In this book, we solve the heat equation partial differential equation by first transforming it into an integral equation. We use exponential temperature profiles which satisfy the boundary conditions and also the initial condition. We also look at cases where ther is natural convection and go ahead and solve for both the transient and steady state solution. We also go ahead an solve the heat equation in cylindrical coordinates. We explain alot of phenomena observed experimentally for example the melting of wax on the sides of a metal rod when heat is applied on one end. For updated information about heat flow, follow the link below:
https://www.slideshare.net/Wasswaderrick3/analytic-solutions-to-the-heat-equation-using-integral-methods-with-experimental-resultspdf
3. TABLE OF CONTENTS
SEMI INFINITE WALL ANALYTICAL SOLUTION TO THE HEAT EQUATION................3
ALTERNATIVE SOLUTION TO THE SEMI-INFINITE WALL PROBLEM ...........................5
HOW DO WE EXPLAIN THE EXISTENCE OF THE FOURIER LAW IN STEADY
STATE?......................................................................................................................................................8
HOW DO WE DEAL WITH CONVECTION AT THE SURFACE AREA OF THE METAL
ROD...........................................................................................................................................................11
EQUAL FIXED TEMPERATURES AT THE END OF AN INSULATED METAL ROD....14
UNEQUAL FIXED TEMPERATURES AT THE END OF AN INSULATED METAL ROD.
....................................................................................................................................................................17
HOW DO WE DEAL WITH OTHER TYPES OF BOUNDARY CONDITIONS?..................20
HOW DO WE DEAL WITH CONVECTION AT THE SURFACE AREA OF THE METAL
ROD FOR FIXED END TEMPERATURE......................................................................................22
WHAT HAPPENS WHEN THE INITIAL TEMPERATURE IS A FUNCTION OF X? .......31
HOW DO WE DEAL WITH CYLINDRICAL CO-ORDINATES?..............................................34
4. SEMI INFINITE WALL ANALYTICAL SOLUTION TO THE
HEAT EQUATION.
The differential equation to be solved is
ππ
ππ‘
= πΌ
π2
π
ππ₯2
Where the initial and boundary conditions are
π» = π»π ππ π = π πππ πππ π
π» = π»β ππ π = β πππ πππ π
π» = π»β ππ π = π πππ πππ π
We postulate:
π =
π β πβ
ππ β πβ
And
π =
π₯
2βπΌπ‘
We get
π2
π
ππ2
+ 2π
ππ
ππ
= 0 (1)
With the transformed boundary and initial conditions
π β 0 ππ π β β
And
π = 1 ππ‘ π = 0
The first condition is the same as the initial condition π = πβ ππ‘ π‘ = 0 and the
boundary condition
π β πβ ππ π₯ β β
Equation 1 may be integrated once to get
ππ
ππ
ππ
= π1 β π2
ππ
ππ
= π2πβπ2
5. And integrated once more to get
π = π3 + π2 β« πβπ2
π π
Applying the boundary conditions to the equation, we get
π = 1 β erf (
π₯
2βπΌπ‘
)
π» β π»β
π»π β π»β
= π β ππ«π (
π
πβπΆπ
)
Or
π»π β π»
π»π β π»β
= ππ«π (
π
πβπΆπ
)
6. ALTERNATIVE SOLUTION TO THE SEMI-INFINITE WALL PROBLEM
The problem of the semi-infinite wall could also be solved as below:
Given the boundary and initial conditions
π» = π»π ππ π = π πππ πππ π
π» = π»β ππ π = β
π» = π»β ππ π = π
And the governing equation
ππ
ππ‘
= πΌ
π2
π
ππ₯2
We assume an exponential temperature profile that satisfies the boundary
conditions:
π β πβ
ππ β πβ
= π
βπ₯
πΏ
We can satisfy the initial condition if we assume that πΏ will have a solution as
πΏ = ππ‘π
Where c and n are constants so that at π‘ = 0, πΏ = 0 and the initial condition is
satisfied as shown below.
π β πβ
ππ β πβ
= π
βπ₯
0 = πββ
= 0
π = πβ ππ‘ π‘ = 0
We then transform the heat governing equation into an integral equation as:
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
= β«
ππ
ππ‘
ππ₯
π
0
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
=
π
ππ‘
β« πππ₯
π
0
You notice that the integral
π
ππ‘
β« πππ₯
π
0
=
π
ππ‘
β« (π β πβ)ππ₯ =
π
ππ‘
β« πππ₯
π
0
β
π(ππβ)
ππ‘
π
0
Since π and πβ are constants independent of time
π(ππβ)
ππ‘
= 0
7. So
π
ππ‘
β« πππ₯
π
0
=
π
ππ‘
β« (π β πβ)ππ₯
π
0
We go ahead and find
π2
π
ππ₯2
=
(ππ β πβ)
πΏ2
β« (
π2
π
ππ₯2
) ππ₯
πΏ
0
= β
(ππ β πβ)
πΏ
(π
βπ
πΏ β 1) =
(ππ β πβ)
πΏ
(1 β π
βπ
πΏ )
π
ππ‘
β« (π β πβ)ππ₯
πΏ
0
=
ππΏ
ππ‘
[(ππ β πβ) (1 β π
βπ
πΏ )]
Substituting into the integral equation, we get
πΌ
(ππ β πβ)
πΏ
(1 β π
βπ
πΏ ) =
ππΏ
ππ‘
(1 β π
βπ
πΏ )
The boundary conditions are
πΏ = 0 ππ‘ π‘ = 0
We find
πΏ = β2πΌπ‘
We substitute in the temperature profile and get
π» β π»β
π»π β π»β
= π
βπ
βππΆπ
You notice that the initial condition is satisfied by the temperature profile
above i.e.,
At π‘ = 0
π» β π»β
π»π β π»β
= π
βπ
βππΆπ
Becomes
π» β π»β
π»π β π»β
= π
βπ
π = πββ
= π
Hence π» = π»β throughout the rod at π‘ = 0
8. Observation.
The two equations
π» β π»β
π»π β π»β
= π
βπ
βππΆπ
And
π» β π»β
π»π β π»β
= π β ππ«π (
π
πβπΆπ
)
Should give the same answer. Indeed, they give answers that are the same with
a small error since the error function is got from tables after rounding off yet in
the exponential temperature profile there is no rounding off.
9. HOW DO WE EXPLAIN THE EXISTENCE OF THE
FOURIER LAW IN STEADY STATE?
The Fourier law states:
π = βππ΄
ππ
ππ₯
Under steady state.
It can be stated as:
ππ
ππ₯
= β
π
ππ΄
Under steady state.
To satisfy the Fourier law under steady state, we postulate the temperature
profile to be:
π» β π»β =
πΈ
ππ¨
πΉπ
βπ
πΉ
πΏ is a function of time π‘ and not distance π₯
We believe that after solving for πΏ, πΏ will be directly proportional to time t so
that πΏ = ππ‘π
sothat at π‘ = β , πΏ = β
And taking the first derivative of temperature with distance x at π‘ = β , we get
ππ
ππ₯
|π‘=β = β
π
ππ΄
π
βπ₯
πΏ = β
π
ππ΄
π
βπ₯
β = β
π
ππ΄
π0
ππ»
ππ
= β
πΈ
ππ¨
Hence the Fourier law is satisfied.
Now let us go ahead and solve for πΏ.
Recall
PDE
ππ
ππ‘
= πΌ
π2
π
ππ₯2
The initial condition is
π = πβ ππ‘ π‘ = 0
The boundary conditions are
10. π = πβ ππ‘ π₯ = β
ππ
ππ₯
|π₯=0 = β
π
ππ΄
The PDE is
The temperature profile that satisfies the conditions above is
π β πβ =
π
ππ΄
πΏπ
βπ₯
πΏ
We transform the PDE into an integral equation
π
ππ‘
β« (π β πβ)ππ₯
πΏ
0
= πΌ β« (
π2
π
ππ₯2
) ππ₯
πΏ
0
And using the temperature profile, we get
β« (
π2
π
ππ₯2
) ππ₯
πΏ
0
=
π
ππ΄
(1 β π
βπ
πΏ )
π
ππ‘
β« (π β πβ)ππ₯
πΏ
0
=
π
ππ΄
πΏ2
(1 β π
βπ
πΏ )
We then substitute into the integral equation
πΌ
π
ππ΄
(1 β π
βπ
πΏ ) = 2πΏ
ππΏ
ππ‘
(
π
ππ΄
(1 β π
βπ
πΏ ))
The boundary conditions are
πΏ = 0 ππ‘ π‘ = 0
πΏ = βπΌπ‘
Substituting into the temperature profile, we get
π β πβ =
π
ππ΄
πΏπ
βπ₯
πΏ
π» β π»β =
πΈ
ππ¨
Γ βπΆπ Γ π
βπ
βπΆπ
You notice that the initial condition is satisfied
12. HOW DO WE DEAL WITH CONVECTION AT THE
SURFACE AREA OF THE METAL ROD
Recall that the temperature profile that satisfies the Fourier law was
π β πβ =
π
ππ΄
πΏπ
βπ₯
πΏ
Recall
PDE
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
The initial condition is
π = πβ ππ‘ π‘ = 0
The boundary conditions are
π = πβ ππ‘ π₯ = β
ππ
ππ₯
|π₯=0 = β
π
ππ΄
We transform the PDE into an integral equation
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
β
βπ
π΄ππΆ
β« (π β πβ)ππ₯
π
0
=
π
ππ‘
β« (π β πβ)ππ₯
π
0
Where:
We are dealing with a cylindrical metal rod.
π = 2ππ πππ π = ππππ ππ‘π¦ ππ πππ‘ππ πππ
And using the temperature profile, we get
β« (
π2
π
ππ₯2
) ππ₯
π
0
=
π
ππ΄
(1 β π
βπ
πΏ )
β« (π β πβ)ππ₯
π
0
=
π
ππ΄
πΏ2
(1 β π
βπ
πΏ )
Substituting in the integral equation above, we get
πΌ β
βπ
π΄ππΆ
πΏ2
= 2πΏ
ππΏ
ππ‘
The boundary condition is
13. πΏ = 0 ππ‘ π‘ = 0
We solve and get
πΏ = β
π΄ππΆπΌ
βπ
(1 β π
ββππ‘
π΄ππΆ )
Substituting in the temperature profile, we get
π β πβ =
π
ππ΄
πΏπ
βπ₯
πΏ
π» β π»β =
πΈ
ππ¨
Γ β
π¨ππͺπΆ
ππ·
(π β π
βππ·π
π¨ππͺ ) Γ π
βπ
βπ¨ππͺ
ππ·
(πβπ
βππ·π
π¨ππͺ )
We notice that the initial condition and boundary conditions are satisfied.
For small time the term
βππ‘
π΄ππΆ
βͺ 1
And using binomial approximation of the exponential, we get
π
ββππ‘
π΄ππΆ = 1 β
βππ‘
π΄ππΆ
Then
(1 β π
ββππ‘
π΄ππΆ ) =
βππ‘
π΄ππΆ
Upon substitution in the temperature profile, we get
π β πβ =
π
ππ΄
Γ βπΌπ‘ Γ π
βπ₯
βπΌπ‘
Upon rearranging, we get
π₯
βπΌπ‘
= ln (
π
ππ΄
βπΌπ‘) β ln (π β πβ)
π₯
βπ‘
= βπΌln(βπ‘) + βπΌ [ln (
π
ππ΄
βπΌ) β ln(π β πβ)]
What we observe is
14. π
βπ
= βπΆπ₯π§(βπ) + βπΆ [π₯π§ (
πΈ
ππ¨βπΆ
(π» β π»β)
)]
That is what we observe for short times.
When the times become big, we observe
π» β π»β =
πΈ
ππ¨
Γ β
π¨ππͺπΆ
ππ·
(π β π
βππ·π
π¨ππͺ ) Γ π
βπ
βπ¨ππͺ
ππ·
(πβπ
βππ·π
π¨ππͺ )
And in steady state (π‘ = β), we observe
π» β π»β =
πΈ
ππ¨
Γ β
π¨ππͺπΆ
ππ·
Γ π
βπ
βπ¨ππͺπΆ
ππ·
πΌ =
π
ππΆ
We finally get
π» β π»β =
πΈ
ππ¨
Γ β
ππ¨
ππ·
Γ π
βπ
βππ¨
ππ·
the heat flow in steady state is given by:
ππ
ππ₯
= β
π
ππ΄
π
βπ₯
βππ΄
βπ
βππ¨
ππ»
ππ
= πΈπ
βπ
βππ¨
ππ·
15. EQUAL FIXED TEMPERATURES AT THE END OF AN
INSULATED METAL ROD.
PDE
ππ
ππ‘
= πΌ
π2
π
ππ₯2
BCs
π» = π»π ππ π = π π < π < β
π» = π»π ππ π = π π < π < β
IC
π» = π»β ππ π = π π β€ π β€ π
we know a Fourier series solution exists given by
π» β π»π
π»β β π»π
=
π
π
β
π
π
β
π=π
πππ (
ππ π
π
) π
β(
ππ
π )
πΆπ
(
π
π
)π
π = π, π, π, β¦
You notice that this solution is not entirely deterministic since it involves
summing terms up to infinity.
There is an alternative solution as shown below:
ππ
ππ‘
= πΌ
π2
π
ππ₯2
BCs
π» = π»π ππ π = π π < π < β
π» = π»π ππ π = π π < π < β
IC
π» = π»β ππ π = π π β€ π β€ π
We assume an exponential temperature profile that satisfies the boundary
conditions:
π β πβ
ππ β πβ
= π
βπ₯
πΏ
(1β
π₯
π
)
16. You notice that the temperature profile above satisfies the boundary
conditions. We can satisfy the initial condition if we assume that πΏ will assume
a solution as
πΏ = ππ‘π
Where c and n are constants so that at π‘ = 0, πΏ = 0 and the initial condition is
satisfied as shown below.
π β πβ
ππ β πβ
= π
βπ₯
0 = πββ
= 0
π = πβ ππ‘ π‘ = 0
We transform the PDE into an integral equation
π
ππ‘
β« (π β πβ)ππ₯
π
0
= πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
β« (
π2
π
ππ₯2
) ππ₯
π
0
= [
(βπ + 2π₯)
πΏπ
Γ π
βπ₯
πΏ
(1β
π₯
π
)
] π
0
=
2
πΏ
β« (π β πβ)ππ₯
π
0
= [
πΏπ
(βπ + 2π₯)
Γ π
βπ₯
πΏ
(1β
π₯
π
)
] π
0
= 2πΏ
Substituting in the integral equation above, we get:
πΌ (
2
πΏ
) = 2
ππΏ
ππ‘
πΏ = β2πΌπ‘
Substituting back πΏ into the temperature profile, we get
π» β π»β
π»π β π»β
= π
βπ
βππΆπ
(πβ
π
π
)
Or
π» β π»π
π»β β π»π
= π β π
βπ
βππΆπ
(πβ
π
π
)
You notice that the initial condition is satisfied.
You notice that when π = β , we reduce to the temperature profile we derived
before
17. π» β π»β
π»π β π»β
= π
βπ
βππΆπ
you notice that in the temperature profile developed, we get an exact solution
to the problem not an approximate as the Fourier series.
18. UNEQUAL FIXED TEMPERATURES AT THE END OF AN
INSULATED METAL ROD.
ππ
ππ‘
= πΌ
π2
π
ππ₯2
The boundary conditions are:
π = ππ ππ‘ π₯ = 0 πππ πππ π‘
π = π1 ππ‘ π₯ = π πππ πππ π‘
The initial condition is
π = πβ ππ‘ π‘ = 0 0 β€ π₯ β€ π
The temperature profile that satisfies the boundary conditions is:
π» β π»β
[
π
π
(π»π β π»β) + (π»π β π»β) (π β
π
π)]
= πβ
π
πΉ
(πβ
π
π
)
For now, we shall have a solution where πΏ is proportional to time t so that at
π‘ = 0, πΏ = 0 and the initial condition will be satisfied.
We then transform the heat governing equation into an integral equation as:
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
= β«
ππ
ππ‘
ππ₯
π
0
Where:
π = πππππ‘β ππ π‘βπ πππ‘ππ πππ
You notice that the integral
β«
ππ
ππ‘
ππ₯
π
0
=
π
ππ‘
β« πππ₯
π
0
=
π
ππ‘
β« (π β πβ)ππ₯
π
0
Since π and πβ are constants independent of time
So, the integral equation becomes:
πΌ β« (
π2
π
ππ₯2
) ππ₯
π
0
=
π
ππ‘
β« (π β πβ)ππ₯
π
0
We go ahead and find
21. HOW DO WE DEAL WITH OTHER TYPES OF BOUNDARY
CONDITIONS?
Consider the following types of boundary conditions and initial condition:
A)
π» = π»π ππ π = π
π π»
π π
= π ππ π = π
π» = π»β ππ π = π
B)
π» = π»π ππ π = π
βπ
π π»
π π
= π(π» β π»β) ππ π = π
π» = π»β ππ π = π
Let us go about solving for the above boundary conditions but let us deal with set A
boundary conditions and then we can deal with set B later.
We start with a temperature profile below:
π β πβ = (ππ β πβ)πβ
π₯
πΏ
(1β
π₯
π
)
[
π₯
π
(π1 β πβ)
(ππ β πβ)
+ (1 β
π₯
π
)]
we take the derivative
π π»
π π
ππ π = π and equate it to 0 and get:
ππ
ππ₯
|π₯=π = (
(π1 β πβ)
π
β
(ππ β πβ)
π
+
(π1 β πβ)
πΏ
)
ππ
ππ₯
|π₯=π = (
(π1 β πβ)
π
β
(ππ β πβ)
π
+
(π1 β πβ)
πΏ
) = 0
We finally get
(π1 β πβ) = (ππ β πβ)(
πΏ
π + πΏ
)
We substitute π1 β πβ into the temperature profile and get
(π» β π»β) = (π»π β π»β)πβ
π
πΉ
(π β
π
π
)
[
π
π
(
πΉ
πΉ + π
) + (π β
π
π
)]
22. So, the temperature profile above satisfies the set A) boundary and initial
conditions and we can go ahead and solve the governing equation using the
temperature profile above.
For set B) boundary conditions, we again start with the temperature profile
below:
π β πβ = (ππ β πβ)πβ
π₯
πΏ
(1 β
π₯
π
)
[
π₯
π
(π1 β πβ)
(ππ β πβ)
+ (1 β
π₯
π
)]
we take the derivative
π π»
π π
ππ π = π and equate it to:
ππ
ππ₯
|π₯=π = (
(π1 β πβ)
π
β
(ππ β πβ)
π
+
(π1 β πβ)
πΏ
)
π π»
π π
|π=π = β
π
π
(π»π β π»β)
We then find the required temperature profile which we can use to solve the
governing equation.
23. HOW DO WE DEAL WITH CONVECTION AT THE
SURFACE AREA OF THE METAL ROD FOR FIXED END
TEMPERATURE
πΌ
π2
π
ππ₯2
β
βπ
π΄ππΆ
(π β πβ) =
ππ
ππ‘
We shall use the integral approach.
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.
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 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 β πβ)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₯
π β 1)
β« (π β πβ)ππ₯
π
0
= βπΏ(ππ β πβ)(π
βπ₯
π β 1)
24. Substituting the above expressions in equation b) above, we get
πΌ β
βπ
π΄ππΆ
πΏ2
= πΏ
ππΏ
ππ‘
We solve the equation above assuming that
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
βπ²π¨
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?
Let us make π₯ the subject of the equation of transient state and get:
π₯2
= [ln (
ππ β πβ
π β πβ
)]2
Γ
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
When the time duration is small and
2βπ
π΄ππΆ
π‘ βͺ 1
25. We use the binomial expansion approximation
π
β2βπ
π΄ππΆ
π‘
= 1 β
2βπ
π΄ππΆ
π‘
Substituting in the equation of π₯2
as the subject, we get
π₯2
= 2πΌ[ln (
ππ β πβ
π β πβ
)]2
Γ π‘
Where:
πΌ =
πΎ
ππΆ
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₯2
against π‘ is a straight-line
graph through the origin as stated by the equation above when the times are
small. The equation is true because that is what is observed experimentally.
How do we measure the heat transfer coefficient?
From,
π₯2
= [ln (
ππ β πβ
π β πβ
)]2
Γ
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
When the times concerned are big, we shall observe the above equation.
Let us call
π΅ = [ln (
ππ β πβ
π β πβ
)]2
Γ
πΎπ΄
βπ
So, the equation above becomes
π₯2
= π΅ (1 β π
β2βπ
π΄ππΆ
π‘
)
Letβs take the derivative of x against time t and get
π₯
ππ₯
ππ‘
= π΅
2β
πππΆ
π
β2βπ
π΄ππΆ
π‘
27. (1 +
(π β πβ)
πβ
)4
β 1 + 4
(π β πβ)
πβ
= 1 + 4
βπ
πβ
= πππ
βπ
πβ
βͺ 1
Simplifying, we get Newtonβs law of cooling i.e.
π πΈ
π π
= π(π + π·)π¨ππΊπ»β
π (π» β π»β)
π πΈ
π π
= ππ¨(π» β π»β)
Where:
π = π(π + π·)ππΊπ»β
π
Substitute for the above parameters of aluminium and get h theoretically and
compare as got experimentally.
How do we deal with metal rods of finite length π ?
The boundary and initial conditions are
π» = π»π ππ π = π
28. βπ
π π»
π π
= π(π» β π»β) ππ π = π
π» = π»β ππ π = π
Let us go about solving for the above boundary conditions
We start with a temperature profile below:
π β πβ = (ππ β πβ)πβ
π₯
πΏ
(1β
π₯
π
)
[
π₯
π
(π1 β πβ)
(ππ β πβ)
+ (1 β
π₯
π
)]
Which says
π = ππ ππ‘ π₯ = 0
π = π1 ππ‘ π₯ = π
π = πβ ππ‘ π‘ = 0
Provided πΏ = 0 ππ‘ π‘ = 0 , then the initial condition above is satisfied
we take the derivative
π π»
π π
ππ π = π and equate it to β
β
π
(π1 β πβ) and get:
ππ
ππ₯
|π₯=π = (
(π1 β πβ)
π
β
(ππ β πβ)
π
+
(π1 β πβ)
πΏ
)
ππ
ππ₯
|π₯=π = β
β
π
(π1 β πβ)
We equate the two and get
(
(π1 β πβ)
π
β
(ππ β πβ)
π
+
(π1 β πβ)
πΏ
) = β
β
π
(π1 β πβ)
We finally get
(π1 β πβ) = (ππ β πβ)(
πΏπ
πΏπ + ππ + βππΏ
)
We substitute π1 β πβ into the temperature profile and get
(π» β π»β) = (π»π β π»β)πβ
π
πΉ
(π β
π
π
)
[
π
π
(
πΉπ
πΉπ + ππ + πππΉ
) + (π β
π
π
)]
This the temperature profile that satisfies the boundary and initial conditions
below
31. πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
We go ahead and substitute for πΏ in the temperature profile below
(π» β π»β) = (π»π β π»β)πβ
π
πΉ
(π β
π
π
)
[
π
π
(
πΉπ
πΉπ + ππ + πππΉ
) + (π β
π
π
)]
When the time is small, πΏ using binomial approximation becomes
πΏ = β
πΎπ΄
βπ
(1 β π
β2βπ
π΄ππΆ
π‘
)
2βπ
π΄ππΆ
π‘ βͺ 1
π
ββπ
π΄ππΆ
π‘
= 1 β
2βπ
π΄ππΆ
π‘
1 β π
β2βπ
π΄ππΆ
π‘
=
2βπ
π΄ππΆ
π‘
πΏ = β2πΌπ‘
We substitute for πΏ in the temperature profile.
What happens when the length is big or tends to infinity?
(π» β π»β) = (π»π β π»β)πβ
π
πΉ
(π β
π
π
)
[
π
π
(
πΉπ
πΉπ + ππ + ππ³ππΉ
) + (π β
π
π
)]
Becomes
(π» β π»β) = (π»π β π»β)πβ
π
πΉ
(π» β π»β) = (π»π β π»β)π
β
π
βπ²π¨
ππ·
(πβπ
βπππ·
π¨ππͺ
π
)
Which is what we got before.
32. WHAT HAPPENS WHEN THE INITIAL TEMPERATURE IS
A FUNCTION OF X?
The governing equation is
πΌ
π2
π
ππ₯2
=
ππ
ππ‘
BCs
π» = π»π ππ π = π π < π < β
π» = π»π ππ π = π π < π < β
IC
π» = β (π) ππ π = π π β€ π β€ π
We assume an exponential temperature profile that satisfies the boundary
conditions:
π β β
ππ β β
= π
βπ₯
πΏ
π = β + ππ π
βπ₯
πΏ β β π
βπ₯
πΏ
The PDE becomes an integral equation given by:
πΌ β« (
π2
π
ππ₯2
) ππ₯
πΏ
0
=
π
ππ‘
β« πππ₯
πΏ
0
You notice the limits of the integral become from 0 π‘π πΏ as this eliminate
exponentials which are functions of πΏ and this will make the integration
simpler.
Let us give an example say
β = π₯
We make T the subject of the formula and get
π = β + ππ π
βπ₯
πΏ β β π
βπ₯
πΏ
β = π₯
π = π₯ + ππ π
βπ₯
πΏ β π₯π
βπ₯
πΏ
34. There are three roots of πΏ but we choose those which reduce to zero when time
is zero.
The two roots that satisfy the above condition are
πΏ1 = β
π
3π
+
1 + πβ3
6π
β(
1
2
) [2π3 + 27π2π + β(2π3 + 27π2π)2 β 4π6)]
3
+
1 β πβ3
6π
β(
1
2
) [2π3 + 27π2π β β(2π3 + 27π2π)2 β 4π6)]
3
Or
πΏ2 = β
π
3π
+
1 β πβ3
6π
β(
1
2
) [2π3 + 27π2π + β(2π3 + 27π2π)2 β 4π6)]
3
+
1 + πβ3
6π
β(
1
2
) [2π3 + 27π2π β β(2π3 + 27π2π)2 β 4π6)]
3
After getting the solution of πΏ ,we go ahead and substitute it into the
exponential temperature profile.
We shall be faced with more scenarios where πΏ is a cubic equation with time
but the root or solution of πΏ to choose is the one for which πΏ = 0 ππ‘ π‘ = 0.
Using this analytical method, we can also go ahead and solve PDES like
ππ
ππ‘
= πΌ
π2
π
ππ₯2
+ π(π₯)
Again, we take limits from 0 π‘π πΏ when solving the integral equation to eliminate
exponentials with a function of πΏ in order to make solving for the solution easy.
35. HOW DO WE DEAL WITH CYLINDRICAL CO-
ORDINATES?
We know that for an insulated cylinder where there is no heat loss by
convection from the sides, the governing PDE equation is
πΌ [
π2
π
ππ2
+
1
π
ππ
ππ
] =
ππ
ππ‘
The boundary conditions are
π = ππ ππ‘ π = 0
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 PDE into an integral equation
πΌ [
π2
π
ππ2
+
1
π
ππ
ππ
] =
ππ
ππ‘
πΌ β« (
π2
π
ππ2
) ππ
πΏ+π1
π1
+ πΌ β« [
1
π
(
ππ
ππ
)]ππ
πΏ+π1
π1
=
π
ππ‘
β« πππ
πΏ+π1
π1
We then go ahead to solve and find πΏ as before.
β« (
π2
π
ππ2
) ππ
πΏ+π1
π1
= [
ππ
ππ
]
πΏ + π1
π1
= β
ππ β πβ
πΏ
[π
β(πβπ1)
πΏ ]
πΏ + π1
π1
=
ππ β πβ
πΏ
(1 β πβ1
)
ππ
ππ
= β
ππ β πβ
πΏ
π
β(πβπ1)
πΏ
β« [
1
π
(
ππ
ππ
)] ππ
πΏ+π1
π1
= β
(ππ β πβ)
πΏ
β«
1
π
π
β(πβπ1)
πΏ ππ
πΏ+π1
π1
β«
1
π
π
β(πβπ1)
πΏ ππ
πΏ+π1
π1
= π’π£ β β« π£
ππ’
ππ
ππ
37. πΌ
(ππ β πβ)
πΏ
(1 β πβ1
) + πΌ(ππ β πβ)(
πβ1
2πΏ + π1
β
1
πΏ + π1
) =
ππΏ
ππ‘
((ππ β πβ)(1 β πβ1) + πβ)
We go ahead and solve for πΏ
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.