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Heat Conduction Equation

Parag Chaware

Department of Mechanical Engineering

Cummins College of Engineering, Pune

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1 Heat Conduction Equation
Other forms of heat conduction equation

2 Thermal Diffusivity

3 Heat Conduction Equation in other Coordinate Systems

Cylindrical Coordinates
Spherical Coordinates

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Heat Conduction Equation
Consider an infinitesimal volume element in a Cartesian coordinate system.
The dimensions of the infinitesimal volume element are dx , dy , and dz in
the respective direction

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Heat Conduction Equation I

According to Fouriers law of heat conduction, the heat flowing into the
volume element from the left (in the x-direction) can be written as,

∂T
q̇x = −k dy dz (1)
∂x
The heat flow out from the right surface (in the x-direction) of the volume
element can be obtained by Taylor series expansion of the above equation.

∂q̇x
q̇x +∆x = q̇x + (2)
∂x
We can write,
∂q̇x
q̇x − q̇x +∆x = − (3)
∂x

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Heat Conduction Equation II
The left side of the above equation represent the net heat flow in the
x-direction.
Putting the value of q̇x in Equation (3)
 
∂ ∂T
q̇x − q̇x +∆x = − −k (4)
∂x ∂x
Similarly for y and z direction we can write
 
∂ ∂T
q̇y − q̇y +∆y = − −k (5)
∂y ∂y
 
∂ ∂T
q̇z − q̇z +∆z = − −k (6)
∂z ∂z
The heat generated by the fluid element is

q̇g dx dy dz (7)

Where q̇g is the heat generated per unit volume in the solid.
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Heat Conduction Equation III
Some heat is entering, some heat is leaving and some heat in generating
in the volume element
Some of the heat will be absorbed by the element.
Rate of change of heat energy within the volume element

∂T
(ρ dx dy dz )cp (8)
∂t
where, cp is the specific heat capacity at constant pressure (J/(kg K)), ρ
is the density (kg/m3 ) of the material, and t is the time (s). An energy
balance on this thin element during a small time interval ‘t’ can be
expressed as

( Rate of heat conduction input + Rate of heat generation)

= (Rate of heat conduction output +
Rate of change of energy content of the element (9)
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Heat Conduction Equation IV

On putting all the values in the above equation,

∂T
(q̇x + q̇y + q̇z ) + q̇g dx dy dz = (q̇x +∆x + q̇y +∆y + q̇z +∆z ) + (ρ dx dy dz )cp
∂t
(10)

((q̇x − q̇x +∆x ) + (q̇y − q̇y +∆y ) + (q̇z − q̇z +∆z )) + q̇g dx dy dz
∂T
= (ρ dx dy dz )cp (11)
∂t
from eqs. (4) to (6) we get,
        
∂ ∂T ∂ ∂T ∂ ∂T ∂T
k + k + k + q̇g = ρcp
∂x ∂x ∂y ∂y ∂z ∂z ∂t
(12)

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Heat Conduction Equation V

Considering the thermal conductivity of the solid is isotropic in nature, the

above relation reduces to,
        
∂ ∂T ∂ ∂T ∂ ∂T ∂T
k + + + q̇g = ρcp
∂x ∂x ∂y ∂y ∂z ∂z ∂t
(13)
or  2
∂2 T ∂2 T

∂ T q̇g ρcp ∂T
2
+ 2 )+ 2 + = (14)
∂x ∂y ∂z k k ∂t
Further,
∂2 T ∂2 T ∂2 T
 
q̇g 1 ∂T
+ + + = (15)
∂x 2 ∂y 2 ∂z 2 k α ∂t

Where,‘α’ is the thermal diffusivity of the material.

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Heat Conduction Equation VI

Heat Conduction equation in Cartesian Coordinates

∂2 T ∂2 T ∂2 T
 
q̇g 1 ∂T
+ + + = (16)
∂x 2 ∂y 2 ∂z 2 k α ∂t

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Other forms of heat conduction equation

Fourier Equation - No source unsteady

1 ∂T
∇2 = ∆T = (17)
α ∂t
Poisson’s Equation - Source Steady
q˙g
∇2 = ∆T + =0 (18)
k
Laplace Equation - No source Steady

∇2 = ∆T = 0 (19)

∆ is Laplacian operator.

∂2 ∂2 ∂2
∆ = ∇ · ∇ = ∆f (~x ) = f + f + · · · + f (20)
∂x12 ∂x22 ∂xn2
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Thermal Diffusivity

The unit of thermal diffusivity is m2 /s signifies the rate at which heat

diffuses in to the medium during change in temperature with time.
The higher value of the thermal diffusivity gives the idea of how fast
the heat is conducting into the medium.
The low value of the thermal diffusivity shown that the heat is mostly
absorbed by the material and comparatively less amount is transferred
for the conduction.

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Cylindrical Coordinates

   
1 d 1 d ∂2 T q̇g 1 ∂T



r dr r ∂T
∂r + r 2 dφ
∂T
∂φ + ∂z 2
+ k = α ∂t

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Spherical Coordinates

      
1 d 1 d 1 d q̇g 1 ∂T
r 2 dr
r 2 ∂T
∂r + r 2 sinθ dθ
sinθ ∂T
∂θ + r 2 sin2θ dφ
∂T
∂φ + k = α ∂t

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