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Identification of a Distributed Process of Heat and Mass Transfer Using a Multilevel Approach
IDENTIFICATION OF A DISTRIBUTED PROCESS OF HEAT AND MASS TRANSFER USING A MULTI LEVEL APPROACH R. Bertin and Z. Srour C .R .A.I., University of Bordeaux I, Talence, France
Abstract. A two -level approach is presented for identification of heat anq mass tranfer process. An example of tunnel-drier with batchwise operation is discussed. The drier model, a system of nonlinear partial differential equations, is decomposed into two coupled subsystems to perform two identifications (i) one at small scale in a laboratory oven (ii) the other at large scale in the overall tunnel. The second identification takes into account as initial guesses the results of the first one to reduce computing time. The results are shown for the two identifications and particularly for a parallel-flow tunnel-drier. This method shows a way to identify a complex distributed drying process on small computers run by acceleration algorithms. This paper is a step towards optimal design. Keywords. drier; distributed parameter systems; modelling; identification ; multi variable systems ; predictor-corrector methods ; optimisation. INTRODUCTION The drying of materials is an unit operation common to many industrial processes. From the economical viewpoint the drying process is an importance factor of cost (Flink, 1977) therefore drying process must be optimized. In order to design and optimize drie r it is desirable to have a mathematical model and to identify the unknown parameters.
in numerous papers which can be found in survey of Polis and Goodson (1976). In this paper, we present the results of the identification of parallel flow tunnel-drier with two phase flows: air and product, for the product a batchwise operation is done. To carry on identification of distributed parameter system by small computers several conditions are necessary for reducing the computational time :
A two-level approach is often used for the drying process to verify the drying model (Bertin and others, 1978 ; Kacki and Niewierowicz, 1977 ; Pierronne, 1977) :
- Initial guesses near optimal values (given by the first identification in this paper)
- First identification from results at small scale in a laboratory oven - Second identification from results at large scale in the drier From conservation laws, i.e. energy and mass balances for our problem, we obtain the overall model of drier. It is a coupled system of partial differential equations, with ones merely hyperbolic (Bertin and others, 1979 ; Nordon and Bainbridge, 1971). In industrial processes hyperbolic systems are numerous ; heat exchangers, chemical and biological reactors, refiner of pulping process, progressive and batch drier. First order hyperbolic system are usually resolved ftom Lax-Wendroff scheme, characteristic method, method of lines or method of weighted residuals. Few authors have dealt with the identification of drying process, on the opposite the identification of hyperbolic system has been shown 1257
- Simulation of the model by algorithms with large step, for example predictor corrector with step size c ontrol (Gear, 1971) - Optimization by algorithm with high speed of convergence, for example conjugate gradient (Seinfeld and Chen, 1974) THE TUNNEL-DRIER The tunnel-drier consists of two compartments: - heating system with blowers and heater - drying sys tern wi th trays and trucks The product is placed on trays. The batchwise operation consists of placing trucks into the drier at time intervals (feed time) about one
R. Bertin and Z. Srour
1258
hundred and five minutes long and overall drying is about sixteen hours long. In this example, we show the cocurrent flow type drier with nine trucks and the product is frui ts (Keey, 1972, 1978). Each pos i tion taken by truck can be considered working as a batch drier. (Fig. 1).
, I
I
~-
G,W+Jll,T +dT.
I
••
t-:';j- --- - - ---
Airflow
Product flow
x + dX
-
- tJDDDD~
Fig . 2. The energy ( ~) and mass ( . . ) transfers between air and product along the drier
Fig. 1. The drier flow diagram shows concurrent flo ws of truck and air
d d) (- + v T + llh w) dt dX [CC; a v
THE MATHEMATICAL MODEL Level 1
The model of product drying is derived from Luikov's model (1966) in capillary porous bodies (System of partial differential equations). In the case of spherical product and with some assumptions, for example parabolic profile of water content W into the product, we obtain an averaging model (Babukha and Shraiber, 1974 ; Bertin and others, 1978). This model derived from averaging over volume of water content Wand temperature product Tp ' gives a macroscopic viewpoint for each particle ; therefore their distributions along the drier will be given by the drier model itself. A product model is often used in engineering papers (Keey, 1972 ; Krisher, 1963) here we use:
E.P-
s dt
s
=
(I+B~
/5)-1 [aCT -T ) a p
s
ilwJ ot
d
d
(ll)
(ii)
(3) X
0
d d G d d L ( - + v - ) (w -)+(- +u-) (W = ) dt dX v dt dX u (Ill)
+
(iii)
0
(4)
(IV) (iV)
In these equations G/v and L/~ respectively represent the dry mass of gas and solid phases per unit of lenght, they have positive value in cocurrent flows and opposite sign in countercurrent flows. Remarks - In batch drying terms (ii) and (iV) are equal to zero. In progressive driers and in stationary conditions (I) to (IV) are equal to zero. The velocity of solid phase is a mean value (for the overall drying in the drier).
u
Level 2 : Plant Model ( 1)
+ llh (p X / a X )""'\ v
(i)
( - + u - ) [(C T ) 1 ] dX dt s p u
Heat and Mass Balances at Product Scale
X X ( p X/a ) C
(l)
~]
The overall one space dimensional model is derived from Eqs (1) to (4) it may be written in normal form :
(2)
In these equations the dependent variables are Tp(x,t) and Ta(x,t) respectively temperature of product and air, W(x,t) and w(x,t) water content of product and air (dry basis). The symbol X indicates functions. Model of the Drier itself For cross circulation drier, ly enthalpy md nass balances applied to both dry gas flow G and dry solid flow L (Bertin and others, 1979 ; Nordon and Bainbridge, 1971) if we suppose, velocities of air v and u of product constant we get: (Fig. 2)
dU at
dU
+ A(x,t,U) dX
=
B (x,t,U)
(5)
Where U is a vector which transposed vector is A is 4 X 4 matrix, B a vector of the same order as U, it depends only of U in this exampie. (Appendix). This mixed initial-boundary value problem gives a solution of the state vector U for (x, t) E 11 X (0, T) with initial and boundary conditions U(O, t)
t>O (for Ta and w only)
( 6)
A distributed process of heat and mass transfer
xEn=
U(x,O)
(7)
(O,l)
product are compared with theoretical values. The parameters are optimized thanks to the criterion J(p) : 4
COMPUTATIONAL ASPECTS
J 259
I
J
2
J(p) = ~ c. ~ ~ e .. k(x.,t.,p) k=l K i=1 j=1 1J J 1
The coupled system of partial differential Eqs. (5) consists of particular equations like hyperbolic equations with single nonzero wave velocity, as the example given for the packed bed reactor by Ritter and others097J). This system is transformed by the method of lines (Ahmad and Carver, 1974) to an ordinary differential system,the method of characteristic was used in (Bertin and others, 1979) In the method of lines we used a mixed scheme (Fig. 3) to take into account this particular system : - At the value x = 0, continuous time integration of the two first equations of Eqs.(5) by multistep Hamming's predictor-corrector method with step size control - The two following steps are run in turn (i) continuous time integration of the two first equations of Eqs.(5) but with step size fixed (at the same values as x = 0 for the same intervals of time Fig. 3). The method is the same Hamming's predictorcorrector method (ii)continuous space integration (with respect to x) of the two last equations of Eqs.(5) by lower order trapezoidal method (with backward finite difference formula for the others unknown derivatives).
- p is the parameter vector posed vector is (a,Bm)
(8)
which trans-
- ck are coefficients to put all partial criteria on the same scale - error eijk gives difference betweep. measurements and model results ; the expression of eijk is :
This criterion is evaluated by the sums of partial criteria related to each dependent variables, k is the indice of each~of th~se criteria. The optimal parameters a and Bm are searched by the discrete conjugate gradient method because of its fast convergence. Primary Identification The product drying is carried out in laboratory oven in the same conditions as the working drier. The experimental results, temperature and water content of product are given from sensors, the others dependent variables, temperature and humidity of air are given values closed to those in the drier. In this case we use as model the subsystem, i.e. the two first equations of Eqs.(5). By identifying we get : A 2 -3 2 a = 66.9W/m. °c Bm = 3.5010 kg/m.s
T
Second Identification
f')@ T +IlT
(j) (i)
T
@
®
I
T -.1T
X ·-.1X
X
X
X+.1X
Pip. 3. The propagation paths of the mixed method This choice of integration methods is necessary to have problems well conditioned in a computational sense (Miranker,1977) because we have stiff equations coming from fast transient heat transfer and slow transient mass transfer. On the other hand they allow to reduce computational effort.
From each position of truck in the tunneldrier we can obtain the experimental values of temperature of air and product and their water content (dry basis) versus time and for a limited number of places in the drier. These values are compared with model values using the criterion J(p). The initial guesses of parameters a and Bm are those of the primary identification. We get by discrete conjugate gradient algorithm given by library subroutine : ~ -3 2 Bm= 5.2510 kg/m.s The results which are presented in Fig. 4 and Fig. 5 are those of a real test during which a loaded test truck, equipped with different sensors, has been run through the drier, step by step numbered J to 9. For computation we have used as boundary conditions the experimental results given by the downstream side of the test truck (Bertin and others, 1979).
IDENTIFICATION The "Model method" (Richalet, 1971) is used as identification method. The parameters to identify are heat transfer coefficient a and mass transfer coefficient Bm' The main dependent variables of the drying process : tempe.... 0:2 ...
".,..0
'!lI....,,.1
1'."~""O""
~n ...... t-oT"'lt-
nf hnt-h
!l;'" !lnrl
CONCLUSIONS Numerical profiles obtained from the theoretical model are in relatively good agreement with experimental data. We find a more large difference, between the two parameter identi-
R. Bertin and Z. Srour
'260
fications, for the mass transfer than for the heat transfer coefficient, it is a consequence of their sensitivities. On the other hand, it is possible using the same method to identify parameters in others types of drier, for example progressive or batch-drier. The method presented in this paper gives a way to identify complex distributed processes of heat and mass transfer on small computers, the algorithms used permit to reduce computational effort. Next step will be to develop them in the way of optimum designing.
d
,
bO O~
at""
I ~I
+
a?
I
bO 3 bO 4
Ql)I
- Dimensionless dependent variables TO
T /T p
P w°
w/w
WO
an
TO
n
W/W
n
X
f
I
b
O
-f
2
la ° (To - TO) + lIh ° f b ol Lap v 2 2J
I
3
SO (WO - wo) m s
- Dimensionless coefficients SO
m
lIhO v
= lIh
= SmZawn /vp s
W
n
W /C T v n s an
(L/G).(v/~)
(J
C /C s
n
WO > I.3/W
I
n
)( 3 (a/a) =
P/P:
I + k WO I
F2
C)(/C s s
I + k WO 2
fI
F 2/3 (I + BiF 1/3) I
f2
FI
f3
WOF - I / 3 (0.622 + w wo)-I n I
f4
F2
fS
F2 F,
-I
5'
=
f
(B~/Bi)3
F -I 2
-2/3
3 FI
-1 / 3
-2/3
2 (m /m 3 )
a
Surface area per unit volume of product (3/Rf)
Bi
Biot number
C
Specific heat
d w
Product desorption coefficient
G
Dry gas flow rate
Z
Lenght of the truck
L
Dry solid flow rate
P
Total air pressure
R f
Dry product external radius
RH
Relative humidity of air
T
Temperature (averaged over volume)
t
Time
u
Truck speed (~ averaged over time)
(m/s)
v
Air velocity
(m/s)
W
Water content in the product dry basis (average over volume)
(kg/kg)
Water content in the air dry basis (average over volume) Longitudinal coordinate
(kg/kg) (m)
O = x/Z
- Dimensionless functions for linking equations b°
WO < I.3/W
Appendix B - Nomenclature
- Dimensionless independent variables to = tv/Z
0.48
FI
F)(
T /T a an
a
=
bO 2
auo
0
w
+
The functions fI to fS can be defined from two functions which take into account of volume shrinkage and specific capacity of product versus WO
The system of Eqs.(S) may be get 1n dimensionless form
auo
n
w
Appendix A
,<» ,
0.4W WO
d
g
(aRf/A) (J/kg.oC)
(kg/s) (m)
(m)
(OC) (s)
- Dimensionless functions WO s
=P
(TO ) d (WO)(0.622 + w WO)/Pw sat surf w n n
TO surf x
A distributed process of heat and mass transfer Greek letters a
B
m
6h
v
Heat transfer coefficient
20 (W / m . C)
Mass transfer coefficient
2 (kg/m .s)
Latent heat of vaporization
A
Thermal conductivity of the product
Ps
Density of bone-dry solids
Air
g
Gas phase
i
Initial or indice
n
Normalized
p
Product
s
Solid phase
Kacki, F. and T. Niewierowicz (1977). A hybrid system of identification of industrial processes of mass and heat transfer. Digital Computer Applications to Process Control, Van Nanta Lemke (Ed.), IFAC and North Holland, 607~615.
(J/Kg)
Subscripts and superscripts a
1261
sat Satured water vapor surf Surface Dimensionless Variable Optimal A, B, C, e, i, j, J, k, p, U are defined from Eqs.(5) to (9). REFERENCES Ahmad, S.Y., and M.B. Carver (1974). A system for the automated solution of sets of implicitly coupled partial differential equations. Proc. Summer Computer Simulation Conference, 104-109. Babukha, G.L., and A.A. Shraiber (1974). Interphase heat transfer in poly-disperse gas suspension flows. Fifth International Transfer Conference Tokyo, 2, 69-73. Bertin, R.,.Pierronne and H.Combarnous(1978). Determination des parametres de sechage de certains fruits compacts. Ann. Technol. Agric. , JJ, 489-500. Bertin, R., F. Pierronne and M. Combarnous (1979). Modeling and simulating a distributed parameter tunnel-drier. J. Food Sci. (to appear). Flink, J.M. (1977). Energy analysis in dehydration processes. Food Technology., 31 77-84. Gear, C.W. (1971). Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, New Jersey.
Keey, R.B.(1972). Drying Principles and Practice. Pergamon Press, Oxfo rd. Keey, R.B. (1978). Introduction to Industrial Drying Operations. Pergamon Press, Oxford. Krisher, O. (1963). Die Wissenschaftlichen Grundlagen der Trockmungstechnik. Springer Verlag, Berlin. Luikov, A.V. (1966). Heat and Mass Transfer in Capillary Porous Bodies. Pergamon Press, Oxford. Miranker, W.L. (1977). The computational theory of stiff differential equations. Report N° 219 7667, Orsay, Univ. Paris
XI. Nordon, P. and N.W. Bainbridge (1971). An analysis of the process of through drying of wet textile materials, Applied polymer Symposium, ~, 1111-1119. John Wiley, N.Y. Pierronne,F. (1977) . Modelisation et identification d'un processus de sechage. Thesis, University of Bordeaux I. Polis, M.P. and R.E. Goodson (1976). Parameter identification in distributed systems: A synthesizing overview. Proceedings of the IEEE, ~, 45-61. Richalet, J., A. Rault and R. Pouliquen (1971). Identification des processus par la methode du modele, Gordon & Breach, London. Ritter, A.B., J.C. Friedly and C.C. Chen (1971). Singular control arcs in the time optimal control of linear hyperbolic systems. Proc. IFAC Symposium on the control of distributed parameters systems, Banff. Seinfeld, J.H. and W.H. Ch en (1974). Estimation of parameters in distributed systems, Proc. AACC Joint automatic control confe~, Austin. Table Values of model parameters T an
87(°C) 3
a
1 .25
C s
6h
2.7
Z
(m/s)
v
R f
W n
v
=
2 0.97910- (m) 1340(J/kg:C)
=
w n
1 (m)
lJ
0.04 77
6 2.510 (J/kg) P
s Bi
=
3 1400(kg/m )
=
1.7910
-2
x a
1262
R. Bertin and Z. Srour
kl
3.78
Cl( g
C g
c
692
2
= 8.42
k2
I070(J/kg.oC) c
c c
3
4
1
=
0
90
80
70
.
\.
"" .
60
"-
...-• - --.......
_....
9
5
Coordinate along the drier X
RH(7.)
40
..
I 30
",.
---.
....
-s--"'2'..... _
20
-- .
......... .. 1
~----
•
• 10
5
9
Coordinate along the drier X Fig. 4. The distribution of temperature T and relative humidity RH in the air versus drier length when time shifts. For ex~le (position nO 1, 5 or 9) the curves are displayed at time 0 mn, 45 mn and 90 mn. Time starts from the moment of the truck arrives at the given position. The symbols for experimental points at 0,45, 90 mn are respectively., • , . and theoretical curves._._, _ _ , ___ .
A distributed process of heat and mass transfe r
J 263
IW 3
2
-.Coo~dinate
along the
drie~
X
70
60
30~ 20
I'll
2
11
3
11
11 Coo~dinate
5
11
6
along the
7
11 drie~
11
8
11
9
X
Fig. 5. The dist~ibution of moistu~e content and tempe~atu~e T of the p~oduct ve~sus drie~ length when time shifts. For each position, the curve sPare displayed at time o mn, 45 mn, and 90 mn starting f~om the arrival of the truck at the given position . The symbols fo~ theoretical curves are the same as in Ng. 4 . The symbol. refers to the experimental data of moisture content W.
Abstract. A two -level approach is presented for identification of heat anq mass tranfer process. An example of tunnel-drier with batchwise operation is discussed. The drier model, a system of nonlinear partial differential equations, is decomposed into two coupled subsystems to perform two identifications (i) one at small scale in a laboratory oven (ii) the other at large scale in the overall tunnel. The second identification takes into account as initial guesses the results of the first one to reduce computing time. The results are shown for the two identifications and particularly for a parallel-flow tunnel-drier. This method shows a way to identify a complex distributed drying process on small computers run by acceleration algorithms. This paper is a step towards optimal design. Keywords. drier; distributed parameter systems; modelling; identification ; multi variable systems ; predictor-corrector methods ; optimisation. INTRODUCTION The drying of materials is an unit operation common to many industrial processes. From the economical viewpoint the drying process is an importance factor of cost (Flink, 1977) therefore drying process must be optimized. In order to design and optimize drie r it is desirable to have a mathematical model and to identify the unknown parameters.
in numerous papers which can be found in survey of Polis and Goodson (1976). In this paper, we present the results of the identification of parallel flow tunnel-drier with two phase flows: air and product, for the product a batchwise operation is done. To carry on identification of distributed parameter system by small computers several conditions are necessary for reducing the computational time :
A two-level approach is often used for the drying process to verify the drying model (Bertin and others, 1978 ; Kacki and Niewierowicz, 1977 ; Pierronne, 1977) :
- Initial guesses near optimal values (given by the first identification in this paper)
- First identification from results at small scale in a laboratory oven - Second identification from results at large scale in the drier From conservation laws, i.e. energy and mass balances for our problem, we obtain the overall model of drier. It is a coupled system of partial differential equations, with ones merely hyperbolic (Bertin and others, 1979 ; Nordon and Bainbridge, 1971). In industrial processes hyperbolic systems are numerous ; heat exchangers, chemical and biological reactors, refiner of pulping process, progressive and batch drier. First order hyperbolic system are usually resolved ftom Lax-Wendroff scheme, characteristic method, method of lines or method of weighted residuals. Few authors have dealt with the identification of drying process, on the opposite the identification of hyperbolic system has been shown 1257
- Simulation of the model by algorithms with large step, for example predictor corrector with step size c ontrol (Gear, 1971) - Optimization by algorithm with high speed of convergence, for example conjugate gradient (Seinfeld and Chen, 1974) THE TUNNEL-DRIER The tunnel-drier consists of two compartments: - heating system with blowers and heater - drying sys tern wi th trays and trucks The product is placed on trays. The batchwise operation consists of placing trucks into the drier at time intervals (feed time) about one
R. Bertin and Z. Srour
1258
hundred and five minutes long and overall drying is about sixteen hours long. In this example, we show the cocurrent flow type drier with nine trucks and the product is frui ts (Keey, 1972, 1978). Each pos i tion taken by truck can be considered working as a batch drier. (Fig. 1).
, I
I
~-
G,W+Jll,T +dT.
I
••
t-:';j- --- - - ---
Airflow
Product flow
x + dX
-
- tJDDDD~
Fig . 2. The energy ( ~) and mass ( . . ) transfers between air and product along the drier
Fig. 1. The drier flow diagram shows concurrent flo ws of truck and air
d d) (- + v T + llh w) dt dX [CC; a v
THE MATHEMATICAL MODEL Level 1
The model of product drying is derived from Luikov's model (1966) in capillary porous bodies (System of partial differential equations). In the case of spherical product and with some assumptions, for example parabolic profile of water content W into the product, we obtain an averaging model (Babukha and Shraiber, 1974 ; Bertin and others, 1978). This model derived from averaging over volume of water content Wand temperature product Tp ' gives a macroscopic viewpoint for each particle ; therefore their distributions along the drier will be given by the drier model itself. A product model is often used in engineering papers (Keey, 1972 ; Krisher, 1963) here we use:
E.P-
s dt
s
=
(I+B~
/5)-1 [aCT -T ) a p
s
ilwJ ot
d
d
(ll)
(ii)
(3) X
0
d d G d d L ( - + v - ) (w -)+(- +u-) (W = ) dt dX v dt dX u (Ill)
+
(iii)
0
(4)
(IV) (iV)
In these equations G/v and L/~ respectively represent the dry mass of gas and solid phases per unit of lenght, they have positive value in cocurrent flows and opposite sign in countercurrent flows. Remarks - In batch drying terms (ii) and (iV) are equal to zero. In progressive driers and in stationary conditions (I) to (IV) are equal to zero. The velocity of solid phase is a mean value (for the overall drying in the drier).
u
Level 2 : Plant Model ( 1)
+ llh (p X / a X )""'\ v
(i)
( - + u - ) [(C T ) 1 ] dX dt s p u
Heat and Mass Balances at Product Scale
X X ( p X/a ) C
(l)
~]
The overall one space dimensional model is derived from Eqs (1) to (4) it may be written in normal form :
(2)
In these equations the dependent variables are Tp(x,t) and Ta(x,t) respectively temperature of product and air, W(x,t) and w(x,t) water content of product and air (dry basis). The symbol X indicates functions. Model of the Drier itself For cross circulation drier, ly enthalpy md nass balances applied to both dry gas flow G and dry solid flow L (Bertin and others, 1979 ; Nordon and Bainbridge, 1971) if we suppose, velocities of air v and u of product constant we get: (Fig. 2)
dU at
dU
+ A(x,t,U) dX
=
B (x,t,U)
(5)
Where U is a vector which transposed vector is A is 4 X 4 matrix, B a vector of the same order as U, it depends only of U in this exampie. (Appendix). This mixed initial-boundary value problem gives a solution of the state vector U for (x, t) E 11 X (0, T) with initial and boundary conditions U(O, t)
t>O (for Ta and w only)
( 6)
A distributed process of heat and mass transfer
xEn=
U(x,O)
(7)
(O,l)
product are compared with theoretical values. The parameters are optimized thanks to the criterion J(p) : 4
COMPUTATIONAL ASPECTS
J 259
I
J
2
J(p) = ~ c. ~ ~ e .. k(x.,t.,p) k=l K i=1 j=1 1J J 1
The coupled system of partial differential Eqs. (5) consists of particular equations like hyperbolic equations with single nonzero wave velocity, as the example given for the packed bed reactor by Ritter and others097J). This system is transformed by the method of lines (Ahmad and Carver, 1974) to an ordinary differential system,the method of characteristic was used in (Bertin and others, 1979) In the method of lines we used a mixed scheme (Fig. 3) to take into account this particular system : - At the value x = 0, continuous time integration of the two first equations of Eqs.(5) by multistep Hamming's predictor-corrector method with step size control - The two following steps are run in turn (i) continuous time integration of the two first equations of Eqs.(5) but with step size fixed (at the same values as x = 0 for the same intervals of time Fig. 3). The method is the same Hamming's predictorcorrector method (ii)continuous space integration (with respect to x) of the two last equations of Eqs.(5) by lower order trapezoidal method (with backward finite difference formula for the others unknown derivatives).
- p is the parameter vector posed vector is (a,Bm)
(8)
which trans-
- ck are coefficients to put all partial criteria on the same scale - error eijk gives difference betweep. measurements and model results ; the expression of eijk is :
This criterion is evaluated by the sums of partial criteria related to each dependent variables, k is the indice of each~of th~se criteria. The optimal parameters a and Bm are searched by the discrete conjugate gradient method because of its fast convergence. Primary Identification The product drying is carried out in laboratory oven in the same conditions as the working drier. The experimental results, temperature and water content of product are given from sensors, the others dependent variables, temperature and humidity of air are given values closed to those in the drier. In this case we use as model the subsystem, i.e. the two first equations of Eqs.(5). By identifying we get : A 2 -3 2 a = 66.9W/m. °c Bm = 3.5010 kg/m.s
T
Second Identification
f')@ T +IlT
(j) (i)
T
@
®
I
T -.1T
X ·-.1X
X
X
X+.1X
Pip. 3. The propagation paths of the mixed method This choice of integration methods is necessary to have problems well conditioned in a computational sense (Miranker,1977) because we have stiff equations coming from fast transient heat transfer and slow transient mass transfer. On the other hand they allow to reduce computational effort.
From each position of truck in the tunneldrier we can obtain the experimental values of temperature of air and product and their water content (dry basis) versus time and for a limited number of places in the drier. These values are compared with model values using the criterion J(p). The initial guesses of parameters a and Bm are those of the primary identification. We get by discrete conjugate gradient algorithm given by library subroutine : ~ -3 2 Bm= 5.2510 kg/m.s The results which are presented in Fig. 4 and Fig. 5 are those of a real test during which a loaded test truck, equipped with different sensors, has been run through the drier, step by step numbered J to 9. For computation we have used as boundary conditions the experimental results given by the downstream side of the test truck (Bertin and others, 1979).
IDENTIFICATION The "Model method" (Richalet, 1971) is used as identification method. The parameters to identify are heat transfer coefficient a and mass transfer coefficient Bm' The main dependent variables of the drying process : tempe.... 0:2 ...
".,..0
'!lI....,,.1
1'."~""O""
~n ...... t-oT"'lt-
nf hnt-h
!l;'" !lnrl
CONCLUSIONS Numerical profiles obtained from the theoretical model are in relatively good agreement with experimental data. We find a more large difference, between the two parameter identi-
R. Bertin and Z. Srour
'260
fications, for the mass transfer than for the heat transfer coefficient, it is a consequence of their sensitivities. On the other hand, it is possible using the same method to identify parameters in others types of drier, for example progressive or batch-drier. The method presented in this paper gives a way to identify complex distributed processes of heat and mass transfer on small computers, the algorithms used permit to reduce computational effort. Next step will be to develop them in the way of optimum designing.
d
,
bO O~
at""
I ~I
+
a?
I
bO 3 bO 4
Ql)I
- Dimensionless dependent variables TO
T /T p
P w°
w/w
WO
an
TO
n
W/W
n
X
f
I
b
O
-f
2
la ° (To - TO) + lIh ° f b ol Lap v 2 2J
I
3
SO (WO - wo) m s
- Dimensionless coefficients SO
m
lIhO v
= lIh
= SmZawn /vp s
W
n
W /C T v n s an
(L/G).(v/~)
(J
C /C s
n
WO > I.3/W
I
n
)( 3 (a/a) =
P/P:
I + k WO I
F2
C)(/C s s
I + k WO 2
fI
F 2/3 (I + BiF 1/3) I
f2
FI
f3
WOF - I / 3 (0.622 + w wo)-I n I
f4
F2
fS
F2 F,
-I
5'
=
f
(B~/Bi)3
F -I 2
-2/3
3 FI
-1 / 3
-2/3
2 (m /m 3 )
a
Surface area per unit volume of product (3/Rf)
Bi
Biot number
C
Specific heat
d w
Product desorption coefficient
G
Dry gas flow rate
Z
Lenght of the truck
L
Dry solid flow rate
P
Total air pressure
R f
Dry product external radius
RH
Relative humidity of air
T
Temperature (averaged over volume)
t
Time
u
Truck speed (~ averaged over time)
(m/s)
v
Air velocity
(m/s)
W
Water content in the product dry basis (average over volume)
(kg/kg)
Water content in the air dry basis (average over volume) Longitudinal coordinate
(kg/kg) (m)
O = x/Z
- Dimensionless functions for linking equations b°
WO < I.3/W
Appendix B - Nomenclature
- Dimensionless independent variables to = tv/Z
0.48
FI
F)(
T /T a an
a
=
bO 2
auo
0
w
+
The functions fI to fS can be defined from two functions which take into account of volume shrinkage and specific capacity of product versus WO
The system of Eqs.(S) may be get 1n dimensionless form
auo
n
w
Appendix A
,<» ,
0.4W WO
d
g
(aRf/A) (J/kg.oC)
(kg/s) (m)
(m)
(OC) (s)
- Dimensionless functions WO s
=P
(TO ) d (WO)(0.622 + w WO)/Pw sat surf w n n
TO surf x
A distributed process of heat and mass transfer Greek letters a
B
m
6h
v
Heat transfer coefficient
20 (W / m . C)
Mass transfer coefficient
2 (kg/m .s)
Latent heat of vaporization
A
Thermal conductivity of the product
Ps
Density of bone-dry solids
Air
g
Gas phase
i
Initial or indice
n
Normalized
p
Product
s
Solid phase
Kacki, F. and T. Niewierowicz (1977). A hybrid system of identification of industrial processes of mass and heat transfer. Digital Computer Applications to Process Control, Van Nanta Lemke (Ed.), IFAC and North Holland, 607~615.
(J/Kg)
Subscripts and superscripts a
1261
sat Satured water vapor surf Surface Dimensionless Variable Optimal A, B, C, e, i, j, J, k, p, U are defined from Eqs.(5) to (9). REFERENCES Ahmad, S.Y., and M.B. Carver (1974). A system for the automated solution of sets of implicitly coupled partial differential equations. Proc. Summer Computer Simulation Conference, 104-109. Babukha, G.L., and A.A. Shraiber (1974). Interphase heat transfer in poly-disperse gas suspension flows. Fifth International Transfer Conference Tokyo, 2, 69-73. Bertin, R.,.Pierronne and H.Combarnous(1978). Determination des parametres de sechage de certains fruits compacts. Ann. Technol. Agric. , JJ, 489-500. Bertin, R., F. Pierronne and M. Combarnous (1979). Modeling and simulating a distributed parameter tunnel-drier. J. Food Sci. (to appear). Flink, J.M. (1977). Energy analysis in dehydration processes. Food Technology., 31 77-84. Gear, C.W. (1971). Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, New Jersey.
Keey, R.B.(1972). Drying Principles and Practice. Pergamon Press, Oxfo rd. Keey, R.B. (1978). Introduction to Industrial Drying Operations. Pergamon Press, Oxford. Krisher, O. (1963). Die Wissenschaftlichen Grundlagen der Trockmungstechnik. Springer Verlag, Berlin. Luikov, A.V. (1966). Heat and Mass Transfer in Capillary Porous Bodies. Pergamon Press, Oxford. Miranker, W.L. (1977). The computational theory of stiff differential equations. Report N° 219 7667, Orsay, Univ. Paris
XI. Nordon, P. and N.W. Bainbridge (1971). An analysis of the process of through drying of wet textile materials, Applied polymer Symposium, ~, 1111-1119. John Wiley, N.Y. Pierronne,F. (1977) . Modelisation et identification d'un processus de sechage. Thesis, University of Bordeaux I. Polis, M.P. and R.E. Goodson (1976). Parameter identification in distributed systems: A synthesizing overview. Proceedings of the IEEE, ~, 45-61. Richalet, J., A. Rault and R. Pouliquen (1971). Identification des processus par la methode du modele, Gordon & Breach, London. Ritter, A.B., J.C. Friedly and C.C. Chen (1971). Singular control arcs in the time optimal control of linear hyperbolic systems. Proc. IFAC Symposium on the control of distributed parameters systems, Banff. Seinfeld, J.H. and W.H. Ch en (1974). Estimation of parameters in distributed systems, Proc. AACC Joint automatic control confe~, Austin. Table Values of model parameters T an
87(°C) 3
a
1 .25
C s
6h
2.7
Z
(m/s)
v
R f
W n
v
=
2 0.97910- (m) 1340(J/kg:C)
=
w n
1 (m)
lJ
0.04 77
6 2.510 (J/kg) P
s Bi
=
3 1400(kg/m )
=
1.7910
-2
x a
1262
R. Bertin and Z. Srour
kl
3.78
Cl( g
C g
c
692
2
= 8.42
k2
I070(J/kg.oC) c
c c
3
4
1
=
0
90
80
70
.
\.
"" .
60
"-
...-• - --.......
_....
9
5
Coordinate along the drier X
RH(7.)
40
..
I 30
",.
---.
....
-s--"'2'..... _
20
-- .
......... .. 1
~----
•
• 10
5
9
Coordinate along the drier X Fig. 4. The distribution of temperature T and relative humidity RH in the air versus drier length when time shifts. For ex~le (position nO 1, 5 or 9) the curves are displayed at time 0 mn, 45 mn and 90 mn. Time starts from the moment of the truck arrives at the given position. The symbols for experimental points at 0,45, 90 mn are respectively., • , . and theoretical curves._._, _ _ , ___ .
A distributed process of heat and mass transfe r
J 263
IW 3
2
-.Coo~dinate
along the
drie~
X
70
60
30~ 20
I'll
2
11
3
11
11 Coo~dinate
5
11
6
along the
7
11 drie~
11
8
11
9
X
Fig. 5. The dist~ibution of moistu~e content and tempe~atu~e T of the p~oduct ve~sus drie~ length when time shifts. For each position, the curve sPare displayed at time o mn, 45 mn, and 90 mn starting f~om the arrival of the truck at the given position . The symbols fo~ theoretical curves are the same as in Ng. 4 . The symbol. refers to the experimental data of moisture content W.