021010 sec8_1-6

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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:
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/22/2011 for the course CHE 120B taught by Professor Zasadinski during the Winter '10 term at UCSB.

### Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online