786 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th15-1.pm5

conductivity k or a low value of heat capacity ρ.c. A low value of heat capacity means

the less amount of heat entering the element would be absorbed and used to raise its

temperature and more would be available for onward transmission. Metals and gases

have relatively high value of α and their response to temperature changes is quite

rapid. The non-metallic solids and liquids respond slowly to temperature changes because

of their relatively small value of thermal diffusivity.

— Thermal diffusivity is an important characteristic quantity for unsteady conduction

situations.

Eqn. (15.13) by using Laplacian ∇^2 , may be written as :

∇^2 t +

q

k

g= ∂t

∂τ

1

α

. ...[15.13 (a)]
Eqn. (15.13), governs the temperature distribution under unsteady heat flow through a
material which is homogeneous and isotropic.
Other simplified forms of heat conduction equation in cartesian co-ordinates :
(i) For the case when no internal source of heat generation is present. Eqn. (15.13) reduces

to

∂

∂

+ ∂

∂

+∂

∂

= ∂

∂τ

2

2

2

2

2

2

t 1

x

t

y

t

z

t

α

. [Unsteady state ∂
∂τ

F ≠

HG

I

KJ

t 0

heat flow with no internal heat generation]

or ∇^2 t =

1

α

.∂

∂τ

t

...(Fourier’s equation) ...(15.14)

(ii) Under the situations when temperature does not depend on time, the conduction then

takes place in steady state ie
..,∂t
∂τ

FHG = 0 IKJ and the eqn. (15.13) reduces to

∂

∂

+∂

∂

+∂

∂

+=

2

2

2

2

2

2 0

t

x

t

y

t

z

q

k

g

or ∇^2 t +

q

k

g= (^0) ...(Poisson’s equation) ...(15.15)
In the absence of internal heat generation, eqn. (15.15) reduces to
∂
∂
+∂
∂
+∂
∂

2
2
2
2
2
2 0
t
x
t
y
t
z
or ∇^2 t = 0 ...(Laplace equation) ...(15.16)
(iii)Steady state and one-dimensional heat transfer
∂
∂
+=
2
2 0
t
x
q
k
g ...(15.17)
(iv)Steady state, one-dimensional, without internal heat generation
∂
∂

2
2 0
t
x
...(15.18)
(v)Steady state, two dimensional, without internal heat generation
∂
∂

• 
  

  ∂
  ∂

  
  

  2
  2
  2
  2 0
  t
  x
  t
  y ...(15.19)
  (vi)Unsteady state, one dimensional, without internal heat generation
  ∂
  ∂
  = ∂
  ∂τ
  2
  2
  t 1
  x
  t
  α

  
  

. ...(15.20)