ert 216 heat & mass transfer sem 2/ 2011-2012portal.unimap.edu.my/portal/page/portal30/lecturer...
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ERT 216 HEAT & MASS TRANSFER
Sem 2/ 2011-2012
Prepared by;
Miss Mismisuraya Meor Ahmad
School of Bioprocess Engineering
University Malaysia Perlis
Conduction 1) Steady state conduction- One
dimension
2) Steady state conduction- Multiple dimension
3) Unsteady state conduction
Concept- The Different Heat transfer problems are often
classified as being steady (steady
state) or transient (unsteady
state).
The term steady implies no
change with time at any point
within the medium, while
transient implies variation with
time or time dependence.
The temp. or heat flux remains
unchanged with time during steady
heat transfer through a medium at
any location, although both quantities
may vary from one location to
another.
Concept- situation Example: Steady State
Heat transfer through the walls of a house is steady when the
conditions inside the house and the outdoor remain constant
for several hours. But even in this case, the temp. on the inner
and outer surface of the wall will be different unless the temp.
on the inner and outer the house are the same.
Example: unsteady state (transient process)
The cooling of an apple in a refrigerator is a transient heat
transfer process since the temp. at any fixed point within the
apple will change with time during cooling.
Concept- Lumped system During transient heat transfer, the temp. normally varies with time as well
as position. In the special case of variation with time but not with position,
the temp. of the medium change uniformly with time. Such heat transfer
system are called lumped systems.
Consider a small hot copper ball coming out of
an oven. Measurement indicates that the temp. of
the copper ball change with time, but it does not
change with position. Thus the temp. of the ball
remains nearly uniform at all times. Thus lump
system analysis is applicable in this case.
Consider a large roast beef in the oven. The
temp. distribution within the roast is not even close
to being uniform. (the outer part of the roast are
well done while centre part is barely warm). Thus
lump system analysis is not applicable in this
case.
Lumped System Analysis
Consider : a body of arbitrary shape of
mass (m), volume (v), surface area
(As), density (ρ) and specific heat (Cp)
initially at a uniform temp. (Ti).
At time t=0, the body is placed into a medium at
temp. T∞, and heat transfer takes place between
the body and its environment, with a heat transfer
coefficient, h.
For discussion, we assume that T∞>Ti (but the analysis is
equal valid for the opposite case). We assume lumped
system analysis to be applicable, so that the temp.
remains uniform within the body at all times and change
with time only, T= T(t).
Lumped System Analysis During differential time interval (dt), the temp. of the body rises by a differential
amount dT. An energy balance of the solid for the time interval (dt) can
be expressed as:
Noting that and since . So that:-
Lumped System Analysis
Integrating from t=0 at T=Ti, to any time t at T=T(t). So that:-
Taking the exponential of both sides and rearranging, obtain:-
Unit b Where:-
So that
Lumped System Analysis
There are 2 observation that can be made:
1) The equation can used to determine the temp. T(t) of a body at time t
or alternatively the time t required for the temp. to reach a specified
value T(t).
2) The temp. of a body approaches the ambient temp. T∞ exponentially.
The temp. of the body change rapidly at the beginning, but rather
slowly later on. A large value of b indicates that the body
approaches the environment temp. in a short time. The larger the
value of exponent b, the higher the rate of decay in temp.
Lumped System Analysis
When the temp. T(t) at time t from , the rate of convection heat
transfer between the body and its environment at that time can be determine from
Newton’s Law of cooling:-
Unit
The total amount of heat transfer between the body and the surrounding medium
over the time interval t=0 to t is simply the change in the energy content of the body:-
Unit
The amount of heat transfer
reaches it upper limit when the
body reaches the surrounding
temp. T∞. There for, the maximum
heat transfer between the body
and its surrounding is:-
Unit
Criteria for Lumped System Analysis The lumped system analysis provides great convenience in heat transfer
analysis.
The 1st step in establishing a criterion for the applicability of the lumped
system analysis is to define a characteristic length, Lc and a dimensionless
Biot number. Bi.
The characteristic length, Lc to be used in the Bi no. for simple
geometries in which heat transfer is 1D, such as a large plane
wall of thickness (2L), along cylinder of radius (ro) and a sphere
of radius (ro) become L (half thickness), ro/2 and ro/3
Where
Criteria for Lumped System Analysis
Biot Number, Bi
Can expressed as:-
The Bi. no is the ratio of the internal resistance of a body to heat
conduction to its external resistance to heat convection. Therefore, a small
Bi no. represent small resistance to heat conduction and thus small
temp. gradient within the body.
OR
Criteria for Lumped System Analysis Lumped system analysis assumes a uniform temp. distribution throughout
the body, which is the case only when the conduction resistance
is zero.
The smaller the Bi no. the more accurate the
lump system analysis. It is generally accepted that lumped
system analysis is applicable if (the 2nd step)
Small bodies with high thermal conductivity are good candidates
for lumped system analysis because the Bi no. small.
Example: Lumped System Analysis Consider Small bodies with high thermal conductivities & low convection
coefficients are most likely to satisfy the criterion for lumped system analysis
So that,
Bi <0.1, satisfy the criterion for lumped system, so the lumped system
analysis can be used.
The Transient Temperature Chart
When the lumped system analysis is not applicable, the variation of temp.
with position as well time can be determined using the transient
temp. chart (Heisler/Grober Charts) for a large plane wall, a
long cylinder, a sphere and a semi-infinite medium respectively.
Transient Heat Conduction in 1D
system
Consider: The variation of temp. with time & position in 1D problems such as those associated with a large plane
wall, a long cylinder and a sphere.
These chart are applicable for 1D heat transfer in those
geometries. Therefore, their use is limited to situation in which:-
1) the body is initially at a uniform temp.
2) All surfaces are subjected to the same thermal conditions, and
3) The body does not involve any heat generation.
These charts can also be used to determine the total
heat transfer from the body up to a specified time, t.
The Transient Temperature Chart
The Transient Temperature Chart
Schematic of the simple geometries in which heat transfer in 1D
Plane wall Cylinder
Sphere
The Transient Temperature Chart (PLATE)
Transient temp. & heat transfer charts for a plane wall of thickness 2L initially at a
uniform temp. Ti subjected to convection from both sides to an environment T∞ with a
convection coefficient of h. The height & the width of the wall are large relatively to its
thickness, and thus heat conduction in the wall can be approximated to be 1D. Also,
there is thermal symmetry about the midplane passing through x=0, & thus the temp.
distribution must be symmetrical about the midplane. Therefore, the value of temp. at
any –x value in –L ≤ x ≤ 0 at any time must be equal to the value at +x in 0 ≤ x ≤ L at
the same time.
This means we can formulate & solve the heat conduction problem in the
positive half domain 0 ≤ x ≤ L and the apply the solution to the other half.
Situation:-
Transient temp. & heat transfer charts for a long cylinder of
radius ro initially at a uniform temp. Ti subjected to
convection from all sides to an environment T∞ with a
convection coefficient of h.
The Transient Temperature Chart (CYLINDER)
Transient temp. & heat transfer charts for a sphere of radius
ro initially at a uniform temp. Ti subjected to convection from
all sides to an environment T∞ with a convection coefficient
of h.
The Transient Temperature Chart (SPHERE)
Solution of 1D transient heat conduction problems
Using the one-term approximation, the solution of 1D transient
heat conduction problems are expressed analytically as:
Where the constant A1 and λ1 are functions of the Bi no. only,
and their value are listed in table against the Bi no. for all three
geometries.
The error involved in one-term approximation is less than 2%
when Fo or >0.2
Solution of 1D transient heat conduction problems
Using the one-term solutions, the fractional heat transfers in different
geometries are expressed as:
Solution of 1D transient heat conduction in semi-infinite solid
The solution of transient heat conduction in a semi-infinite solid with constant
properties under various boundary condition at the surface are given as follow:-
1) Specified Surface Temp., Ts= constant
2) Specified Surface Heat Flux, qs= constant
3) Convection on the Surface, qs(t) = h[T∞- T(0,t)]
4) Energy Pulse at surface, es= constant
Where erfc(ɳ) is the complementary error function of argument ɳ.
Summary Summary of the solutions for 1D transient conduction in plane
wall of thickness 2L, a cylinder of radius, ro and a sphere of
radius ro subjected to convection from all surface
Coefficient used in the one-
term approximate solution of
transient 1D heat conduction
in plane walls, cylinders and
sphere (Bi = hL/k for a plane
wall of thickness, 2L and Bi
=hro/k for a cylinder or
sphere of radius ro)
Term- transient heat conduction Bi no. measure of the relative magnitudes of
the 2 heat transfer mechanism: convection at the
surface & conduction through the solid. A small
value of Bi no. indicates that the inner resistance
of the body to heat conduction is small relative to
the resistance to convection between the surface
& the fluid. As a result temp. distribution within
the solid become fairly uniform and lumped system
analysis become applicable (Bi<0.1)
Fourier no. (ד or Fo) measured of heat conducted through a body relative
to heat stored. Thus, a large value of the Fourier no. indicates faster
propagation of heat through a body.
Solution of 2D transient heat conduction problems
Using a superposition principle called the product solutions, the charts (the
transient temp. chart used to determine the temp. distribution & heat transfer
in 1D heat conduction problem) can also be used to construct solutions for the
2D transient heat conduction problems encountered in geometries (a
short cylinder, a long rectangular bar or a semi-ifinite cylinder or plate)
even 3D problem associated with geometries (rectangular prism or a
semi-infinite rectangular bar), provided that all surfaces of the solid are
subjected to convection to same fluid at temp. T∞ with the same convection
heat transfer coefficient, h & the body involved no heat generation.
Transient Heat Conduction in
Multidimensional Systems
The solution in such multidimensional geometries can be expressed as the
product of solution for the 1D geometries whose intersection is the
multidimensional geometry.
Solution of 2D transient heat conduction problems
The total heat transfer to or from a multidimensional geometry
can also be determined by using the 1D value. The transient
heat transfer for 2D geometry formed by the intersection of two
1D geometries 1 & 2 is:
Transient Heat Conduction in
Multidimensional Systems
Transient heat transfer for 3D body formed by the intersection
of three 1D bodies1, 2 & 3 is given by:
Most heat transfer problems encountered in practice are
transient (unsteady state) in nature, but they are usually
analyzed under some presumed steady conditions since steady
processes are easier to analyze and they provide the answers
to our questions.
Conclusion
Unsteady-state heat transfer analysis is obviously of
significantly practical interest because of the large no.
of heating and cooling process that must be
calculated in industrial applications.
To analyze transient heat-transfer problem solving the
general heat-conduction equation by separation-of-
variables method (similar to the analytical treatment used
for the 2D steady state problem).
Conclusion Example:
Heat transfer through the walls & ceiling of a typical house is never
steady since the outdoor conditions such as the temp. the speed and
direction of the wind, the location of the sun and so on, change
constantly. The conditions in a typical house are not so steady either.
Therefore, it is almost impossible to perform a heat transfer analysis of
the house accurately. That’s why we really need heat transfer analysis.
If the purpose of a heat transfer analysis of a house is to determine
the proper size of heater, so that, we need to know the max. rate of
heat loss from the house, which is determine by considering in heat
loss from the house during steady state operation under worst
conditions. Therefore, we can get the answer to our question by doing
a heat transfer analysis under steady conditions.