searation of variables technique. The exact solution is an infinite series exansion and is available in texts such as that of Carslaw and Jaeger (959). A better insight into the hysics of the roblem can be obtained by examining certain limiting cases of this solution. First, let:, N (8) N is a function of and is a function only of. Inserting Equation 8 into Equation 7 and rearranging: N / / N (9) Because the left hand side of Equation 9 deends only on and the right hand side only on, for the equality to hold both sides must be equal to a constant,. Equation 7 searates into two ordinary differential equations: 0; C ex (0a) N N 0; cos N C C sin (0b), C ex C cos C sin (0c) Equation 0(c) is the general solution of Equation 7. For this articular roblem, the initial and boundary conditions must be met. Starting with Equation 7(b): 0, 0 C ex C; or C 0 (a) 8
3 ChE 0B in whichcc gives: C C, ex  cos. Imosing the second boundary condition by inserting Equation (b) into Equation 7(c) Cex sin Cex Bicos (a) sin Bi cos (b) Bi tan (c) There are an infinite number of solutions to Equation (b) which are known as eigenvalues, n. The solution can be written as an infinite series in which the Cn are chosen to match the initial condition, Equation 7a, and the n are the solutions of Equation (b). C, ex cos () n n n n For large n tan n 0 and n n. However, these terms do not contribute much to the solution because they are damed by the ex n factor in Equation. Often, the first term of Equation is the only significant term in the solution (see Equation 5). For our uroses, it is best to examine two limiting cases: (finite Bi ), Case I: Case II: Bi< Bi When Bi <, convection from the samle to the cryogen is slow comared to conduction. For Bi convection is fast comared to conduction and the surface temerature of the samle is always the same as the cryogen. Case I: Bi < I, Convection Limited For Bi, will also be much less than one. Exanding tan in a Taylor series for small gives Bi tan 0 (4a) The term 0 means that the error in this aroximation is of order. For small enough (for = 0.5, the error is less than 0%) only the linear term is imortant and Bi or (4b) Bi 8
4 ChE 0B The second eigenvalue,, can be seen from Figure to be aroximately equal to. It is useful to comare this second term in the solution to the first 0 : e e 5 50 x Bi e e Clearly, only the first term in the solution is significant for reasonable. Therefore, the solution to Equation 7 in this aroximation is, C ex Bi cosbi / (6) (5) The cosine term above can also be exanded in a Taylor series: / Bi cosbi (7) For Bi sufficiently small there are no satial gradients within the samle. Matching the initial condition dictates that C and the final aroximate solution for small Bi is: ex Bi (8a) or, in terms of the hysical arameters of the roblem ht Tx, ttc T0 TCex Cd (8b) Because the internal temerature of the secimen is osition indeendent and the only mechanism of heat transfer is convection, it is simle to generalize this aroach to cover arbitrary shaes. The average cooling rate is then roortional to: dt A ht 0 T C (9) dt V C It is necessary to recognize two consequences of Equation 9 Bi. First, the cooling rate (Equation 9) is roortional to A, or the inverse first ower of the characteristic dimension of the object (d the half V R R thickness of a slab, for a cylinder, and for a shere). Second, the cooling rate is indeendent of the thermal conductivity of the secimen; the only hysical roerty of the samle that is imortant is the "thermal density," C. The cooling rate is also linearly deendent on the temerature difference between the secimen and the liquid cryogen, and the heat transfer coefficient, h. 84
5 ChE 0B Case II: Bi Conduction Limited For Bi, Equation b becomes tan, n n, n 0,,... (0) Physically, as Bi, the secimen surface temerature must aroach the cryogen temerature, Tc Tc. This gives us a new boundary condition,, 0, to relace Equation 7c and Equation 0 now reads: cos n 0 () which gives the same eigenvalues as Equation 0. The first term in the solution for, is then;, C ex cos Because the conductionlimited solution (Equation a) varies with osition, the initial condition cannot be matched by a single term as in Equation 8: the full infinite series solution (Equation b) is necessary to match this condition exactly. The full solution in terms of the hysical arameters of the roblem is: n n kt n x Tx, ttc T0 Tcn 0 ex cos Cd d n (b) However, by the same reasoning as used in Equation 5, the first term of the infinite series solution dominates all others for 0: 4 kt x Tx, ttc T0 Tcex cos 4 Cd d (c) To comare Equation c with Equation 8. it is necessary to take its satial average. This involves integrating Equation (c) over x from zero to d, and dividing by d, which gives the following: 8 kt Tx, t T T T ex 4 Cd c 0 c (a) Hence, the average cooling rate for the slab is roortional to: dt k T0 TC dt d C and the average cooling rate for arbitrary shaes is roortional to: (a) 85
6 dt A k T 0 T C dt V C ChE 0B (b) In comarison to Case I, the cooling rate is roortional to the thermal conductivity of the slab, k, and the A inverse square of the characteristic dimension,, of the slab. The heat transfer coefficient, h, does not aear V in the cooling rate. Conduction Limited Convection Limited Characteristic Time C kt V A ht C V A Characteristic Scaling V A V A Satial deendent T Satial indeendent T Material deendent T Distribution Material indeendent T distribution 86
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