Distributed Parameter Identification in Drying Equations

Copyright © IFAC 3rd Symposium Control of Distributed Pa rameter Systems T oulouse . France . 1982

DISTRIBUTED PARAMETER IDENTIFICATION IN DRYING EQUATIONS R. Bertin and S. Boverie Laboratoire d 'Automatique C.R.A.I. Universite de Bordeaux I, 33405 Talence Cedex, France

Abstract. The purpose of this paper is to present an identification method of nonlinear coefficients from drying measurements. This study from examples shows the hardness of distributed parameter identification in drying and particularly for foodstuff. The nonlinear coefficient to identify depends on function of state variables, in our case we identify the mass diffusivitycoeff~ cient. The identification method, using variational approach and Lagrangian formulationconsis~ in calculating the discretized gradient for minimizing a least squares criterion. The gradient is obtained from solutions of partial differential state and adjoint equations. The identified function is found in discretized points for the case of different foodstuff. Keywords. drying; modelling; identification; partial differential equations.

INTRODUCTION with often in literature (Kacki and Niewierowick, 1977; Bertin and Srour, 1980; Pierronne, 1977; Srour, 1980). In this paper we dealt with the identification of the mass diffusivity coefficient which is a function of water content and temperature of the product. One of the chosen products is beetroot: because it is difficult to get distributed measurements for small products (Vaccarezza and other~ 1974; Vaccarezza and Lombardi, 1974; Chu and Hustrulid, 1968). Another way is to use only average values for the overall product to carry on the identification.

The drying is common to many industries and it is an important factor of cost; then drying processes must be optimized. Therefore, it is necessary to know the mathematical model and to identify unknown parameters. (Kacki and Niewierowicz, 1977; Bertin and Srour, 1980; Pierronne, 1977, Srour, 1980). The problem comes from the fact that drying is a complex operation. Then mathematical models consist of numerous equations (Luikov, 1966) with different coefficients which are difficult to measure. The Luikov's model often used is :

r at*

p

= 'V .

(am' 'YW)

+ 'V .

(am 6 .

aT

C* -----E s p at*

'V .( A. 'VT ) + £

p

'V

T ) p

t,h p -aw v s at*

(1 )

MODELLING For our example we take the Luikov's model to deduce a simple model, taking the coefficient 6 equal to zero and taking an average value over the product volume of the equation (2) seen above with boundary conditions at the surface :

(2)

Whe re W is water content, T temperature of product, t* time , p p s' C*p ' E , 6 , t, h , A are different coefficients, v a the mass diffusivity coefficient~ A simple form, used in modelling of foodstuff drying consists in taking only the first equation where 6 is put to zero or by adding a s e cond equation for the average temp erature of the product . We take this s e cond form in this paper. The drying process identification is not dealt

(3) (4 )

a (T a -T sur f)+(1- E) t, h v m(t*) Where ; , 0., £ , bh are constant, m(t*)is v the mass flux, Tsurf surface tempera97

R. Bertin and S. Boverie

98

regarding our drying model :

ture, Ta Air temperature. We obtain : aw = at*

'V.

{

(a . 'VW) withW=Weqat the m surface aT

p

R C* ---.-J2 s v p at*

=

Cl

(5)

(T -T )+6h p R aw a surf v s v at * (6)

Where R is the ratio volume-surface, v W average moisture content and T av~ rage temperature over product vo~ume. We use uniform initial conditions and an estimation of surface temperature by taking E equal to zero with a parabolic profile for a sphere :

(i) We define a leaS:. squares criterion between measured and theoretical values J (a), a is the function to identify. (ii)From state equation in our case -- we find a variational formulation e(y, p, t; a) (iii)From these formulae we pose the ---Lagrangian :

~(y,p;a)

+~:e(y,p,t;a)

= J(a)

dt

(iv) From equalling the Lagrangian Gderivative / y, to zero; we obtain the adjoint equation. (v) Then the gradient can be calculed from the LagrangianGrlerivative/a. For the example of drying in dimensionless form we get : -criterion with z = ~ where vation operator :

Dimensionless form of the model To simplify calculations on computer and to obtain a simple form of the model, we normalize variables and we study the model in space Q and tIE [O,TJ with.

Lf '"

L

J (a)

J:

T" 1 -'V. (a. 'Vy)

o

(x, t) €

xJ 0 , T [

Q

+

t

EJO,T[ Ysurf = Yeq

Initial conditions

y(x,O)=y =1 o u (0)

= uo

With dimensionless variablest[0,1] where TF final time, R characteristic length:

a = a TF/R2 and y m

Meas Q

1

-

Yi

'Z u ) 2 dt

o

(z- -

y

z_)2 dt

(9)

y

- From the Lagrangian we deduce the adjoint equations : aP1 - ~ - a

(y, u) 6P1- A·Y(y,u)p2 / MeaS Q k

-

2 B1

2. (z i:.1 Yi

(t)

- 2 BM(z_(t) - '"z_(t) y y

y dx

The normalized mass diffusivity coefficientaro identify is a function of state variables y and u. We suppose that: a (y,u)=b(y) c(u) where c(u) is a known function (Arrhenius formula).

u

) 2 dt

z

Where z values are qot from calculated model values and i from measurements; B1 , B2 , BM' are constant coefficients.

t=t*/T , x=x*/R, y=W/W o , u=Tp /T ao F (subscript 0 : initial value) and we pose :

BM]

(8)

Boundary condition

(z

-

Yi

obser-

T

(7)

A(y, u)

(z

i=10

ce

t

6 *(x-x. ) ~

)/Meas r.

(10)

aP2 u - ~ - A· (y, u)P2 + C(t)

~ - 2 B2 (Zu(t) - ~u (t»

(11)

RESULTS OF IDENTIFICATION The given identification method has been studied in different papers (Chavent, 1980; Burger and Chavent, 1979; Chavent and Lemonnier, 1973). We only give the different steps and results

with

C (t)

The superscrit .y indicates derivative

Distribut ed Parameter Id e ntification in Drying Equ a ti ons

99

/ '1 .

real case of drying process.

The superscrit .u indicates derivative / u.

The next step would be to perform the identification using only average values as in the real experiments of shelled corn drying.

Boundary condition; Pl I

J

r

= 0

on L. = r x 0 , T [ Final conditions Pl (x,T)= 0

REFERENCES

and P2(T) = 0 From Lagrangian formulation we obtain the gradient

aJ -

ab

T

=

J

0

c(u)

(12)

In the studied examples we suppose a one dimensional approach, i. e. for a sphere we take radial coordinate and for a semi-infinite plane-shaped solid coordinate along thickness. For identifying the mass diffusivity function in discretized points (piecewise linear function), we integrate state and adjoint equations and calculate the gradient using conjugued gradient algorithm. In a first step, to test the method we have used a known mass diffusivity coefficient (Chu and Hustrulid, 1968) see results on Figure 1 for shelled corn drying. The results are given for product temperature assumed to be air temperature. In a second step using distributed measurements given in reference (Vaccarezza and others, 1974; Vaccarezza and Lombardi, 1974), we identify the mass diffusivity function in several steps, Fig. 2-3, i.e. to obtain the "true" function; we give first a constant value given by reference (Vaccarezza and others, 1974; Vaccarezza and Lombardi, 1974) and after different linear functions as initialguesses to verify the results. We get results in good agreement and a better gradient with last initial condition (Initial condition 4).

BERTIN R. and SROUR Z. (1980). Ident~ fication of a distributed process of heat and mass transfer using a multilevel approach, Proc. 5th IFAC Symp on identification and system parameter identification 1979, 1257-1263 Pergamon Press. BURGER J. and CHAVENT G. (1979). Ide~ tification de parametres repartis dans les equations aux derivees partielles, R.A.I.R.O. Automatique / Systems Analysis and contro~ 22, 2, 115 - 126. CHAVENT G. (1980). Identification of distributed parameter systems : about the output least square method its implementation and identifiability. Proc. Vth IFAC symposium on identification and sy~ tern parameter estimation Darmstadt 1979, Pergamon Press. CHAVENT G. and LEMONNIERP.(1973). Iden tification de la non linearite d'une equation parabolique quasilineaire, report Laboria nO 45. KACKI F. and NIEWIEROWICZ T. (1977). A hybrid system of identification of industrial processes of mass and heat transfer, Digital Computer Applications to Process contrd. Van Nanta Lemke (Ed), IFAC and North Holland, 607 - 615. LUIKOV A. V. (1966). Heat and mass transfer in capillary porous bodies, Pergamon Press. PIERRONNE F. (1977). Modelisation et identification d'un processus de sechage, These Universite Bordeaux 1.

CONCLUSIONS This paper deals with an identification method to identify a nonlinear coefficient, the mass diffusivity coefficient, in distributed drying processes. Two drying examples have been carried on. The first one concerns the drying of shelled corn for testing the method with true function which is known. The second one using smooth experimental distributed measurements from the drying of sugar beetroot permits to know the identification results in a

CHU SHU-TUNG and HUSTRULID A.

(1968). Numerical solution of diffusion equations, Transactions of the ASAE, 11, 5, 705 - 708.

SROUR z. (1980). Identification et optimisation d'un processus de sechage, These Universite Paul Sabatier de Toulouse. VACCAREZZA L.M., LOMBARDI J.L. (1974) Heat transfer effect on drying rate of food dehydratation, The Canadian Journal of Chemical--Engineerin~~, 576 - 579.

<:

am x 10

. ---0

10

o>-

Identified function True function

o o

o

0>::x::u MM

4

Iteration

dimensionless state variable Y

0

:UN HN

":1>-

M

t"'

e.. •

• :s: 3

t=o hr

-------- - -- -------------. ' ....- .....,...,. ______ Identified values ~ .. ,--

2

Iteration 15

experimenta 1 results.

,./

Dimensionless water content

o

,I

•

2/11

4/11

am x 10

6/11

,t

Y-Y eq

8/11

I

10

.---.

4

Iteration

Identified function True function

I.

,'1 /

..... . -

""

; I·

,~/

,"

, .'\

...

......

Dimensionless space variable x

.d

.~

//

:u

!>': tJ t-'·H ::l

(1)e.. r1"'

o

\

?" III ::l

Hlo.

',t=2hr \

~.--

,r

'"

en El s:: 0

Ul t-'. III en

'

,\, \\

,,

,

I1 ' .' I

/,'

\\

'1r1"

s::

tr'1 (1) (1)

"'\

Cl)

,, .\\I

r1"El '1 0 o
\',.

o

0.5

":10.

Fig.l - Identification of the diffusion coefficient from distributed values. Drying of shelled corn.

o.s::

Fig. 2 - Identification of the diffusion coefficient from distributed values Drying of beetroot Ta = 81° C

'1 r1" t-'. (1) ::l

OUl

::T ::l III

o

t-'.

f-''1

I 'oD 0. - '1

W'<

L-__

o

~

____________

~t-'. ~~~~___~

-.I::l I Ul

2/11

4/11

6/11

8/11

Y-Y eq

.. .

::l

',0 , \ ' It.. " \,., \ \ \ .

....

n

0-

\\

t=3 hr

Cl It)

::l

\\

/'

0.5

J>.:S:

. >-

t-'. t"'

--- " .. ' ..,. ' ,~ " ,. ',

0.5

-.10

~ tD

o en

\

•• r1 ' " "

15

~

J I

/.

I

I ,

3

2

I

r

1

.

I

I ••

.I I

"' \ , t=l hr

/,/

'oDt"'

Wo

~ Hl

Vl

Cl

o <:

It)

....

.. . It)

Distributed Parameter Identification in Drying Equations

., r .5

101

Norm 11 ~i 11)( 1 cl.

aJ

31)xfO

5

t IT27

I

Dimensionless water content

0

4

i

Y-Y eq

\

3

-.5 2 - 1

Initial condition

Initial condition

4

4

Iteration number

o

10

20

30

b(y)Initial Conditions

2.2~

____________________

~

2.1

2.2

2.1 le f le 2

IC 3 IC 4

2

-

le le --- le -.-- le

2.

~----

Dimensionless water content

o

1 Y-Y eq

1 IT:30 2 IT:30 3 IT:30 4 IT:27

Dimensionless water content

o

Fig. 3 - Identification of the diffusion coefficient from distributed values. Drying of beetroot Ta = 81°C

1 Y-Y eq