Here are the key claims, findings, and conclusions from the text:
Claims:
- Transient state heat transfer in a cylindrical metal rod with one end heated can be modeled mathematically. - The temperature profile along the rod depends on whether the rod is semi-infinite or finite.
- There is convection at the heated end of the rod as investigated experimentally in addition to conduction along the rod.
Findings: - For a semi-infinite rod, the temperature profile decays exponentially with distance from the heated end.
- For a finite rod with no flux at the far end, the temperature profile involves hyperbolic cosine functions.
- For a finite rod with convection at the far end, the temperature profile also depends on hyperbolic sine functions. - The models match experimental measurements of melting wax particles along aluminum rods. We deduce from experiment that thereβs convection at the hot end and that the heat transfer coefficient at the far end varies with length of the metal rod.
Conclusions:
- The integral transform approach allows deriving temperature profiles satisfying boundary conditions.
- There is convection at the hot end, and the heat transfer coefficient at the far end varies with length L as investigated experimentally.
- The simple models provide good agreement with experimental temperature profiles in aluminum rods. In summary, the text presents mathematical models of heat transfer in metal rods that agree with experiments and provide insight into the temperature profiles by considering different boundary conditions. The key findings relate to the functional forms of the temperature profiles in semi-infinite versus finite rods.
3. TABLE OF CONTENTS
PREFACE ............................................................................................................................................3
WHAT DO WE OBSERVE EXPERIMENTALLY WHEN HEATING A CYLINDRICAL
METAL ROD AT ONE END WITH WAX PARTICLES ALONG ITS SURFACE AREA? ..4
HOW DO WE DEAL WITH NATURAL CONVECTION AT THE SURFACE AREA OF A
SEMI-INFINITE METAL ROD FOR FIXED WALL TEMPERATURE...................................6
SO HOW DO WE PRODUCE THE SEMI-INFINITE OBSERVED ROD SOLUTION?.16
HOW DOES HEAT FLOW MANIFEST ITSELF FOR FINITE METAL RODS? ...............26
CASE 1: CONVECTION AT THE END OF A FINITE METAL ROD...............................26
HOW DO WE INVESTIGATE THE NATURE OF ππ³ EASILY?.....................................31
DERIVATION OF THE GENERAL EXPRESSION FOR HEAT TRANSFER
COEFFICIENT ππ³........................................................................................................................39
CASE 2: ZERO FLUX AT THE END OF THE METAL ROD?..........................................44
HOW DO WE DEAL WITH CYLINDRICAL COORDINATES? .............................................50
HOW DO WE DEAL WITH CYLINDRICAL CO-ORDINATES FOR A SEMI-INFINITE
RADIUS CYLINDER? .....................................................................................................................51
HOW DO WE DEAL WITH NATURAL CONVECTION AT THE SURFACE AREA OF A
SEMI-INFINITE CYLINDER FOR FIXED END TEMPERATURE.......................................54
REFERENCES..................................................................................................................................58
4. PREFACE
In this book we go ahead and investigate the nature of heat conduction in a
metal rod heated at one end while the other end is free. We do this by sticking
wax particles along the surface area of the metal rod at known distances x from
the hot end and then record the time taken for each wax to melt since the
introduction of the flame at the hot end. We first of all look at the case of heat
conduction in a semi-infinite metal rod and solve the heat equation analytically
using the integral transform approach and compare the solution got in the
transient state to experimental observations. We make deductions and
conclusions from both the solution and the experimental values.
We then look at the case of a finite length metal rod heat conduction with
convection at the free end. We use the hyperbolic function solutions known in
literature to interpret experimental data. One fact that we get to learn from the
experimental values is that the heat transfer coefficient βπΏ at the end of the
metal rod is not a constant but varies with length L as shall be shown later. We
note that in deriving the solution for the convection boundary condition, the
solution derived should reduce to the semi-infinite rod solution as the length
tends to infinity.
We then look at the case of zero flux at the end of a finite metal rod and also
derive the governing equation.
Finally, we use the integral approach to solve the heat equation in cylindrical
co-ordinates for radial heat conduction and use the same techniques we used
before to solve for observed phenomena.
5. WHAT DO WE OBSERVE EXPERIMENTALLY WHEN
HEATING A CYLINDRICAL METAL ROD AT ONE END
WITH WAX PARTICLES ALONG ITS SURFACE AREA?
The situation we are talking about looks as below:
First of all, let us call the distance π₯ to be the distance of the wax particle from the hot
end and π‘ to be the time taken for the wax to melt since the introduction of the flame
at the hot end.
For a semi-infinite rod(π = β), it is observed that a graph of π₯ against time π‘ is a curve
as shown below for an aluminium rod of radius 2mm:
A semi-infinite cylindrical rod means that the length of the metal rod extends to
infinity but the radius is finite.
The graph below is for an aluminium rod of length 75cm and radius 2mm and it can
be treated as a semi-infinite metal rod.
7. HOW DO WE DEAL WITH NATURAL CONVECTION AT
THE SURFACE AREA OF A SEMI-INFINITE METAL ROD
FOR FIXED WALL TEMPERATURE
A semi-infinite cylindrical rod means that the length of the metal rod extends to
infinity but the radius of the metal rod is finite. The governing heat equation is:
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
We shall use the integral transform approach to solve the heat equation above.
The boundary and initial conditions are
π» = π»π ππ π = π πππ πππ π
π» = π»β ππ π = β
π» = π»β ππ π = π
Where: π»β = ππππ πππππππππππ
First, we assume a temperature profile that satisfies the boundary conditions as:
π β πβ
ππ β πβ
= π
βπ₯
πΏ
where πΏ is to be determined and is a function of time t and not x.
for the initial condition, we assume πΏ = 0 at π‘ = 0 seconds so that the initial
condition is satisfied i.e.,
Since at π‘ = 0, πΏ = 0 we get
π β πβ
ππ β πβ
= π
βπ₯
0 = πββ
= 0
Hence
π = πβ
Which is the initial condition.
The governing partial differential equation is:
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
8. Let us change transform the heat equation into an integral equation as below:
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
β
βπ
π΄ππΆ
β« (π β πβ)ππ₯
π
0
=
π
ππ‘
β« (π)ππ₯
π
0
β¦ β¦ . . π)
π2
π
ππ₯2
=
(ππ β πβ)
πΏ2
π
βπ₯
πΏ
β« (
π2
π
ππ₯2
) ππ₯
π
0
=
β(ππ β πβ)
πΏ
(π
βπ
πΏ β 1)
But for a semi-infinite cylindrical rod, π = β, upon substitution, we get
β« (
π2
π
ππ₯2
) ππ₯
π
0
=
(ππ β πβ)
πΏ
β« (π β πβ)ππ₯
π
0
= βπΏ(ππ β πβ)(π
βπ
πΏ β 1)
But π = β, upon substitution, we get
β« (π β πβ)ππ₯
π
0
= πΏ(ππ β πβ)
π = (ππ β πβ)π
βπ₯
πΏ + πβ
β« (π)ππ₯
π
0
= βπΏ(ππ β πβ)(π
βπ
πΏ β 1) + πβπ
Substitute π = β and get
π
ππ‘
β« (π)ππ₯
π
0
=
ππΏ
ππ‘
(ππ β πβ) +
π
ππ‘
(πβπ)
Since πβ πππ π are constants
π
ππ‘
(πβπ) = 0
π
ππ‘
β« (π)ππ₯
π
0
=
ππΏ
ππ‘
(ππ β πβ)
Substituting the above expressions in equation b) above, we get
9. πΌ β
βπ
π΄ππΆ
πΏ2
= πΏ
ππΏ
ππ‘
We solve the equation above with initial condition
πΏ = 0 ππ‘ π‘ = 0
And get
πΏ = β
πΌπ΄ππΆ
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
Substituting for πΏ in the temperature profile, we get
π» β π»β
π»π β π»β
= π
βπ
β
π²π¨
ππ·
(πβπ
βπππ·
π¨ππͺ
π
)
From the equation above, we notice that the initial condition is satisfied i.e.,
π» = π»β ππ π = π
The equation above predicts the transient state and in steady state (π‘ = β) it
reduces to
π» β π»β
π»π β π»β
= π
ββ(
ππ·
π²π¨
)π
What are the predictions of the transient state?
For transient state the governing solution is:
π β πβ
ππ β πβ
= π
βπ₯
β
πΎπ΄
βπ
(1βπ
β2βπ
π΄ππΆ
π‘
)
10. Let us make π₯ the subject of the equation of transient state and get:
π₯ = [ln (
ππ β πβ
π β πβ
)] Γ β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
To measure the value of h, we use trial and error method in Microsoft excel and
choosing (
2βπ
π΄ππΆ
= 0.005) , plotting a graph of π₯ against β(1 β πβ0.005π‘) for a semi-
infinite aluminium metal rod of radius 2mm gave a straight-line graph with a
negative intercept as shown below for all times contrary to the equation
above i.e.,
π₯ = βπ + [ln(
ππ β πβ
π β πβ
)] Γ β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
Let:
π = β(1 β πβ0.005π‘) = β(1 β π
β2βπ
π΄ππΆ
π‘
)
π = βπ + [π₯π§ (
π»π β π»β
π» β π»β
)(β
π²π¨
ππ·
)] Γ π
11. Varying the radius of the aluminium metal rod to π = 1ππ the graph looked as
below:
From the graph above, it is observed that the intercept c is directly proportional
to radius squared.
i.e.
π = πππππππ
The heat transfer coefficient is calculated from
y = 1.515x - 0.0528
RΒ² = 0.9967
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2
X
Y
A Graph of X against Y for radius 2mm semi-
infinite aluminium rod
Series1
Linear (Series1)
y = 1.3556x - 0.0139
RΒ² = 0.987
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.05 0.1 0.15
X
Y
A Graph of X against Y for radius 1mm semi-
infinite aluminium rod
Series1
Linear (Series1)
12. π = β(1 β πβ0.005π‘) = β(1 β π
β2βπ
π΄ππΆ
π‘
)
2βπ
π΄ππΆ
= 0.005
h for aluminium was found to be
β =
6.075π
π2πΎ
Using the graph and the equation below:
π₯ = [ln (
ππ β πβ
π β πβ
)] Γ β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
) β π½π2
From the gradient of the graph of x against β(1 β π
β2βπ
π΄ππΆ
π‘
)above the ratio (
ππβπβ
πβπβ
)
was measured and was found to be:
(
π» β π»β
π»π β π»β
) β π. πππππ
From the graph above it is deduced that ππ is a constant temperature
independent of time.
To account for the intercept in the graph above for a semi-infinite rod, we have
to postulate that thereβs convection at the hot end of the metal rod as below
i.e.,
16. We notice that π½ is directly proportional to the thermal conductivity k.
The above expression shows that the temperature at π₯ = 0 varies with time also
and is not fixed until steady state is achieved
π
ππ
= π·ππ
And we finally get:
ππ β πβ
ππ β πβ
=
πΏ
πΏ + π½π2
For a semi-infinite rod,
π» β π»β
π»π β π»β
= πβ
π
πΉ
Substituting the expression of (ππ β πβ) we get:
π» β π»β
π»π β π»β
= (
πΉ
πΉ + π·ππ
)πβ
π
πΉ
As the general solution.
17. SO HOW DO WE PRODUCE THE SEMI-INFINITE OBSERVED ROD
SOLUTION?
As got before:
ππ β πβ
ππ β πβ
= (
πΏ
πΏ + π½π2
)
From
(
π» β π»β
π»π β π»β
) = πβ
π
πΉ
Substituting the expression of (ππ β πβ) we get:
π» β π»β
π»π β π»β
= (
πΉ
πΉ + π·ππ
)πβ
π
πΉ
Continuing with
π» β π»β
π»π β π»β
= πβ
π
πΉ
Let us make π₯ the subject of the formula:
π₯ = [ln(
ππ β πβ
π β πβ
)] Γ πΏ
Where:
πΏ = β2πΌπ‘
For small time.
Or generally
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
But we know from the above that:
(ππ β πβ) = (ππ β πβ)(
πΏ
πΏ + π½π2
)
π₯ = [ln(ππ β πβ) β ln(π β πβ)] Γ πΏ
ππ(ππ β πβ) = ππ(ππ β πβ) + ππ(
πΏ
πΏ + π½π2
)
18. Upon substitution of ππ(ππ β πβ) in the equation of π₯ above, we get
π₯ = [ππ(ππ β πβ) + ππ(
πΏ
πΏ + π½π2
) βln(π β πβ)] Γ πΏ
We get
π = πΉ[ππ (
π»π β π»β
π» β π»β
) + ππ(
πΉ
πΉ + π·ππ
)]
Let us manipulate the equation above and get:
π = πΉππ (
π»π β π»β
π» β π»β
) + πΉππ(
πΉ
πΉ + π·ππ
)
Factorizing out πΏ in the denominator we get:
π₯ = πΏππ (
ππ β πβ
π β πβ
) + πΏππ[
πΏ
πΏ
(
1
1 +
π½π2
πΏ
)]
π₯ = πΏππ (
ππ β πβ
π β πβ
) + πΏππ[(1 +
π½π2
πΏ
)β1
]
Since
π½π2
πΏ
βͺ 1 πππ π‘
We can use the binomial first order approximation
(1 + π₯)π
β 1 + ππ₯ πππ π₯ βͺ 1
(1 +
π½π2
πΏ
)β1
= 1 β
π½π2
πΏ
πππ
π½π2
πΏ
βͺ 1
And we get:
π₯ = πΏππ (
ππ β πβ
π β πβ
) + πΏππ[(1 β
π½π2
πΏ
)]
Again, we can expand the natural log as below:
ln(1 β π₯) β βπ₯ πππ π₯ βͺ 1
ππ [(1 β
π½π2
πΏ
)] = β
π½π2
πΏ
πππ
π½π2
πΏ
βͺ 1
19. Upon substitution we finally get
π = πΉππ (
π»π β π»β
π» β π»β
) β π·ππ
Which is what we got before.
π = πΉππ (
π»π β π»β
π» β π»β
) β π·ππ
Where:
π½ =
ππ
ππ
ππ‘
πΆπ
We notice that the intercept above is proportional to the square of the radius as
demonstrated from experiment.
Looking at the general solution:
π = πΉππ (
π»π β π»β
π» β π»β
) + πΉππ(
πΉ
πΉ + π·ππ
)
Plotting a graph of π against πΉ[ππ (
π»πβπ»β
π»βπ»β
) + ππ(
πΉ
πΉ+π·ππ
)] was found to give a
straight-line graph through the origin as stated by the equation above.
Let us call π = πΉ[ππ (
π»πβπ»β
π»βπ»β
) + ππ(
πΉ
πΉ+π·ππ
)]
Where:
π·ππ
= π. ππππ for an aluminium rod of radius 2mm and πΎ = 238
π
ππΎ
, β = 6
π
π2πΎ
π =
2700
ππ
π3, πΆ = 900
π½
πππΎ
and
πβπβ
ππβπβ
= 0.21981
And
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
Then a graph of x against p is a straight-line graph through the origin for a
semi-infinite rod as shown below for a semi-infinite aluminium rod of radius
2mm
20. Calling the solution below the approximated solution:
π = πΉππ (
π»π β π»β
π» β π»β
) β π·ππ
Or
π₯ = [ln (
ππ β πβ
π β πβ
)] Γ β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
) β π½π2
What are the lessons we have learnt?
ο· We have learnt that in the approximated solution, we can measure off h
in the transient state.
ο· We have learnt that knowing the thermo-conductivity and other physical
parameters of the metal rod, in the approximated solution, we can
measure off the ratio
π»βπ»β
π»πβπ»β
ο· The intercept in the approximated solution can help us learn how its
nature varies with the radius of the rod. We can use the intercept to
measure the value of π·.
y = 1.0768x
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2 0.25
x
p
A Graph of x against p
21. The general solution is given by:
π = πΉππ (
π»π β π»β
π» β π»β
) + πΉππ(
πΉ
πΉ + π·ππ
)
You notice that the initial condition is still satisfied.
From the general solution, we get:
π» β π»β
π»π β π»β
= (
πΉ
πΉ + π·ππ
)π
β
π
πΉ
For the initial condition; At π‘ = 0, you get
π» β π»β
π»π β π»β
= π Γ π
βπ
π = π
Hence
π» = π»β
Considering the approximated equation below:
π = [π₯π§ (
π»π β π»β
π» β π»β
)] Γ βππΆπ β π·ππ
What that equation says is that when you stick wax particles on a long metal
rod (π = β) at distances x from the hot end of the rod and note the time t it
takes the wax particles to melt, then a graph of π₯ against βπ‘ is a straight-line
graph with an intercept as stated by the equation above when the times are
small. The equation is true because that is what is observed experimentally.
Looking at the approximate solution for a semi-infinite metal rod:
π₯ = βπ½π2
+ [ln(
ππ β πβ
π β πβ
)] Γ β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
Let:
π = β(1 β πβ0.005π‘)
A graph of x against Y looked as below:
22. Since the graph above is a straight-line graph, it shows that ππ IS NOT a
function of time as stated by the equation above of π₯ against Y.
Another point to note is that from experiment ππ was found to be independent
of radius of the metal rod.
For aluminium
πΎ = 238
π
ππΎ
, β = 6
π
π2πΎ
Another way to measure ππ 1 is to consider the steady state equation and plot
the graph of
π β πβ
ππ β πβ
= (
πΏ
πΏ + π½π2
)π
ββ(
βπ
πΎπ΄)π₯
π₯π§(π» β π»β) = ππ(π»π β π»β) + ππ(
πΉ
πΉ + π·ππ
) β β(
ππ·
π²π¨
)π
And
πΏ = β
πΎπ΄
βπ
Upon substitution
y = 1.515x - 0.0528
RΒ² = 0.9967
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2
X
Y
A Graph of X against Y
Series1
Linear (Series1)
23. π₯π§(π» β π»β) = ππ(π»π β π»β) + ππ(
βπ²π¨
ππ·
βπ²π¨
ππ·
+ π·ππ
) β β(
ππ·
π²π¨
)π
A graph of ln(π β πβ) against x gives an intercept [ππ(π»π β π»β) + ππ(
β
π²π¨
ππ·
β
π²π¨
ππ·
+π·ππ
)] from
which ππ can be measured.
Also knowing the thermo-conductivity, from the gradient of the above graph the
heat transfer coefficient can be measured off.
From experiment, using an aluminium rod of radius 2mm and using a
thermoconductivity value of πππ πΎ
ππ²
β , The heat transfer coefficient h of
aluminium was found to be π πΎ
πππ²
β .
Therefore, for a semi-infinite rod, the equation obeyed for small times is:
π = [π₯π§ (
π»π β π»β
π» β π»β
)] Γ βππΆπ β π·ππ
Looking at the steady state solution below:
π β πβ
ππ β πβ
= (
πΏ
πΏ + π½π2
)π
ββ(
βπ
πΎπ΄
)π₯
OR
π» β π»β
π»π β π»β
= (
βπ²π¨
ππ·
βπ²π¨
ππ·
+ π·ππ
)π
ββ(
ππ·
π²π¨
)π
In most cases
β
π²π¨
ππ·
β« π·ππ
So, we observe:
24. π» β π»β
π»π β π»β
= π
ββ(
ππ·
π²π¨
)π
Which is the usual solution we know.
From experiment, using a flame and candle wax on the aluminium rod, the
ratio below was found to be
(
π» β π»β
π»π β π»β
) β π. πππππ
So, when can we apply the semi-infinite rod solution?
Using the steady state equation of heat conduction
π β πβ
ππ β πβ
= π
ββ(
βπ
πΎπ΄
)πΏ
= πβππΏ
Where:
π = β(
βπ
πΎπ΄
)
And
From literature [1] the limiting length for use of semi-infinite model is got when
π β πβ
ππ β πβ
= 0.01
The corresponding value of ππΏ = 4.6
Hence as long as π³ =
π.π
π
= π. πβ(
π²π¨
ππ·
) the equation can be applied accurately.
25. NB
A particular method we can use to predict which temperature profile to use in
solving the heat equation is by looking at the steady state equation below:
From
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
In steady state
ππ
ππ‘
= 0
So, the governing equation becomes:
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) = 0
The general solution of the equation above is
(π β πβ) = πΆ1πβππ₯
+ πΆ2πππ₯
Where:
π = β
βπ
πΎπ΄
For the semi-infinite case: The boundary conditions are:
π» = π»π ππ π = π
π» = π»β ππ π = β
The second boundary condition makes πΆ2 = 0
And the other boundary condition:
π» = π»π ππ π = π
Leads to
π β πβ
ππ β πβ
= πβππ₯
From what we learned earlier is that
π =
1
πΏ
26. From now onwards we are going to use the fact that the temperature profile
below satisfies the heat equation
(π β πβ) = πΆ1πβππ₯
+ πΆ2πππ₯
Or
(π» β π»β) = πͺππ
βπ
πΉ + πͺππ
π
πΉ
27. HOW DOES HEAT FLOW MANIFEST ITSELF FOR FINITE
METAL RODS?
CASE 1: CONVECTION AT THE END OF A FINITE METAL ROD
The boundary and initial conditions are:
π» = π»π ππ π = π
βπ
π π»
π π
= ππ³(π» β π»β) ππ π = π
π» = π»β ππ π = π
This same analysis can be extended to a metal rod with convection at the other
end of the metal rod where the temperature profile is given by: [2]
π β πβ
ππ β πβ
=
cosh[π(πΏ β π₯)] + (
βπΏ
ππ
) sinh[π(πΏ β π₯)]
cosh ππΏ + (
βπΏ
ππ
) π ππβππΏ
Or
π β πβ
ππ β πβ
=
cosh [
(πΏ β π₯)
πΏ
] + (
βπΏπΏ
π
) sinh[(
πΏ β π₯
πΏ
)]
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
) π ππβ
πΏ
πΏ
To show that the initial condition is satisfied we see from the above that ππ‘ π‘ =
0, πΏ = 0.
π β πβ
ππ β πβ
=
cosh [
(πΏ β π₯)
πΏ ] + (
βπΏπΏ
π ) sinh[(
πΏ β π₯
πΏ )]
cosh
πΏ
πΏ
+ (
βπΏ
π
)π ππβ
πΏ
πΏ
Becomes:
π β πβ
ππ β πβ
=
cosh [
(πΏ β π₯)
πΏ
]
cosh
πΏ
πΏ
=
π
(πΏβπ₯)
πΏ + π
β(πΏβπ₯)
πΏ
π
πΏ
πΏ + π
βπΏ
πΏ
π
β(πΏβπ₯)
πΏ = πβ
(πΏβπ₯)
0 = πββ(πΏβπ₯)
= 0
Similarly
28. π
βπΏ
πΏ = π
βπΏ
0 = πββπΏ
= 0
So, we are left with
π β πβ
ππ β πβ
=
π
(πΏβπ₯)
πΏ
π
πΏ
πΏ
= π
βπ₯
πΏ = π
βπ₯
0 = πββπ₯
= 0
Hence at π‘ = 0, π = πβ and hence the initial condition.
To explain the transient state provide we have to get the expression for
(ππ β πβ) from:
As we learned earlier in the semi-infinite case, we use
βπ
ππ»
ππ
|π=π = ππ(π»π β π»π)
Recall the compact temperature profile is:
π β πβ
ππ β πβ
=
cosh[π(πΏ β π₯)] + (
βπΏ
ππ
) sinh[π(πΏ β π₯)]
cosh ππΏ + (
βπΏ
ππ
) π ππβππΏ
Where:
π =
1
πΏ
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
As shall be shown later
ππ
ππ₯
|π₯=0 = β(ππ β πβ)(
ππ‘ππβππΏ +
βπΏ
π
1 +
βπΏ
ππ
π‘ππβππΏ
)
βπ
ππ
ππ₯
|π₯=0 = π(ππ β πβ) (
ππ‘ππβππΏ +
βπΏ
π
1 +
βπΏ
ππ
π‘ππβππΏ
)
32. HOW DO WE INVESTIGATE THE NATURE OF ππ³ EASILY?
For convection boundary condition, the temperature profile obeyed is:
π β πβ
ππ β πβ
=
(
πΏ
(π½π2 (
π‘ππβ
πΏ
πΏ
+
βπΏπΏ
π
1 +
βπΏπΏ
π
π‘ππβ
πΏ
πΏ
) + πΏ)
)
(
cosh [
(πΏ β π₯)
πΏ
] + (
βπΏπΏ
π
) sinh [(
πΏ β π₯
πΏ
)]
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
)π ππβ
πΏ
πΏ
)
To investigate ππ³ easily, we use this simple experiment:
Where:
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
As shall be shown later when we solve the heat equation analytically using the
integral transform to get πΏ.
For a wax particle at
π₯ = πΏ
As shown in the diagram above with convection allowed:
The temperature profile obeyed is:
33. π β πβ
ππ β πβ
=
(
πΏ
(π½π2 (
π‘ππβ
πΏ
πΏ
+
βπΏπΏ
π
1 +
βπΏπΏ
π
π‘ππβ
πΏ
πΏ
) + πΏ)
)
(
1
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
)π ππβ
πΏ
πΏ
)
Mathematically, what form should the equation of βπΏ against length L take on?
First of all, we know that when the length L becomes zero(i.e., there is no metal
rod), the flux is due to only the flame and is given by
π = βπ(ππ β πβ)
The power then is
βππ΄(ππ β πβ) =
ππ
ππ‘
π»πΆ
The above is the condition to be satisfied.
The flux at x=L, is given by:
π = βπΏ(ππΏ β πβ)
Upon substituting for (ππΏ β πβ), we get
π = ππ³
(
πΉ
(π·ππ (
ππππ
π³
πΉ
+
ππ³πΉ
π
π +
ππ³πΉ
π
ππππ
π³
πΉ
) + πΉ)
)
(
π
ππ¨π¬π‘
π³
πΉ
+ (
ππ³πΉ
π
)ππππ
π³
πΉ
) (π»π β π»β)
We know that when L=0, the flux should reduce to
π = ππ(π»π β π»β)
Making the first guess that
βπΏ = βππβππΏ
Where:
35. π = β0 (
1
(1 + 1)
) (ππ β πβ)
We finally get:
π =
π
π
ππ(π»π β π»β)
Which doesnβt satisfy the condition above.
Now choosing
βπΏ =
πΎ
πΏπ
Which means that βπΏ is inversely proportional to length L to power n.
As length πΏ β 0, βπΏ β β
Upon substituting in the flux equation for L=0, we end up with:
π = βπΏ
(
πΏ
(π½π2 (
π‘ππβ
0
πΏ
+
βπΏπΏ
π
1 +
βπΏπΏ
π
π‘ππβ
0
πΏ
) + πΏ)
)
(
1
cosh
0
πΏ
+ (
βπΏπΏ
π
) π ππβ
0
πΏ
) (ππ β πβ)
π = βπΏ (
πΏ
(π½π2 (
βπΏπΏ
π
) + πΏ)
) (ππ β πβ)
π = βπΏ (
1
(π½π2 (
βπΏ
π
) + 1)
)(ππ β πβ)
But ππ πΏ β β, βπΏ β β, π π π½π2
(
βπΏ
π
) + 1 β π½π2
(
βπΏ
π
)
We get:
π = βπΏ
(
1
(π½π2 (
βπΏ
π
))
)
(ππ β πβ)
36. We get
π =
π
π½π2
(ππ β πβ)
But
π
π½π2
= βπ
So, we end up with:
π = βπ(ππ β πβ)
Which is the required equation hence βπΏ takes on the form
βπΏ =
πΎ
πΏπ
From experiment, it was found that π = 1.
Going back to
π β πβ
ππ β πβ
=
(
πΏ
(π½π2 (
π‘ππβ
πΏ
πΏ
+
βπΏπΏ
π
1 +
βπΏπΏ
π
π‘ππβ
πΏ
πΏ
) + πΏ)
)
(
1
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
)π ππβ
πΏ
πΏ
)
We go ahead and rearrange the equation above to get a quadratic equation in
βπΏ and investigate the nature of βπΏ by varying the length of the metal rod and
noting the time taken for the wax to melt.
Calling,
π β πβ
ππ β πβ
=
1
π΅
And
π½π2
= π
Upon rearranging, we get:
ππ³
π
[
πΏ2
π
π2
π ππβ (
πΏ
πΏ
) +
πΏ3
π2
π ππβ (
πΏ
πΏ
) π‘ππβ (
πΏ
πΏ
)] + ππ³ [
πΏπ
π
πππ β (
πΏ
πΏ
) +
πΏπ
π
π ππβ (
πΏ
πΏ
) π‘ππβ (
πΏ
πΏ
) +
2πΏ2
π
π ππβ (
πΏ
πΏ
) β
π΅πΏ2
π
π‘ππβ (
πΏ
πΏ
)] + [ππ ππβ (
πΏ
πΏ
) + πΏπππ β (
πΏ
πΏ
) β π΅πΏ]
37. From experimental values, it was found that βπΏ varies inversely with length
taking on the form below:
ππ³ =
ππ³π
ππ
Where:
π = β
βπ
πΎπ΄
Taking natural logs, we get:
ππ(βπΏ) = ln(
βπΏ0
π
) β ln(πΏ)
For aluminium rods of radius 2mm, the graph looked as below:
ππ³π = πΊ Γ
π²
π
Where:
π = ππππ π ππ£ππ‘π¦
π = 0.006671
βπΏ =
βπΏ0
ππ
y = -1.001x + 5.1071
RΒ² = 0.9781
0
1
2
3
4
5
6
7
8
9
10
-4 -3 -2 -1 0
Ln(hL)
Ln(L)
A graph of Ln(hL) against Ln(L) for AL rods radius
2mm
Series1
Linear (Series1)
38. ππ³ = πΊ Γ
π²
π³
Γ β
π²
πππ
The emissivity π can be taken to be independent of nature of metal.
For aluminium rods of radius 1mm, the graph looked as below:
Looking at the solution at π₯ = πΏ
π β πβ
ππ β πβ
=
(
πΏ
(π½π2 (
π‘ππβ
πΏ
πΏ
+
βπΏπΏ
π
1 +
βπΏπΏ
π
π‘ππβ
πΏ
πΏ
) + πΏ)
)
(
1
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
)π ππβ
πΏ
πΏ
)
Rearranging the equation above, we get
(π½π2 (
π‘ππβ
πΏ
πΏ
+
βπΏπΏ
π
1 +
βπΏπΏ
π
π‘ππβ
πΏ
πΏ
) + πΏ) = (
ππ β πβ
π β πβ
) Γ (
πΏ
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
) π ππβ
πΏ
πΏ
)
Calling
π¦ = (π½π2 (
π‘ππβ
πΏ
πΏ
+
βπΏπΏ
π
1 +
βπΏπΏ
π
π‘ππβ
πΏ
πΏ
) + πΏ)
y = -1.078x + 5.364
RΒ² = 0.9919
0
1
2
3
4
5
6
7
8
9
10
-4 -3 -2 -1 0
Ln(hL)
Ln(L)
A graph of Ln(hL) against Ln(L) for AL rods radius
1mm
Series1
Linear (Series1)
39. And
π₯ = (
πΏ
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
)π ππβ
πΏ
πΏ
)
Plotting a graph of y against x for aluminium rods of radius 1mm looked as
below with:
βπΏ = π Γ
πΎ
πΏ
Γ β
πΎ
2βπ
ππ³ = π. ππππππ Γ
π²
π³
Γ β
π²
πππ
The flux at π₯ = πΏ is given by:
π = ππ³(π» β π»β)
It can be shown that after substituting for temperature (π» β π»β) and ππ³, the
maximum possible flux got is when length L tends to zero and is given by
ππππ = ππ(π»π β π»β)
Which is the flux of the hot flame.
y = 4.3301x
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
y
x
A graph of y against x
40. DERIVATION OF THE GENERAL EXPRESSION FOR HEAT TRANSFER
COEFFICIENT ππ³.
Recall for cylindrical rods the expression was:
βπΏ = π Γ
πΎ
πΏ
Γ β
πΎ
2βπ
OR
βπΏ = 0.006671 Γ
πΎ
πΏ
Γ β
πΎ
2βπ
How could we arrive to that expression from a general expression?
The general expression is given by:
ππ³π¨π³(π»π³ β π»β) = πΊ Γ
ππ²
ππ
Γ πΈ
OR
ππ³π¨π³(π»π³ β π»β) = π. ππππππ Γ
ππ²
ππ
Γ πΈ
Where:
π =
ππΏ β πβ
π
π = πππππ’ππ‘ππ£π πππ ππ π‘ππππ
π΄πΏ = πππππ’ππ‘πππ ππππ ππ‘ πππππ‘β πΏ
π = β
βπ
πΎπ΄
πππ ππ¦πππππππππ πππ π = β
2β
πΎπ
For cylindrical rod
π =
πΏ
πΎπ΄
41. For cylindrical metal rods,
π΄πΏ = π΄
So, we have
βπΏπ΄(ππΏ β πβ) = 0.006671 Γ
πΎ
2β
β
2β
πΎπ
Γ πΎπ΄(
ππΏ β πβ
πΏ
)
Upon simplification, we get the expected expression:
βπΏ = 0.006671 Γ
πΎ
πΏ
Γ β
πΎ
2βπ
OR
ππ³ = πΊ Γ
π²
π³
Γ β
π²
πππ
We can extend the above analysis to cylindrical co-ordinates heat conduction
knowing their conductive resistance.
42. Let us solve the heat equation to get the expression for πΏ.
We use this temperature profile which satisfies the initial condition to solve the
heat equation and get πΏ as shown below:
Boundary and initial conditions are:
π» = π»π ππ π = π
βπ
π π»
π π
= ππ³(π» β π»β) ππ π = π
π» = π»β ππ π = π
The governing temperature profile is:
π β πβ
ππ β πβ
=
cosh [
(πΏ β π₯)
πΏ
] + (
βπΏπΏ
π
) sinh[(
πΏ β π₯
πΏ
)]
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
) π ππβ
πΏ
πΏ
The governing equation is
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
Let us change this equation into an integral equation as below:
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
β
βπ
π΄ππΆ
β« (π β πβ)ππ₯
π
0
=
π
ππ‘
β« (π)ππ₯
π
0
β¦ β¦ . . π)
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
β
2β
πππΆ
β« (π β πβ)ππ₯
π
0
=
π
ππ‘
β« (π)ππ₯
π
0
β« (
π2
π
ππ₯2
)ππ₯
π
0
= [
ππ
ππ₯
]
π
0
=
(ππ β πβ)
πΏ
(
β
βπΏπΏ
π
+ (π ππβ
πΏ
πΏ
+
βπΏπΏ
π
πππ β
πΏ
πΏ
)
πππ β
πΏ
πΏ
+
βπΏπΏ
π
π ππβ
πΏ
πΏ
)
β« (π β πβ)ππ₯
π
0
= |βπΏ(ππ β πβ) (
sinh [
(πΏ β π₯)
πΏ
] +
βπΏπΏ
π
cosh [
(πΏ β π₯)
πΏ
]
πππ β
πΏ
πΏ
+
βπΏπΏ
π
π ππβ
πΏ
πΏ
)|
π
0
43. β« (π β πβ)ππ₯
π
0
= πΏ(ππ β πβ) (
β
βπΏπΏ
π
+ (π ππβ
πΏ
πΏ
+
βπΏπΏ
π
πππ β
πΏ
πΏ
)
πππ β
πΏ
πΏ
+
βπΏπΏ
π
π ππβ
πΏ
πΏ
)
π β πβ
ππ β πβ
=
cosh [
(πΏ β π₯)
πΏ
] + (
βπΏπΏ
π
) sinh[(
πΏ β π₯
πΏ
)]
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
) π ππβ
πΏ
πΏ
π =
cosh [
(πΏ β π₯)
πΏ
] + (
βπΏπΏ
π
)sinh[(
πΏ β π₯
πΏ
)]
cosh
πΏ
πΏ
+ (
βπΏπΏ
π
)π ππβ
πΏ
πΏ
(ππ β πβ) + πβ
π
ππ‘
β« (π)ππ₯
π
0
=
π
ππ‘
[πΏ(ππ β πβ) (
β
βπΏπΏ
π
+ (π ππβ
πΏ
πΏ
+
βπΏπΏ
π
πππ β
πΏ
πΏ
)
πππ β
πΏ
πΏ
+
βπΏπΏ
π
π ππβ
πΏ
πΏ
)] +
π(ππβ)
ππ‘
π(ππβ)
ππ‘
= 0
Upon substitution of all the above in the heat equation, we get:
πΌ
(ππ β πβ)
πΏ
(
β
βπΏπΏ
π
+ (π ππβ
πΏ
πΏ
+
βπΏπΏ
π
πππ β
πΏ
πΏ
)
πππ β
πΏ
πΏ
+
βπΏπΏ
π
π ππβ
πΏ
πΏ
) β
2β
πππΆ
πΏ(ππ β πβ)(
β
βπΏπΏ
π
+ (π ππβ
πΏ
πΏ
+
βπΏπΏ
π
πππ β
πΏ
πΏ
)
πππ β
πΏ
πΏ
+
βπΏπΏ
π
π ππβ
πΏ
πΏ
) =
π
ππ‘
[πΏ(ππ β πβ)(
β
βπΏπΏ
π
+ (π ππβ
πΏ
πΏ
+
βπΏπΏ
π
πππ β
πΏ
πΏ
)
πππ β
πΏ
πΏ
+
βπΏπΏ
π
π ππβ
πΏ
πΏ
)]
We notice that the term(ππ β πβ) (
β
βπΏπΏ
π
+(π ππβ
πΏ
πΏ
+
βπΏπΏ
π
πππ β
πΏ
πΏ
)
πππ β
πΏ
πΏ
+
βπΏπΏ
π
π ππβ
πΏ
πΏ
) is common and can be
eliminated and what this signifies is that the nature of (ππ β πβ) doesnβt matter
and so we get:
πΌ
πΏ
β
2β
πππΆ
πΏ =
ππΏ
ππ‘
We go ahead and solve for πΏ provided πΏ = 0ππ‘ π‘ = 0 and get the expression
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
So, the final solution for the finite metal rod with convective flux at the end of
the metal rod is:
44. π» β π»β
π»π β π»β
=
(
πΉ
(π·ππ (
ππππ
π³
πΉ
+
ππ³πΉ
π
π +
ππ³πΉ
π
ππππ
π³
πΉ
) + πΉ)
)
(
ππ¨π¬π‘ [
(π³ β π)
πΉ
] + (
ππ³πΉ
π
)π¬π’π§π‘ [(
π³ β π
πΉ
)]
ππ¨π¬π‘
π³
πΉ
+ (
ππ³πΉ
π
)ππππ
π³
πΉ
)
The solution reduces to the semi-infinite rod solution when the length L of the
metal rod tends to infinity.
Using the above solution, it was shown experimentally that the ratio
π» β π»β
π»π β π»β
= π. πππππ
As for the semi-infinite metal rod.
The above completes our analysis.
45. CASE 2: ZERO FLUX AT THE END OF THE METAL ROD?
In reality, it is hard to achieve zero flux.
The boundary and initial conditions are:
π» = π»π ππ π = π
π π»
π π
= π ππ π = π
π» = π»β ππ π = π
The governing equation is
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
Recall that the temperature profile we are going to use is:
(π β πβ) = πΆ1πβππ₯
+ πΆ2πππ₯
Or
(π» β π»β) = πͺππ
βπ
πΉ + πͺππ
π
πΉ
First of all, to satisfy the boundary conditions above, the temperature profile
becomes [2]:
π β πβ
ππ β πβ
=
πππ₯
1 + π2ππΏ
+
πβππ₯
1 + πβ2ππΏ
46. Or
The equation above can be rearranged to get [3]
π β πβ
ππ β πβ
=
cosh[π(πΏ β π₯)]
cosh ππΏ
In terms of πΉ we get
π β πβ
ππ β πβ
=
cosh[
(πΏ β π₯)
πΏ
]
cosh
πΏ
πΏ
Or using the first equation, we get:
π β πβ
ππ β πβ
=
π
π₯
πΏ
1 + π
2πΏ
πΏ
+
π
βπ₯
πΏ
1 + π
β2πΏ
πΏ
Let us examine the initial condition,
It can be shown that after solving the heat equation πΏ will take on the form:
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
At
π‘ = 0, πΏ = 0 πππ π =
1
πΏ
= β
Upon substitution in
π β πβ
ππ β πβ
=
πππ₯
1 + π2ππΏ
+
πβππ₯
1 + πβ2ππΏ
We get
π β πβ
ππ β πβ
=
πβπ₯
1 + π2βπΏ
+
πββπ₯
1 + πβ2βπΏ
For a given π₯
We get:
47. π β πβ
ππ β πβ
=
πππ₯
1 + π2βπΏ
β
πππ₯
π2ππΏ
= πβπ(πΏβπ₯)
= πββ(πΏβπ₯)
= 0
Since
(πΏ β π₯) > 0
Hence the initial condition is satisfied.
Getting back to business, we noticed that in the semi-infinite rod solution there
was convection at the hot end of the rod. So, to solve for what is observed in
the finite metal rod solution with zero flux at the end of the rod, we have to use
that fact as stated below:
βπ
ππ»
ππ
|π=π = ππ(π»π β π»π)
Recall the compact temperature profile for zero flux at the end of a finite metal
rod is:
π β πβ
ππ β πβ
=
cosh[π(πΏ β π₯)]
cosh ππΏ
Where:
π =
1
πΏ
ππ
ππ₯
|π₯=0 = β(ππ β πβ)ππ‘ππβππΏ
βπ
ππ
ππ₯
|π₯=0 = π(ππ β πβ)ππ‘ππβππΏ
β0(ππ β ππ ) = π(ππ β πβ)ππ‘ππβππΏ
β0(ππ) β β0(ππ ) = π(ππ )ππ‘ππβππΏ β π(πβ)ππ‘ππβππΏ
Collecting like terms we get:
ππ (πππ‘ππβππΏ + β0) = β0(ππ) + π(πβ)ππ‘ππβππΏ
ππ =
β0(ππ) + π(πβ)ππ‘ππβππΏ
(πππ‘ππβππΏ + β0)
Subtracting πβ from both sides we get:
48. ππ β πβ =
β0(ππ) + π(πβ)ππ‘ππβππΏ
(πππ‘ππβππΏ + β0)
β πβ
Upon simplification, we get:
ππ β πβ
ππ β πβ
=
β0
πππ‘ππβππΏ + β0
But
π =
1
πΏ
Upon substitution and simplification, we get
ππ β πβ
ππ β πβ
=
πΏ
π
β0
π‘ππβ
πΏ
πΏ
+ πΏ
But from the semi-infinite rod solution, we have
π
β0
= π½π2
Upon substitution, we get:
ππ β πβ
ππ β πβ
=
πΏ
π½π2π‘ππβ
πΏ
πΏ
+ πΏ
(ππ β πβ) = (ππ β πβ)(
πΏ
π½π2π‘ππβ
πΏ
πΏ
+ πΏ
)
Upon substitution in the temperature profile
π β πβ
ππ β πβ
=
cosh[
(πΏ β π₯)
πΏ
]
cosh
πΏ
πΏ
We get:
(π» β π»β) = (π»π β π»β)(
πΉ
π·ππππππ
π³
πΉ
+ πΉ
)(
ππ¨π¬π‘[
(π³ β π)
πΉ
]
ππ¨π¬π‘
π³
πΉ
)
49. The above temperature profile satisfies the initial condition and the boundary
conditions provided the temperature at the hot end varies with time.
Let us now solve the heat equation using the above temperature profile:
Recall the compact temperature profile is:
π β πβ
ππ β πβ
=
cosh[
(πΏ β π₯)
πΏ
]
cosh
πΏ
πΏ
The boundary and initial conditions are:
π» = π»π ππ π = π
π π»
π π
= π ππ π = π
π» = π»β ππ π = π
The governing equation is
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
Let us change this equation into an integral equation as below:
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
β
βπ
π΄ππΆ
β« (π β πβ)ππ₯
π
0
=
π
ππ‘
β« (π)ππ₯
π
0
β¦ β¦ . . π)
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
β
2β
πππΆ
β« (π β πβ)ππ₯
π
0
=
π
ππ‘
β« (π)ππ₯
π
0
β« (
π2
π
ππ₯2
) ππ₯
π
0
= [
ππ
ππ₯
]
π
0
= (ππ β πβ)
tanh(
πΏ
πΏ
)
πΏ
β« (π β πβ)ππ₯
π
0
= (ππ β πβ)πΏtanh(
πΏ
πΏ
)
π =
cosh [
(πΏ β π₯)
πΏ
]
cosh
πΏ
πΏ
(ππ β πβ) + πβ
50. π
ππ‘
β« (π)ππ₯
π
0
=
π
ππ‘
[πΏ(ππ β πβ) tanh (
πΏ
πΏ
)] +
π(ππβ)
ππ‘
π(ππβ)
ππ‘
= 0
Upon substitution of all the above in the heat equation, we get:
πΌ(ππ β πβ)
tanh (
πΏ
πΏ
)
πΏ
β
2β
πππΆ
(ππ β πβ)πΏ tanh (
πΏ
πΏ
) =
π
ππ‘
[πΏ(ππ β πβ) tanh (
πΏ
πΏ
)]
We notice that the term (ππ β πβ)tanh(
πΏ
πΏ
) is common and can be eliminated and
what this signifies is that the nature of (ππ β πβ) doesnβt matter and so we get:
πΌ
πΏ
β
2β
πππΆ
πΏ =
ππΏ
ππ‘
We go ahead and solve for πΏ provided πΏ = 0ππ‘ π‘ = 0 and get the expression
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
So, the final solution for the finite metal rod with zero flux at the end of the
metal rod is:
(π» β π»β) = (π»π β π»β)(
πΉ
π·ππππππ
π³
πΉ
+ πΉ
)(
ππ¨π¬π‘[
(π³ β π)
πΉ
]
ππ¨π¬π‘
π³
πΉ
)
The solution reduces to the semi-infinite rod solution when the length L of the
metal rod tends to infinity.
Using the above solution, it was shown experimentally that the ratio
π» β π»β
π»π β π»β
= π. πππππ
As for the semi-infinite rod.
51. HOW DO WE DEAL WITH CYLINDRICAL
COORDINATES?
We know that for an insulated cylinder where there is no heat loss by
convection from the sides, the governing PDE equation is
πΆ
π
π
ππ
(π
ππ»
ππ
) =
ππ»
ππ
In steady state
ππ
ππ‘
= 0
We end up with
π
ππ
(π
ππ
ππ
) = 0
We can then integrate once to get
β« (
π
ππ
(π
ππ
ππ
)) ππ = β«(0)ππ
And get
π
ππ
ππ
= πΆ1
Therefore
ππ
ππ
=
πΆ1
π
We can go ahead and find the temperature profile as a function of radius r.
52. HOW DO WE DEAL WITH CYLINDRICAL CO-ORDINATES
FOR A SEMI-INFINITE RADIUS CYLINDER?
The governing PDE is:
πΆ
π
π
ππ
(π
ππ»
ππ
) =
ππ»
ππ
The boundary conditions are
π = ππ ππ‘ π = π1
π = πβ ππ‘ π = β
The initial condition is:
π = πβ ππ‘ π‘ = 0
The temperature profile that satisfies the conditions above is
π β πβ
ππ β πβ
= π
β(πβπ1)
πΏ
We transform the equation above into an integral equation and take integrals
with limits from π = π1 to π = π = β.
πΌ
π
π
ππ
(π
ππ
ππ
) =
ππ
ππ‘
We take integrals and get
β« [
πΌ
π
π
ππ
(π
ππ
ππ
)]ππ
π
π1
=
π
ππ‘
β« (π)ππ
π
π1
β« [
πΌ
π
π
βr
/
(π
ππ
ππ
)]βr
/
π
π1
=
π
ππ‘
β« (π)ππ
π
π1
You notice that
π
ππ
cancels out with π π and we get:
πΌ
ππ
ππ
=
π
ππ‘
β« (π)ππ
π
π1
But
ππ
ππ
= β« (
π2
π
ππ2
)ππ
π
π1
So, the PDE becomes:
53. πΆ β« (
ππ
π»
πππ
)π π
πΉ
ππ
=
π
ππ
β« (π»)π π
πΉ
ππ
We then go ahead to solve and find πΏ as before.
ππ
ππ
= β
ππ β πβ
πΏ
π
β(πβπ1)
πΏ
β« (
π2
π
ππ2
) ππ
π
π1
= [
ππ
ππ
]
π
π1
= β
(ππ β πβ)
πΏ
[π
β(πβπ1)
πΏ ]
π = β
π1
=
(ππ β πβ)
πΏ
π = (ππ β πβ)π
β(πβπ1)
πΏ + πβ
β« πππ
π =β
π1
= β« ((ππ β πβ)π
β(πβπ1)
πΏ )ππ
π =β
π1
+ β« πβππ
π =β
π1
= πΏ(ππ β πβ) + πβ(π β π1)
π
ππ‘
β« πππ
π
πΏ
=
ππΏ
ππ‘
(ππ β πβ) +
π(πβ(π β π1))
ππ‘
But
π(πβ(π β π1))
ππ‘
= 0
Since πβ, π , π1 are constants independent of time.
So
π
ππ‘
β« πππ
π
πΏ
=
ππΏ
ππ‘
(ππ β πβ)
substituting all the above in the integral equation, we get
πΌ β« (
π2
π
ππ2
)ππ
π
π1
=
π
ππ‘
β« (π)ππ
π
π1
πΌ
πΏ
(ππ β πβ) =
ππΏ
ππ‘
(ππ β πβ)
Divide through by (ππ β πβ) and get
πΌ
πΏ
=
ππΏ
ππ‘
πΌ
πΏ
=
ππΏ
ππ‘
The boundary conditions are:
πΏ = 0 ππ‘ π‘ = 0
54. πΏ = β2πΌπ‘
We substitute πΏ in the temperature profile and get:
π β πβ
ππ β πβ
= π
β(πβπ1)
πΏ
π» β π»β
π»π β π»β
= π
β(πβππ)
βππΆπ
You notice that the initial condition is satisfied for the above temperature
profile.
We can also go ahead and look at situations where there is natural convection
and other situations where the radius r is finite and not infinite.
55. HOW DO WE DEAL WITH NATURAL CONVECTION AT
THE SURFACE AREA OF A SEMI-INFINITE CYLINDER
FOR FIXED END TEMPERATURE
The governing equation is:
πΆ
π
π
ππ
(π
ππ»
ππ
) β
ππ·
π¨ππͺ
(π» β π»β) =
ππ»
ππ
π = 2ππ
π΄ = 2πππ
Where:
π = βπππβπ‘ ππ ππ¦ππππππ
The boundary conditions are:
π = ππ ππ‘ π = π1
π = πβ ππ‘ π = π = β
The initial condition is:
π = πβ ππ‘ π‘ = 0
The temperature profile that satisfies the boundary conditions above is:
π β πβ
ππ β πβ
= π
β(πβπ1)
πΏ
πΌ
π
π
ππ
(π
ππ
ππ
) β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
56. We transform the PDE into an integral equation and take the limits to be from
π = π1 to π = π = β
πΌ β« [
πΌ
π
π
ππ
(π
ππ
ππ
)]ππ
π =β
π1
β
β
πππΆ
β« (π β πβ)ππ
π =β
π1
=
π
ππ‘
β« (π)ππ
π =β
π1
β« [
πΌ
π
π
βr
/
(π
ππ
ππ
)]βr
/
π =β
π1
β
β
πππΆ
β« (π β πβ)ππ
π =β
π1
=
π
ππ‘
β« (π)ππ
π =β
π1
You notice that
π
ππ
cancels out with π π and we get:
πΌ
ππ
ππ
β
β
πππΆ
β« (π β πβ)
π =β
π1
ππ =
π
ππ‘
β« (π)ππ
π =β
π1
But
ππ
ππ
= β« (
π2
π
ππ2
)ππ
π
π1
So, the PDE becomes:
πΆ β« (
ππ
π»
πππ
)π π
πΉ
ππ
β
π
π ππͺ
β« (π» β π»β)
πΉ=β
ππ
π π =
π
ππ
β« (π»)π π
πΉ
ππ
β
πππΆ
β« (π β πβ)ππ
π =β
π1
=
πΏβ
πππΆ
(ππ β πβ)
From the derivations above, we get:
β« (
π2
π
ππ2
) ππ
π
π1
=
(ππ β πβ)
πΏ
π
ππ‘
β« πππ
π
π1
=
ππΏ
ππ‘
(ππ β πβ)
Upon substitution of the above expressions in the integral equation, we get:
πΌ
πΏ
β
β
πππΆ
πΏ =
ππΏ
ππ‘
πΌ
πΏ
β
β
πππΆ
πΏ =
ππΏ
ππ‘
57. πΌ β
β
πππΆ
πΏ2
= πΏ
ππΏ
ππ‘
The boundary conditions are:
πΏ = 0 ππ‘ π‘ = 0
The solution of the equation above is
πΏ = β
πΌπππΆ
β
(1 β π
β
2βπ‘
πππΆ)
πΌ =
πΎ
ππΆ
And get
πΏ = β
πΎπ
β
(1 β π
β
2βπ‘
πππΆ)
The temperature profile becomes:
π β πβ
ππ β πβ
= π
β(
πβπ1
πΏ
)
Substitute for πΏ and get
π» β π»β
π»π β π»β
= π
β(
πβππ
βπ²π
π
(πβπ
β
πππ
π ππͺ)
)
You notice that the initial condition is satisfied by the above temperature
profile.
What do we observe for short time in transient state?
For short time, the exponential is small and it becomes:
π
β
2βπ‘
πππΆ = 1 β
2βπ‘
πππΆ
After using binomial expansion of the exponential in the above
And
(1 β π
β
2βπ‘
πππΆ) =
2βπ‘
πππΆ
58. πΏ = β
πΎπ
β
(1 β π
β
2βπ‘
πππΆ)
Becomes
πΏ = β2πΌπ‘
Where:
πΌ =
πΎ
ππΆ
Let us make r the subject of the temperature profile above:
π β πβ
ππ β πβ
= π
β(
πβπ1
πΏ
)
π β π1
πΏ
= [ln(
ππ β πβ
π β πβ
)]
(π β π1) = πΏ [ln (
ππ β πβ
π β πβ
)]
Where:
πΏ = β2πΌπ‘
For small time.
We can get an expression for ππ as we did before.
We can extend the above analysis to finite radius metal rods and even to
rectangular metal rods.
59. REFERENCES
[1] C.P.Kothandaraman, "HEAT TRANSFER WITH EXTENDED SURFACES(FINS)," in Fundamentals of Heat
and Mass Transfer, New Delhi, NEW AGE INTERNATIONAL PUBLISHERS, 2006, p. 131.
[2] C. E. W. R. E. W. G. L. R. James R. Welty, "HEAT TRANSFER FROM EXTENDED SURFACES," in
Fundamentals of Momentum, Heat, and Mass Transfer 5th Edition, Corvallis, Oregon, John Wiley &
Sons, Inc., 2000, pp. 236-237.
[3] C. E. W. R. E. W. G. L. R. James R. Welty, "HEAT TRANSFER FROM EXTENDED SURFACES," in
Fundamentals of Momentum, Heat and Mass Transfer 5th Edition, Oregon, John Wiley & Sons, Inc.,
2008, p. 237.