Fuzzy pid based Temperature Control of Electric Furnace for Glass Tempering Process

To model the delay time: 
The process involves the flow of ordinary glass at a constant velocity v, through a perfectly 
insulated furnace of length L. The glass is initially at a constant temperature T0, uniformly, 
throughout the entire furnace length. At time t = 0, the temperature of the glass coming in at 
the furnace inlet (z = 0) is changed to
( ) we are now interested in investigating how the 
temperature at the furnace outlet (z = L) responds to such an input change [34]. 
It is easy to see that any temperature changes implemented at the furnace inlet will not be 
registered instantaneously at the outlet. This is because it will take a finite amount of time for 
an individual glass element at the new temperature, to traverse the distance L from the inlet to 
the outlet so that its new temperature can be observed. 
Since the glass velocity is v, and it is assumed constant, the time for each glass element to 
traverse the required distance from the furnace inlet to the outlet is L/v. Since the furnace is 
also assumed to be perfectly insulated, there will be no heat losses, or any other changes for 
that matter, experienced by each glass element in the process. Thus, any changes implemented 
in the inlet will be preserved intact, to be observed at the outlet after the time required to 
traverse the entire pipe length has elapsed [34]. 
Mathematical Model and Analysis:
 
Consider an element of glass of thickness whose boundaries are arbitrarily located at the 
points Z and along the length of the furnace. 
For a furnace of constant, uniform cross-sectional area A, an energy balance over such an 
element gives [34]:

---

Fuzzy PID Based Temperature Control of Electric Furnace for Glass Tempering Process
M.Sc. Thesis, Addis Ababa University, December 2016 
27 
(
) 
(
) 
--------------------------3.6 
Where: 
and
are, respectively, the glass density, and specific heat capacity 
T* is the usual, reference temperature 
By dividing through by , and taking the limits as , Eq. (6) becomes: 
( )
( )
-------------------------------------------------------------------------3.7 
The system whose dynamic behavior is represented by PDE's is known as a distributed 
parameter system. Thus the process currently under consideration is one of such systems. 
Let us now define the deviation variable: 
( ) ( ) ( )
-------------------------------------------------------------------------3.8 
Where we recall that T(O) = T0 is the initial glass temperature at t = 0 along the furnace length. 
In terms of this deviation variable, the process model in Eq. (3.7) is 
( )
( )
----------------------------------------------------------------------------3.9 
To develop a transfer function model for this process, we must now take Laplace transforms of 
the expression in Eq. (3.9). From the definition of the Laplace transform, we have [34]: 
( ) ∫
( )
-----------------------------------------------------3.10 
Since the first term is simply the Laplace transform (with respect to time) of the indicated time 
derivative. For the second term in Eq. (3.10), interchanging the order of integration and 
differentiation (an exercise which can be shown to be valid in this case) gives: 
∫
( )
∫
( )
( )
--------------------------3.11 
So that Eq. (3.7) becomes: 
( )
( )
----------------------------------------------------------------3.12 

---

Fuzzy PID Based Temperature Control of Electric Furnace for Glass Tempering Process
M.Sc. Thesis, Addis Ababa University, December 2016 
28 
The solution of the ODE in Eq. (3.12) is easily obtained as: 
( )
(
) 
---------------------------------------------------------------------------3.13 
We may evaluate the arbitrary constant C1 by using the entrance condition at z = 0, which is 
( ) ( ). 
So that Eq. (14) becomes
( ) ( ) 
(
) 
-------------------------------------------------------------------3.14 
Eq. (3.14) represents the relationship between the Laplace transform of the inlet temperature 
and the Laplace transform of the temperature at any other point z along the length of the 
furnace. Observe further that the term z/v is the ratio of the distance traveled in arriving at this 
point, to the flow velocity, a direct measure of the time taken to traverse the indicated distance. 
We are particularly concerned with the situation at the furnace exit; i.e., at z = L. In this case, 
Eq. (3.14) becomes: 
( ) ( ) 
(
) 
---------------------------------------------------------------3.15 
And if we define as D the time it takes for a glass element to traverse the entire length of the 
furnace, i.e.: 
Then we have
: 
----------------------------------------------------------------------------------------3.16 
( ) ( ) 
) -------------------------------------------------------------3.17 
Thus the transfer function relating the process input (changes in the inlet temperature, y(0,s)) 
to the process output (changes in the exit temperature, y(L,s)) is clearly seen from Eq. (13.17) 
to be given by:
( )
-----------------------------------------------------------------------------3.18 
We may now use Eq. (3.18) to investigate how this process will respond to changes in the input 
function u(t) = y(0,t). This task is seen to be very easy once we recall the effect of the 
translation function of Laplace transforms. We have, for any input u(t) that 

---

Fuzzy PID Based Temperature Control of Electric Furnace for Glass Tempering Process
M.Sc. Thesis, Addis Ababa University, December 2016 
29 
( )
( )
-----------------------------------------------------------------------3.19 
Which, when converted back to the time domain gives the result: 
Y (L, t) = u (t- D) --------------------------------------------------------------------------3.20 
It indicating that the output is exactly the same as the input, only delayed by D time units.