 Open Access
 Total Downloads : 94
 Authors : Gokhan Koyunlu
 Paper ID : IJERTV5IS110321
 Volume & Issue : Volume 05, Issue 11 (November 2016)
 DOI : http://dx.doi.org/10.17577/IJERTV5IS110321
 Published (First Online): 29112016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Modeling of the Dynamic Drying System
Gokhan Koyunlu Faculty of Engineering Nile University of Nigeria Abuja/Nigeria
Abstract Differentialalgebraic equations (DAEs) naturally come up in many engineering problems , as well as numerical and analytical difficulties. In this paper , Paper drying unit was modelled by using DAEs in terms of fundamental equations of thermodynamic . Meanwhile, the index of DAE seems to be main numerically difficult. Hence , in order to solve it , the original DAE was transformed into an equivalent DAE with lower index.
Keywords Differential Algebraic Equations, Paper Drying unit, thermodynamics
I. INTRODUCTION
explicit index1. So z can be gotten as a function of y(t) by inverse function theorem.
1.2 Index Reduction
Another way for dealing with the problem of instability is to build up a new equation system by carrying out index reduction on the original DAE system. In order to understand index reduction, we can advise to look at the definition of the differentiation index, see [3]. This index gives the number of times m that depicts how many times we will differentiate following the equation system.
In this paper , a model for the dynamic drying
system ,which is currently used in the pulp industry, has been mathematically developed. Here, The model of Dynamic Drying System is used for only paper drying system cylinders . Dynamic Drying System has four inputs which are Steam, Paper in comes from press size, and Supply air comes from heat recovery system, leakage comes from air system. The Dynamic Drying System has four outputs. One is Exhaust, the other is condense . Lastly paper out is the last point for paper producing with ignoring coating, which means that we are not interested predrying unit. By this modeling , we are able to control total mass of flow and steam .
1.1 DIFFERENTIAL ALGEBRAIC EQUATIONS (DAEs)
Adams, Runge and Kutta has developed the numerical solution of ordinary differential equations. The theory for Differential Algebraic Equations (DAEs) has not been studied as the same as ODEs at the past . It was firstly studied by Gear and Petzold [1]. Guzel,N and Bayram, M [2`] has been introductorily depicted DAE as follows
(, , ) = 0 (1.1)
= (, )
0 = ()
with x being the independent variable. When we are able to rearrange equations, then it will be called the primary ODE. The principle of index reduction is to differentiate the equation system. It will give us a new set of equations, so that a new equation system with index one lower can be set up using algebraic manipulations. This reduction will be continued in following steps for lowering the index of the system. Finally this reduction enable us the use of methods for lower index problems.
1.7 Modelling of Paper Drying Unit
Ghosh [3] has been giving fundamentals of paper drying , Actually there are two main drying units , one is pre drying , the other is drying . Hereby, we assume that pre drying unit is ignored
`
`
Here
is singular. The rank and structure of this
Condense
Condense
Exhaust
Exhaust
Paper out
Paper out
Paper in
Paper in
Jacobian matrix may be dependent on the solution of y(t), and we will always assume that it is independent of t in order to make essay
1.1 Semi Explicit DAEs
Supply Air
Supply Air
Leakage Air
Leakage Air
= (, , ) (1.2) 0= (, , )
The index is 1 if
is nonsingular, Thats why
this is a special case of (1.1) . We can divide into differential variables x(t) and algebraic variables z(t) if DAE has semi
Figure 1.1 The Open Scheme of Drying Model System.
Input parameters
The mass flow main stream
The mass flow paper in The mass flow of the supply air The mass flow of the leakage air
Output parameters
The mass flow Exhaust
The mass flow Condense
The mass flow Paper out
2.0 Mass and Energy Balance For The Dynamic Drying System
Mass Balance for the Dynamic Drying System
= + + +
= + = + (2.7)
The Differential Algebraic Equations defined by above Eqs. have the following dynamic variables:
= [, ] (2.8)
And the following algebraic variables:
= [, , ] (2.9)
By applying the following DAEs
y' (t) f ( y, z, t) 0 g( y, z, t)
Where y(t) contains the differential variables and z(t) the algebraic variables. We can write the dynamic drying system as follows.
()
(2.1)
Energy Balance for the Dynamic Drying System
+ +
= [ + + ]
+ 0 0 1 ] = (, , ) (2.10)
[0 0 [0 0
= + + +
(2.2)
Now, Eqs. (2.1) and (2.2) are not sufficient for our
and
0 1 0 1 1 0
0 0 0 .y h pv h pv 0.z g( y, z, t)
drying part modeling. Hence, we attempted to write this
1 f f s s
modeling in terms of differential algebraic forms as follows
V 0 0 v v 0
the following algebraic equations
Total mass of flow and steam in system
f s
i.e.
(2.11)
M = Mf+Ms (2.3)
1 1 0
Total Energy in system
g h

pv h pv 0
z
f f s s
The following equations ,which are given in [4] and [5] , are used to formulate on the basis internal energy , U(t) , and specific energy u(t) ;
vf
vs 0
(2.12)
U (t) =M (t).u (t) (2.4)
The last column of (2.12) is 0 . Therefore is
u (t) =h(t)p(t).v(t) (2.5)
where h(t) is enthalpy , p(t) is pressure , v(t) is
singular. The DAE systems index is greater than 1, expected the integration can cause difficulties.
we are
specific volume
= + = +
( ) + ( ) (2.6) Total volume of Flow and Steam
2.3 Index Reduction of DAE for the Dynamic Drying System
Now , our aim is that we would like to reduce the index of DAE such that we can write the DAE in terms of Ordinary Differential Equation.
y(t) = f(y, t)
0 = g(y, z, t) (2.13)
h h h
m (1
pin ) m
(1
sup ) m
(1
leak )
pin h
sup h
leak h
s s s
With the following algebraic variables:
m (1 hexh ) m (1 hcond1 ) m
(1 hpout ) dM 1 dU
exh h cond1 h
pout
h dt h dt
T
T
y = M, Ms , Mf , U
s
s
zT m
(2.14)
(2.15)
s s s s
(2.22)
Eqs. (2.19)(2.20) and (2.22) form ODE which can be written in the form
Firstly we will differentiate the mass balance for the dynamic drying system Eq. (2.3)
A( y,t).y B( y,t).y C
(2.23)
dM dM f dM s
y y0 for t t0
(2.24)
dt dt dt
(2.16)
y'T
dM , dM s , dM f , dU
dt dt dt dt
And we will differentiate the total volume of the dynamic drying system Eq. (2.7)
(2.25)
T
T
y M ,Ms , M f ,U
0 dV v . dMs M . dvs v . dM f M
dvf
dt s dt
s dt
f dt
f dt
(2.17)
and
= [ ] (2.26)
And by applying the chain rule for differentiation
where
0
0
0
0
1 1
1 0
dMs
dvs dp
dM f dvf dp
vs v f
0 v .
M . .
v . M . .
A 0 (h p.v ) (h p.v ) 1
s dt
s dp dt
w dt f dp dt
s s f f
(2.18)
Differentiation and reduction of the total energy content in the Dynamic Drying System Eq.(2.6)
1 0 0
1
hs (2.27)
0 0 0 0
0
dp . dvs
dp dv f
0
dU dM w
dp dhf
dvf
dt dp dt dp
B
(hw p.vw ) M f . .
p. v f
dp dh dv dp dh dv
dt dt dt
dp dp
0 ( f p. f v ) s p. s v 0
s
s
dt dp dp f dt dp dp
dM dp dh dv
0 0 0 0
(h pv ) s M .
s p. s v
s s dt
s dt dp dp s
(2.19)
(2.28)
Rewriting Eqs. [2.1] and [2.2] yields
CT
0, 0, 0, m
(1 h pin ) m
(1 hsup ) m
(1 hleak ) m
(1 hexh )
pin h
sup h
leak h
exh h
m (m

m m
) m m m

m dM
m (1 hcond1 ) m
s s s s
(1 hpout )
s pin
sup
leak exh cond1
cond 2
pout dt
cond1 h
pout h
m 1 (m h


m h

m h
) (m h

m h

m h
(2.20)

dU )
s s
(2.29)
s
s
hs
pin pin
sup sup
leak leak exh exh cond1 cond1
pout pout
dt
CONCLUSION
By using these equations , we can eliminate and we can write as following
(2.21)
ms ,
We used Differential Algebraic Equations for Simple Paper Drying Modeling in Pulp Industry . By the way , we are able to control the mass of steam , flow and energy in the system
REFERENCES

Asher, M. and Petzold, R., Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations. USA, 1998

Guzel, N. and Bayram, M., Numerical Solution of Differential Algebraic Equations with Index 2, Applied Mathematics and Computation, Vol 174. pp12791289, 2006

Ghosh, A.K., Fundamentals of Paper Drying, Evaporation Condensation and Heat Transfer, pp. 536581, In Tech, Crotia

Guggenheim, E.A., Thermodynamics, North Holland Publisching Company, Amsterdam, 1959

Venkanna, B. and Swati, B., Basic Thermodynamics , PHI Learning Pvt. Ltd, 2010