英文学术论文范例

1、Lebensm.-Wiss. u.-Technol., 28.38-42 ( 1995 Mathematical Model for Computer Simulation of Moisture Transfer in Multiple Package Systems G. C. P. Rangarao, U. V. Chetana, and P. Veerraju Centre for Food Packaging. Central Food Technological Research Institute, Mysore 570013 (India) (Received May 24.

2、1994, accepted July 27. 1994) A mathematical model was developed for moisture tran,fer ill nudtiple package systems wherein different food products or components are packed in individual primal 3 packages and fitrther unitized in a seconda O package. Tile Guggenhein-Anderson-de Boer (GAB) equation f

3、or moisture sorption isotherm was differentiated with respect to time and incorporated in the generalized moisture transport equation for the multiple package system resulting ill a system simultaneous 6rst order differential equations relating changes in water activities of the products to storage

4、tittle. These equations are solved hv the Runge-Ktttta method, and the corresponditlg moisture changes in tile products are obtained by using the original isotherm equations. The computer programme developed on tile basis of the model was used to simulate moisture changes in a 2-component multiple p

5、ackage system )doughnut mix. The predictions were in close agreement with tile eaperimental values. Introduction Loss or gain of moisture plays a critical role in deciding the quality and shelf-life of foods. Moisture transfer in a packaged food depends on the water activity of the food, temperature

6、 and humidity conditions of storage, and the permeability of the package to water vapour. The relation- ships between moisture sorption properties of foods, perme- ability of the film to moisture, and shelf-life were brought out by Heiss (1), which were used in predicting moisture transfer under ste

7、ady state conditions of temperature and humidity. Further modifications were made by Karel et al. (2), and computer solutions were introduced at a later stage (3,4). Labuza and Contreras-Medellin 15) reviewed the work on prediction of moisture protection requirements of foods. Most of the work repor

8、ted on moisture transfer in packaged foods relates to the more common situation of a single product packed in a primary package. Though multiple package systems wherein different food products or compo- nents are packed in individual primary packages and further unitized in a secondary package are q

9、uite common, very little information is available on moisture transfer in such systems. Multiple package systems are frequently used when two or more complementary components of a food product are sold together as in case of instant mixes. Assorted food items are also packed in a similar way. Someti

10、mes, one of the components may be without the primary package in which case the component of smaller quantity is usually packed in a small sachet and inserted into the former and then both packed together in an outer package. Multiple package systems are often used for incorporating in-package desic

11、cants in moisture-sensitive foods. Also, multiples of primary packages containing the same product are unitized in a secondary package to facilitate convenient sale of multiple units. 0023-6438/95/010038 + 05 $08.00/0 In all these multiple package systems, the packaging vari- ables are very many com

12、pared to those of a single package, and it can be very time consuming and expensive to carry out shell-life tests based on moisture changes under a variety of packaging and environmental conditions. Hence, computer simulation is the alternative approach for predic- tion of moisture changes and shelf

13、-life and also for the design of an optimum packaging system for moisture- sensitive foods. Hirata et al. (6,7) reported a method for prediction of moisture changes in dried, roasted, and sea- soned laver double packaged with a desiccant using a generalized mathematical model. However, they used emp

14、irical equations to describe the moisture sorption iso- therms of the product. If the sorption isotherm equation selected is of the Guggenheim-Anderson-de Boer (GAB) type, wherein water activity cannot be expressed as a direct function of moisture, direct calculation of water activity from moisture

15、will not be possible as per their model. In that case, iteration method has to be employed thus making the computational procedure less efficient. Besides, the compu- tational procedure used for solving the differential equations does not employ a standard numerical method. This study was designed t

16、o develop a generalized mathematical model for computer simulation of moisture transfer in various cases of multiple package systems, by incorporating the widely accepted GAB equation for moisture sorption iso- therms in the moisture transport equations for the packaging system. Mathematical Model F

17、ood moisture sorption isotherms, which show the relation between water activity and moisture content of the food can be approximated by a variety of mathematical relationships (8). In general, it can be represented as: () 1995 Academic Press Limited 38 Iwt/vol. 28 1995) No. 1 m = flaw) Eqn 11 where

18、m is the percent moisture content on dry basis and a w is the water activity. Differentiating Eqn 1 with respect to time, t, c3m c3f aw Eqn 2 3t 9aw c)t Though there are about 75 equations to describe the iso- therms (9), the GAB equation was selected for this work because of its known advantages su

19、ch as its sound theoret- ical background and its wide range of applicability from zero to 0.9 water activity (10) and for a large variety of foods (11, 12). It may also be noted that the GAB equation was recommended by the COST-90 group (European Coop- eration in the Field of Scientific and Technica

20、l Research) to fit moisture sorption isotherms (13). The standard GAB equation is given by: moCka w m = (1 - kaw) (1 - kaw + Cka w) where Eqn 3 m o is the moisture content corresponding to the saturation of all primary adsorption sites by one water molecule (formerly called the monolayer in the BET

21、theory) C is the Guggenheim constant relating to interaction energy between water and food, and k is the constant correcting the properties of the multi- layer molecules with respect to the bulk liquid. Differentiating Eqn 3 with respect to time, m moCk 1 - k2a w 2 (1 - C) a w t (l-kaw)2l -kaw(l -C)

22、 2 t Eqn 4 The scheme devised for the generalized multiple package system in this study is represented in Fig. 1. It consists of n food products each packed in multiple number of primary packages of r 1, r 2, r 3 . rn numbers, which in tum are packed together in a secondary external package. Let W I

23、, W 2, W 3 . W n represent the weights of dry matter in the products; Mi, M2, M3 . M, the quantities of moisture in the foods; m 1, m 2, m 3 . m n, their respective percent moisture contents on dry basis; and awl, aw2, aw3, ., awn, their water activities. Let P1, P2, P3 . P, be the water vapour perm

24、eabilities of the films used for the primary packs, and A I, A2, A 3 . An, the corresponding surface areas. The water activity inside the external package is represented by aw and the water activity outside by awE which is equal to the external relative humidity divided by 100. Let PE and A E be the

25、 permeability of the film and the surface area corresponding to the external package respec- tively. The moisture transport through a thin packaging film is described by a pseudo-steady state equation based on Ficks and Henrys laws (14), and for a primary package, it is given by: OM - PiAiPo (aw - a

26、wl) Eqn 5 Ot for i = 1, 2, 3 . n, where Po is the saturated water vapour pressure at the temperature of storage. aWE = RH/100 AE PE A1 Pl A2 P2 A n P. aw 1 2 rl 1 2 r 2 1 2 r n Fig. 1 Schematic representation of the generalized multiple package system Since the major moisture transport mechanisms th

27、rough packaged foods are water vapour diffusion through the package barrier, and within the food, with the former controlling the phenomenon (14), Eqn 5 can be considered to represent the rate of change in the quantity of moisture in the food product. Moisture transport across the external package c

28、an also be written as: M - = PEAEPo (awE - aw) Eqn 6 The rate of moisture transferred through the external pack- age should be equal to the sum of the rates of moisture transferred through the unit packages, since the amount of moisture present in the micro-environment between the primary packages a

29、nd the secondary package is relatively negligible.Hence: M n Mj Eqn 7 Substituting Eqns 5 and 6 in Eqn 7 and simplifying: PEAEawE + rjPjAja w j=l a w = , Eqn 8 PEAE + 17jAj j=l Substituting Eqn 8 in Eqn 5 and representing it in terms of rate of change in percent moisture content: 39 Iwt/vol. 28 (199

30、5) No. I tl 100 PiAiPo PEAE (awe - awl) + Y, rjPjA l (awj - awl) j=1 pl t)t W i PEAE + ,)PjA/ .i=l Eqn 191 From Eqns 4 and 9, 100 PoPiAi PEAE tawE - awi ) + t)PA i 2=1 qawi = (awj - awl) (1 - kiawi)2 1 - kiawi(l - Ci)2 )t WimoiCik i IPEAE + t)PjAy) 1 - ki-awi- I I - Ci) I I for/= 1,2,3 . n. Eqn 10 E

31、qn 10 consists of a system of simultaneous first order differential equations which can be solved for awl values by the Runge-Kutta method (15) or any other suitable numer- ical method. The changes in the moisture contents of the food products corresponding to the changes in the water activities as

32、obtained by solving Eqn 10 can be calculated by using Eqn 3. The model developed relates to the most generalized form of a multiple package system. The adaptability of the model for certain special cases is discussed in the following paragraphs. Case 1 With a single food product (n = I) packed in mu

33、ltiple primary packages which are again unitized in a secondary package, i = j = 1, and Eqn 191 reduces to oqml 100 PoPIAIPEAE(awE - awl) = Eqn 111 oqt W l (PEAE + QPIAt) Case 2 If the multiple package system consists of one product without primary packaging and the remaining products with individua

34、l primary packages, the moisture transport equa- tions represented by 9 can be modified as follows: Let i be the index number of the product without primary package and k represent the index number for the remaining products. The absence of primary package for the ith product is equivalent to having

35、 a primary package of infinite permeability and finite surface area. Hence: k = 1 Pi = oo r i = 1 to n and : i Now, Eqn 9 can be rewritten for the ith product as: 100poPiAi PEAE (awe - awl ) + Yt)PA i (awj - awl ) Ow, qmi = 1=i+1 ;)t i-I n W i PEAE + ,jPA) + ,PiAi + ,jPjAj j=l j=t+l i-I lOOPoPiAi PE

36、AE (awE - awl) + rjP)Aj (awj -awi) n j = I + y,jPjAj (a w - awl j=t+l i-I n WiPi I-i + 1=1 4- A i 4- )= Pi -t J Since Pi = ,x, the above equation becomes: m i l O0 Po i- 1 - - - - PEAE (awe - awi) + rjpiAj (awj - awl) 3t W i , .i = I + tliPjA j (awj -awi) j=i+l Eqn 12 Since i is the index number ass

37、igned to the product without a primary package in this case, awi, the water activity of this product inside an imaginary package of infinite permeabil- ity, is equal to the water activity of the micro-environment lying between the secondary package and the other primary packages. The moisture transp

38、ort equation for all other products kept in individual primary packages can be similarly obtained: c)m k 1 O0 poPk A t (awi - awk) = Eqn 13 t Wk Thus in a multiple package system consisting of only two components of a food product (n = 2), one without (i = 1 ) and the other with (k = 2) primary pack

39、aging, the reduced governing equations are: c)m I 100 Po - - - - PEAE (awe - awl) + P2A2 (aw2 - awl) c)t W 1 Eqn 14 )m2 IOOpoP2A2 (awl - wl - aw2) = Eqn 15 c)t W The moisture transport equations 1 I, 12, 13, 14, and 15 obtained in all these cases can be combined with Eqn 4 to result in differential

40、equations of the form Eqn 10 which can be solved in the same way. Though the above models were derived utilizing the GAB equation for the moisture sorption isotherms of the food products, any other suitable isotherm equation can be incorporated to obtain a system of first order differential equation

41、s of the type 10. Materials and Methods A 2-component multiple package system of doughnut mix was used to carry out the storage experiments to test the model. The doughnut mix formulated and prepared at the Central Food Technological Research Institute, Mysore, consists of two components-the flour m

42、ix and the dry yeast which are packed separately and mixed only at the time of use. The flexible packaging films, vi-., low density polyethylene (LDPE) of thickness 85 Ixm and 55 tm and high density polyethylene (HDPE) of thickness 55 txm, used in the experimental multiple package system, were obtai

43、ned from the local market. The moisture content of the food materials was determined by drying in a vacuum oven at 70C for over 12h until constant weights (within -+0.5 mg) were obtained for con- secutive weighings. The results were expressed as percent of moisture on dry weight basis. For determini

44、ng equilibrium moisture contents at various humidities, samples of the food material weighing 5 _ 0.5 g in duplicate were exposed in Petri dishes to water activities 40 Iwt/vol. 28 1995J No. I Table 1 Characteristics of the doughnut mix Component Component 1 (flour mix) Component 2 (dry yeast) Initi

45、al GAB parameters for moisture moisture sorption isotherm (g/100g of solids m o C k I 1.56 5.4517 21.5348 0.8119 8.34 3.7432 10.3771 1.0224 ranging from 0. I1 to 0.92 at 38 - IC by using salt solutions (16) until equilibrium was reached. The moisture sorption data so obtained were processed using a

46、polyno- mial curve-fitting program (17) modified to yield the GAB parameters on the basis of the transformed GAB equation as suggested by Schfir and Rtiegg (18). The GAB parameters obtained and the initial moisture contents of the components of the doughnut mix are given in Table 1. The water vapour

47、 permeabilities of the flexible films used tbr the packages were determined by packing anhydrous calcium chloride inside same-sized packages as used in the storage experiments in quadruplicate and finding the rate of moisture gain when kept at 90 _+ 2% r.h. and 38 z 1 C. The details of the multiple

48、package system used for storage experiments were described in Table 2. The two compo- nents, vi:., flour mix (10 - 0.0005g) and dry yeast (1 - 0.0005 g) were weighed and packed in individual primary packages which were again weighed after heat-sealing the closures. The primary packages were arranged

49、 side by side while packing in the secondary package in such a way that the primary packages do not overlap each other and full surface areas of the pouches are available for moisture transfer. Five sets in duplicate of such multiple packs were kept in a humidity chamber maintained at 90 - 2c r.h. a

50、nd 38 1 C. Samples were withdrawn at five intervals spread over a total period of 25 days, secondary packages cut open, and the weights of the individual primary packages noted. The changes in weights of the primary packages were used in calculating the moisture in the food materials. Computer Simul

51、ation Based on the mathematical model detailed in the preceding section, a computer programme which we named as MULT- PACK was written in Turbo PASCAL (Version 5.0). Pro- gramme development and simulation were carried out on PCL-4AT6, an lntel-80486-based micro-computer. The pro- gramme listing of M

52、ULTPACK is available from the authors on request. To make the programme as general as possible, the equa- tions representing (i) the most general case of different products with individual primary packages packed together in a secondary pack and (ii) the special case of one of the products packed wi

53、thout primary packaging, were incorpo- rated into the programme. The case of packing a single or multiple primary packages of the same product in a second- ary pack comes under (i). After inputting the required data pertaining to the food products and the packaging system, the initial values of wate

54、r activities of the food products are calculated by an iteration process from initial moisture values and the GAB constants. The next step involves solving the differential equations involved in the Runge-Kutta numerical method (151. Based on the package system under consideration, an appropriate se

55、t of equations are chosen to compute the water activities at required time intervals. The corresponding moisture levels are calculated using the respective GAB constants. The results are output in tabular as well as graphic form. Results and Discussion The mathematical model developed is based on in

56、corpora- tion of the first derivative with respect to time of the standard GAB equation for the moisture sorption isotherm in the generalized moisture transport equation for the multiple package system to result in a system of first order differential equations. Solution of these equations using the

57、 initial boundary conditions and the Runge-Kutta numerical method facilitates prediction of moisture transfer among the various foods packed in the multiple package system. Suitable equations for the special case of a single product packed in multiple primary packages and unitized in a secondary pac

58、kage, or of a product with one of the compo- nents packed without the primary package were also deduced. Though the generalized moisture transport equa- tion for the multiple package system used is similar to that of Hirata et al. (7), our step involving differentiation of the sorption isotherm equa

59、tion makes its incorporation into the Table 2 Description of the muhiple package system used in the study Primary packages 1 2 Secondary package Material packed component l component 2 - Weight of material packed (g) 10.0 1.0 - Packaging film LDPE HDPE LDPE Film thickness (m) 85 55 55 Pouch dimensions: Length (cm) 8.0 6.0 15.0 Breadth (cm) 6.0 4.0 8.0 Water vapour permeability of 4.22 3.96 5.17 the film (I 0