Introduction to
Conduction
Chap. 2
Introduction to Conduction
Fouriers law is applicable to transient, multidimensional conduction
in complex geometries.
Fouriers law: What are its origins? What form does it take for
different geometries?, How does its proportionality constant (the
thermal conductivity) depend on the physical nature of the medium?
the heat conduction equation governs the temperature distribution in
a medium.
The solution to this equation provides knowledge of the temperature
distribution, which may then be used with Fouriers law to determine
the heat flux
The Conduction Rate Equation
Fouriers law is phenomenological: it is developed from observed
phenomena rather than being derived from first principles
where k, the thermal conductivity (W/m K), is an important property
of the material
the heat rate in x direction
the heat flux in x direction
the minus sign is necessary because heat is always transferred in the
direction of decreasing temperature.
the direction x of is normal to the cross-sectional area A.
General Conduction Rate Equation
The direction of heat flow will always be normal to a surface of
constant temperature, called an isothermal surface.
the heat flux is a vector quantity
general statement of the conduction rate equation (Fouriers law )
where is the three-dimensional del operator and T(x, y, z) is the scalar
temperature field
= + + =
Where is the three-dimensional del operator and T(x, y, z) is the
scalar temperature field
Alternative form of Fouriers law
An alternative form of Fouriers law is
The heat flux vector normal to an isotherm in a 2-D
coordinate system
qn is the heat flux in a direction n, which is normal to an isotherm
n is the unit normal vector in that direction.
in Cartesian coordinates, the general expression for q is
It is also implicit in Equation that the medium in which the conduction occurs is
isotropic. For such a medium, the value of the thermal conductivity is
independent of the coordinate direction.
The Thermal Properties of Matter
Thermal Conductivity k
The temperature dependence of the thermal
conductivity of substances.
Thermal conductivity k of some Thermal conductivity k of some Thermal conductivity k of some
solids. liquids. gases.
Thermophysical properties
Include two distinct categories, transport and thermodynamic properties
The transport properties include the diffusion rate coefficients
k (W/mK), the thermal conductivity (for heat transfer), and ,
, (m2/s) the kinematic viscosity (for momentum transfer)
Thermodynamic properties pertain to the equilibrium state of a system.
Density (kg/m3) or specific volume v (m3/kg)
cp , specific heat (kJ/kg C)
The product cp (J/m3 K), commonly termed the volumetric heat capacity, measures the ability of
a material to store thermal energy
thermal diffusivity , = k/cp (m2/s). It measures the ability of a material to conduct thermal
energy relative to its ability to store thermal energy.
The Heat Diffusion Equation
Differential control volume, dx dy dz, for conduction analysis in Cartesian coordinates.
Within the medium there may also be an energy source Eg term associated with
the rate of thermal energy generation qg
If the material is not experiencing a change in phase, (latent energy effects are
not pertinent), and it only changes the sensible (thermal) energy, then the
energy storage term is Est
Energy balance ( 1st Law of Thermodynamics)
On a rate basis, the general form of the conservation of energy requirement is
Ein + Eg - Eout = Est
Equation is the general form, in Cartesian coordinates, of the heat diffusion equation.
This equation, often referred to as the heat equation, provides the basic tool for heat
conduction analysis
+ + + = . + =
From its solution, we can obtain the temperature distribution as a function of time
T(x,y,z,t)
Therefore, this equation, states that at any point in the medium the net rate of energy
transfer by conduction into a unit volume plus the volumetric rate of thermal energy
generation must equal the rate of change of thermal energy stored within the volume.
Simplified versions of the heat diffusion equation
If the thermal conductivity k is constant, the heat equation is
2 2 2
+ + + = 2 + =
2 2 2 1 1
+ + + = 2 + =
where k/.cp is the thermal diffusivity
Heat diffusion equation under steady-state condition
under steady-state conditions, there can be no change in the amount of energy
storage
+ + + = 0 . + = 0
Moreover, if the heat transfer is one-dimensional (1D) (e.g., in the x-direction)
and there is no energy generation, the heat diffusion equation reduces to
=0
The important implication of this result is that, under steady-state, one-
dimensional conditions with no energy generation, the heat flux is a constant in
the direction of transfer
=
=0
Heat diffusion equation expressed in cylindrical
coordinates
+ +
= + +
= = =
1 1
. + = + 2 + + =
Heat equation expressed in spherical coordinates
+ +
1 1
= + +
sin
1 1
= =
=
sin
1 2
1 1
. + = + 2 + 2 sin + =
2 sin 2 sin
Boundary and Initial Conditions
Because the heat equation is second order in
the spatial coordinates, two boundary
conditions must be expressed for each
coordinate needed to describe the system.
Because the equation is first order in time,
however, only one condition, termed the
initial condition, must be specified.
Three kinds of boundary conditions commonly
encountered in heat transfer are summarized
in the table.
The conditions are specified at the surface x=
0 for a one-dimensional system.
Boundary Condition of the first kind:
Dirichlet condition
The first condition corresponds to a situation for which the surface is
maintained at a fixed temperature Ts. It is commonly termed a
Dirichlet condition, or a boundary condition of the first kind.
It is closely approximated, for example, when the surface is in contact
with a melting solid or a boiling liquid. In both cases, there is heat
transfer at the surface, while the surface remains at the temperature
of the phase change procesos.
Boundary Conditions of the second kind:
Neumann condition
The second condition corresponds to
the existence of a fixed or constant
heat flux at the surface.
This heat flux is related to the
temperature gradient at the surface
by Fouriers law,
It may be realized by bonding a thin
film electric heater to the surface.
A special case of this condition
corresponds to the perfectly
insulated, or adiabatic, surface for
which (T/x)x=0 =0
Boundary Conditions of the third kind:
Convection
The boundary condition of the third kind corresponds to the existence
of convection heating (or cooling) at the surface and is obtained from
the surface energy balance.
Summary
In the general formulation of Fourier.s law (applicable to any geometry),
what are the vector and scalar quantities?
Why is there a minus sign on the right-hand side of the equation?
What is an isothermal surface?
What can be said about the heat flux at any location on this surface?
What form does Fourier.s law take for each of the orthogonal directions of Cartesian, cylindrical, and spherical coordinate
systems?
In each case, what are the units of the temperature gradient?
Can you write each equation from memory?
An important property of matter is defined by Fourier.s law . What is it?
What is its physical significance? What are its units?
What is an isotropic material?
Why is the thermal conductivity of a solid generally larger than that of a liquid?
Why is the thermal conductivity of a liquid larger than that of a gas?
Why is the thermal conductivity of an electrically conducting solid generally larger than that of a nonconductor?
Why are materials such as beryllium oxide, diamond, and silicon carbide (see Table A.2) exceptions to this rule?
Summary
Is the effective thermal conductivity of an insulation system a true manifestation of the efficacy with which heat is transferred
through the system by conduction alone?
Why does the thermal conductivity of a gas increase with increasing temperature?
Why is it approximately independent of pressure?
What is the physical significance of the thermal diffusivity? . How is it defined and what are its units?
What is the physical significance of each term appearing in the heat equation?
Cite some examples of thermal energy generation.
If the rate at which thermal energy is generated per unit volume, , varies with location in a medium of volume V, how can the rate
of energy generation for the entire medium, , be determined from knowledge of
For a chemically reacting medium, what kind of reaction provides a source of thermal energy q>0
What kind of reaction provides a sink for thermal energy q<0
To solve the heat equation for the temperature distribution in a medium, boundary conditions must be prescribed at the surfaces
of the medium..
What physical conditions are commonly suitable for this purpose?
