TITLE.PM5

HEAT TRANSFER 783

dharm

\M-therm\Th15-1.pm5

By comparing eqns. (15.4) and (15.5), we find that I is analogus to, Q, dV is analogous to dt

and R is analogous to the quantity
dx
kA

F

HG

I

KJ. The quantity

dx

kA

is called thermal conduction resist-

ance (Rth)cond. i.e.,

(Rth)cond. = dx
kA
The reciprocal of the thermal resistance is called thermal conductance.
It may be noted that rules for combining electrical resistances in
series and parallel apply equally well to thermal resistances.
The concept of thermal resistance is quite helpful white making calculations for flow of
heat.

15.2.4. General heat conduction equation in cartesian coordinates

Consider an infinitesimal rectangular parallelopiped (volume element) of sides dx, dy and
dz parallel, respectively, to the three axes (X, Y, Z) in a medium in which temperature is varying
with location and time as shown in Fig. 15.2.
Let, t = Temperature at the left face ABCD ; this temperature may be assumed
uniform over the entire surface, since the area of this face can be made
arbitrarily small.
dt
dx
= Temperature changes and rate of change along X-direction.

Then,

∂

∂

F

HG

I

KJ

t

x

dx = Change of temperature through distance dx, and

t +

∂

∂

F

HG

I

KJ

t

x

dx = Temperature on the right face EFGH (at distance dx from the left face

ABCD).

Further, let, kx, ky, kz = Thermal conductivities (direction characteristics of the material)

along X, Y and Z axes.

Qx

Z

Y

X

A (X, Y, Z)

y

x

z

O

A(x, y, z)

C

B

F

Q = q dx.dy.dzgg

Qy

dx

dy

Q(z + dz)

Elemental volume

(rectangular

parallelopiped)

dz

E

Q(x + dx)

H

Q(y + dy)

D

Qz

G

Fig. 15.2. Elemental volume for three-dimensional heat conduction analysis—Cartesian co-ordinates.

Fig. 15.1

t 1 Q t 2

R=th dxkA