Heat conduction equation
Unlike temperature, heat transfer has
direction as well as magnitude, and thus it is
a vector quantity.
For example, saying that the temperature on
the inner surface of a wall is 18°C describes
the temperature at that location fully.
But saying that the heat flux on that surface
is 50 W/m2 immediately prompts the
question “in what direction?” We can
answer this question by saying that heat
conduction is toward the inside (indicating
heat gain) or toward the outside (indicating
heat loss).
Heat conduction equation
The various distances and angles involved when describing the
location of a point in different coordinate systems.
Heat conduction equation
Steady versus Transient Heat Transfer
The term steady implies no change with
time at any point within the medium, while
transient implies variation with time or
time dependence
For example, heat transfer through the walls
of a house will be steady when the
conditions inside the house and the
outdoors remain constant for several hours
The cooling of an apple in a refrigerator, on
the other hand, is a transient heat transfer
process
In the special case of variation with time but not with position, the
temperature of the medium changes uniformly with time. Such heat
transfer systems are called lumped systems
Heat conduction equation
Multidimensional Heat Transfer
The rate of heat conduction
through a medium in a
specified direction (say, in the
x-direction) is proportional to
the temperature difference
across the medium and the area
normal to the direction of heat
transfer, but is inversely
proportional to the distance in
that direction. This was
expressed in the differential
form by Fourier’s law of heat
conduction for one-
dimensional heat conduction.
Heat conduction equation
Multidimensional Heat Transfer
To obtain a general relation for Fourier’s
law of heat conduction, consider a
medium in which the temperature
distribution is three-dimensional
The heat flux vector at a point P on this
surface must be perpendicular to the
surface, and it must point in the
direction of decreasing temperature. If n
is the normal of the isothermal surface
at point P, the rate of heat conduction at
that point can be expressed by Fourier’s
law as:
Heat conduction equation
Multidimensional Heat Transfer
In rectangular coordinates, the heat
conduction vector can be expressed
in terms of its components as:
where 𝑖Ԧ, 𝑗Ԧ and 𝑘 are the unit vectors,
and 𝑄ሶ 𝑥 , 𝑄ሶ 𝑦 𝑎𝑛𝑑𝑄ሶ 𝑧 are the magnitudes
of the heat transfer rates in the x, y, and
z directions, which again can be
determined from Fourier’s law as:
Heat conduction equation
One-dimensional Heat Conduction Equation in a Large Plane Wall
Consider a thin element of thickness x in a
large plane wall. Assume the density of the
wall is ρ, the specific heat is C, and the area
of the wall normal to the direction of heat
transfer is A.
An energy balance on this thin element
during a small time interval t can be
expressed as:
Heat conduction equation
One-dimensional Heat Conduction Equation in a Large Plane Wall
The change in the energy content of the
element and the rate of heat generation
within the element can be expressed as:
Substituting we get:
Heat conduction equation
One-dimensional Heat Conduction Equation in a Large Plane Wall
Noting that the area A is constant for a plane
wall, the one-dimensional transient heat
conduction equation in a plane wall becomes:
Variable conductivity:
Heat conduction equation
One-dimensional Heat Conduction Equation in a Large Plane Wall
The thermal conductivity k of a material, in
general, depends on the temperature T (and
therefore x), and thus it cannot be taken out of
the derivative. However, the thermal
conductivity in most practical applications can
be assumed to remain constant at some average
value. The equation above in that case reduces
to:
Constant conductivity:
𝑘
where the property ∝= is the thermal diffusivity of
ρ𝐶
the material and represents how fast heat propagates
through a material
Heat conduction equation
One-dimensional Heat Conduction Equation in a Large Plane Wall
The above equation reduces to the following
forms under specified conditions:
(1) Steady-state:
(2) Transient, no heat
generation:
(3) Steady-state, no heat
generation:
Heat conduction equation
Heat Conduction Equation in a Long Cylinder
Variable conductivity:
Constant conductivity:
(1) Steady-state:
(2) Transient, no heat generation:
(3) Steady-state, no heat generation:
Heat conduction equation
Heat Conduction Equation in a Sphere
Variable conductivity:
Constant conductivity:
(1) Steady-state:
(2) Transient, no heat generation:
(3) Steady-state, no heat generation:
Heat conduction equation
Combined One-Dimensional Heat Conduction Equation
An examination of the one-dimensional transient heat conduction
equations for the plane wall, cylinder, and sphere reveals that all three
equations can be expressed in a compact form as:
where n = 0 for a plane wall, n = 1 for a cylinder, and n = 2 for a
sphere. In the case of a plane wall, it is customary to replace the
variable r by x. This equation can be simplified for steady-state or no
heat generation cases as described before
Heat conduction equation
General heat conduction equation-Multidimensional
Rectangular Coordinates
In the case of constant thermal conductivity:
(Fourier-Biot equation)
𝒈ሶ 𝟏 𝝏𝑻
𝛁𝟐 𝑻 + = 𝛁 = 𝐋𝐚𝐩𝐥𝐚𝐜𝐢𝐚𝐧 𝐨𝐩𝐞𝐫𝐚𝐭𝐨𝐫
𝒌 ∝ 𝝏𝒕
Heat conduction equation
General heat conduction equation-Multidimensional
Cylindrical Coordinates
Spherical Coordinates
Heat conduction equation
Boundary and initial conditions
We need to specify 2 boundary
conditions for 1-dimensional
problems, 4 boundary conditions for 2-
dimensional problems, and 6 boundary
conditions for 3-dimensional problems
Heat conduction equation
Boundary and initial conditions
Specified Temperature Boundary Condition
Specified Heat Flux Boundary Condition
The heat flux in the positive x-direction anywhere
in the medium, including the boundaries, can be
expressed as:
For a plate of thickness L subjected to heat flux of 50
W/m2 into the medium from both sides, the specified
heat flux boundary conditions can be expressed as:
Heat conduction equation
Boundary and initial conditions
Insulated Heat Flux Boundary Condition
Thermal Symmetry Heat Flux Boundary Condition
Two surfaces of a large hot plate of thickness L
suspended vertically in air will be subjected to the
same thermal conditions. Therefore, the center plane
can be viewed as an insulated surface, and the
thermal condition at this plane of symmetry can be
expressed as:
Heat conduction equation
Boundary and initial conditions
Convective Boundary Condition
The convection boundary condition
is based on a surface energy balance
expressed as:
Heat conduction equation
Boundary and initial conditions
Radiation Boundary Condition
The radiation boundary condition on a
surface can be expressed as:
HEAT GENERATION IN A SOLID
Heat generation is usually expressed per unit volume of the medium,
and is denoted by 𝑔,ሶ whose unit is W/m3. For example, heat generation
in an electrical wire of outer radius r0 and length L can be expressed as:
where I is the electric current and Re is the electrical
resistance of the wire
HEAT GENERATION IN A SOLID
Consider a solid medium of surface area As,
volume V, and constant thermal conductivity k,
where heat is generated at a constant rate of 𝑔ሶ
per unit volume.
Heat is transferred from the solid to the surrounding medium at T, with
a constant heat transfer coefficient of h. All the surfaces of the solid are
maintained at a common temperature Ts. Under steady conditions, the
energy balance for this solid can be expressed as:
HEAT GENERATION IN A SOLID
Disregarding radiation the heat transfer rate can also be expressed
from Newton’s law of cooling as:
Solving for the surface temperature Ts gives:
HEAT GENERATION IN A SOLID
Consider heat transfer from a long solid
cylinder with heat generation. The heat
generated within this inner cylinder must be
equal to the heat conducted through the outer
surface of this inner cylinder. That is, from
Fourier’s law of heat conduction:
