Page 324 - Bird R.B. Transport phenomena

308 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

A reasonably good description of the system may be obtained by approximating the
true physical situation by a simplified model:

True situation Model

1. T is a function of x, y, and z, but the 1. Г is a function of z alone,
dependence on z is most important.
2. A small quantity of heat is lost from the 2. No heat is lost from the end or from the
fin at the end (area 2BW) and at the edges.
edges (area (2BL + 2BL).
3. The heat transfer coefficient is a function 3. The heat flux at the surface is given by
of position. q z — h(T - T ), where h is constant and
a
T depends on z.
The energy balance is made over a segment Az of the bar. Since the bar is stationary, the
terms containing v in the combined energy flux vector e may be discarded, and the only
contribution to the energy flux is q. Therefore the energy balance is

2BWq \ - 2BWq \ - h(2W^z)(T - T ) = 0 (10.7-1)
a
z z+Az
z z
Division by 2BW Az and taking the limit as Az approaches zero gives
dq 7 и
~ = £ (T - T a) (10.7-2)
az D
We now insert Fourier's law (q z = —kdT/dz), in which к is the thermal conductivity of
the metal. If we assume that к is constant, we then get
p_ h _ ^ (10.7-3)
T
=
{J
az k£>
This equation is to be solved with the boundary conditions
B.C.I: atz = 0, T=T w (10.7-4)
B.C. 2: at z = L, Щ- = 0 (10.7-5)
dz
We now introduce the following dimensionless quantities:
T - T
a
© = — =pr = dimensionless temperature (10.7-6)
С = j = dimensionless distance (10.7-7)

N 2 = -r-r- = dimensionless heat transfer coefficient 2 (10.7-8)
kB
The problem then takes the form

= l and ^ = 0 (10.7-9,10,11)

2 2 2
The quantity N may be rewritten as N = (hL/k)(L/B) = Bi(L/B), where Bi is called the Biot
number, named after Jean Baptiste Biot (1774-1862) (pronounced "Bee-oh"). Professor of physics at the
College de France, he received the Rumford Medal for his development of a simple, nondestructive test
to determine sugar concentration.