Identification of the Diffusion Coefficient of a Drying Process

Copyright © IFAC Identification and System Parameter Estimation 1982 , Washington D.C., USA 1982

IDENTIFICATION OF THE DIFFUSION COEFFICIENT OF A DRYING PROCESS R. Bertin and S. Boverie Laboratoire d 'Automatique C.R.A.I., University of Bordeaux I, 33405 Talence Cedex, France

Abstract. A procedure for identifying the mass diffusion coefficient of adrYlng process is proposed. Two simple models are used for the drying of shelled corn, where the mass diffusion coefficient is a function of state variables i.e. moisture content and temperature of the product. The given identification method, uses a v~riational approach with Lagrangian formulation. Results are get from two models for the example of drying shelled corn. The "free" fonction of the mass diffusion coefficient to identify is given in different cases and particularly in the case of real experiments where we have only average values over volume for the water content of shelled corn. Keywords.

drying; modelling; identification; partial differential equations.

INTRODUCTION The most popular mathematical model (paY>tial differential equations) used for drying and particularly for shelled corn drying is the "diffusion" model (Chu and Hustrulid, 1968; Hamdy and Barre, 1969; Vaccarezza and others, 1974; Brunovska and others, 1977):

show the different steps of the method. We pose: g C ~3 with boundary r Let the mathematical model ay at

aW at = v.(am· v W)

v . ( a. vy) =OinQ

on

y lr = Ye

Where hi is water content (dry basis) of the product and a the diffusion coefficient. For practical uti~ization it is necessary to know the mass diffusion coefficient which is a function of water content Wand temperature of the product (which in this simple model is assumed to be air temperature). We give in this paper an other model , also used by different authors (Vaccarezza and Lombardi, 1974; Hamdy and Barre, 1970) which takes into account effect of average temperature of product by adding an ordinary differential equation.

L

nxJO,T[

r x ] 0, T[

y(x,O) = yo(x) The functions y ,y are known and a is the parameter we want t8 identify a E A parameter space, a = a(y). We can know certain measurements i( x,t) on the process. In our case, we suppose to get some average point observation i.e. average value on small domain surrounding the point or average measurements for the overall domain n. We define the observation operator ~€)f:

1 (x,t)

The example of shelled corn has been chosen (i) to test the identification method (ii) because much works have been done to characterize the drying rate (iii) several authors give value of mass diffusion coefficient by fitting experimental data with different models (Chu and Hustrulid, 1968; Hamdy and Barre, 1969, 1970). The aim of this paper is to show a different approach to identify the diffusion coefficient.

and

=

'e9

z( x,t)

(x,t) (x,t)~ n x ]O,T[ =

~y(x,t)

The criterion to minimize can be taken for the identification problem as : 2

J(a) = I lz

-111 where

II Il is a norm

in~

We introduce A dspace of admissible parameters AadCA/the~ the identification problem is : Min J(a) a€.A ad

IDENTIFICATION METHOD This method has been studied by different authors (Chavent, 1980; Burger and Chavent, 1979) we use to simplify a simple model to

For obtaining the variational formulation we take v as a function of n +ffi, equal to zero on r, we mUltiply the state equation 217

R. Bertin and S. Boverie

218

by v

~{

e(y,t,v; a) = (" )

-~v.(a.VY)VdX

v dx

0.

j.

a) = i Iz -

~ 112

+J \

(t,y,p;a) dt

( y: >i

x ]O,T[ lfi(y) = OYi

1

The gradient is given for this example by

rr

a ~ ca = T Vy, vp dt dx 6a aa ~0. ~O

6J

Then

aJ - "

aa

J

I}

E.

The adjoint variable p must be chosen in :

y( s , t)E:[ x}l, TD

For the example chosen we get using Green's formula : if(y,p;a) =

~

JT

i =1

•

0

~

a Vy. Vpdq -

Vy t: Y ,V P

~a

'y.".p d'

Va E Aad C A

<== P ,

ADJOINT EQUATION AND GRADIENT If y = y (a) is the solution of the state equation we have y 6 Y then :

;1 (y

(a), pi a)=

11

Gy differentiating J 6J = -a'«! ay 6y

+

z-~

2

11

=J (a)

(a~(Gateaux

T Vy, vp

dt

0

DRY I NG PROBL EM

in which ~ . are the boundary and initial condition. 1

P = { p: 0. x] 0, T[-+ [R 1p (s, t) = 0

= 1

w . w·1

o we define by Y set of solutions :

Y=

dx

x

0.

All functions and parameters are chosen in order to make sense to all integrals in the last equation. LAGRANGIAN FORMULATION The Lagrangian is composed of the criterion and the variational form of the state equation :

:;t (y,p;

o

In drying of foodstuff there are two mathematical models which describe with accuracy the drying of foodstuff (shelled corn for our example). The first one is the "diffusion" model used for modell ing shelled corn dryi ng Mode 1 I

t

~ = v. (a . v W) at m

W

Weq at the surface

W

Wo

t* = 0

Where Wis the water content (dry basis) of the product, am the diffusion coefficient. For this model I we only can use the mathematical results see~ above. The second model (Model II) we take into account the effect of temperature of product Model 11 :

aW a t*

v . (am . v

= a (T a - Tsurf)\

Ya E Aad derivative);

+

a:t a a 6a

For pEP we choose p to make the part with 6y equal to zero; we obtain by this way the adjoint equation :

W)

Tsurf = T p

+

6 hv Ps Rv

aW a t*

alp Ps RV C*P (Bi / 5 a) - at*

k

ap -at p

a(y) 6p

= -

22: i =1

=0

p(x,T) where x

on

80undary cond it ion

(z.-i.) 1

1

Meas

w·

1

r xJO,T[

0

is the characteristic function

~ i

of subdomain wi surrounding the point i :

Initial condition

W

Weq at the su rface

hi

~oPo

T

P

J

t*

0

In this model Tp is the product temperature, T its average value for the overall volume, RP the ratio volume to surface of the product (~rain) Ps , C*p , a, 6hv , Si are coefficients, Ta air temperature, Tsur f surface temperat~ Waverage value of water-content, t* time.

Identification of the Diffusion Coefficient. of a Drying Process

We give now the dimension~ess form of model 11 only because Model I is a particular case of Model 11 with T 1:

~

ay _ v. (a, Vy)

=0

Y(x,t)E

It

xJO,T[

at

at Boundary condition! ysurf = Yeq

¥ Er

J

s

x 0, T[

u(O) = Uo {x E It Where dimensionless variables and parameters are get by t

t*/TF

t* E. [O,TFJ

tfc[o,1j

x

x*/R;

x* ~ [O,RJ

x~

u = Tp/Ta 0

2

[0, 1J

(subscript 0 : initial value)

y Meas

It JIt y dx

The dimensionless diffusion coefficient is function of the two state variables y and u, we suppose that : a(y, u) = b(y) c(u) Where c (u) is a know function (type Arrhenius formula). Now using the result of the last paragraph we obtain by the same way the criterion :

J(a)=B1~1; -z )2dt + B2 ) f'T(z u-~u)2 dt i =1 y i Yi 0

jT

o

+ BM 0 (Zy _Zy)2dt Where B1 , B2 , BM are constant coefficients. From the lagrangian using the same method as in the last paragraph we get the adjoint equations : aP1

- at - a(y,u) l'I P1 [

A'YCY",u) P2/ Meas

It

k

::-2B1

~ (z (t)-z (t)) Xw i -2B M(z-(t)-£-(t)) i=1 Yi Yi Meas wi y y /Meas D

aP2

- ~ - A· u (Y,u)P2+C(t)= -2B 2(zu(t) - zu(t)) C( t)

=

1

a' u vy. vp 1

The superscript The superscript

J

T

=

c(u) vy . VP1 dt

0

RESULTS

y(x,O) = Yo = 1 and

a = am TF/R

~

E'£ JO,T[

Initial condition:

y = W/W o

Boundary condition: P1\r =0 on L = r xJO,T[ Final conditions: P1(x,T) =0 and P2(T) = 0 Then the gradient :

ab

Vt

~ = A (y, u)

219

dx

u indicates derivative / u.

Y indicates derivative / y.

In the studied example i.e. shelled corn, we suppose the grain as a sphere, then we use one dimension approach. For obtaining the "free" function of the mass diffusion coefficient to identify in discretized points it is necessary to integrate the state and adjoint equations. We use implicit scheme versus time and finite difference versus space. The function to identify is approximated as a continuous piecewise linear function of y. The algorithm of minimization used is the conjugued 9radient. For testing the results obtained from this identification method, we have chosen the function to identify i.e. the mass diffusion coefficient equal to the function given by Chu and Hustrulid (1968). They have found this function by fitting the model I to experime~ tal data. By this way the simulation of drying model I using the results of Chu and Hustrulid (1968) permits (i) to get distributed values taken as "data" in the identification (ii) to have "true" value of the function to test the identification method (iii) then to compare identified and true values (iv) to get average values for the overall volume as in the real experiments (where it is very difficult to have measurements in distributed points inside the product). The results of the different identifications using model I ("diffusion" model) are shown on Fig. 1 to Fig. 6.The figure 7 shows the result of identified value using model 11 where the mass diffusion coefficient: a = b(y) c(u) with c(u) = d exp(-B/u) The coefficients d and B are constant. From these curves we can see the good agreement between state variables computed and "experimental" values distributed or averaged over the grain volume. The different tests on the identification method consist to give several guesses for initial condition of the mass diffusion coefficient and to compare the results with true values one in two cases. The first case takes into account only distributed va 1ues, the second one on 1y average va 1ues of state variables. The best results are obtained with linear function as initial guess and a better result with distributed values than with average values for the overall grain. COMClUS IONS The identification method presented there permits to find the mass diffusion coefficient in different cases. The different results

R. Bertin and S. Boverie

220

show that the identified mass diffusion coefficient is in good agreement with the truevalue in case of the distributed values if they are numerous. The results show that with only average values over the grain volume, which is the case in drying shelled corn, it is possible to get results in agreement with true values. We have applied this identification method to two models but others examples of drying can be carried on with this method. REFERENCES Brunovska , A., Brunovsky, P., Ilavsky, J., (1977). Estimation of the diffusion coefficient from sorption measurements. Chemical Engineering Sciences, 32, 717~. Burger, J., and CHAVENT, G., (1979). Identification de parametres r~partis dans les equations aux derivees partielles, R.A.I. R.D. Automatique / Systems jl.nalysis----ancrcontrol, 13, 2, 115 - 126. Chavent, G., (1980). Identification ofdistributed parameter systems : about the output least square method its imple~entation and identifiability. Proc. Vth IFAC symposium on identification and system parameter estlmatlon Darmstadt 1979, Pergamon Press. 0.9

708.

--

Hamdy, M.Y., and BARRE, H.J., (1970). Analysis and hybrid simulation of deep bed drying of grain, Transaction of the ASAE, ll, 6, 752 - 757. Hamdy M.Y. , and BARRE M.J. (1969). Evaluating film coefficient in single kernel dryin~, Transaction of the ASAE, .!i, 205 - 208. Vaccarezza L.M., Lombardi, J.L., and Cherife , J., (1974). Kinetics of moisture movement during air drying of sugar beetroot, J. Fd technol. 9, 317 327. Vaccarezza L.M., Lombardi J.L. (1974). Heat transfer effect on drying rate of food dehydratation, The Canadian Journal of Chemical Engineerlng 52, 576 - 579.

Dimensionless state variable y

~--r-_

Experimental value Final value

Dimensionless space variable

x

Chu Shu-Tung and Hustrulid A. (1968). Numerical solution of diffusion equations, Transactions of the ASAE, 11, 5, 705 -

0.5

+-------------~~--+-~--~~--~---.

1 .8.6

1

.9

t

.4.2

0 .2.4.6

&8

Average dimensionless state variable y Fig. 1 - Identification of the diffusion coefficient from distributed values. Model I drying of shelled corn

.8 .7

.5

Dimensionless time ~O----------------~on.c5----------------~-----t~~

Identification of the Diffusion Coefficieht of a Drying Process ClJ

Norm 11 ab 11

X

221

8 10

J

0.25

0.2

0.15

0.1

1IIJlI'.:-o:--

tion number

Iteration number 1-_~==~==~~ o 10 15 5

am x 10 10

.-.

Fig. 2 - Identification of the diffusion coefficient from distributed values. Model I drying of shelled corn. 1010 I_I Identified function Identified function True function am x __ True function

4

Iteration 0 3

4

Iteration : 0

3

2

2

Dimensionless water content

Dimensionless water content 2/11 4/11 6/11 8/11

4 3

am x 10 10

o 2/11 4/11

6/11 8/11

4

Iteration

3

2

2

Dimensionless water content

o '--2'-1-1-4-/1-1-6-/-1-1-8-/1~1--1""'· Y-Yeq Fig. 3- Identification of the diffusion coefficient from distributed values. Model I drying of shelled corn

Iteration 15 ~

d

.

.h Dimensionless water content

o L-----------PY-yeq 2/11 4/11 6/11 8/11

1

Fig. 4 - Identification of the diffusion coefficient from distributed values. Model I drying of shelled corn

R. Bertin and S. Boveri e

222

0-. 4

10 am x 10

Identi fied value True value

._e

4

Identi fied value True value

3

3

2

2

o

3 m Iterati on 20

o

Y-Y eq

2/11 4/11 6/11 8/11 a x 10 10

~

3

Iterati on 10

2

/.:7 .. - . Identi fied value .,'''' True value

e_e

I~~'

o

Y-Y eq 2/11 4/11 6/11 8/11 Fig. 5 - Identi ficatio n of the diffusi on coeffi cient from average values . Model 1 drying of shelled corn

o

•

bey)

Identi fied value True value

__' __'--'-T, -r'--'-- '______ _ ~

2/11 4/11 6/11 8/11

Y-Y eq

Fig. 6 - rdenti ficatio n of the diffusi on coeffi cient from average values. r40de 1 1 dryi ng of she 11 ed corn

.--, Identi ficatio n value True value

15 Iterati on 0

10

5

Y-Y eq

am x 10 10

/_./

2

2/11 4/11 6/11 8/11

Iterati on 15

Dimensionless water conten t

Fig.? - Identi ficatio n of the diffus ion coeffi cient from distrib uted values . Model 11 drying of shelled corn