In this chapter we show how a number of
heat conduction problems are solved by an
analogous procedure:
◦ (i) an energy balance is made over a thin slab or shell perpendicular to the direction of the heat flow, and this balance leads to a first-order
differential equation from which the heat flux distribution is obtained
◦ (ii) then into this expression for the heat flux, we substitute Fourier's law of heat conduction, which gives a first-order differential equation for the
The integration constants are then
determined by use of boundary conditions for
the temperature or heat flux at the bounding
surfaces.
We select a slab (or shell), the surfaces of which are
normal to the direction of heat conduction, then for steady-state (i.e., time-independent) systems:
These three terms can be added to give the
"combined energy flux" e.
Note that in non-flow systems (for which v is zero)
the e vector simplifies to the q vector, which is given by Fourier's law.
The energy production term in Eq. 10.1-1 includes:
◦ (i) the degradation of electrical energy into heat
◦ (ii) the heat produced by slowing down of
neutrons and nuclear fragments liberated in the fission process
◦ (iii) the heat produced by viscous dissipation
After Eq. 10.1-1 has been written for a thin slab or
shell of material, the thickness of the slab or shell is allowed to approach zero.
This procedure leads ultimately to an expression
for the temperature distribution containing
constants of integration, which we evaluate by use of boundary conditions.
The commonest types of boundary conditions
are:
◦ The temperature may be specified at a surface
◦ The heat flux normal to a surface may be given (this is equivalent to specifying the normal component of the temperature gradient
◦ At interfaces the continuity of temperature and of the heat flux normal to the interface are required
◦ At a solid-fluid interface, the normal heat flux component may be related to the difference between the solid surface temperature, To and the "bulk" fluid temperature, Tb
Newton's law of cooling
The first system we consider is an electric wire of
circular cross section with radius R and electrical conductivity k, ohm-1 cm-1.
Through this wire there is an electric current with
current density, I, amp/cm2.
The transmission of an electric current is an
irreversible process, and some electrical energy is converted into heat (thermal energy).
The rate of heat production per unit volume is
given by the expression:
Assume that the temperature rise in the wire is not
so large that the temperature dependence of either the thermal or electrical conductivity need be
considered.
The surface of the wire is maintained at
temperature To.
How to find the radial temperature distribution
within the wire?
resulting from electrical
For the energy balance we take the system to
be a cylindrical shell of thickness ∆r and
length L.
Since v = 0 in this system, the only
We now substitute these quantities into the energy balance.
Division by 2 L∆r and taking the limit as ∆r goes
to zero gives:
Then, we get:
The integration constant C, must be zero because
of the boundary condition that:
Then, we get:
This states that the heat flux increases linearly with
We now substitute Fourier's law in the form
q
r= -k(dT/dr),
Finally we get:
There is, after all, a pronounced similarity between
the heated wire problem and the viscous flow in a circular tube. Only the notation is different:
There are many examples of heat conduction problems in the electrical industry. The minimizing of temperature rises inside electrical machinery prolongs insulation life.
A copper wire has a radius of 2 mm and a
length of 5 m. For what voltage drop would
the temperature rise at the wire axis be 10°C,
if the surface temperature of the wire is 20°C?
Given
Repeat the analysis in section 10.2, assuming
that
T
ois not known, but that instead the
heat flux
at
the wall is given by Newton's "law
of cooling" (Eq. 10.1-2). Assume that the heat
transfer coefficient h and the ambient air
temperature T
airare known.
We consider the flow of an incompressible
Newtonian fluid between two coaxial
cylinders as shown in Fig. 10.4-1. The
surfaces of the inner and outer cylinders are
maintained at
T = T
oand
T = T
b,
respectively.
We can expect that T will be a function of r
Modification of a portion of the flow system in Fig. 10.4-1, in which the curvature of the bounding surfaces is
As the outer cylinder rotates, each cylindrical
shell of fluid "rubs" against an adjacent shell
of fluid
This friction between adjacent layers of the
fluid produces heat; that is, the mechanical
energy is degraded into thermal energy
The volume heat source resulting from this
"viscous dissipation," which can be
designated by S
vappears automatically in the
shell balance when we use the combined
If the slit width b is small with respect to the
radius R of the outer cylinder, then the problem can be solved approximately by using the somewhat simplified system depicted in Fig. 10.4-2. That is, we ignore curvature effects and solve the problem in Cartesian coordinates. The velocity distribution is then vz = vb(x/b), where vb = ΩR.
We now make an energy balance over a shell of
thickness ∆x, width W, and length L. Since the fluid is in motion, we use the combined energy flux vector e
