1) The document derives the one-dimensional heat conduction equation for plane walls, cylinders, and spheres through an energy balance on a thin volume element. 2) For all three geometries, the heat conduction equation can be written compactly as (1/r^n)∂(r^n ∂T/∂r) + ġ = ρC∂T/∂t, where n=0 for a plane wall, n=1 for a cylinder, and n=2 for a sphere. 3) The heat conduction equation is also derived for several special cases including steady-state and transient conditions with and without heat generation.

1. Heaven’s Light is our Guide
Rajshahi University of Engineering & Technology
Department of Mechanical Engineering

2. PAGE 1
One DimensionalHeat conductionEquation – Plane wall
Consider a thin element of thickness ∆xin a large plane wall as shown in
figure 1.1
Let,
The density of the wall = ρ
The specific heat = C, and
The area of the wall normalto the direction of heat transfer = A
Fig.1.1: One dimensional heat conduction through a volume element in a
large plane wall
An energy balance on this thin element during a small time interval ∆t can
be expressed as
(
𝑅𝑎𝑡𝑒 𝑜𝑓
ℎ𝑒𝑎𝑡
𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
𝑎𝑡 𝑥
) - (
𝑅𝑎𝑡𝑒 𝑜𝑓
ℎ𝑒𝑎𝑡
𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
𝑎𝑡 𝑥 + ∆𝑥
)+ (
𝑅𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡
𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒
𝑒𝑙𝑒𝑚𝑒𝑛𝑡
) = (
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑒
𝑜𝑓 𝑡ℎ𝑒 𝑒𝑛𝑒𝑟𝑔𝑦
𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒
𝑒𝑙𝑒𝑚𝑒𝑛𝑡
)
̇Q̇ x - Q̇ x+∆x + Ġ element =
∆𝐸 𝑒𝑙𝑒𝑚𝑒𝑛𝑡
∆𝑡
………………………………… (1.1)

3. PAGE 2
The change in the energy content and the rate of heat generation can be
expressed as
∆E element =Et+∆t – Et = mCA∆x(Tt+∆X –Tt)
Ġ element =ġV element =ġA∆x
Substituting into Equation 1.1, weget
Q̇ X - Q̇ x+∆x + ġA∆x = ρCA∆x
𝑇𝑡+∆𝑡−𝑇𝑡
∆𝑡
Dividing by A∆x , taking the limit as ∆x 0 and ∆t 0 and fromFourier’s
law:
1
𝐴
𝜕
𝜕𝑥
(𝐾𝐴
𝜕𝑇
𝜕𝑥
) + ġ = ρC
𝜕𝑇
𝜕𝑡
The area A is constant for a plane wall, the one dimensional transientheat
conduction equation in a plane wall is
Variable conductivity:
𝜕
𝜕𝑥
(𝑘
𝜕𝑇
𝜕𝑥
) + 𝑔̇ = 𝜌𝐶
𝜕𝑇
𝜕𝑡
Constantconductivity:
𝜕2 𝑇
𝜕𝑥2
+
𝑔̇̇
𝑘
=
1
𝛼
𝜕𝑇
𝜕𝑡
wherethe property α =
𝑘
𝜌𝐶
is the thermal diffusivity of the material and
represents how fastheat propagates through a material.
The one dimensionalheat conduction equation may be reduces to the
following forms under special conditions
(1)Steady state:
𝑑2 𝑇
𝑑𝑥2
+
𝑔̇̇
𝑘
= 0
(2) Transient, no heat generation:
𝜕2 𝑇
𝜕𝑥2
=
1
𝛼
𝜕𝑇
𝜕𝑡
(3)Steady state, no heat generation:
𝑑2 𝑇
𝑑𝑥2
= 0

4. PAGE 3
One Dimensional Heat Conduction Equation – Long Cylinder
Consider a thin cylindricalshell element of thickness ∆r in a long cylinder as
shown in Figure1.2
Assume,
The density of the cylinder = ρ
The specific heat = C, and
The length = L
The area of the cylinder normal to the direction of heat transfer, A =2𝜋rL
wherer is the value of the radius.
Fig.1.2: One dimensional heat conduction through a volume element in a
long cylinder
An energy balance on this thin cylindricalshell element during a small time
interval ∆t can be expressed as
(
𝑅𝑎𝑡𝑒 𝑜𝑓
ℎ𝑒𝑎𝑡
𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
𝑎𝑡 𝑥
) - (
𝑅𝑎𝑡𝑒 𝑜𝑓
ℎ𝑒𝑎𝑡
𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
𝑎𝑡 𝑥 + ∆𝑥
)+ (
𝑅𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡
𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒
𝑒𝑙𝑒𝑚𝑒𝑛𝑡
) = (
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑒
𝑜𝑓 𝑡ℎ𝑒 𝑒𝑛𝑒𝑟𝑔𝑦
𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒
𝑒𝑙𝑒𝑚𝑒𝑛𝑡
)
Q̇ r - Q̇ r+∆r + Ġ element =
∆𝐸 𝑒𝑙𝑒𝑚𝑒𝑛𝑡
∆𝑡
………………………………… (1.2)

5. PAGE 4
The change in the energy content and the rate of heat generation can be
expressed as
∆E element =Et+∆t – Et = mC(Tt+∆X –Tt) = ρCA∆r(Tt+∆t –Tt)
Ġ element =ġV element =ġA∆r
Substituting into Equation 1.1, weget
Q̇ r - Q̇ r+∆r + ġA∆x = ρCA∆r
𝑇𝑡+∆𝑡−𝑇𝑡
∆𝑡
Dividing by A∆r , taking the limit as ∆x 0 and ∆t 0 and from Fourier’s
law:
1
𝐴
𝜕
𝜕𝑟
(𝐾𝐴
𝜕𝑇
𝜕𝑟
) + ġ = ρC
𝜕𝑇
𝜕𝑡
The area varies with the independent variable r according to A = 2𝜋rL, the
one dimensional transient heat conduction equation in cylinder becomes
Variable conductivity:
1
𝑟
𝜕
𝜕𝑟
(𝑟𝑘
𝜕𝑇
𝜕𝑟
) + 𝑔̇ = 𝜌𝐶
𝜕𝑇
𝜕𝑡
Constantconductivity:
1
𝑟
𝜕
𝜕𝑟
(𝑟
𝜕𝑇
𝜕𝑟
) +
𝑔̇̇
𝑘
=
1
𝛼
𝜕𝑇
𝜕𝑡
wherethe property α =
𝑘
𝜌𝐶
is the thermal diffusivity of the material and
represents how fastheat propagates through a material.
The one dimensionalheat conduction equation may be reduces to the
following forms under special conditions
(1)Steady state:
1
𝑟
𝑑
𝑑𝑟
(𝑟
𝑑𝑇
𝑑𝑟
) +
𝑔̇̇
𝑘
= 0
(2) Transient, no heat generation:
1
𝑟
𝜕
𝜕𝑟
(𝑟
𝜕𝑇
𝜕𝑟
) =
1
𝛼
𝜕𝑇
𝜕𝑡
(3)Steady state, no heat generation:
𝑑
𝑑𝑟
(𝑟
𝑑𝑇
𝑑𝑟
) = 0

6. PAGE 5
One Dimensional Heat Conduction Equation – Sphere
Consider a spherewith density ρ, specific heat C, and outer radius R. The
area of the sphere normalto the direction of heat transfer, A =4𝜋r2
where
r is the value of the radius.
Fig.1.3: One dimensional heat conduction through a volume element in a
sphere
An energy balance on this thin sphericalelement during a small time
interval ∆t can be expressed as
(
𝑅𝑎𝑡𝑒 𝑜𝑓
ℎ𝑒𝑎𝑡
𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
𝑎𝑡 𝑥
) - (
𝑅𝑎𝑡𝑒 𝑜𝑓
ℎ𝑒𝑎𝑡
𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
𝑎𝑡 𝑥 + ∆𝑥
)+ (
𝑅𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡
𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒
𝑒𝑙𝑒𝑚𝑒𝑛𝑡
) = (
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑒
𝑜𝑓 𝑡ℎ𝑒 𝑒𝑛𝑒𝑟𝑔𝑦
𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒
𝑒𝑙𝑒𝑚𝑒𝑛𝑡
)
Q̇ r - Q̇ r+∆r + Ġ element =
∆𝐸 𝑒𝑙𝑒𝑚𝑒𝑛𝑡
∆𝑡
………………………………… (1.3)
The change in the energy content and the rate of heat generation can be
expressed as
∆E element =Et+∆t – Et = mC(Tt+∆X –Tt) = ρCA∆r(Tt+∆t –Tt)
Ġ element =ġV element =ġA∆r
Substituting into Equation 1.1, weget
Q̇ r - Q̇ r+∆r + ġA∆x = ρCA∆r
𝑇𝑡+∆𝑡−𝑇𝑡
∆𝑡

7. PAGE 6
Dividing by A∆r , taking the limit as ∆x 0 and ∆t 0 and fromFourier’s law:
1
𝐴
𝜕
𝜕𝑟
(𝐾𝐴
𝜕𝑇
𝜕𝑟
) + ġ = ρC
𝜕𝑇
𝜕𝑡
The area varies with the independent variable r according to A = 4𝜋r2
, the
one dimensional transient heat conduction equation in sphere becomes
Variable conductivity:
1
𝑟2
𝜕
𝜕𝑟
(𝑟2
𝑘
𝜕𝑇
𝜕𝑟
) + 𝑔̇ = 𝜌𝐶
𝜕𝑇
𝜕𝑡
Constantconductivity:
1
𝑟2
𝜕
𝜕𝑟
(𝑟2 𝜕𝑇
𝜕𝑟
) +
𝑔̇̇
𝑘
=
1
𝛼
𝜕𝑇
𝜕𝑡
wherethe property α =
𝑘
𝜌𝐶
is the thermal diffusivity of the material and
represents how fastheat propagates through a material.
The one dimensionalheat conduction equation may be reduces to the
following forms under special conditions
(1)Steady state:
1
𝑟2
𝑑
𝑑𝑟
(𝑟2 𝑑𝑇
𝑑𝑟
) +
𝑔̇̇
𝑘
= 0
(2) Transient, no heat generation:
1
𝑟2
𝜕
𝜕𝑟
(𝑟2 𝜕𝑇
𝜕𝑟
) =
1
𝛼
𝜕𝑇
𝜕𝑡
(3)Steady state, no heat generation:
𝑑
𝑑𝑟
(𝑟2 𝑑𝑇
𝑑𝑟
) = 0

8. PAGE 7
Combined One Dimensional Heat Conduction Equation
The one dimensionaltransient heat conduction equations for the plane
wall, cylinder and spherereveals that all three equations can be expressed
in a compact formas
1
𝑟 𝑛
𝜕
𝜕𝑟
(𝑟 𝑛 𝜕𝑇
𝜕𝑟
) + 𝑔̇ = 𝜌𝐶
𝜕𝑇
𝜕𝑡
Where n=o for a plane wall
n=1 for a cylinder
n=2 for a sphere
In the case of a plane wall, it is customary to replace the variable r by x.
The End