TITLE.PM5

784 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th15-1.pm5

If the directional characteristics of a material are equal/same, it is called an ‘‘Isotropic
material’’ and if unequal/different ‘Anisotropic material’.
qg = Heat generated per unit volume per unit time.
Inside the control volume there may be heat sources due to flow of electric current in electric
motors and generators, nuclear fission etc.
[Note. qg may be function of position or time, or both].
ρ = Mass density of material.
c = Specific heat of the material.
Energy balance/equation for volume element :
Net heat accumulated in the element due to conduction of heat from all the coordinate
directions considered (A) + heat generated within the element (B) = Energy stored in the element (C)
...(1)
Let, Q = Rate of heat flow in a direction, and
Q′ = (Q.dτ) = Total heat flow (flux) in that direction (in time dτ).
A. Net heat accumulated in the element due to conduction of heat from all the directions
considered :
Quantity of heat flowing into the element from the left face ABCD during the time interval
dτ in X-direction is given by :

Heat influx. Qx′ = – kx(dy.dz)∂

∂

t

x

.dτ ...(i)
During the same time interval dτ the heat flowing out of the right face of control volume
(EFGH) will be :

Heat efflux. QQ

x

′=′+xdx x ∂ Qdxx

∂

()+ ()′ ...(ii)

∴ Heat accumulation in the element due to heat flow in X-direction,

dQ Q Q

x

xx x′= ′− ′+∂ Qdxx

∂

L ′

NM

O

QP

() [Subtracting (ii) from (i)]

=−

∂

∂

′

x

()Qdxx

=−∂

∂

− ∂

∂

L

NM

O

x QP

kdydzt

x

x(. ) .ddxτ

= ∂

∂

∂

∂

L

NM

O

x QP

k t

x

x dx dy dz d...τ ...(15.6)

Similarly the heat accumulated due to heat flow by conduction along Y and Z directions in

time dτ will be :

dQy′ =

∂

∂

∂

∂

L

NM

O

x QP

k t

y

y dx dy dz d...τ ...(15.7)

dQ

z

k t

z

zz′=∂ dx dy dz d

∂

∂

∂

L

NM

O

QP

...τ ...(15.8)

∴ Net heat accumulated in the element due to conduction of heat from all the co-ordinate
directions considered