Modeling of the Dynamic Drying System

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 index-1. 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 pre-drying 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

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

2. Guzel, N. and Bayram, M., Numerical Solution of Differential Algebraic Equations with Index 2, Applied Mathematics and Computation, Vol 174. pp1279-1289, 2006

3. Ghosh, A.K., Fundamentals of Paper Drying, Evaporation Condensation and Heat Transfer, pp. 536-581, In Tech, Crotia

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

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