Auxiliary problem. The boundary value problem that defines a Green's function, which includes a differential equation with an impulsive
heating term and homogeneous boundary conditions.
See ``What is a Green's Function?''
Boundary condition. A specified value (or relation) at a surface of a body. See Dirichlet, Neumann, and convection conditions.
Cartesian coordinates. See rectangular coordinates.
Causality relation. The requirement that the transient GF is zero for any time before
the heat pulse is released:
In control theory, a function
is said to be ``causal'' if the response only appears after the effect (heat pulse)
that causes it.
Conductivity. See thermal conductivity.
Convection at boundary, described by
Here h is the heat transfer coefficient and
outward normal at the body surface. The homogeneous convection boundary
Delta function. See Dirac delta function.
Dirac delta function. Symbol
is zero everywhere except at
in such a way that the integral over the volume is
unity. Also called the unit impulse function. See Properties of Dirac delta function.
Dirichlet boundary condition. Specified temperature at a boundary.
The homogeneous Dirichlet boundary condition is
energy generation. Symbol g, units [Watts/meter3 or Joule/meter
], energy deposited in a body per unit volume per unit time, for
example from electric heating, microwave absorption, nuclear reaction, etc.
erf See error function.
erfc See error function (complementary).
Error function. Symbol erf(z), is defined
Error function (complementary). Related to the error
Fin. A solid body exposed to a fluid for the purpose of exchanging heat with the fluid.
Green's function. A fundamental solution of a linear differential equation
satisfying homogeneous boundary conditions. (other names include
influence function, impulse response, source solution). See What is
Green's function solution equation. Formal solution to a
boundary value problem in the form of one or more integrals,
each of which contains a Green's
function and a non-homogeneous term (``driving term''). The
non-homogeneous terms may be boundary
conditions, initial conditions, or volume energy generation.
Heat equation. Also called the transient heat conduction equation. Describes the
movement of heat by diffusion (molecule-to-molecule transport) in a solid
(or motionless) medium. In vector notation,
Here T is temperature, k is conductivity, g is energy generation,
thermal diffusivity, r is spatial coordinate, and t is time.
Heat flux. Energy per unit time per unit area. Units Watts/meters2.
Heat transfer coefficient. Symbol h, units [W/m2/K].
Relates the surface temperature and surface heat flux with a surrounding fluid
according to Newton's law of cooling:
Helmholtz equation. Given by
When m2 is real this equation describes steady heat
with ``side'' heat losses (fin losses).
When m2 is imaginary the Helmholtz equation is the heat equation
in Fourier-transform space (also called the thermal-wave equation).
Finally, replace -m2 by real +m2 to give the
wave equation in Fourier-transform space.
Homogeneous equation. An equation in which every non-zero term contains
variable. For example, in a homogeneous heat equation every term contains T (there
is no energy generation term).
Homogeneous boundary condition. A boundary condition defined by an equation
in which the temperature appears
in every non-zero term. The following boundary
conditions are homogeneous:
type 1, T = 0;
Homogeneous body. A body composed of the same material all the way through.
Initial condition. Temperature distribution at t=0.
Insulated boundary. A boundary with no heat flow, defined by
on the boundary. See also Neumann boundary condition.
Laplace equation. Also called the steady heat conduction equation.
Multiplicative property. For transient heat conduction, many rectangular
and cylindrical 2-D and 3-D GF may be constructed by multiplying 1-D GF.
See Beck (1992, section 4.5) for restrictions.
Neumann boundary condition. The specified heat flux boundary condition,
is the outward normal on the body surface at ri.
The homogeneous Neumann condition is the insulated boundary,
Pseudo Green's function. A GF modified for use in a body with all boundaries of type
2 (specified heat flux); for these problems the usual GF does not exist.
The defining auxiliary equation for the pseudo GF has an additional term.