A steel wall of infinite thickness is initially at 100
degrees Fahrenheit. One of its surfaces
is suddenly changed to 1,000 degrees F.
Determine the temperature of the wall 4 inches below the surface after 4
hours. What is the amount of heat flux
at that time? How much heat has already
passed into the wall till that time?
Steel has specific heat of 0.12 BTU per pound per degree
Fahrenheit, thermal conductivity of 24 BTU per hour per foot per degree
Fahrenheit, and density of 488 pounds per cubic foot.
Solution:
A wall of infinite thickness and at a uniform original
temperature is subject to surroundings with constant temperature Ts. It is assumed that there is no contact
resistance between the medium and the surface it contacts, so that the face
temperature of the wall is also Ts.
This differs form ordinary quenching in which there is a very definite
contact resistance. The group k over (c
rho) is the thermal diffusivity consisting only of the properties of the
conducting material. Calling this group
alpha, the conduction equation is represented by partial t over partial theta
is equal to alpha partial square t over partial x square. The boundary conditions for an infinite wall
heated on face are that, when x is equal to x and theta is equal to zero, t is
equal to t0 and, when x is equal to zero and theta is equal to zero,
t is equal to Ts, where t0 is the initial uniform temperature of the
solid.
Solution to this equation is given as Y is equal to f1
of chi, where chi is given by x over (2 square root of alpha
theta). Further, Y is the
dimensionless temperature and is give by the ratio of Ts minus T and
Ts minus T0.
For this case, chi is calculated to be 0.13, and error
function, f1, is 0.146.
Plugging in the values of Ts and T0, we get T as
869 degrees Fahrenheit.
Amount of flux at that time is given by k times (Ts
minus T0) over square root of (pi alpha theta). Its value is found to be 9,518 BTU per hour
per square foot.
Total quantity of heat that has passes through the body till
that time is give by 2 times k times (Ts minus T0)
times square root of (theta over pi alpha).
Its value is found to be 76,140 BTU per square foot. The general
equation for conduction is represented by partial t over partial theta is equal
to alpha partial square t over partial x square.