Heat transfer takes place between these bodies and

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Unformatted text preview: ach geometry is placed in a large medium that is at a constant temperature T and kept in that medium for t 0. Heat transfer takes place between these bodies and their environments by convection with a uniform and constant heat transfer coefficient h. Note that all three cases possess geometric and thermal symmetry: the plane wall is symmetric about its center plane (x 0), the cylinder is symmetric about its centerline (r 0), and the sphere is symmetric about its center point (r 0). We neglect radiation heat transfer between these bodies and their surrounding surfaces, or incorporate the radiation effect into the convection heat transfer coefficient h. The variation of the temperature profile with time in the plane wall is illustrated in Fig. 4–12. When the wall is first exposed to the surrounding Ti at t 0, the entire wall is at its initial temperature Ti. But medium at T the wall temperature at and near the surfaces starts to drop as a result of heat transfer from the wall to the surrounding medium. This creates a temperature cen58933_ch04.qxd 9/10/2002 9:12 AM Page 217 217 CHAPTER 4 T h Initially T = Ti 0 T h T h Lx (a) A large plane wall Initially T = Ti T h T Initially T = Ti 0 r 0 h ro ro r (b) A long cylinder (c) A sphere gradient in the wall and initiates heat conduction from the inner parts of the wall toward its outer surfaces. Note that the temperature at the center of the wall remains at Ti until t t2, and that the temperature profile within the wall remains symmetric at all times about the center plane. The temperature profile gets flatter and flatter as time passes as a result of heat transfer, and eventually becomes uniform at T T . That is, the wall reaches thermal equilibrium with its surroundings. At that point, the heat transfer stops since there is no longer a temperature difference. Similar discussions can be given for the long cylinder or sphere. The formulation of the problems for the determination of the onedimensional transient temperature distribution T(x, t) in a wall results in a partial differential equation, which can be solved using advanced mathematical techniques. The solution, however, normally involves infinite series, which are inconvenient and time-consuming to evaluate. Therefore, there is clear motivation to present the solution in tabular or graphical form. However, the solution involves the parameters x, L, t, k, , h, Ti, and T , which are too many to make any graphical presentation of the results practical. In order to reduce the number of parameters, we nondimensionalize the problem by defining the following dimensionless quantities: Dimensionless temperature: (x, t) Dimensionless distance from the center: X Dimensionless heat transfer coefficient: Bi Dimensionless time: T(x, t) T Ti T x L hL k t L2 (Biot number) (Fourier number) The nondimensionalization enables us to present the temperature in terms of three parameters only: X, Bi, and . This makes it practical to present the solution in graphical form. The dimensionless quantities defined above for a plane wall can also be used for a cylinder or sphere by replacing the space variable x by r and the half-thickness L by the outer radius ro. Note that the characteristic length in the definition of the Biot number is taken to be the FIGURE 4–11 Schematic of the simple geometries in which heat transfer is one-dimensional. Ti t = t1 t=0 t = t2 t = t3 T 0 h Initially T = Ti t→ Lx T h FIGURE 4–12 Transient temperature profiles in a plane wall exposed to convection from its surfaces for Ti T . cen58933_ch04.qxd 9/10/2002 9:12 AM Page 218 218 HEAT TRANSFER half-thickness L for the plane wall, and the radius ro for the long cylinder and sphere instead of V/A used in lumped system analysis. The one-dimensional transient heat conduction problem just described can be solved exactly for any of the three geometries, but the solution involves infinite series, which are difficult to deal with. However, the terms in the solu0.2, keeping the first tions converge rapidly with increasing time, and for term and neglecting all the remaining terms in the series results in an error under 2 percent. We are usually interested in the solution for times with 0.2, and thus it is very convenient to express the solution using this oneterm approximation, given as Plane wall: Cylinder: Sphere: T(x, t) T Ti T T(r, t) T Ti T T(r, t) T Ti T (x, t)wall (r, t)cyl (r, t)sph A1e A1e A1e 2 1 2 1 2 1 cos ( 1x/L), J0( 1r/ro), 0.2 0.2 sin( 1r /ro) , 1r /ro 0.2 (4-10) (4-11) (4-12) where the constants A1 and 1 are functions of the Bi number only, and their values are listed in Table 4–1 against the Bi number for all three geometries. The function J0 is the zeroth-order Bessel function of the first kind, whose value can be determined from Table 4–2. Noting that cos (0) J0(0) 1 and the limit of (sin x)/x is also 1, these relations simplify to the next ones at the center of a plane wall, cylinder, or sphere: Center of plane wall (x Center of cylinder (r Center of sphere (r 0): 0): 0): 0, wall 0, cyl 0, sph To Ti To Ti To Ti T T T T T T A1e A1e A1e 2 1 (4-13) 2 1 (4-14) 2 1 (4-15) Once the Bi number is known, the above relations can be used...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.

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