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8/13/2019 Lecture 10_Unsteady Conduction II_CHBE 351
http://slidepdf.com/reader/full/lecture-10unsteady-conduction-iichbe-351 1/14
2-D Unsteady-State Conduction X-1
THE SEMI-INFINITE SOLID
In this section we will examine three important cases for a semi-infinite wall which is initially
at a uniform temperature, T i. The exposed surface is suddenly subject to either a new constant
temperature, or a constant heat flux or a convection heat transfer. In all three cases we would
like to determine how the temperature changes in the solid.
The equation for this case is
t
T
x
T
12
2
subject to:
Initial condition: iT xT )0,( (the temperature is constant at T i at t =0 at all x locations)
Boundary conditions: x iT T (same for all three cases)
0 x sT T or ""
0
os
x
qq x
T k
or ),0(0
t T T h x
T k
x
The solution can be found by a method called a similarity variable method. According to this,
the two independent variables, t and x, are combined to result a single and the PDE is
8/13/2019 Lecture 10_Unsteady Conduction II_CHBE 351
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2-D Unsteady-State Conduction X-2
transformed into an ODE. The similarity variable in this case is 2/1)4/( t x . Using this the
PDE becomes
T T 2
2
2
The solutions in these cases are
Case 1 Constant Surface Temperature: sT t T ),0(
t
xerf
T T
T t xT
si
s
2
),(
t
T T k t q is
s
)()("
Case 2 Constant Surface Heat Flux: ""os qq
t
xerfc
k
xq
t
x
k
t qT t xT oo
i
24exp
)/(2),(
"22/1"
"" )( os qt q
Case 3 Surface Convection: ),0(0
t T T h x
T k
x
k t h
t xerfc
k t h
k hx
t xerfc
T T T t xT
i
i
2exp
2),(
2
2
),0()("t T T ht qs
Where )( erf is the Gaussian error function defined as
o
duuerf )exp(2
)( 2
)( erfc is the complementary error function defined as
)(1)( erf erfc
Appendix B.2 in your textbook includes the values of the )( erf and )( erfc functions.
Note: For case 1 the temperature at the surface is constant and the heat flux decreases with
time, 2/1" )( t t qs . In case 2, the flux is constant while the temperature at the surface increases
8/13/2019 Lecture 10_Unsteady Conduction II_CHBE 351
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2-D Unsteady-State Conduction X-3
monotonically with 2/1t T s . Finally in case 3, the flux decreases with time since T s
approaches T . Selected profiles of temperature for this case (surface convection) are plotted
below.
An interesting application of case 1 is when 2 semi-infinite solids, initially at uniform
temperatures i AT , and i BT , are placed into contact. If the contact resistance is negligible, the
requirement of thermal equilibrium requires both T’s and fluxes at the interface to be the same.
Thus,
"
,
"
, Bs As qq or 2/1
,
2/1
, )()(
t
T T k
t
T T k
B
ibs B
A
i As A
Solving for T s.
2/12/1
,
2/1
,
2/1
B A
i B Bi A A
sck ck
T ck T ck T
Since the surface temperature is constant upon contact, the equations for case one can be used to
determine the profiles that are plotted in the figure below.
8/13/2019 Lecture 10_Unsteady Conduction II_CHBE 351
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2-D Unsteady-State Conduction X-4
EXAMPLE: In laying water mains, utilities must be concerned with the possibility of freezing
during cold periods. Although the problem of determining the temperature in soil as a function
of time is complicated by changing surface conditions, reasonable estimates can be based on the
assumption of constant surface temperature over a prolonged period of cold weather. Consider
the water main in the schematic below which is buried in soil initially at 20oC and is suddenly
subjected to a constant surface temperature of -15oC for 60 days.
(a.) Calculate and plot the temperature history at the burial depth of 0.68m for thermal
diffusivity of smand /0.3,38.1,0.110 27
(b.) For sm /1038.1 27 , plot the temperature distribution over the depth m x 0.10
for times of 1,5,10,30 and 60 days.
(c.) For sm /1038.1 27 , show that the heat flux from the soil decreases with increasing
time by plotting ),0(" t q x as a function of time for the 60-day period. On this graph, also
plot the heat flux at the depth of the buried main, ),68.0(" t mq x
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2-D Unsteady-State Conduction X-5
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2-D Unsteady-State Conduction X-6
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2-D Unsteady-State Conduction X-7
FINITE-DIFFERENCE METHODS
Analytical methods are restricted to simple geometries and boundary conditions. Thus in most
cases we have to use numerical methods to obtain solutions. One such method is the finite-
difference.
DISCRETIZATION OF THE HEAT EQUATION: THE EXPLICIT METHOD
Consider a 2-D system under transient conditions with constant properties and no internal
generation. The heat diffusion equation reduces to
t
T
y
T
x
T
12
2
2
2
To obtain the finite-difference form of this equation, we may use the same approximations as
before to the spatial derivatives. Again m, n subscripts are used to designate the x and y locations
of the discrete nodal points.
To discretize this equation in time, we use the integer p, so that
t pt
Thus the total time is subdivided into time intervals of t at which we would like to perform the
calculations. These time interval are t pt pt t t )1(,,....3,2,
Then the finite-difference approximation to the time derivative can be expressed as
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2-D Unsteady-State Conduction X-8
t
T T
t
T p
nm
p
nm
nm
,
1
,
,
Where (p+1) refers to the new time and p to the previous time interval.
This finite-difference scheme is called a forward-difference and the method is called explicit
method because all temperatures are evaluated at the previous time ( p).
The overall equation for an interior node can be written
2
,1,1,
2
,,1,1,
1
,
)(
2
)(
21
y
T T T
x
T T T
t
T T p
nm
p
nm
p
nm
p
nm
p
nm
p
nm
p
nm
p
nm
Assuming ( ) y x and 2
x
t Fo
, then
p
nm
p
nm
p
nm
p
nm
p
nm
p
nm T FoT T T T FoT ,1,1,,1,1
1
, )41(
Thus we can calculate ),( y xT at the new time t p )1( in a straightforward manner.
If the system is 1-D unsteady
t
T
x
T
12
2
The finite-difference equation becomes
p
m
p
m
p
m
p
nm T FoT T FoT )21(11
1
,
It is now evident why this method is called explicit. Since the initial conditions is known
),(0 y xT t
It becomes easy to calculate ),( y xT at later times. This method is not very accurate, but you can
improve the accuracy if you use small values for t and y x , . However, in this case the
computational time increases.
This explicit method frequently becomes unstable (the solution diverges). To prevent this
instability from happening, the x and t should satisfy a requirement, that is: The coefficient
of the temperature p
nm
T ,
of the equation for that specific node (m,n) should be positive, thus
1-D: 021 Fo or 2/1Fo or
2/12
x
t
2-D: 041 Fo or 4/1Fo or
4/12
x
t
8/13/2019 Lecture 10_Unsteady Conduction II_CHBE 351
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2-D Unsteady-State Conduction X-9
The above finite-difference equations can be derived by using the energy method (see the
example below for a boundary node with convection)
Assumptions: 1-D conduction, no heat generation, unsteady-state and constant physical
properties.
The energy equation for the control volume reduces to
st in E E
or t
T T xcAT T
x
kAT T Ah
p
o
p
o p
o
p p
o
1
12
or solving for 1 p
oT we get
p
o
p
o
p p
o
p
o T T T x
t T T
xc
t hT
12
1 22
However, BiFo x
t
k
xh
xc
t h22
22
p
o
p p
o T BiFoFo BiT T FoT 2212 1
1
The finite-difference Biot number is
k
xh Bi
The stability criterion defined before requires
0221 BiFoFo or 2/1)1( BiFo
8/13/2019 Lecture 10_Unsteady Conduction II_CHBE 351
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2-D Unsteady-State Conduction X-10
From each equation you may get a different criterion for stability. You compare all of them and
take the most restrictive. Table 5.3 of your textbook gives you the forward finite-difference
equation for various cases.
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2-D Unsteady-State Conduction X-11
EXAMPLE 5.6 : A fuel element of a nuclear reactor is in the shape of a plane wall of thickness
2 L = 20 mm and is convectively cooled at both surfaces, with h = 1100 W/m2 • K and T =
250°C. At normal operating power, heat is generated uniformly within the element at a
volumetric rate of1
q = 107W/m3. A departure from the steady-state conditions associated with
normal operation will occur if there is a change in the generation rate. Consider a sudden change
to 2q = 2 107 W/m3, and use the explicit finite-difference method to determine the fuel
element temperature distribution after 1.5 s. The fuel element thermal properties are k = 30 W/m
• K and a = 5 106m2/s.
Assumptions: 1-D conduction, no heat generation, unsteady-state and constant physical
properties.
Using a space increment of mm x 2 we can write the explicit finite difference equation
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2-D Unsteady-State Conduction X-12
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2-D Unsteady-State Conduction X-13
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2-D Unsteady-State Conduction X-14
EXAMPLE (Textbook 5.94): A one dimensional slab of thickness 2 L is initially at a uniform
temperature Ti. Suddenly, electric current is passed through the slab causing a uniform
volumetric heating q (W/m3). At the same time, both
outer surfaces ( L x ) are subjected to a convection
process at T with a heat transfer coefficient.
Write the finite difference equation expressing
conservation of energy for node 0 located on the outer surface at L x . Rearrange your
equation and identify any important dimensionless coefficients.