MOISTURE DIFFUSIVITY ESTIMATION FROM TEMPERATURE MEASUREMENTS: INFLUENCE OF MEASUREMENT ACCURACY

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1 12th Intenational Dying Symposium (IDS2), Noodwijkehout, The Nethelands, August 28-31, 2. MOISTURE DIFFUSIVITY ESTIMATION FROM TEMPERATURE MEASUREMENTS: INFLUENCE OF MEASUREMENT ACCURACY Gligo H. Kanevce, Ljubica P. Kanevce and Vangelce B. Mitevski 1 Geoge S. Dulikavich 2 1. Faculty of Technical Sciences, St. Kliment Ohidski Univesity, Bitola, Macedonia, Tel/Fa: +389 () , kanevce@osi.net.mk 2. Multidisciplinay Analysis, Invese Design, and Optimization (MAIDO), Depatment of Mechanical and Aeospace Engineeing, UTA Bo 1918 The Univesity of Teas at Alington, Alington, TX , U.S.A. Tel/Fa: +1(817) /51, gsd@mae.uta.edu Keywods: dying, moistue diffusivity, paamete estimation, invese appoach ABSTRACT In this pape a method of moistue diffusivity estimation on the basis of a dying body tempeatue esponse is analyzed by using an invese appoach. The method takes advantage of the inteelation between the heat and mass tanspot pocesses within the dying body as well as fom its suface to the suoundings. As a esult, the low accuacy local moistue content o standad dying cuve measuements can be substituted by this accuate and easy to pefom method that equies only single themocouple tempeatue measuements. The Levenbeg-Maquadt pocedue is applied fo estimation of the moistue content and tempeatue dependent moistue diffusivity. Numeical epeiments have been conducted to investigate sensitivity of this method to the heat and mass tansfe coefficients accuacy. In ode to simulate eal measuements, a nomally distibuted eo was added to the numeical tempeatue esponse. An analysis of the influence of the dying ai velocity, dying ai tempeatue, dying body dimension and dying time on the moistue diffusivity estimation that enables the design of the pope epeiment is pesented as well. INTRODUCTION Invese appoach to paamete estimation in last few decades has become widely used in vaious scientific domains. While in the classical diect poblem the cause is given and the effect is detemined, the invese poblem involves estimation of the cause fom the knowledge of the effect. The solution of

2 invese poblems often equies the solution of thei diect poblems. Theefoe, the ability of an invese poblem method is often closely elated to diect poblem method of solution. Thee ae seveal methods fo descibing diect poblem of the comple simultaneous heat and moistue tanspot pocesses within dying mateial. In the appoach poposed by Luikov (1972) the moistue and tempeatue fields in the dying body ae epessed by a system of two coupled patial diffeential equations. The system of equations incopoates coefficients, which ae functions of tempeatue and moistue content, and must be detemined epeimentally. Fo many pactical calculations the influence of the tempeatue and moistue content on all the tanspot coefficients ecept fo the moistue diffusivity is small and can be neglected. Fo many dying pocesses, including the dying pocesses consideed in this pape, the influence of the themal diffusion is small and can be ignoed. In this case, the Luikov's moistue tanspot equation is the same as the Fick's second law equation, whee concentation has been conveted to moistue content on a dy basis. The moistue diffusivity has the same meaning in both of these appoaches. It accounts fo vaious types of possible dying pocesses including molecula (liquid) diffusion, vapo diffusion, suface diffusion, hydodynamic flow, Knudsen flow, and othe consideations. An effective moistue diffusivity, which lumps all possible moistue tanspot mechanisms into a single measuable paamete, is often used to chaacteize the dying behavio egadless of the dominating mechanism (Feng et al., 1999). The moistue diffusivity dependence on moistue content and tempeatue eets a stong influence on the dying pocess calculation. This effect can not be ignoed fo the majoity of pactical cases. All the coefficients ecept fo the moistue diffusivity can be elatively easily detemined by epeiments (Kaathanos et al., 1996; Rahman, 1995). A numbe of methods fo the epeimental detemination of the moistue diffusivity eist such as: soption kinetics methods, pemeation methods, concentation-distance methods, dying methods, adiotace methods, and methods based on the techniques of electon spin esonance and nuclea magnetic esonance. Thee is no standad method fo the epeimental detemination of the moistue diffusivity. The adoption of a genealized method fo moistue diffusivity estimation would be of geat impotance; howeve, this does not seem pobable in the nea futue (Zogzas et al., 1996). The application of the moistue diffusivity estimation methods based on the epeimental dying cuves in elation to the analytical solution of the diffeential diffusion equation seems to be the most popula epeimental pactice (Zogzas and Maoulis, 1996; Feng et al., 1999). Numeical solutions of the Fick's law diffeential diffusion equation with constant (Daud et al., 1997) o moistue and tempeatue dependent (Zogzas and Maoulis, 1996) diffusivity also have been used fo the moistue diffusivity estimation. The main idea of the pesent method is to make use of the inteelation between the heat and mass (moistue) tanspot pocesses within the dying body and fom its suface to the suoundings. Then, the moistue diffusivity can be estimated on the basis of an accuate and easy to pefom single themocouple tempeatue measuement by using an invese appoach. Kanevce, Kanevce and Dulikavich (2a, 2b) and Dantas et al. (2) ecently analyzed this idea of the moistue diffusivity estimation by tempeatue esponse of a dying body. In this pape, the solution of the invese poblem of estimating the moistue content and tempeatuedependent moistue diffusivity is pesented. The pesent paamete estimation poblem is solved by using the Levenbeg-Maquadt method of minimization of the least-squaes nom, by using simulated epeimental data with andom eos. Instead of actual tempeatue measuements, the tempeatue esponse duing convective dying is obtained fom the numeical solution of the non-linea onedimensional Luikov's equations. In ode to simulate eal measuements, a nomally distibuted eo was added to the numeical tempeatue esponse. As a epesentative dying body, a mitue of bentonite and quatz sand with known themophysical popeties has been chosen. The objective of this pape is an analysis of the sensitivity of this method of moistue diffusivity estimation to the heat and mass tansfe coefficients accuacy. An analysis of the influence of the dying ai velocity, dying ai tempeatue, dying body dimension and dying time on the moistue diffusivity

3 estimation that enables the design of the pope epeiment is pesented as well. In ode to ealize this analysis the sensitivity coefficients and the sensitivity mati deteminant wee calculated (Kanevce, Kanevce and Dulikavich, 2b) fo the chaacteistic dying egimes and dying body dimensions. MATHEMATICAL MODEL OF DRYING In the case of an infinite flat plate of thickness 2L, if the shinkage of the mateial duing dying can be neglected, the esulting system of equations fo the tempeatue, T(, t), and moistue content, X(, t), can be epessed as T T X cρ s = k + ερs H t ( 1) t X t = X T D + Dδ ( 2) Hee, t,, c, k, H, ε, δ, D, ρ s ae time, distance fom the mid-plane of the plate, heat capacity, themal conductivity, latent heat of vapoization, atio of wate evapoation ate to the eduction ate of the moistue content, themo-gadient coefficient, moistue diffusivity, and density of the dy plate mateial, espectively. As initial conditions, unifom tempeatue and moistue content pofiles ae assumed The bounday conditions on the fee plate suface ( = L) ae t = T(,) = T, X(,) = ( 3) X T k Dρ s X = L = L + j q H(1 ε) j + Dδρ s T m = L = + j m = ( 4) In the case of convective dying of the sample, the convective heat flu, j q (t), and mass flu, j m (t), on the suface of evapoation ae ( T T ) jq = h a = L jm = hd( C= L Ca ) ( 5) whee h is the heat and h D the mass tansfe coefficient, T a is the dying ai bulk tempeatue, and C a is the concentation of wate vapo in the dying ai. The wate vapo concentation of the ai in equilibium with the fee suface of the body is calculated by C = L = a(t = L,X= L) ps(t = L) / 461.9/(T = L + 273) ( 6) Hee p s is the satuation pessue and a is the wate activity calculated fom the wate soption isothems. The poblem is symmetical, and bounday conditions on the symmety suface of the plate ae

4 T = =, X = = ( 7) The system of equations (1) and (2) with the initial (3) and the bounday conditions (4) and (7) has been solved numeically. In ode to appoimate the solution an eplicit pocedue has been used (Kanevce, Kanevce and Dulikavich, 2a). MOISTURE DIFFUSIVITY ESTIMATION The poposed method of the moistue diffusivity estimation by tempeatue esponse of a dying body was tested fo a model mateial (Kanevce, Kanevce and Dulikavich, 2a) which was a mitue of bentonite and quatz sand with known themophysical popeties (Kanevce et al., 198; Kanevce, 1998). Fom the epeimental and numeical eaminations of the tansient moistue and tempeatue pofiles (Kanevce, 1998) it was concluded that fo the calculations in this study, the influence of the themal diffusion is small and can be ignoed. It was also concluded that the Luikov s system of simultaneous patial diffeential equations could be used by teating the tanspot coefficients as constants ecept fo the moistue diffusivity. The appopiate mean values fo the model mateial ae: the density of dy solid, ρ s = 1738 kg/m 3, heat capacity, c = 155 J/(kgK), themal conductivity, k = 2.6 W/(mK), latent heat of wate vapoization, H = J/kg, atio of wate evapoation ate to the eduction ate of the moistue content, ε =.5 and themo-gadient coefficient, δ =. The epeimentally obtained desoption isothems of the model mateial ae pesented by the empiical equation a = 1 ep( (T+ 273) (T + 273) X ) ( 8) Any othe type of equation, coesponding to the Smith, BET, GAB model can be used in the pesented method. The following empiical function of tempeatue and moistue content can descibe the epeimentally obtained elationship fo the moistue diffusivity (Kanevce et al., 198) 1 D1 T D = ( 9) 2 D2 + X 33 whee D 1 and D 2 ae constants. The values fo the model mateial ae: D 1 = and D 2 =. Ahenius-type equation fo the moistue diffusivity can be also used in the pesented method. Fo the invese poblem investigated hee, values of D 1 and D 2 ae egaded as unknown paametes and all othe quantities figuing in diect poblem fomulation wee assumed to be known. Fo the estimation of these paametes, we conside available the tansient single themocouple tempeatue measuements. The estimation methodology used is based on minimization of the odinay least squae nom E(P) = [Y T(P)] [Y T(P)] T ( 1) Hee, Y T = [Y 1,Y 2,,Y M ] is a vecto of measued tempeatues and T = [T 1 (P), T 2 (P), T M (P)] is a vecto of estimated tempeatues at the measuement location at time t i (i = 1, 2,, M), while P T = [P 1, P 2,, P N ] is the vecto of unknown paametes, M is total numbe of measuements, and N is the total numbe of unknown paametes (M N).

5 A vesion of Levenbeg-Maquadt method was applied fo the solution of the pesented paamete estimation poblem (Maquadt, 1963). This method is quite stable, poweful, and staightfowad. It has been applied to a vaiety of invese poblems and belongs to damped least squae methods (Beck and Anold, 1977). The solution fo P is achieved using the following iteative pocedue P + 1 = P + [(J ) T J + µ Ι] 1 (J ) T [Y T(P )] ( 11) whee Ι is the identity mati, µ is the damping paamete, and J epesents sensitivity mati. T1 T... 1 P 1 PN J =. ( 12) TM TM... P1 PN The tem µi damps instabilities due to ill-conditioned chaacte of the poblem. Nea the initial guess, the poblem is geneally ill-conditioned and damping paamete is chosen lage making tem µi lage as compaed to tem J T J. So, the mati J T J is not equied to be non-singula at the beginning of iteations and the pocedue tends towads a slow-convegent steepest descent method. As the iteation pocess appoaches the conveged solution, the damping paamete deceases, and the Levenbeg-Maquadt method tends towads Gauss method. In fact, this method compomises between the steepest descent and Gauss method choosing µ so as to follow the Gauss method to as lage an etend as possible, while etaining a bias towads the steepest descent diection to pevent instabilities. The pesented iteative pocedue stops if the nom of gadient of E(P) is sufficiently small, o if the atio of the nom of gadient of E(P) to the E(P) is small enough, o if the changes in the vecto of paametes ae vey small. RESULTS AND DISCUSSION The pesented method allows a moistue and tempeatue dependent heat and mass tansfe coefficients to be used. The best appoach should be if the appopiate elations ae obtained by compaison of the known with individual epeiments ealized on the same epeimental set-up fo the moistue diffusivity estimation. In this case the lage value fo the mass tansfe Biot numbe indicates that the intenal esistance contols the dying pocess. Consequently, the moistue content of the suface of the body is pactically constant, nea equilibium moistue content, vey shot afte the beginning of the dying (Figue 4). In conclusion, constant heat and mass tansfe coefficients ae taken fo the puposes of this analysis. The coesponding mean values have been calculated fom the Nesteenko's elations (Luikov, 1972) fo heat and mass Nusselt numbes in dying conditions. In ode to investigate influence of the bounday conditions, deteminant of the sensitivity mati J T J with nomalized elements M T T = i T [ J J] i m,n Pm Pn, m, n = 1,, N ( 13) Pm Pn i= 1 and the elative sensitivity coefficient, D 1 T i / D 1, i = 1,2,,M, wee calculated (Kanevce, Kanevce, Dulikavich, 2b). The sensitivity coefficients analysis was caied out fo the plate of thickness 2L, with initial moistue content of X(, ) =.2 kg/kg and initial tempeatue T(,) = 2 O C. In ode to

6 investigate the influence of the bounday conditions, the dying ai bulk tempeatue, T a, and velocity, V a, wee petubed. The elative humidity of the dying ai was ϕ =.12. The test cases ae shown in the Table 1. Table 1. Dying ai conditions Thickness 2L 3 mm 6 mm T a [ C] Dying ai V a [m/s] Tansfe coefficients h[w/m 2 K] h D 1 2 [m/s] A1 AA B1 BB C1 CC A2 AA B2 BB C2 CC Table 1 also contains the coesponding convective heat, and mass tansfe coefficients h and h D, obtained fo an 8 mm long plate. The test cases wee epeated fo two diffeent thicknesses of the dying body, 2L = 3 mm and 6 mm. Figue 1 shows the deteminant of the sensitivity mati J T J with nomalized elements and Figue 2 the elative sensitivity coefficient fo all test cases depicted in Table 1. Since the sample dying object epesented by a flat plate is vey thin, a single themocouple was located in the mid-plane of the infinite flat plate. Two plateaus can be seen on the pesented sensitivity deteminant cuves. The fist plateau coesponds to the moment (Kanevce, Kanevce, Dulikavich, 2a) when the body moistue content is nealy equal to the equilibium. Afte that, small evapoation ate and fast dying ate decease accompanied by fast body tempeatue incease occu. The second plateau coesponds to the moment when nealy equilibium tempeatue was obtained. Deteminant C1 B1 C2 A1 B2 A Deteminant CC1 CC2 BB1 BB2 AA1 AA time [s] time [s] Figue 1. Sensitivity Deteminants The tendency of inceasing deteminant as well as sensitivity coefficient with dying ai velocity can be eplained by the inceasing of the mass tansfe Biot numbe and the moistue content gadients inside the body. The heat tansfe Biot numbe and the tempeatue gadients (Figue 4) ae vey small in all the cases. The mass tansfe Biot numbe changes duing the dying pocess as a function of the local moistue content and tempeatue change. The value of this Biot numbe is vey high, pactically infinite, at the beginning of the dying, and tends to low values at the end of the dying. Unde these conditions

7 mostly the moistue diffusivity and dimensions of the body goven the pocess of dying. The same conclusion emains when plate thickness incease. The influence of the dying ai tempeatue also can be eplained though the mass tansfe Biot numbe and the moistue content gadients inside the body. Highe dying ai tempeatue leads to a highe dying body tempeatue and lowe moistue content (highe dying ate) that lead to highe moistue diffusivity, and consequently lowe mass tansfe Biot numbe. That leads to a tendency of deceasing deteminant and sensitivity coefficient with incease in dying ai tempeatue. Sensitivity Coefficient C1 C2 B1 A1 time [s] A2 B2 Sensitivity Coefficient CC1 BB1 CC2 AA2 AA1 BB2 time [s] Figue 2. Sensitivity Coefficients Sensitivity Deteminant Case C time [s] Figue 3. The Chaacteistic Times Sensitivity Coefficient Fo the estimation of the moistue diffusivity available the tansient eadings of the tempeatue senso located in the mid-plane have been consideed. Instead of actual tempeatue measuements, the tempeatue esponse duing convective dying is obtained fom the numeical solution of the non-linea one-dimensional Luikov's equations, by teating the values and epessions fo the mateial popeties as known. In ode to simulate eal measuements, a nomally distibuted eo with zeo mean and standad deviation of.5 o C was added to the numeical tempeatue esponse. In ode to investigate influence of the consideed duation of tempeatue esponse paametes D 1 and D 2 wee estimated (Kanevce, Kanevce, Dulikavich, 2b) fo the test case (C2). Five dying times, coesponding to (Figue 3 and Figue 4) wee consideed: the end of dying and the maimum deteminant value (6 s), the maimum sensitivity coefficient value (31 s), the fist deteminant value plateau and zeo sensitivity coefficient value (272 s), the maimum negative sensitivity coefficient value (22 s), and the dying time below that (14 s). Vey good ageement was achieved fo all the cases with dying time equal o longe than that coesponding to the fist plateau on the deteminant cuve. The esults showed the tendency of inceasing D 1 accuacy with deteminant. The mean accuacy of D 2 was vey high fo all the cases. Fo the dying times shote than that coesponding to the fist plateau, the poblem is ill-posed and local minimums wee obtained depending on the initial guesses.

8 X [kg/kg] t=272 s t=22 s t=14 s [mm] t= s t=31 s t=6 s T [C] t=6 s t=31 s t=272 s t=22 s t=14 s t= s [mm] Figue 4. Tansient Moistue Content and Tempeatue Pofiles (Case C2) In ode to investigate influence of heat and mass tansfe coefficients petubations on the moistue diffusivity estimation accuacy, test cases pesented in Table 2 have been analyzed. Case C2 has been chosen owing to the shotest dying time and case CC1 because of its highest values of sensitivity coefficient and deteminant. The selected chaacteistic dying times enable accuate estimating of the moistue diffusivity if "eact" values fo heat and mass tansfe coefficients ae used as input data. The tempeatue esponse had constant time-step of 2 s, so diffeent numbe of measued tempeatues has been taken fo the cases with diffeent dying time. The numbe of the space gid points has been 41 in all the dying pocess calculation schemes. The initial guess fo all the cases was the same and it was vey fa fom the eact values of paametes : D 1init = and D 2init =.1. To calculate moistue diffusivity values in Table 2, besides the "eact" values, 1% inceased values fo one o both heat and mass tansfe coefficients wee used simulating data with measuement eos. Fo the compaison, the eact paamete values ae also shown. The elative eo has been calculated as ε 1 (%) = 1 (D 1 -D 1e. )/D 1e. The D 2 accuacy depends on the X 2 value and is changing fom the beginning to the end of dying, ε 2 = ((X 2 + D 2 )- X 2 ) / X 2 = D 2 / X 2. The mean elative eo of D 2 has been calculated with the mean moistue content value X m as ε 2 (%) = 1 D 2 /X m 2. As a mean moistue content value duing the dying, X m =.1 kg/kg has been used fo all the cases. Table 2. Influence of heat and mass tansfe coefficients' petubations Case t[s] h[w/m 2 K] h D 1 2 [m/s] D ε 1 [%] D 2 ε 2 [%] C C C C C C CC Eact values 9.. It can be seen that fo the dying time of 272 s fo Case C2 the obtained esults ae significantly diffeent fom eact values, especially fo D 2. The othe test cases show that petubations of heat and mass tansfe coefficients poduce educed eos of estimated moistue diffusivity paametes if whole duation of dying is taken in the analysis.

9 CONCLUSIONS The method fo estimation of moistue diffusivity on the basis of themal tansient esponse of a dying body by using invese appoach is pesented. The Levenbeg-Maquadt method is applied fo evaluation of unknown paametes in moistue diffusivity dependence on moistue and tempeatue. The esults obtained with simulated measuements show good ageement between evaluated and eact paamete values and confim the validity of the poposed method. Tendency of inceasing deteminant as well as sensitivity coefficient was obtained with dying ai velocity and dying body thickness incease and with dying ai tempeatue decease. This can be eplained by the incease of the mass tansfe Biot numbe and the moistue content gadients inside the body. The heat tansfe Biot numbe and the tempeatue gadients ae vey small. Unde these conditions, mostly the moistue diffusivity and dimensions of the body goven the pocess of dying. The influence of the dying ai velocity inceasing on the deteminant and dying time leads to the computational time deceasing. The opposite influence of the dying ai tempeatue and dying body thickness on the deteminant and dying time suggests that additional eseach needs to be done in defining its optimal values concening the minimal computational time. Concening the sensitivity of this method to the heat and mass tansfe coefficient accuacy, it can be concluded that petubations (simulated eos) in heat and mass tansfe coefficients poduce educed eos of estimated moistue diffusivity paametes. NOTATION a wate activity - c heat capacity J/K/kg db C concentation of wate vapo in ai kg/m 3 D moistue diffusivity m 2 /s h heat tansfe coefficient W/m 2 /K h D mass tansfe coefficient m/s I identity mati - j m mass flu kg/m 2 s j q heat flu W/m 2 J sensitivity mati - k themal conductivity W/m/K L flat plate thickness m p s satuation pessue Pa P vecto of unknown paametes - t time s T tempeatue T vecto of estimated tempeatue V velocity m/s distance fom the mid-plane m X moistue content (dy basis) kg/kg db Y vecto of measued tempeatue Geek Symbols - δ themo-gadient coefficient 1/K H latent heat of vapoization J/kg ε wate evapoation ate atio - µ damping paamete - ρ density kg/m 3 ϕ elative humidity - C C C

10 Subscipts Supescipts a ai iteation numbe s dy solid T tanspose LITERATURE Beck J. V. and Anold K. J., 1977, Paamete Estimation in Engineeing and Science, John Wiley&Sons., Inc., New Yok Dantas L. B, Olande H. R. B. and Cotta R. M., 2, Paamete Estimation in Moist Capillay Poous Media by Using Tempetatue Measuements, Int. Symp. on Invese Poblems in Eng. Mech. (ISIP 2), eds: Tanaka, M. and Dulikavich, G. S., Mach 7-1, Nagano, Japan. Daud, R. W., Ibahim, M. H. and Talib, Z., M., 1997, Paamete Estimation of Fick's Law Dying Equation, Dying Technology, Vol.15, No. 6-8, pp Feng, H., Tang, J., Cavaliei, R. P., 1999, Combined Micowave and Spouted Bed Dying of Diced Apples: Effect of Dying Conditions on Dying Kinetics and Poduct Tempeatue, Dying Technology, Vol.17, No. 1, pp Kanevce G. H., 1998, Numeical Study of Dying, IDS'98, Poceedings of the 11th Intenational Dying Symposium, Vol. A, Halkidiki, Geece, pp Kanevce G. H., Kanevce Lj. P., Dulikavich G. S., 2a, Moistue Diffusivity Estimation by Tempeatue Response of a Dying Body, Int. Symp. on Invese Poblems in Eng. Mech. (ISIP 2), eds: Tanaka, M. and Dulikavich, G. S., Mach 7-1, Nagano, Japan. Kanevce G. H., Kanevce Lj. P, Dulikavich G. S, 2b, Influence of Bounday Conditions on Moistue Diffusivity Estimation by Tempeatue Response of a Dying Body, Int. Symp. on Invese Poblems in Heat Tansfe, National Heat Tansfe Confeence, August 2-22, Pittsbugh, PA, USA Kanevce G. H, Stefanovic M., Pavasovic V., 198, Epeimental Detemination of the Diffusivity of Moistue within Capillay Poous Bodies, Dying'8, Vol. 1, Hemisphee, pp Kaathanos, V. T., Maoulis, Z. B., Mainos-Kouis, D. and Saavacos, D. G., 1996, Higothemal and Quality Popeties Applicable to Dying. Data Souces and Measuement Techniques, Dying Technology, Vol.14, No. 6, pp Luikov A. V., 1972, Teplomassoobmen, Enegia, Moscow, Russia Maquadt D. W., 1963, An Algoithm fo Least Squaes Estimation of Nonlinea Paametes, J. Soc. Ind. Appl. Math., pp Rahman, S., 1995, Food Popeties Handbook, CRC Pess, Inc., Boca Raton, New Yok. Zogzas, N. P. and Maoulis, Z. B., 1996, Effective Moistue Diffusivity Estimation fom Dying Data. A Compaison Between Vaious Methods of Analysis, Dying Technology, Vol. 14, No. 7-8, pp Zogzas, N. P. Maoulis, Z. B. and Mainos-Kouis, D., 1996, "Moistue Diffusivity Data Compilation in Food Poducts," Dying Technology, Vol. 14, No. 1, pp