APPLICATION OF INVERSE CONCEPTS TO DRYING

APPLICATION OF INVERSE CONCEPTS TO DRYING by Ljubica KANEVCE, Gligor KANEVCE, and George DULIKRAVICH Original scientific paper UDC: 536.24:66.021.4 BIBLID: 0354-9836, 9 (2005), 2, 31-44 This pa per deals with the ap pli ca tion of in verse ap proaches to es ti ma tion of dry ing body pa ram e ters. Simultaneous estimation of the thermophysical prop er ties of a dry ing body as well as the heat and mass trans fer co ef fi - cients, by us ing only tem perature measurements, is analysed. A mathematical model of the dry ing pro cess has been de vel oped, where the mois ture con tent and tem per a ture fields in the dry ing body are ex pressed by a sys tem of two cou pled par tial dif fer en tial equa tions. For the es ti ma tion of the un - known pa ram e ters, the transient readings of a single temperature sensor lo - cated in an in fi nite flat plate, ex posed to con vec tive dry ing, have been used. The Levenberg-Marquardt method and a hy brid op ti mi za tion method of minimization of the least-squares norm are used to solve the pres ent pa ram - e ter es ti ma tion prob lem. An anal y sis of the in flu ence of the dry ing air ve loc - ity, dry ing air tem per a ture, dry ing body di men sion, and dry ing time on the thermophysical prop er ties es ti ma tion, that en ables the de sign of the proper ex per i ments by us ing the so-called D-op ti mum cri te rion was con ducted. In or der to per form this anal y sis, the sen si tiv ity co ef fi cients and the sen si tiv ity ma trix de ter mi nant were cal cu lated for the char ac ter is tic dry ing re gimes and the dry ing body di men sions. Key words: inverse approach, drying, thermophysical properties, heat and mass transfer coefficients Introduction In verse ap proach to pa ram e ter es ti ma tion in the last few de cades has be come widely used in var i ous sci en tific dis ci plines. Kanevce, Kanevce, and Dulikravich [1-5] and Dantas, Orlande, and Cotta [6-8] re cently ana lysed ap pli ca tion of in verse ap proaches to es ti ma tion of dry ing body pa ram e ters. There are sev eral meth ods for de scrib ing the com plex si mul ta neous heat and mois ture trans port pro cesses within dry ing ma te rial. In the ap proach pro posed by Luikov [9], the dry ing body mois ture con tent and tem per a ture field are ex pressed by a sys tem of two cou pled par tial dif fer en tial equa tions. The sys tem of equa tions in cor po rates co ef fi - cients that are func tions of tem per a ture and mois ture con tent, and must be de ter mined ex - per i men tally. All the co ef fi cients, ex cept for the mois ture diffusivity, can be rel a tively eas ily de ter mined by ex per i ments. The main prob lem in the mois ture diffusivity de ter mi - na tion by clas si cal or in verse meth ods is the dif fi culty of mois ture con tent mea sure ments. 31

THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 31-44 The main idea of the pres ent method is to make use of the in ter re la tion be tween the heat and mass (mois ture) trans port pro cesses within the dry ing body and from its sur - face to the sur round ings. Then, the mois ture diffusivity can be es ti mated on the ba sis of ac cu rate and easy to per form sin gle ther mo cou ple tem per a ture mea sure ments by us ing an in verse ap proach. The ob jec tive of this pa per is an anal y sis of the pos si bil ity of si mul ta neous es ti - ma tion of sev eral thermophysical prop er ties of a dry ing body and the heat and mass trans - fer co ef fi cients. In or der to per form this anal y sis, the sen si tiv ity co ef fi cients and the sen - si tiv ity ma trix de ter mi nant were cal cu lated. The pro posed method of si mul ta neous es ti ma tion of sev eral thermophysical prop er ties of the dry ing body and the re lated heat and mass trans fer co ef fi cients us ing the tran sient read ings of a sin gle tem per a ture sen sor was tested for a model ma te rial. It was a mix ture of ben ton ite and quartz sand with known thermophysical prop er ties and neg li gi ble shrink age ef fect. Physical problem and mathematical formulation The phys i cal prob lem in volves an in fi nite flat plate of thick ness 2L ini tially at uni form tem per a ture and uni form mois ture con tent (fig. 1). The sur faces of the dry ing body are in con tact with the dry ing air, thus re sult ing in a con vec tive bound ary con di tion for both the tem per a ture and the mois ture con tent. The prob lem is sym met ri cal rel a tive to the mid-plane of the plate. Figure 1. Scheme of the drying experiment In this case of an in fi nite flat plate, if the shrink age of the ma te rial can be ne - glected (r s = const), the un steady tem per a ture field, T(x, t), and mois ture con tent field, X(x, t), in the dry ing body are ex pressed by the fol low ing sys tem of cou pled non lin ear par tial dif fer en tial equa tions for en ergy and mois ture trans port: cr s T t x k T x er H X sd (1) t 32

Kanevce, Lj., Kanevce, G., Dulikravich, G.: Application of Inverse Concepts to Drying X t x D X D T d (2) x x Here, t, x, c, k, H, e, d, D, and r s are time, distance from the mid-plane of the plate, specific heat, thermal conductivity, latent heat of vaporization, ratio of water evaporation rate to the reduction rate of the moisture content, thermo-gradient coefficient, moisture diffusivity, and density of dry solid, respectively. As ini tial con di tions, uni form tem per a ture and mois ture con tent pro files are as - sumed: t = 0 T(x, 0) = T 0, X(x, 0) = X 0 (3) The tem per a ture and the mois ture con tent bound ary con di tions on the sur faces of the dry ing body in con tact with the dry ing air are: T k jq DH ( 1 e) jm 0 x xl X T Drs Ddrs jm 0 x x xl xl (4) The con vec tive heat flux, j q (t), and mass flux, j m (t), on these sur faces are: j h( T T ) q a xl j h ( C C ) m D xl where h is the heat transfer, and h D is the mass transfer coefficient, T a is the temperature of the drying air, and T x=l is the temperature on the surface of the drying body. The water vapor concentration in the drying air, C a, is calculated from: a (5) C a j p ( Ta ) 4619. ( 273) s T a (6) where jis the relative humidity of the drying air and p s is the saturation pressure. The water vapor concentration of the air in equilibrium with the surface of the body exposed to convection is calculated from: C xl a( TxL, X xl ) ps ( TxL ) 4619. ( T 273) xl (7) The wa ter ac tiv ity, a, or the equi lib rium rel a tive hu mid ity of the air in con tact with the con vec tion sur face at tem per a ture T x=l and mois ture con tent X x=l is cal cu lated from ex per i men tal wa ter sorp tion iso therms. The bound ary con di tions on the mid-plane of the dry ing plate are: 33

THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 31-44 T x X 0, 0 (8) x x0 x0 The prob lem de fined by eqs. (1-8) is re ferred to as a di rect prob lem when ini tial and bound ary con di tions as well as all pa ram e ters ap pear ing in the for mu la tion are known. The ob jec tive of the di rect prob lem is to de ter mine the tem per a ture and mois ture con tent fields in the dry ing body. Estimation of parameters For the in verse prob lem of in ter est here, the mois ture diffusivity to gether with other thermophysical prop er ties of the dry ing body as well as the heat and mass trans fer co ef fi cients are re garded as un known pa ram e ters. The es ti ma tion meth od ol ogy used is based on the minimization of the or di nary least square norm: T E( P) [ Y T( P)] [ Y T( P)] (9) Here, Y T = [Y 1, Y 2,, Y imax ] is the vector of measured temperatures, T T = [T 1 (P), T 2 (P), T imax (P)] is the vector of estimated temperatures at time t i (i = 1, 2,, imax), P T = = [P 1, P 2,, P N ] is the vector of unknown parameters, imax is the total number of measurements, and N is the total number of unknown parameters (imax N). A hy brid op ti mi sa tion al go rithm OPTRAN [10] and the Levenberg-Marquardt method [11, 12] have been uti lized for the minimization of E(P) rep re sent ing the so lu tion of the pres ent pa ram e ter es ti ma tion prob lem. The Levenberg-Marquardt method is a quite sta ble, pow er ful, and straight for - ward gra di ent search minimization al go rithm that has been ap plied to a va ri ety of in verse prob lems. It be longs to a gen eral class of damped least square meth ods. The so lu tion for vec tor P is achieved us ing the fol low ing it er a tive pro ce dure: P r+1 = P r +[(J r ) T J r + m r I] 1 (J r ) T [Y T(P r )] (10) where r is the number of iterations, I is identity matrix, m is the damping parameter, and J is the sensitivity matrix defined as: T P J Ti P 1 1 max 1 T P T P 1 N i max N (11) 34

Kanevce, Lj., Kanevce, G., Dulikravich, G.: Application of Inverse Concepts to Drying Near the ini tial guess, the prob lem is gen er ally ill-con di tioned so that large damp ing pa ram e ters are cho sen thus mak ing term mi large as com pared to term J T J. The term mi damps in sta bil i ties due to the ill-con di tioned char ac ter of the prob lem. So, the ma trix J T J is not re quired to be non-sin gu lar at the be gin ning of it er a tions and the pro ce - dure tends to wards a slow-con ver gent steep est de scent method. As the it er a tion pro cess ap proaches the con verged so lu tion, the damp ing pa ram e ter de creases, and the Levenberg-Marquardt method tends to wards a Gauss method. In fact, this method is a com pro mise be tween the steep est de scent and Gauss method by choos ing m so as to fol - low the Gauss method to as large an ex tent as pos si ble, while re tain ing a bias to wards the steep est de scent di rec tion to pre vent in sta bil i ties. The pre sented it er a tive pro ce dure ter - mi nates if the norm of gra di ent of E(P) is suf fi ciently small, if the ra tio of the norm of the gra di ent of E(P) to E(P) is small enough, or if the changes in the vec tor of pa ram e ters are very small. An al ter na tive to the Levenberg-Marquardt al go rithm is the hy brid op ti mi sa tion pro gram OPTRAN [10]. OPTRAN in cor po rates six of the most pop u lar op ti mi sa tion al - go rithms: the Davidon-Fletcher-Powell gra di ent search [13], se quen tial qua dratic pro - gram ming (SQP) al go rithm [14], Pshenichny-Danilin quasi-new to nian al go rithm [15], a mod i fied Nelder-Mead (NM) sim plex al go rithm [16], a ge netic al go rithm (GA) [17], and a dif fer en tial evo lu tion (DE) al go rithm [18]. Each al go rithm pro vides a unique ap proach to op ti mi sa tion with vary ing de grees of con ver gence, re li abil ity and ro bust ness at dif fer - ent stages dur ing the it er a tive op ti mi sa tion pro ce dure. The hy brid optimiser OPTRAN in cludes a set of rules and switch ing cri te ria to au to mat i cally switch back and forth among the dif fer ent al go rithms as the it er a tive pro cess pro ceeds in or der to avoid lo cal min ima and ac cel er ate con ver gence to wards a global minimum. The pop u la tion ma trix was up dated ev ery it er a tion with new de signs and ranked ac cord ing to the value of the ob jec tive func tion, in this case the or di nary least square norm. As the op ti mi sa tion pro cess pro ceeded, the pop u la tion evolved to wards its global minimum. The op ti mi sa tion prob lem was com pleted when one of sev eral stop ping cri te - ria was achieved: the maximum number of iterations or objective function evaluations was exceeded, the best design in the population was equivalent to a target design, or the optimisation program tried all six algorithms, but failed to produce a non- -negligible decrease in the objective function. The last cri te rion usu ally in di cated that a global min i mum had been found. Results and discussions The pro posed method of si mul ta neous es ti ma tion of the thermophysical prop er - ties of a dry ing body as well as the heat and mass trans fer co ef fi cients, by us ing only tem - per a ture mea sure ments, was tested for a model ma te rial, in volv ing a mix ture of ben ton ite and quartz sand. From the ex per i men tal and nu mer i cal ex am i na tions of the tran sient mois ture and tem per a ture pro files [19] it was con cluded that for prac ti cal cal cu la tions, the in flu ence of the thermodiffusion is small and can be ig nored. Con se quently, dwas 35

THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 31-44 uti lized in this pa per. It was also con cluded that in this case, the sys tem of two par tial dif - fer en tial eqs. (1) and (2) can be used by treat ing all the co ef fi cients ex cept for the mois - ture diffusivity as con stants. The ap pro pri ate mean val ues for the model ma te rial are [19]: density of the dry solid, r s = 1738 kg/m 3, heat capacity, c = 1550 J/Kkg (db), thermal conductivity, k = 2.06 W/Km, latent heat of vaporization, H = 2.3110 6 J/kg, phase conversion factor, e = 0.5, and thermo-gradient coefficient, d = 0. The fol low ing ex pres sion can de scribe the ex per i men tally ob tained re la tion ship for the mois ture diffusivity [20]: 10 12 2 T 273 D 90. 10 X (12) 303 The ex per i men tally ob tained desorption iso therms of the model ma te rial [19] are pre sented by the em pir i cal equa tion: 6 0. 91 0. 005( T 273) 3. 91 a 1 exp[ 15. 10 ( T 273) X ] (13) where the water activity, a, represents the relative humidity of the air in equilibrium with the drying object at temperature T and moisture content X. For the di rect prob lem so lu tion, the sys tem of eqs. (1) and (2) with the ini tial con di tions (3) and the bound ary con di tions (4) and (8) was solved nu mer i cally with the ex per i men tally de ter mined thermophysical prop er ties. An ex plicit fi nite-dif fer ence pro - ce dure was used. For the in verse prob lem in ves ti gated here, val ues of the mois ture diffusivity, D, den sity of the dry solid, r s, heat ca pac ity, c, ther mal con duc tiv ity, k, phase con ver sion fac tor, e, heat trans fer co ef fi cient, h and mass trans fer co ef fi cient, h D, are re garded as un - known. Here the mois ture diffusivity of the model ma te rial has been rep re sented by the fol low ing func tion of tem per a ture and mois ture con tent: 2 T 273 D D X X 303 where D X and D T are constants. Thus, the vec tor of un known pa ram e ters is: D T (14) P T = [D X, D T, r s, c, k, e, h, h D ] (15) For the es ti ma tion of these un known pa ram e ters, the tran sient read ings of a sin - gle tem per a ture sen sor lo cated in the mid-plane of the sam ple was con sid ered (fig. 1). 36

Kanevce, Lj., Kanevce, G., Dulikravich, G.: Application of Inverse Concepts to Drying The relative sensitivity coefficient, P m T i /P m, i = 1, 2,..., imax, m = 1, 2,, N, anal y sis was car ried out in or der to de fine the in flu ence of the dry ing air ve loc ity and tem per - a ture and dry ing body di men sion [2-5]. Fol low ing the con clu sions of these pre vi ous works the se lected dry ing air bulk tem per a ture, speed, and rel a tive hu mid ity were T a = 80 C, v a, = 10 m/s and j = 0.12, re spec tively, and the plate thick ness, 2L = 4 mm. The ini tial mois - ture con tent was X(x, 0) = 0.20 kg/kg and ini tial tem per a ture T(x, 0) = 20 C. Fig ure 2 shows the relative sensitivity coefficients for temperature with respect to all unknown parameters, m = 1, 2,, 8. Figure 2. Relative sensitivity coefficients It can be seen that the rel a tive sen si tiv ity co ef fi cients with re spect to the phase con ver sion fac tor, e, and the ther mal con duc tiv ity, k, are very small. This in di cates that e and k can not be es ti mated in this case. This also in di cates that the in flu ence of the phase con ver sion fac tor and the ther mal con duc tiv ity on the tran sient mois ture con tent and tem - per a ture pro files is very small in this case. This can be ex plained by the very small heat transfer Biot num ber (Bi = hl/k = 0.08) and con se quently very small tem per a ture gra di - ents in side the body dur ing the dry ing. For these rea sons, the phase con ver sion fac tor and the ther mal con duc tiv ity were treated as known quan ti ties for the ex am i na tion de scribed be low. The rel a tive sen si tiv ity co ef fi cients with re spect to the dry ma te rial den sity, r s, and the con vec tion heat trans fer co ef fi cient, h, are lin early-de pend ent. This makes it im - pos si ble to si mul ta neously es ti mate r s and h. Due to these rea sons and to the fact that the den sity of the dry ma te rial can be rel a tively eas ily de ter mined by a sep a rate ex per i ment, the den sity of the dry ma te rial was as sumed as known for the in verse anal y sis. 37

THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 31-44 Thus, it ap pears to be pos si ble to es ti mate si mul ta neously the mois ture diffusivity pa ram e ters, D X and D T, the heat ca pac ity, c, the con vec tion heat trans fer co ef - fi cient, h, and the mass trans fer co ef fi cient, h D, by a sin gle ther mo cou ple tem per a ture re - sponse in a thin dry ing plate. De ter mi nant of the in for ma tion ma trix J T J with nor mal ized el e ments: i max T Ti Ti [ J J] m, n Pm Pn, m, n, N i P m P 1 1 n (16) has been calculated in order to define drying time. Figure 3 presents transient variations of the determinant of the information matrix if (D X, D T, c, h, h D ), (D X, D T, c, h), (D X, D T, h, h D ), and (D X, D T, h) are simultaneously considered as unknown parameters. Elements of these determinants of the information matrix were defined [21] for a large, but fixed number of transient temperature measurements (101 in this case). Figure 3. Determinant of the information matrix The du ra tion of the dry ing ex per i ment (the dry ing time) cor re spond ing to the max i mum de ter mi nant value was used for the com pu ta tion of the un known pa ram e ters. The max i mum de ter mi nant of the in for ma tion ma trix cor re sponds to the dry ing time when nearly equilibrium mois ture con tent and tem per a ture pro files have been reached, as can be seen in figs. 4 and 5. The tran sient read ings of the sin gle tem per a ture sen sor lo cated in the mid-plane of the sam ple were ob tained by sim u lated ex per i ments. The ex per i men tal data were ob - tained from the nu mer i cal so lu tion of the di rect prob lem pre sented above, by treat ing the val ues and ex pres sions for the ma te rial prop er ties as known. In or der to sim u late real 38

Kanevce, Lj., Kanevce, G., Dulikravich, G.: Application of Inverse Concepts to Drying Figure 4. Transient moisture content and temperature profiles Figure 5. Volume-averaged moisture content and temperature changes during the drying mea sure ments, a nor mally dis trib uted er ror with zero mean and stan dard de vi a tion, s of 0.2, 0.5, 1.0, and 1.5 ºC was added to the nu mer i cal tem per a ture re sponse (fig. 6). Ta ble 1 shows the es ti mated pa ram e ters for s= 0.2, 0.5, 1.0, and 1.5 C. For com par i son, the val ues of ex act pa ram e ters and the val ues es ti mated with er ror less (s = = 0) tem per a ture data are shown in this ta ble. Table 1 also shows the rel a tive er rors of the es ti mated pa ram e ters, e. From the ob tained re sults it ap pears to be pos si ble to es ti mate si mul ta neously the mois ture diffusivity pa ram e ters, D X and D T, the heat ca pac ity, c, the con vec tion heat 39

THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 31-44 trans fer co ef fi cient, h, and the mass trans fer co ef fi cient, h D, by a sin gle ther mo cou ple tem - per a ture re sponse. The ac cu racy of com put ing the pa ram e ters strongly de pends on the noise, s, (cases A1 to A5). Sim i lar ac cu racy has been ob tained if D X, D T, h, and h D were si mul ta - neously es ti mated (case A6). But, in the case when only the heat trans fer co ef fi cient, h, was si mul ta neously es ti mated with the mois ture Figure 6. Simulated temperature diffusivity pa ram e ters, D X and D T, the rel a tive response at x = 0 with a noise s = 1.5 C er rors of the com puted pa ram e ters were within one per cent (case A7). The very high mass trans fer Biot num ber and the very small heat trans fer Biot num ber can ex plain this. From this rea son, the tem per a ture sen si tiv ity co ef fi cient with re spect to the con vec tion mass trans fer co ef fi cient h D is very small rel a tive to the tem per a ture sen si - tiv ity co ef fi cient with re spect to the con vec tion heat trans fer co ef fi cient, h. Table 1. Estimated parameters Case D X 10 12 [m 2 /s] D T [ ] c [J/kgK] h [W/m 2 K] h D 10 2 [m/s] A1 Es ti mated 8.99 10.0 1551 83.1 9.29 s = 0 C e[%] 0.1 0.0 0.1 0.0 0.0 A2 Es ti mated 9.04 10.00 1550 83.2 9.12 s = 0.2 C e[%] 0.4 0.0 0.0 0.1 1.8 A3 Es ti mated 9.06 10.10 1551 83.3 8.88 s = 0.5 C e[%] 0.7 1.0 0.1 0.2 4.4 A4 Es ti mated 9.45 9.44 1597 83.5 8.62 s = 1.0 C e[%] 5.0 5.6 3.0 0.5 7.2 A5 Es ti mated 9.76 9.03 1627 83.8 8.31 s = 1.5 C e[%] 8.4 9.7 5.0 0.8 10.5 A6 Es ti mated 9.87 8.79 83.2 8.28 s = 1.5 C e[%] 9.7 12.1 0.1 10.9 A7 Es ti mated 8.93 9.95 83.0 s = 1.5 C e[%] 0.8 0.5 0.1 Exact values 9.00 10.0 1550 83.1 9.29 40

Kanevce, Lj., Kanevce, G., Dulikravich, G.: Application of Inverse Concepts to Drying To over come this prob lem, in this pa per the mass trans fer co ef fi cient was re lated to the heat trans fer co ef fi cient through the anal ogy be tween the heat and mass trans fer pro cesses in the bound ary layer over the dry ing body [22]: D hd 095 k. a where D a and k a are the moisture diffusivity and thermal conductivity in the air, respectively. The obtained relation is very close to the well-known Lewis relation. By using the above relation between the heat and mass transfer coefficients, they can be estimated simultaneously, if only the heat transfer coefficient is regarded as an unknown parameter (tab. 2). a h (17) Table 2. Estimated parameters using the relation (17) Case D X 10 12 [m 2 /s] D T [ ] c [J/kgK] h [W/m 2 K] h D 10 2 [m/s] B4 Es ti mated 8.83 10.20 1600 83.4 9.33 s = 1.0 C e[%] 1.9 2.0 3.2 0.4 0.4 B5 Es ti mated 8.82 10.17 1620 83.5 9.34 s = 1.5 C e[%] 2.0 1.7 4.5 0.5 0.5 B6 Es ti mated 8.89 10.02 83.0 9.28 s = 1.5 C e[%] 1.2 0.2 0.1 0.1 Exact values 9.00 10.0 1550 83.1 9.29 The rel a tive er rors of the com puted heat ca pac ity, c, in the cases B4 and B5 are nearly the same as in the cases A4 and A5, re spec tively, but the rel a tive er rors of the com - puted D X, D T, and h D are much lower. From the ob tained re sults in the case B6, it ap pears to be pos si ble to es ti mate si mul ta neously with very high ac cu racy, the mois ture diffusivity pa ram e ters, D X and D T, and the heat and mass trans fer co ef fi cients, h and h D, by a sin gle ther mo cou ple tem per a ture re sponse even in the case with the rel a tively high noise of 1.5 C. Conclusions A method of si mul ta neous es ti ma tion of the mois ture diffusivity to gether with the other thermophysical prop er ties of a dry ing body as well as the heat and mass trans fer co ef fi cients by us ing only tem per a ture mea sure ments, has been an a lyzed in this pa per. 41

THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 31-44 The Levenberg-Marquardt and the hy brid op ti mi sa tion method were ap plied for eval u a tion of the un known pa ram e ters. The two mois ture diffusivity pa ram e ters and the heat ca pac ity of the dry ing body to gether with the heat and mass trans fer co ef fi cients were si mul ta neously es ti - mated. The ob tained re sults show good agree ment be tween the eval u ated and ex act val - ues of pa ram e ters and con firm the va lid ity of the pro posed method. An anal y sis of the in flu ence of the tem per a ture mea sure ments er rors on the ac - cu racy of the es ti mated pa ram e ters was also pre sented. Nomenclature a water activity c heat capacity, [J/Kkg(db)] C concentration of water vapor, [kg/m 3 ] D moisture diffusivity, [m 2 /s] h heat transfer coefficient, [W/m 2 K] h D mass transfer coefficient, [m/s] H latent heat of vaporization, [J/kg] I identity matrix j m mass flux, [kg/m 2 s] j q heat flux, [W/m 2 ] J sensitivity matrix k thermal conductivity, [W/Km] L flat plate thickness, [m] p s saturation pressure, [Pa] P vector of unknown parameters t time, [s] T temperature, [ C] T vector of estimated temperature, [ C] v velocity, [m/s] x distance from the mid-plane, [m] X moisture content (dry basis), [kg/kg(db)] Y vector of measured temperature, [ C] Greec letters d thermo-gradient coefficient, [1/K] e phase conversion factor relative error, [%] s standard deviation m damping parameter r density, [kg/m 3 ] j relative humidity Subscripts a s drying air dry solid 42

Kanevce, Lj., Kanevce, G., Dulikravich, G.: Application of Inverse Concepts to Drying References [1] Kanevce, G. H., Kanevce, L. P., Dulikravich, G. S., Mois ture Diffusivity Es ti ma tion by Tem - per a ture Re sponse of a Dry ing Body, in: In verse Prob lems in En gi neer ing Me chan ics II (Eds. M. Tanaka, G. S. Dulikravich), Elsevier, Am ster dam, 2000, pp. 43-52 [2] Kanevce, G. H., Kanevce, L. P., Dulikravich, G. S., In flu ence of Bound ary Con di tions on Moisture Diffusivity Es ti ma tion by Tem per a ture Re sponse of a Dry ing Body, Pro ceed ings, 34 th ASME Na tional Heat Trans fer Conference, Pitts burgh, PA, USA, Au gust 20-22, 2000, ASME pa per NHTC2000-12296 [3] Kanevce, G. H., Kanevce, L. P., Mitrevski, V. B., Dulikravich, G. S., Mois ture Diffusivity Estimation from Temperature Measurements: Influence of Measurement Accuracy, in: Pro - ceedings (Eds. P. J. A. M. Kerkhof, W. J. Coumans, G. D. Mooiweer), 12 th In ter na tional Dry - ing Sym po sium IDS 2000, Noordwijkerhout, The Neth er lands, 2000, pa per No. 337 [4] Kanevce, G. H., Kanevce, L. P., Dulikravich, G. S., Si mul ta neous Es ti ma tion of Thermophysical Prop er ties and Heat and Mass Trans fer Co ef fi cients of a Dry ing Body, in: In verse Prob lems in En gi neer ing Me chan ics III (Eds. M. Tanaka, G. S. Dulikravich), Elsevier, Am - ster dam, 2002, pp. 3-12 [5] Kanevce, G. H., Kanevce, L. P., Dulikravich, G. S., Orlande H. R. B., Es ti ma tion of Thermophysical Prop er ties of Moist Ma te ri als un der Dif fer ent Dry ing Con di tions, Inverse Prob lems in Sci ence and En gi neer ing, 13 (2005) 4, pp. 341-353 [6] Dantas, L. B., Orlande, H. R. B.. Cotta, R. M., De Souza, R., Lobo, P. D. C., In verse Anal y sis in Moist Cap il lary Po rous Me dia, Pro ceed ings on CD, 15 th Brazilian Congress of Mechanical En gi neer ing, Sao Paulo, Brazil, No vem ber 7-11, 1999 [7] Dantas, L. B., Orlande, H. R. B., Cotta, R. M., Lobo, P. D. C., Pa ram e ter Es ti ma tion in Moist Cap il lary Po rous Me dia by Us ing Tem per a ture Mea sure ments, in: In verse Prob lems in En gi - neering Mechanics II (Eds. M. Tanaka, G. S. Dulikravich), Elsevier, Am ster dam, 2000, pp. 53-62 [8] Dantas, L. B., Orlande, H. R. B., Cotta, R. M., Ef fects of Lat eral Heat Losses on the Pa ram e - ter Es ti ma tion Prob lem in Moist Cap il lary Po rous Me dia, in: In verse Prob lems in En gi neer - ing Me chan ics III (Eds. M. Tanaka, G. S. Dulikravich), Elsevier, Am ster dam, 2002, pp. 13-22 [9] Luikov, A. V., Heat and Mass Trans fer (in Rus sian), Energia, Mos cow, 1972 [10] Dulikravich, G. S., Mar tin, T. J., Den nis, B. H., Fos ter, N. F., Multidisciplinary Hy brid Con - strained GA Op ti mi za tion, Chap ter 12, in: Evo lu tion ary Al go rithms in En gi neer ing and Com puter Sci ence: Re cent Ad vances and In dus trial Ap pli ca tions (Eds. K. Miettinen, M. M. Makela, P. Neittaanmaki, J. Periaux), Jyvaskyla, Fin land, May 30 June 3, 1999, John Wiley & Sons, Ltd., pp. 231-260 [11] Marquardt, D. W., An Al go rithm for Least Squares Es ti ma tion of Non lin ear Pa ram e ters, J. Soc. Ind. Appl. Math., 11 (1963), pp. 431-441 [12] Beck, J. V., Ar nold, K. J., Pa ram e ter Es ti ma tion in En gi neer ing and Sci ence, John Wiley & Sons., Inc., New York, 1977 [13] Fletcher, R., Powell, M. J. D., Rap idly Con ver gent De scent Method for Minimization, Com - puter Jour nal, 6 (1963), pp. 163-168 [14] Rao, S., En gi neer ing Op ti mi za tion: The ory and Prac tice, 3 rd ed., John Wiley Interscience, New York, 1996 [15] Pshenichny, B. N., Nu mer i cal Meth ods in Extremal Prob lems (in Rus sian), Mir, Mos cow, USSR 1969 [16] Nelder, J. A., Mead, R., A Sim plex Method for Func tion Minimization, Com puter Jour nal, 7 (1965), pp. 308-313 [17] Goldberg, D. E., Ge netic Al go rithms in Search, Op ti mi za tion and Ma chine Learn ing, Ad di - son-wes ley Publ. Comp., Cam bridge, MA, USA, 1989 43

THERMAL SCIENCE: Vol. 9 (2005), No. 2, pp. 31-44 [18] Storn, R., Dif fer en tial Evo lu tion A Sim ple and Ef fi cient Heu ris tic for Global Op ti mi za tion over Con tin u ous Spaces, Jour nal of Global Op ti mi za tion, 11 (1997), 4, pp. 341-359 [19] Kanevce, G. H., Nu mer i cal Study of Dry ing, Pro ceed ings, 11 th In ter na tional Dry ing Sym po - sium IDS'98, Halkidiki, Greece, Au gust 19-22, 1998, Vol. A, pp. 256-263 [20] Kanevce, G. H., Stefanovi}, M., Pavasovi}, V., Ex per i men tal De ter mi na tion of the Diffusivity of Mois ture within Cap il lary Po rous Bod ies, Pro ceed ings, Drying'80, Hemi - sphere Publ. Comp., 1980, Vol. 1, pp. 128-131 [21] Ozisik, M. N., Orlande, H. R. B., In verse Heat Trans fer: Fun da men tals and Ap pli ca tions, Tay lor and Fran cis, New York, 2000 [22] Kanevce, G. H., Kanevce, L. P., Dulikravich, G. S., An In verse Method for Dry ing at High Mass Trans fer Biot Num ber, Pro ceed ings, HT03 ASME Sum mer Heat Trans fer Con fer ence, Las Ve gas, NV, USA, July 21-23, 2003, ASME pa per HT20003-40146 Authors' addresses: Lj. Kanevce, Faculty of Technical Sciences St. Kliment Ohridski University Bitola, Macedonia G. Kanevce Macedonian Academy of Sciences and Arts Skopje, Macedonia G. Dulikravich Department of Mechanical and Materials Engineering Florida International University 10555 West Flagler St., EC 3474 Miami, Florida 33174, U.S.A. Corresponding author (Lj. Kanevce): E-mail: kanevce@osi.net.mk Paper submitted: June 25. 2005 Paper revised: July 9, 2005 Paper accepted: August 18, 2005 44