TITLE.PM5

HEAT TRANSFER 785

dharm

\M-therm\Th15-1.pm5

= ∂

∂

∂

∂

L

NM

O

QP

+ ∂

∂

∂

∂

L

N

M

O

Q

P +

∂

∂

∂

∂

L

NM

O

x QP

k t

x

dx dy dz d

y

k t

y

dx dy dz d

z

k t

z

x ...τττy ... z dx dy dz d...

= ∂

∂

∂

∂

F

HG

I

KJ

+ ∂

∂

∂

∂

F

HG

I

KJ

+ ∂

∂

∂

∂

F

HG

I

KJ

L

N

M

O

Q

x k P

t

xy

k t

yz

k t

z

x y z dx dy dz d...τ ...(15.9)

B. Total heat generated within the element (Qg′) :
The total heat generated in the element is given by :
Qg′=q dx dy dz dg(..)τ ...(15.10)

C. Energy stored in the element :
The total heat accumulated in the element due to heat flow along coordinate axes (eqn. 15.9)
and the haet generated within the element (eqn. 15.10) together serve to increase the thermal
energy of the element/lattice. This increase in thermal energy is given by :

ρτ(..). .dx dy dz c ∂t d

∂τ ...(15.11)

[Q Heat stored in the body = Mass of the body × specific heat of the body material

× rise in the temperature of body].

Now, substituting eqns. (15.9), (15.10), (15.11), in the eqn. (1), we have

∂

∂

∂

∂

F

HG

I

KJ

+ ∂

∂

∂

∂

F

HG

I

KJ

+ ∂

∂

∂

∂

F

HG

I

KJ

L

N

M

O

Q

P +=

∂

x ∂τ

k t

xy

k t

yz

k t

z

x y z dx dy dz d...ττρ τq dx dy dz dg(..) (..). .dx dy dz c t d

Dividing both sides by dx.dy.dz.dτ, we have

∂

∂

∂

∂

F

HG

I

KJ

+ ∂

∂

∂

∂

F

HG

I

KJ

+ ∂

∂

∂

∂

F

HG

I

KJ

+=∂

x ∂τ

k t

xy

k t

yz

k t

z

x y z qcg ρ..t ...(15.12)

or, using the vector operator ∇, we get

∇.(k∇t) + qg = ρ.c.∂

∂τ

t

This is known as the general heat conduction equation for ‘non-homogeneous ma-

terial’, self heat generating’ and ‘unsteady three-dimensional flow’. This equation estab-

lishes in differential form the relationship between the time and space variation of temperature at

any point of solid through which heat flow by conduction takes place.

General heat conduction equation for constant thermal conductivity :

In case of homogeneous (in which properties e.g., specific heat, density, thermal conductiv-

ity etc. are same everywhere in the material) and isotropic (in which properties are independent of

surface orientation) material, kx = ky = kz = k and diffusion equation eqn. (15.12) becomes

∂

∂

+∂

∂

+∂

∂

+= ∂

∂τ

= ∂

∂τ

2

2

2

2

2

2

t 1

x

t

y

t

z

q

k

c

k

g ρ tt

α

. ..
...(15.13)

where α =
k
ρ.c

=Thermal conductivity

Thermal capacity

The quantity, α =

k

ρ.c is known as thermal diffusivity.

— The larger the value of α, the faster will the heat diffuse through the material and its

temperature will change with time. This will result either due to a high value of thermal