021010 sec8_1-6

021010 sec8_1-6 - ChE 120B Lumped Parameter Models for Heat...

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ChE 120B Lumped Parameter Models for Heat Transfer and the Blot Number Imagine a slab that has one dimension, of thickness 2d, that is much smaller than the other two dimensions; we also assume that the slab is homogeneous with temperature and position-independent physical parameters. At time t = 0, the slab is placed into contact with a liquid of constant temperature, T c . Heat transfer from the slab to the fluid is governed by Newton's law of cooling:  s c Qh A TT  ( 1 ) in which Q is the heat flux. A is the surface area for heat transfer, T s is the surface temperature of the specimen, and 21 hW cm K  is the heat transfer coefficient. Within the specimen itself, heat transfer is by conduction T Qk A x (2) 11 kW cm K is the thermal conductivity of the specimen. At the boundary of the sample, the heat flux given by the two expressions must be equal: s sc T hT T k x ( 3 ) s T x is the temperature gradient at the specimen surface. Equation 3 is used as a one boundary condition in solving for the transient temperature distribution within the specimen,   , Tx t : ( 4 ) 2 2 ; s T Tk tx C p    3 gc m is the density and CJg K the heat capacity of the specimen. is known as the thermal diffusivity and has units of cm 2 -sec -1 . If the sample is cooled simultaneously from both sides, as is typically the case, the symmetry of the problem gives a second boundary condition: 0, 0 T t x ( 5 ) The specimen is initially assumed to be at a uniform temperature T 0 : 0 ,0 T ( 6 ) To simplify the problems and to help in identifying important parameters, we define the following dimensionless variables: 8-1
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ChE 120B 2 0 ; c c TT x t t dT Td   which reduces the problem and the initial and boundary conditions to the following: 2 2   ,0 1,  Initial Condition (7a) 0, 0, Boundary Condition @0 x (7b) 1, , hd k   Boundary Condition x (7c) The dimensionless group 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 problem can be solved by a separation of variables technique. The exact solution is an infinite series expansion and is available in texts such as that of Carslaw and Jaeger (1959). A better insight into the physics of the problem can be obtained by examining certain limiting cases of this solution. First, let:
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021010 sec8_1-6 - ChE 120B Lumped Parameter Models for Heat...

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