ERT 216
HEAT & MASS TRANSFER
Sem 2/ 2014-2015
Dr Akmal Hadi Ma’ Radzi
School of Bioprocess Engineering
University Malaysia Perlis
Conduction
1) Steady state conduction- One dimension
2) Steady state conduction- Multiple dimension
3) Unsteady state conduction
Steady State
ConductionMULTIPLE
Dimension
Point of view
“Steady state conduction-1D”  Heat transfer was calculated in system
in which temp. gradient and area could be expressed in term of 1 space
coordinate
Now, our aim to analyze more general case of 2D heat flow. For steadystate with no heat generation, the Laplace Equation applies and
assuming constant thermal conductivity.
 Laplace Equation
The solution obtained by some analysis which are;
1)Analytical,
2)Graphical or
3)Numerical
General Analysis
The objective of heat transfer analysis  predict heat flow
or the temperature
From Laplace Equation 
The general solution give:
the temp. in a 2D body as a function of the
2 independent space coordinates x and y.
Then the heat flow in the x and y direction
calculated from the Fourier equation:
and
This heat flow quantities are directed either in the x direction or in the y direction.
The total heat flow at any point in the material is the resultant
of the qx and qy at that point. Thus the total heat flow vector is
perpendicular to the lines of constant temp. in the materials.
If the temp. distribution in the material is known, we may
easily establish the heat flow
1) Analytical Analysis
System:
For the rectangular plate as shown (3 sides of the plate are maintained at
the constant temp. T1 and upper side has some temp. distribution T=f(x)).
Temp. distribution could be simple as constant
temp. or more complex (sin-wave distribution)
After follow the steps and consider some method also used the boundary
condition, the final solution for this system using analytical analysis is:
1) Analytical Analysis
Conclusion:
Extensive study of “Analytical technique” used
in conduction heat transfer requires a
background in the theory of Orthogonal
Function (Fourier series, Bessel function and
etc.) which is involved a complex series. That’s
why the analytical solution are not always
possible to obtain  very cumbersome and
difficult to use.
2) Graphical Analysis
Consider: 2D system with inside surface is maintained at T1 and outer
surface temp. is maintained at T2.
The element (curvilinear-square)
To calculated the heat transfer sketched the isotherms and heat –flow lanes to
form groupings of curvilinear
The heat flow across this curvilinear section is given by Fourier ‘s Law
(assuming unit depth of material):
2) Graphical Analysis
Note: The heat
flow will be same through each section within heat flow lane
and the total heat flow will be the sum of the heat flows through all the lanes.
If the sketch 
So that,
to the
, the heat flow is proportional
across the element .
Because the heat flow is constant, the
across each element must be same within
the same heat flow lane. Thus,
across an
element is given by:
N: The no. of temp. increment
between inner & outer surfaces.
Total heat transfer 
M: The no. of heat-flow lanes.
“Conduction Shape Factor, S” 
2) Graphical Analysis
Conclusion
To calculate hate transfer:
Need only construct the curvilinear-square plots and count the
no. of temp. increments and heat-flow lanes. Make sure to
construct the plot so that
and the lines are perpendicular.
The accuracy of graphical analysis  depend on entirely on
the skill of the person sketching the curvilinear-squares. It may
not be expected to be used for the solution of many practical
problems.
2) Graphical Analysis
The Conduction Shape factor, S
In 2D system where only 2 temp. limits, S define as
The values of S based on
geometries (refer table).
For 3D wall (like furnace), separate shape factors are used to calculate the heat
flow through the edge and corner sections.
When all the interior dimensions > 1/5 of the wall
thickness;
Conduction shape factors, S based on geometries:
Example 1
Example 2
Example 3
3) Numerical Analysis
Consider, 2D body that
is to be divided into
equal increment in
both the x and y
direction

Where:
m: location indicating x increment
n: location indicating y increment
Aim to establish the temp. at any of these nodal points within the body using
Laplace equations 
Finite differences are used to approximate differential increments in the temp. &
space coordinates
Note:
The smaller we choose finite increments, the
more closely the true temp.
will be approximated.
distribution
3) Numerical Analysis
The temp. gradient may be written as follows:
and
So that;
x direction 
y direction 
3) Numerical Analysis
Thus the finite difference approximation for Laplace equation becomes;
General
Equation
If
, then 
For case: K constant , so the heat flow may all be expressed in term of temp.
differential. The net heat flow into any node is 0 at steady state conditions.
If consider heat generation happen in system, so add the term
general equation and obtain:
If
, then 
into
3) Numerical Analysis
To utilize the numerical analysis, equation 
must be written for each node within the material.
Example:
Equation for nodes 1, 2, 3 and 4:
The solution of these equations:
Once the temp. are determine, the heat flow can be calculated from:
The
is taken at the boundaries
3) Numerical Analysis
Physical situation:
(to get the temp. distribution and energy balance on each node)
1)Interior node
2)Convection boundary node
3)Convection at a corner section
3) Numerical Analysis
“Interior node”
So that, the heat flow may be calculated at either
the 500 °C face or the three 100 °C faces.
Note: The value of heat flow at 500 °C face and 100 °C face should be nearly
the same. As general practice, it is usually best to take the arithmetic average of
the 2 values for use in the calculations.
3) Numerical Analysis
When the solid is exposed to some convection boundary condition, the temp. at
the surface must be computed differently from the method given before.
“Convection boundary node”
Consider this boundary 
The energy balance on node (m, n) is;
If
, then 
3) Numerical Analysis
If the corner section exposed to a convection boundary condition.
Consider this boundary 
The energy balance for the corner section is;
If
, then 
“Convection at corner section”
3) Numerical Analysis
Summary of nodal formula for finite-difference calculation. (Dashed lines indicate
element volume)
3) Numerical Analysis
3) Numerical Analysis
3) Numerical Analysis
Numerical analysis is simply means of approximating a
continuous temp. distribution with finite nodal
elements. The more nodes taken, the closer the
approximation but more equations mean more
cumbersome solutions. Fortunately, computer and even
programmable calculators have the capability to obtain
these solutions very quickly.
In practical problem, there are several solution technique:
1) Matrix notation
2) Software packages
3) Gauss-Seidel iteration
3) Numerical Analysis
Numerical Formulation in term of resistance elements
Used resistance concept for writing the heat transfer between nodes.
Designating our node interest with subscript i and adjoining nodes with
subscript j (shown in figure- general conduction node solution).
At steady-state the net heat input to node I must be 0 
: the heat delivered to node i by heat generation, radiation etc
: the form of convection boundaries, internal conduction etc
Gauss-Seidel iteration
# iterative technique: more efficient solution to
the nodal equations than a matrix inversion
From heat flow in terms of resistances,
and temp. of the adjoining nodes , as:
For
may solve for the temp. i,
, and no heat generation 
Note: Convection boundary may be
converted to insulated surface by setting
Bi=0
Assignment 3
(submit: 25 Jan 2011 before 1200)
Book: J.P. Holman
1)3-3
2)3-8
3)3-11
4)3-53
5)3-59
6)3-74
Just give the temp. equation and energy
balance for each node.