The heat flux q and rate of heat flow [Q\dot] are related by
q =
d (Q/A)
dt
=
×
Q_{A}
=
×
Q
A
= k ÑT
Therefore
×
Q
= k A ÑT
In Cartesian coordinates for a wall of span X_{2}X_{1} = L
subject to Dirichlet conditions such that T2  T_{1} = DT,
such that the flow of heat is along the positive x direction,
×
Q
= k A
dT
dx
= k A
DT
L
= 
DT
R
where the thermal resistance R
is defined as
R =
L
A k
In Cylindrical coordinates for a shell of span R_{2}  R_{1} = L,
subject to Dirichlet conditions such that T2  T_{1} = DT,
such that the flow of heat is along the positive r direction
×
Q
= 
DT
R
where the thermal resistance R is defined as
R =
ln(R_{2}/R_{1})
2 pk L
In Spherical coordinates for a shell of span R2  R_{1} = L,
subject to Dirichlet conditions such that T2  T_{1} = DT,
such that the flow of heat is along the positive r direction
×
Q
= 
DT
R
where the thermal resistance R is
R =
R_{2}R_{1}
4 pk r_{a} r_{b}
Since at a solidfluid interface q = [Q\dot]/A = h (T_{s}  T_{¥})
the convective thermal resistance there is
R =
1
h A
Thermal resistances are analogous to electrical resistances
and for heat flowing through a series of resistances, the
overall thermal resistance is obtained by simple addition.
To illustrate, consider a plane wall subjected to
convective heat exchange at surfaces X_{1} and X_{2},
with heat exchange coefficients and fluid temperatures,
respectively given by h_{1}, T_{1,¥}, h_{2}, T_{2,¥},
the rate of heat flow is given by
×
Q
= 
DT
R_{o}
= 
T_{b,¥}  T_{a,¥}
R_{a} + R_{L} + R_{b}
where R_{a} = 1/h_{a} A, R_{L} = L/k A and R_{b} = 1/h_{b} A.
Sometimes, the concept of overall heat transfer coefficient U
is used. It is related to the overall thermal resistance by
U =
1
A R_{o}
An important related concept which will be useful
later on is the Biot number Bi.
It is a measure of the relative magnitude of internal
and external resistances to heat flow and is defined as
Bi =
h L
k
The above ideas are useful when studying steady state conduction
through composite walls or shells. A direct application is
the notion of critical radius of insulation.
Consider a long pipe carrying a hot fluid and coated with insulation
of thermal conductivity k.
Taking the derivative of the overall thermal resistance
with respect to the outer radius of insulation for constant
h at the outer radius of insulation yields
d R_{o}
dr
=
1
2 pr
(
1
k

1
hr
)
Note the overall thermal resistance has an extremum at
r_{c} =
k
h
Taking the second derivative one can show the extremum is minimum.
For problems involving nonzero internal heat generation,
the governing differential equations are simply obtained by
maintaining the term g in the equation.
Specifically, for Cartesian coordinates
Often, analytical solutions are not possible and one
must use numerical approximation techniques. The most
commonly used numerical approximation methods are:
finite differences, finite volumes and finite element methods.
They are all based on the conversion of the original
problem into a discrete, algebraic form.
Standard techniques of numerical algebra are then
used to obtain the desired approximation.
In the finite difference method
one starts with the differential formulation
and by a process of discretization transforms the problem into a
system of interlinked simultaneous algebraic equations that then must be
solved in order to determine an approximation to the desired solution.
Discretization consists of first introducing a mesh of
nodes by subdividing the solution domain into a finite number
of sub domains and then approximating the derivatives in the boundary value problem
by means of appropriate finite difference ratios which can be obtained
from a truncated Taylor series expansion Finite difference method)
or by integration over small volume elements (finite volume method).
A system of interlinked simultaneous algebraic equations is obtained that then must be
solved in order to determine an approximation to the desired solution.
Consider the following steady state heat conduction problem
inside a homogeneous, isotropic flat wall of span L = X_{2}X_{1},
constant thermal conductivity, with internal heat generation
k
d^{2} T
dx^{2}
= g
and subjected to Dirichlet conditions
T(x=X_{1}) = 0
T(x=X_{2}) = 0
Introduce a mesh in [X_{1},X_{2}] by dividing the interval into
N equal subintervals of size h.
This produces two boundary mesh points
x_{1}=X_{1} and x_{N+1}=X_{2}, and N1 interior mesh points
x_{i}, i=2,..,N1 for a total of N+1 mesh points.
Since the values T(x_{1}) = 0 and
T(x_{N+1}) = 0 are known, the values
T(x_{i}), i=2,..,N are to be determined.
From the Taylor expansions for T(x_{i+1}) and T(x_{i1}) the following
centered difference formula is obtained
d^{2} T
dx^{2}
_{xi} =
1
h^{2}
[T(x_{i+1})  2 T(x_{i}) + T(x_{i1})] +
h^{2}
12
d^{4} T
dx^{4}
(x_{i})
for some x_{i} Î (x_{i1},x_{i+1}).
If the higher order terms are discarded from the above formula and the
approximations are introduced in the original differential equation, the
result, after slight rearrangement and with T_{i} = T(x_{i}), is
 T_{i1} + 2 T_{i}  T_{i+1} = h^{2} (
g
k
)
This is a tridiagonal system of simultaneous linear algebraic equations
for the unknown approximate values of T_{xi} = T_{i}, for i=2,3,...,N.
In matrix notation the system becomes
Au = b
where A is a tridiagonal matrix, a sparse, positive definite, diagonally
dominant matrix with very nice properties and, for the specific boundary value
problem considered above, given (in augmented form) by
A =
æ ç ç ç ç ç
ç ç ç ç ç è
1
0
0
0
0
...
0
1
2
1
0
0
...
0
0
1
2
1
0
...
0
..
..
..
..
..
...
0
..
..
..
..
..
...
0
0
0
0
1
2
1
0
0
0
0
0
1
2
1
0
0
0
0
0
...
1
ö ÷ ÷ ÷ ÷ ÷
÷ ÷ ÷ ÷ ÷ ø
Moreover, u is the (column) vector
of unknowns given in this case by
u =
æ ç ç ç ç ç
ç ç ç ç è
T_{1}
T_{2}
T_{3}
...
...
T_{N}
T_{N+1}
ö ÷ ÷ ÷ ÷ ÷
÷ ÷ ÷ ÷ ø
and b is the (column) forcing vecto,
again, given in this case by
f = h^{2} (
g
k
)
æ ç ç ç ç ç
ç ç ç ç è
0
1
1
...
...
1
0
ö ÷ ÷ ÷ ÷ ÷
÷ ÷ ÷ ÷ ø
An identical procedure is used to obtain the corresponding algebraic set
of equations for problems in polar coordinates.
In the finite element method,
one starts with the variational formulation
and then, by a process of finite element function representation
transforms the problem into a
system of interlinked simultaneous algebraic equations.This system must be
solved in order to determine an approximation to the desired solution.
Consider again the problem of steady state heat conduction
through a flat wall of span L = X_{2}X_{1} = 1  0 = 1
and of uniform, isotropic material of
constant thermal conductivity, with internal energy generation
and subject to Dirichlet boundary conditions.
To keep things simple, assume the boundary conditions
are homogeneous, i.e.
T(x=0) = 0
T(x=1) = 0
The variational formulation is obtained by introducing a
function v(x) satisfying v(0)=0 and ò_{0}^{1} v^{2} dx < ¥,
but otherwise rather arbitrary and
then integrating the differential equation, i.e.
ó õ
1
0

d^{2} T
dx^{2}
v dx =
ó õ
1
0
g
k
v dx
Integrating the left hand side by parts readily yields the variational
or weak form
a(T,v) = (f,v)
where
a(T,v) =
ó õ
1
0
dT
dx
dv
dx
dx
and
(f,v) =
ó õ
1
0
f v dx
where f = g/k.
In the Galerkin finite element method,
one seeks an approximate solution
directly using the variational statement of the problem
a(T,v) = (f,v).
One starts by introducing a set of basis functions
{f_{i}(x)}_{i=1}^{N} satisfying the stated boundary conditions.
The number N is the number of finite elements in the system.
The approximate solution is then expressed
as a simple linear combination of the basis functions by
T_{N}(x) =
N å
i=1
T_{i} f_{i}(x)
Note that when using this representation, the quantities T_{i} are
simply the numerical coefficients of the basis functions in the representation
and are independent of the variable x. Most importantly, these
coefficients turn out to be the values of the computed approximation
at the nodal locations of the finite element mesh.
Furthermore, the test function v(x) in this case is selected as a simple
linear combination of the basis functions,
v(x) =
N å
j=1
f_{j}(x)
The above choices for T_{N}(x) and v(x) are introduced
into the variational statement in order to obtain the discrete
system.
The finite elements are introduced when the given domain
x Î [X_{1},X_{2}] is subdivided into a collection of N contiguous subdomains
[x_{1}=X_{1},x_{2}], [x_{2},x_{3}],...,[x_{i},x_{i+1}], ..., [x_{N},x_{N+1}=X_{2}].
The points x_{1}, x_{2},x_{3},...,x_{i},...,x_{N+1} are
called nodes and each contiguous pair of them, [x_{i},x_{i+1}],
represents the boundaries of the element that spans the space
h_{i} = x_{i+1}x_{i} between the nodes (nodal spacing).
In this case, each element has two nodes
one located at each end of the element. The location of all
these points is given with reference to a global system of coordinates.
If nodes are numbered based on their position is terms of global coordinates
one obtains global node numbers.
For simplicity, assume all finite elements are of equal size
(uniformly spaced mesh of nodes)
so that h_{1} = h_{2} = ... = h_{i} = h.
In the local description of the element bounded by nodes [x_{i},x_{i+1}], the
positions measured with respect to the global system of coordinates are transformed
into coordinates [x_{1},x_{2}] = [1,+1] referred to a system of coordinates
with its origin in the center of the element, i.e.
The affine or linear transformation
is simplest, i.e.
x(x) =
2 x  x_{i}  x_{i+1}
h_{i}
This is called the standard element.
The inverse transformation is
x(x) =
1x
2
x_{i} +
1+x
2
x_{i+1} = y_{1}(x) x_{i} + y_{2}(x) x_{i+1}
Note that the functions y_{1}(x) and y_{2}(x)
can be readily written as functions of the global coordinate x
by substituting the affine mapping for x, i.e.
y_{1}(x) =
1
2
( 1
2 x  x_{i}  x_{i+1}
h_{i}
)
y_{2}(x) =
1
2
( 1+
2 x  x_{i}  x_{i+1}
h_{i}
)
Global basis functions f_{i}(x)
(i.e. basis functions in the global coordinate system)
for each nodal location x_{i} where elements e_{i1} and
e_{i} meet, are readily obtained from the above
expressions for the standard element as follows
f_{i}(x) =
ì ï í
ï î
y_{2}(x)
ifx_{i1} £ x £ xi
y_{1}(x)
ifx_{i} £ x £ x_{i+1}
0
otherwise
The role of the local basis functions is to generate
the value of the approximated quantity inside the element from the
values at the nodes by interpolation.
For instance, for an arbitrary interior element
e_{i} where the index i denotes the global node number.
If the values of the required solution at x_{i} and
x_{i+1} are u(x_{i}) = u_{i} and u(x_{i+1}) = u_{i+1}, respectively,
the approximate values of T(x), T(x)^{ei} inside the element
are calculated by simple interpolation as follows
T^{ei}(x) = T_{i} f_{1}^{i} + T_{i+1} f_{2}^{i}
Here, the local finite element basis functions
for element e_{i},
f_{1}^{i} and f_{2}^{i} are defined as
f_{1}^{i} =
x_{i+1}  x
h_{i}
f_{2}^{i} =
x  x_{i}
h_{i}
where h_{i} = x_{i+1}  x_{i} is the nodal spacing
(and, in this case, also the element size)
and all the positions are measured in the global coordinate system.
Note that the indices 1 and 2 on the local basis functions
refer to the local node numbers for the element.
Associated with each node of the finite element mesh, there
is a global basis function f_{i}(x), i=1,2,...N
which is readily constructed once the local finite element
basis functions are specified.
The approximate solution on the entire domain T_{h}(x)
is represented as a linear combination of the global
finite element basis functions, i.e.
The specific relationships between local and global
finite element basis functions are obtained as follows.
For node 1,
f_{1} =
ì ï í
ï î
f_{1}^{1} =
x_{2}x
x_{2}x_{1}
ifx Î e_{1} = [x_{1},x_{2}]
0 elsewhere
for node i,
f_{i} =
ì ï ï ï í
ï ï ï î
f_{2}^{i1} =
xx_{i1}
x_{i}x_{i1}
ifx Î e_{i1}=[x_{i1},x_{i}]
f_{1}^{i} =
x_{i+1}x
x_{i+1}x_{i}
ifx Î e_{i}=[x_{i},x_{i+1}]
0 elsewhere
and for node N+1
f_{N+1} =
ì ï í
ï î
f_{2}^{N+1} =
xx_{N}
x_{N+1}x_{N}
ifx Î e_{N}=[x_{N},x_{N+1}]
0 elsewhere
Finally, as mentioned before,
in the Galerkin formulation of the finite element method,
the test functions v(x) are selected as
v(x) =
N+1 å
i=1
f_{i}(x)
Introducing all the above into the variational formulation of the problem
and rearranging (with, for simplicity X_{1}=0 and X_{2}=1,
yields the Galerkin finite element method equations as
T_{j1}
ó õ
x^{j+1}
x_{j1}
d f_{j1}
dx
d f_{j}
dx
dx + T_{j}
ó õ
x^{j+1}
x_{j1}
(
d f_{j}
dx
)^{2} dx + T_{j+1}
ó õ
x^{j+1}
x_{j1}
d f_{j+1}
dx
d f_{j}
dx
dx =
ó õ
x_{j+1}
x_{j1}
f f_{j} dx
for all j=1,3...N+1, or more succinctly as
N å
i=1
T_{i} a(f_{i}, f_{j}) = (f, f_{j})
for all i,j=1,2,...,N+1. Here
a(f_{i}, f_{j}) =
ó õ
1
0
d f_{i}
dx
d f_{j}
dx
dx
and
(f, f_{j}) =
ó õ
1
0
f f_{j} dx
for i,j=1,2,...,N+1.
Using matrix notation, the above is simply written as
Ku = F
Solving the algebraic problem yields the values of the
nodal values u_{i}, i=1,2,...,N+1. The matrix K
is called the finite element stiffness matrix,
the colum vector F is the force vector and
the column vector u is the vector of unknown
nodal values of u.
Extended surfaces are widely used to enhance heat exchange.
By attaching fins to a hot wall, heat is more easily dissipated
and the wall temperature is reduced. The enhanced heat dissipation
takes place through the extended surface of the fin.
Consider a thin fin attached to a hot flat wall at x=0. The fin
has cross sectional area A(x) and perimeter P(x).
Since the fin is thin, only temperature variations along the length
of the fin will be considered.
Consider a differential element in the fin of thickness
Dx located at some distance x from the base of the fin.
Under steady state conditions, of the heat entering the element
a fraction is dissipated by convection through the extended surface
and the rest is transferred further down along the fin.
The result of writing the down the differential thermal energy balance
and computing the limit as Dx ® 0 is
d
dx
(A(x)
dT
dx
) 
h Px
k
(T  T_{¥}) = 0
Once A(x), P(x), h, k, and T_{¥} are specified,
together with suitable boundary conditions, the differential
equation can be solved to obtain the temperature distribution
along the fin T(x).
The heat transfer enhancement effectiveness of a fin is
assessed by computing its fin efficiency. This is defined
as the ratio of the actual amount of heat convected through the
extended surface of the fin to the convected heat that would
obtain if the fin were at the base temperature throughout.
File translated from
T_{E}X
by
T_{T}H,
version 3.85. On 23 Sep 2009, 17:13.