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Drying modelling and water diffusivity in carrots and potatoes
Journal
of Food
Engineering 22 ( 1994) 329-348 8 1994 Elsevier Science Limited
Printed in Great Britain. All rights reserved 0260-8774/94/$7.00 ELSEVIER
Drying Modelling and Water Ditisivity Potatoes Antonio
in Carrots and
Mulet
Food Technology Department, Universidad PolitCcnica de Valencia, Camino de Vera s/n, 0707 1 Valencia, Spain
ABSTRACT Modelling is a useful way to validate mechanisms of drying and to establish properties, and thus can be used to determine the reliabilityof calculated values of the eflective di~ivi~ (De). Di~ive models are considered for the desc~.pt~on of convective drying of p~~~cul~te vegetables (carrots and potatoes). Four models with different degrees of complexity are d&cussed. The degree of complexity depends on consideration of shrinkage, heat tranqer and particle temperature variation during drying, and the D, dependence on moisture and temperature simultaneously. Expe~.ments have been carried out to testthe models and the assumed bounder conditions. Expe~ment~l results show that the eflect of shrinkage cannot be neglected in establishing reliable valuesfor 0,. The effect of the external resistance, which is affected by the velocityof the air used in drying, also appears to be an important factor for testing De values. Two methods are proposed to carry out this test, both of which could be readily applied if ihe dying curve is available. Expe~ments on drying carrots and potatoes are used to illustratethe procedures.
NOTATION
adw
De G
r T t W Y
Constants Effective diffusivity (m2 /s) Air flow rate (kg/m2h) Half-thickness (m) Temperature (“C, K) Time(s) Moisture content (kg water/kg dry solid) Dimensionless moisture ( W- rY,)/( WC- We) 329
A. Mulet
330
Subscripts c
1” P
Critical Equilibrium Local Particle
INTRODUCTION Modelling is essentially a way of representing processes or phenomena to explain the observed data and to predict behaviour under different conditions. Two kinds of models are generally considered: one tries to interpret phenomena from physical laws, and the other represents these phenomena to account for the experimental data and to generalize regardless of the physical reality. Often the bounda~es between the two kinds of model are ill defined, mostly because physical laws are, in themselves, models. In fact, physical phenomena are regulated by laws that we do not know, and ‘physicai laws’ deal with the abstractions of our models. The essence of scientific activity is to establish more realistic models, or, in other words, to attain a better description of reality. Complexity is the key characteristic of reality, and sapling is the key process in modelling. Thus modelling is science with a great deal of art and sometimes luck. A considerable amount of work has been carried out in recent years (Jayaraman & Das Gupta, 1992) on the drying of fruits and vegetables. To simplify and delimit the subject to some extent, we will concentrate mostly on drying of particulate foods by convective drying. For this particular kind of drying, models based on physical laws should explain two drying periods; the first showing a constant rate of drying, and the second with a falling rate. The first period does not always appear, and this poses a problem in defining the starting point for the second one. In this case, the initial characteristics of the material are assumed as the starting point. The first drying period, or constant rate period, persists until the transpo~ mechanisms inside the particle are able to feed the surface with enough water to maintain the constant mass flow from the surface. Usually, the first period is associated with the existence of free water at the surface, which is a wet surface. Nevertheless, it has been shown that with only partia~y wet surfaces the first drying period is detected in some cases (Maneval & Whitaker, 1988). During this period a moisture
gradient builds up in the particle and this should be considered in determining the falling rate period starting point. Strong ar~me~ts took place in the past about the transport mechanisms describing water migration during the second drying period, or the falling rate period (Tsotsas, 1992). Nowadays, although we recognize the complexity of the physical phenomena and the fact that no simple model can account far real behaviour, two types of models are commonly used. One is based on the diffusional transport of water (the diffusion model), and the other is based mostly on capillary transport (the receding front model). The diffusion model was formulated by Lewis (1922 ) and further developed by Sherwood (1929). The transport on capiIlary porous media was mathemati~~y foliated by Luikov f 1966). A third kind of model is based on the complete ~onse~ation equations ~W~taker, 3977), and gives rna~erna~~~y complex formu~a~ons with the added problems linked to the lack of ~owledge of model parameters. In addition to these rn~h~s~~ models, more empirical des~~p~~~s of the dreg prucess are often used. These empirical models do not pretend to describe how the process takes place but rather give an outline of what happens. This is done through the use of drying curves, which may or may not be standard~ed. Despite the fact that these modeis do not provide a physical desc~ptiun of the phenamena taking place, they are often useful for design purposes in excessively complex cases. The ~haracte~stic drying curve is useful for this purpose even with shrinking food materials (Ratti & Crapiste, 1992). The use of mechanistic models for design purposes is preferred wherever possible (Tsotsas, 1992). In this case, the drying kinetics of a single particle is the best choice for scaling-up. The scaling-up procedure is done using estab~shed engineering procedures. Among the mechanistic models the d~sion~ ones seem to be the most popular, They are ease to formulate and normally provide reasonable resuhs. The only difficulty is that along the way a number of simphfying assump~ons are admitted. Therefore we should talk about effective di~~i~ty (l>,)” This di~sivity includes the effects of the known hypetheses as well as the unknown phenomena not being modeled, Under these cir~u~tances, the question of the vale of the values computed is raised as well as that of what is realIy happe~g inside the particle, Many so-called ‘m~ch~sti~’ models have such safe assumptions that they are in effect empirical models. The mudel shoutd be viewed as a descriptive tool of the phenomena taking place. Its use as an analytical tool makes it valuable for understanding meehanisms in some cases, although it can be of very limited usefi&ress in other cases.
A. Mulet
332
The next step will be to validate the models through experimentation. The experiments needed depend on the kind of model; thus varying degrees of detail will be necessary. Until recently, the measurements commonly used were the weight of the samples or the particle temperature. The water distribution inside the particle was not measured becaue of the absence of techniques; it was computed with the use of a model. The total water content was obtained by weighing the sample, and the accuracy of the model was evaluated by integrating the profile of moisture and comparing it with the sample weight. This type of procedure is disconcerting because we gain insight into the behaviour of the solid only through the integration of the model to match the water content. Of course, this matching can be done with different models and assumptions; thus, testing models can be a difficult task. Consequently, the need arises for more powerful experimental tools which can show water profiles inside particles. Two techniques are now very promising - NMR imaging (McCarthy & Perez, 1990; Schrader & Lichtfield, 1992) and neutron transmission (Ketelaars et al., 1992). These could be an additional and interesting way of testing hypotheses in modelling. In addition, the NMR technique could be used for determining water binding (Wang & Brennan, 1992). These newer techniques will bring a better understanding of the phenomena. One of the main conclusions arising from the use of these techniques is that, in general, the diffusional model does not appear to describe the process inside the particle accurately. Nevertheless, the use of effective diffusivity of water in drying will be a useful way of representing the phenomena involved in the description of the overall drying rate and thus is useful for design. The diffusion-like process taking place in drying is then described through water diffusivity, a transport property which will be examined more closely below.
COMPLEXITY
OF THE MODELS
The complexity needed in a model, or in other words the level of detail, depends on the target to be reached. The end use of the model will establish the degree of complexity. In general, a model could be built to describe microscopic particle or macroscopic level behaviour. The lower the scale of description the greater the complexity, because a more detailed description of the phenomena is involved. This means also that, at the microscopic level, more theoretical models are considered ,and at the other extreme, at the
Drying modelling and waterdi@sivity in vegetables
333
macroscopic level, lumped parameters and empirical approaches are commonly used. The ma~ematical form of the model fo~ulation also influences the different approaches used to identify the numerical value of the parameters and thus the effective diffusivity. Microscopic level descriptions are mainly useful for understanding and describing the drying mechanisms. Particle level could be reached by proper integration of microscopic models. The properties measured are then averages within the particle. The kind of water more or less bound to the material changes during the drying operation. At the beginning, the free water is taken out and the remaining water is then bound more strongly to the solid matrix. Of course, there is no si~c~t change in the kind of water being evaporated. Thus, when observing the effective d~si~ty it is difficult to establish different zones within the falling rate period. Often, only one zone is reported in the falling rate period and then only one value of the effective diffusivity identified, regardless of the amount of water still present in the solid, or the character of the remaining water. As pointed out above, the use of more precise measurements, such as those provided by NMR imaging, could throw light on this problem (Wang & Brennan, 1992). At the macroscopic level, a description of the drying env~o~ent is needed. Flow distribution, air velocity, air mois~re, bed characte~stics, etc., should be taken into account. A general way of attaining this level of description could be by integration of particle level models together with the description of the drier characteristics. This approach is preferred by some workers (Tsotas, 1992) in the scaling-up process. It constitutes a more general approach than direct modelling at macroscopic level. From these considerations, it seems appropriate to consider the particle level description as reasonable for most applications. At this level, the effective diffusivity should be carefully evaluated and special attention paid to bounda~ conditions of the model. At particle level, the macroscopic phenomena taking place are well known. The main factors affecting the drying rate are related to the external and internal resistance to mass and heat transfer. If the boundary conditions are not in accordance with the physical reality, e.g. if we admit negligible external resistance when this is not the case, the results are of little value. Most foods shrink when drying, thus moving boundaries need to be taken into account. This makes the model difficult to solve and often only approximate solutions are possible because of anisotropic shrinkage.
334
A. Mulet
The external resistance is linked to the flow around the particle. Thus, shrinkage will affect this resistance considerably. In some instances, the effect of this phenomenon on the ratio of external to internal resistances can be very important. In this case, in some circumstances, the value of the effective diffusivity in the solid could be misleading because it is affected by the external resistance. Shrinkage in food drying appears to be an important aspect to consider in a model. If shrinkage is neglected, as is often the case in the literature, the simple di~sional model still gives a good description of the experimental data. Ahhough it is useful for describing the drying rates, the effective diffusivity reflects this phenomenon. Consequently, it is difficult to view this di~usivity as linked only to water transport. Thus this effective diffusi~ty is not strictly speaking a transport property. In simplified diffusional models the effective diffusivity is identified as the value that gives the better fit to experimental data. Within the better fit, many phenomena known to a greater or lesser extent are included, so the models lose some physical meaning, It appears that effective diffusivity could lose its meaning if the model is not carefully written and solved. It seems that this transport property is not very selective of the phenomena it is supposed to represent, the main reason being that it cannot be directly measured. Although it is very useful to consider shrinkage, for the reasons already mentioned, and also because it makes the values of D, more meaningful, it could be that the added complexity does not compensate for the advantages of simplified models for most of the common applications. It is useful to consider models with different degrees of complexity to evaluate the effort required to obtain the additional complexity more effectively. Four different models will be considered (Table 1) to show the effects of shrinkage, and the drying of carrot cubes (Mulet et al., 1989a,b) and potatoes (Rossello; et al., 1992) will be examined. The main characteristics of the models are shown in Table 1. In all four models it is assumed that external resistance to heat and mass transfer is negligible; thus the external surface of the solid is at equilibrium. It is also assumed that the geometry (shape) remains unchanged. To describe the volume variation of the carrots with change in moisture content, a relationship from literature was used (Madarro et al., 198 1). The first model (A) is the simplest and can be solved by variable separation. Models B, C and D, with moving boundaries, need numerical tec~iques (e.g. finite differences) to solve the partial di~erential equation representing the mass balance. The heat balance ordinary differen-
Drying model~i~g and water di~~vi~
Characteristics
Model Model Model Model
A B C D
in vegetables
335
TABLE 1 of Four Drying Models
Solid temperature
Difusivity
Volume
Constant (air) Constant (air)
Constant Constant
Constant V=fi w 1
D=.flqJ
V=fi w 1 V=fiWi
T,=fM T,=f(4
D=f( T,,. W,)
tial equation is solved in models C and D by a numerical integration method (Runge-Kutta), A full description of the model definition and solution has been published (Mulet et al., 1989a,b) for the models A, B and D. Model C is a simplified version of model D, where the effective diffusivity is considered to vary only with the particle tempera~re during the experiment. In drying model D the diffusivity is considered to vary as suggested by Crank (1975), according to D-exp[a+(b/T)+
cW]
0)
By introducing the heat transfer in the modelling (models C and D), the temperature influence on L) could be established from a single experiment, as the particle temperature varies from the initial value to nearly the drying air tempera~re. A method for testing the models will be by predicting the drying curves at various air temperatures. The agreement between the results given by the model and those obtained experimentally will constitute a way of testing the model. Experiments were carried out at various temperatures with carrot cubes of 1 cm and an air flow rate of around 8000 kg/m* h. The results are shown in Fig. 1 for models A and B. The effective diffusivities obtained for the first diffusional period by using three of the models are: Model A:D,=5.08
x lOI_“exp( -2964/T)
Model B:D, = 3.84 x lo’-‘exp(
(2)
- 3185/T )
Model D:n, = exp[ - 0.97 - (3460/T ) + 0.059WJ
(4)
336
A. Mulet
When analysing the experiments with models A and B it is observed (Fig. 1) that in both cases the effective diffusivity (D,) varies linearly with l/K Also, the temperature influence (slope) is similar in both cases. The main difference lies in the nume~ca~ value of f),. From these data it can be seen that it is important to take shrinkage into account when studying experimental data to obtain values of 0, for shrinking materials. Those values are different even if the temperature influence is well described. Model D is difficult to compare with models A and B, as it also takes into consideration the local moisture content of the particle. Nevertheless, by considering only the tem.perature influence on D,, a higher influence of the local temperature than in models A and B (higher activation energy) can be observed (eqns (2)-( 4)). A feature of the models presented, which has not yet been outlined, is the amount of work necessary to compute II,. The emphasis is nowadays placed mainly on obtaining good experimental results and defining properly the models and boundary conditions. The computational time required is becoming a less important feature as a result of the increased speed of computers. Whereas in models A and B an experiment needs to be performed to obtain each of the values of D,(T), like those shown in Fig. 1, in model D only one experiment is needed. This is because model D considers, as already mentioned, the temperature change of the particle during drying. This lower amount of expe~mental work is somewhat offset by greater computational complexity. To test the models further, the first diffusional period was simulated
1.0 1
0.8
-
0.6
-
Oa4
:
e^ 3
4 x Q ” 0.2
-
l/T x l@ (K)
Fig. 1.
Influence of air temperature on the effective di~sivity. Drying of l-cm carrot cubes at an air flow rate of G = 8000 kg/m2 h.
Drying modelling and water diffusivity in vegetables
337
by introducing the expression for D, in the models. The results for all three models were acceptable when compared with the experimental ones. The root mean square deviations (~SD) between experiments and predictions are shown in Table 2. Although model B appears to give on average the best agreement, it should be noted that ~dividual values of D, were computed by fitting experimental data obtained at each drying air temperature and then used for calculating the FWSD. This was done only at 50°C in model D; for all the other temperatures reported the predictions were made without using any information on drying experiments carried out at those drying air temperatures. This is evidence that heat transfer and the effect of the temperature on D, are reasonably well integrated in model D. The higher RMSD at 30 and 7O”C, although similar to that of model A, could be explained by the fact the these temperatures are never attained within the particle that dried at 50°C. An assumption that should also be examined as a possible source of the higher deviation is the use of a constant value of the heat transfer coefficient. From the above results, it can be concluded that model D is more precise for analysing a particular experiment; also it needs little experimental work, and it is possible to extrapolate with little risk to surrounding temperatures. Model B is simpler but requires more experimental work to describe a wide range of temperatures adequately. The target to be attained will help in choosing the model. Other points to be taken into account are the simplicity of the experiments and the computations difficulties in model D. When defin~g an effective di~sivi~ by considering it as a water transport property, it is implicidy assumed that its value is not dependent on particle shape and size. It is interesting to test how the three models considered behave when considering different particle sizes. This test is
Agreement
TABLE 2 (RMSD x 10’) Between Experimental and Computed Particle Size r= 5 X 10-j m
Moisture
Content,
Temperature “C
Model A Model B Model D
30
40
50
60
70
Average
6.2 2.6 9.0
4.4 26 2.5
6.4 2.5 1.3
11.5 2.7 1.5
10.6 2.0 5.4
7.8 2.5 3.9
338
A. Mulet
carried out easily with models A and B. This is not the case for model D because r>, is moisture dependent and varies within the particle. The De values were computed for various particle sizes, and are shown in Table 3. It can be observed that with model B reasonably constant values of D, are obtained, whereas with model A large variations are observed. For cubes smaller than 10 mm {r= 5 mm), the effect of size increased in model B; this could be explained by the fact that smaller particles lose their shape more easily than larger ones. By using model D, where the function I), was identified by employing only one experiment, it can be seen (Table 4) that the RMSD values are similar for all the sizes, and also are similar to those obtained previously for the testing temperature (50°C). These results obtained with model D are fairly good, not only because of the fit to the experimental data but also because the II, function was obtained with an experiment performed at another temperature (30°C higher) and particle size,
TABLE 3
Effective Diffusivities in Carrots at 60°C (G= 8000 kg/m” h) r
D, (IOh
x 10,’
M
3hj
Motel A
Motel E
4-O 5-O
4%
2-7
6.0
6-O 7.5
6.6 7%
3.2 3-3 3.6
63 (1.3)
3.2 (O-4)
Average
TABLE 4
Agreement
r
x 10.’ 4.0 5.0 6.0 7.5
Average
of Experimental
Data and Predictions with Model D; Experiments and G= 8000 kg/m’ with Carrots
at 60°C
RMSD x 10’ 2-7 2.6 2.6 1.5 2.3
Drying modelling and water di~ivi~
in vegetables
339
Again, model D gives more general and accurate results, although model B is similar in some ways. For design purposes, model B presents the advantage of being easier to handle. Where moisture profiles inside the particle have to be established, model D should be chosen. The effective d~usivi~ is strongly influenced by shrinkage, and this effect cannot be neglected unless great care is taken. Both models seem to deal with this adequately. Nevertheless, the more complex the model for diffusivity, the more realistic it should be. Data on the drying of carrots showed that the influence of temperature on drying was well described even with a s~pli~ed model. It has also been found that model B gives results which in some cases are comparable with those furnished by the more complex model D. At this point, it is interesting to try an intermediate model, C, where the diffusivity was assumed to be moisture independent, although it varied with the particle temperature during the drying process. To test another raw material, potatoes were chosen, and experiments at 30,40,50,60,70,80 and 90°C were carried out, with an air flux of 10 000 kg/m2 h. With models B and C, again, the effect of the air temperature is similar when plotting the values of D, for all the experiments. A straight line is obtained in an Arrhenius plot, the slope being similar for both models. The numerical values for D, obtained with models B and C and the corresponding RMSD values are shown in Table 5. From the RMSD values it can be observed that model C gave a better fit when the values of rs?, were identified for each experiment. These values are similar to those obtained when only the experiment at the higher temperature is used (figures in parentheses). As was previously mentioned, with model C a temperature dependence of De is identified. To check the reliabi~tv of the procedure, this dependence was evaluated in all cases for the model: lnL),=6+P/(T+273.16)
(5)
The results are shown in Table 6. It can be observed that the effect of temperature represented by p is surprisingly constant, thus indicating that the influence of this variable is fairly well established with the model considered. 8 is also constant. These results show that the use of model C could be convenient for establis~ng the temperature effects by using only one experiment. Another point to be raised when examining the D, values is the ratio of internal to external resistance to mass transfer. The internal resistance depends on temperature, the material being dried and the kind of water being extracted, which can be detected in some cases by different drying
340
A. Mulet
TABLE 5 Effective Diffusivities and Agreement with Experimental Data According and C; drying of 1-cm Potato Cubes at G= 10 000 kg/m* h
T-r%)
Model B
Fidel
to Models B
C
D, x IO7 (m’/h)
RMSD x 10’
D, x 1O7(ml/h)
RMSD x lo”
3.98 5.60 6.79 9.19 10.0 15.4 16.3
1.2 1.3 1.8 15 1.8 2.5 l-6
511 7.73 9.91 14.23 15.84 21.15. 2955
45 (10.9) 4.9 (185) 4.3 (10.5) 7.5 (24.0) 5.4 (4.9) 76 (197) 2@2 (2@2)
30 40 50 60 70 80 90
TABLE 6 Values for the Parameters 8 and /3 in Model C (eqn (5)) 0
T (“C) 30 40 50 60 70 80 90 Average (SD)
P
- 4.093 -4011 - 4.073 - 4005 -4,173 -4.144 - 4.054
-3151.0 -3151-o -3151.0 -3151-o -3151.0 -3151.0 -3151.5
- 4.079 (0,063)
- 3151.0 (0,2)
periods. The external resistance depends on the boundary layer that develops around the particle. All the factors affecting the boundary layer will influence this resistance, these factors mainly being air velocity, shape and size of the particle and fluid properties. The external resistance can be evaluated approximately through existing correlations, although it is very difficult to find a correlation to cover all the possible situations encountered, and so reliability decreases. This is why the influence of external resistance in drying is often determined by indirect methods. This evaluation is crucial because otherwise the value of D, is meaningless. Although this seems evident, in the experiments reported in the literature it is very often simply assumed to be negligible or is ignored. Then, according to the model structure and bounda~ conditions, the values of De are reliable to a greater or lesser extent.
Drying mode&g
and waterdi#usivityin vegetables
341
Up to this point, the question is still how to check D, data from the literature because they are highly dependent on the model and experimental conditions. Two major points appear to need careful checking: one related to shrinkage, which we have briefly discussed above, and another related to the external resistance influence, which merits further consideration. It is common to find data on D, that should be similar to other data, but in fact differ by more than experimental error. To date, screening of D, data has not been carried out, although an effort has been made recently to gather data on 0, (Voilley & Vidal, 1990). To gather valuable data, checking should be considered for the aspects related to the model complexity and experimental procedures. Otherwise, the data are representative of only a particular experimental set-up. TESTING DATA In the literature, drying curves from which values for De are derived are often shown. As we have mentioned above, these values often need to be checked to assess their validity regarding the experimental set-up for at least external resistance. One approach used in the past (Mulet et aZ., 1987; Rossello et al., 1992) was to increase the air velocity until the value D, remained constant. The kind of variation shown in Fig. 2 observed for carrots was also found for potatoes. Of course, this testing is valid for a type of material, a shape and a size, assuming constant air flow characteristics. This procedure has several disadvantages. The most important is the fact that it is very time consuming, and much experimentation is needed to determine the air velocity at which external resistance becomes negligible, In addition, as mentioned above, this is dependent on size and shape, which means that for a different shape or size it is necessary to repeat at least part of the experiments. To determine the ranges in which each resistance is predominant, correlations from the literature were used to establish the external mass transfer coefficient. For this purpose, the flux density was computed both expe~menta~y and with the mass transfer models. The results for the drying of potatoes are shown in Fig. 3. It can be observed that the data at lower flow rates correspond well to the model with external resistance prevailing (Skelland, 1974), and at higher ones to the diffusion model. There is a large transition zone which is difficult to establish and predict. The method outlined above could be used to check an experimental situation, but is useless for testing the data in the literature. In this latter case, what is available, at best, is a drying curve from which D, is derived.
A. Mulet
342
2
4
6
G x l@
8
fkg~m’hf
Fig. 2. Infiuence of air flow rate (G) on the effective diffusivity D,. Drying of l-cm carrot cubes at 30°C.
2 "8
6.0 -
‘M 5 5
4.0 I
# -a--,
2’
A.._*_-__
*__-__-_-‘_a_*_
2.0 -
0.0 0
t 20000
30000
G Fig. 3. Mass flux densities:
experimental, externaf and internal Potato, l-cm cube, at 50°C.
(kg&? resistance
h) predictions.
Drying modelling and waterdi~ivi~
in vegetables
343
Consequently, a new way of checking D, data should be found (Berna et al., 1992). For the sake of illustration and simplicity we will consider model A. In this case (considering only one term of the series development), the dimensionless moisture content of the particle is given by Y(t) = exp[ - (3Jr2/4)(L),/L2ft] If the value of D, is obtained by plotting ln Y(t) against t, a straight line should be found, the slope (a)being L),. These results are similar to those obtained by other numerical methods (Karathanos, 1990). However, if L), is not constant, no linear reiations~p will appear. By plotting the slope (a, dlnY/dt), which is proportional to O,, vs t, it can be observed that there is really no constant range as model A should predict. Instead a smooth variation is observed, as in Fig. 4. This smooth variation constitutes a first check ~dicat~g a dependence on other variables, the most likely being the moisture content, according to the literature. This fact was pointed out by Sherwood (1931), who observed that ‘the diffusion constant decrease with the moisture concentration’. By plotting against this variable, a similar variation is observed, thus pointing to the moisture content as a variable to be taken into account to clarify variations in D,. Because the slope (a),mentioned previously, will represent an effective diffusivity only in the case where internal resistance prevails, if this is not the case the influence of the external resistance is also present and does not really represent 0,. Consequently, it is expected that the slope of the curves will be different according to the prevailing resistance. To check this point, experiments at different air flow rates were carried out -1
0 G=10500
-6
0
2000
4000
6000
8000 t
Fig. 4.
10000
-I 12000
(s)
Test of the constancy for De.Drying of carrot cubes at 40°C and C= 10 500 kg/ m2 h.
A. Mulet
344
-5 -
l
G=4000
*
G=6000
*
G=8000
l
-6. 0.0
,
G=lOOOO . , 0.2
.
+
.
GrlOSOO , . 0.4
++ .
,
.
.
0.6
,
++ .
0.8 Y
Fig. 5. Drying of carrot cubes at various air flows.
and the results are plotted in Fig. 5. It can be observed that, effectively, there is a different slope. This fact could be used to establish whether the external resistance influences the value of 0, obtained. It is observed that if external resistance is important the influence of Y is negligible or slightly positive, for values of Y larger than 0.3. To test previous results on carrot drying according to this method, and to understand this change in slope better, the results of several experiments run at different air velocities (Fig. 2) were considered. In Fig. 2 it can be observed that when the slope a variation is positive for the higher values of !P the values of D, vary with air velocity. It can also be observed that, for negative variation, the prevailing resistance is the internal one. Agreement exists between these results and those previously found. According to these results, because the slope a is calculated in a single experiment, it can readily be seen whether the experiment was carried out with the external resistance being negligible or not, and then identify whether the value of D, was affected by air velocity in each case. If it is affected, the figure obtained is related to the experimental conditions, and is not very useful for other situations. The slope of the curves shown in Fig. 5 indicates that for the particular case examined internal resistance always prevails at the end of the drying period. The steep change in slope that occurs at about Y = 0.1 could be explained by the existence of more strongly bound water. In fact, Wang and Brennan (1992) showed, using NMR techniques, that in the case of potatoes, for values of Y lower than 0.2, two species of more strongly
Drying model&
and water difhsivity in vegetables
345
bound water start to prevail, over the one (less strongly bound) that prevailed from the beginning of the drying operation. Another approach to testing the influence of external resistance on drying has been developed. It is also based on the use of empirical correlations to compute the mass flux from the particles being dried. After stating the basic mass flux equations and ad~tting some s~plif~ng assumptions, using the Pasternak and Gauvin correlation (1960) for the calculation of the external mass transfer coefficient, and admitting the volume change given by Madarro et al. ( 198 1), the following expression was found (Berna, er al., 1992): dlnY/dt=(a+
bY))0.5/Y
(7)
This indicates that the plot of (W (dlnY/dt)) vs Y should be a straight line if external resistance prevails. By examining the results shown in Fig. 5 and replotting them according to eqn (7) (Fig. 6), it can be observed that a linear variation appears for the whole range of dimensionless moisture content for the lower air flow rates. The same procedure was applied to the drying of potatoes, and similar results were obtained. It was observed that a linear relationship also appears for low flow rates and a parabola-like shape for high flow rates.
CONCLUSIONS Various approaches are possible for modelling the drying operation with varying degrees of comple~ty. At present, di~siou~ models seem to be among the most commonly used in the convective drying of food particulates for engineering applications. The complexity of drying foodstuffs arises from the diversity of materials and shrinkage. Thus, it is difficult to make generalizations. The models are also mathematically cumbersome if shrinkage is considered. These aspects give rise to a diversity of models with degrees of complexity which vary according to the end use of the model. Of course, the values of the effective diffusivities for the same product and experimental conditions will vary according to the model. The boundary conditions are where most of the simplifying assumptions are made. Two of the boundary conditions commonly differentiate the models. One is related to shrinkage (moving or static boundaries) and the other to the equilibrium (attained or not) at the outer surface. This latter assumption, related to the external resistance being negligible, is often made but seldom tested.
346
0.0
0.2
0.4
0.6
0.8
1.0
F
Fig. 6. Influence of external resistance on drying. Drying of l-cm carrot cubes at 40°C.
It has been seen that the influence of temperature on the value of D,, during the first diffusional period, is the same regardless of the consideration of shrinkage by the model. What makes the difference is the value of r>,, which differs according to the model. Although this has been tested only in potatoes and carrots, for other vegetable materials the same phenomena are expected. In any case, shrinkage should be accounted forto attain reliable values of D,, otherwise they are dependent on size and experimental conditions, and thus are meaningless as transport properties. With models which take into account heat transfer and particle temperature variation during drying, the temperature influence on D, can be identified by using only one experiment. The errors in the prediction of the drying curves at different air temperatures are similar to those obtained identifying D, at each particular temperature by using model B (shrinkage). Ident~ying a De vs Y relations~p does not seem to offer great advantage in en~eering applications. Three ways have been developed to test data to assess whether the effective diffusivity is influenced by the drying air velocity. One consists of carrying out experiments with increasing air velocity until a constant value for D, is obtained. The others are based on examining the drying curves. One of the latter is based on plotting dln Y / dt, a magnitude which is linked to D, vs Y. If the slope is very low or positive at high moisture contents, it can be assumed that the external resistance is important. In the second of the latter methods, a different shape is also found according to the prevailing resistance. The use of these methods could help in screening De data from the literature if drying curves are available.
Drying modelling and water difisivity in vegetables
347
This work was ~ancia~y supported by IMP~A/~ICYT (RE 3473). The co~aboration of A. Berna, C. Rossell6, S. Simal and J. Garcia is acknowledged. REFERENCES Berna, A., Garcia, J. & Mulet, A. (1992). AINIA Report. Valencia, AINIA. Crank, J. (1975). The Mathematics ofthe Di@sion, 2nd edn. Oxford University Press, London, 4 14 pp. Jayaraman, KS. & Das Gupta, D.K. (1992). Dehydration of fruits and vegetables. Recent developments in principles and techniques. Drying Technology, lO( l), l-50. Karathanos, V.T., Villalobos, G. & Saravacos, G.D. (1990). Comparison of two methods of estimation of the effective moisture diffusivity from drying data. 1. FoodScience, 55(l), 218-23. Ketelaars, A.A.J., Lauf,O., Coumans, J.W. & Kerhof, P.J.A.M. (1992). Determination of diffusion coefficients from experimental moisture concentration ~;yhlil;in clay. In Drying ‘92, ed. A.S. Mujumdar. Elsevier, Amsterdam, pp. Lewis, W.K. (1921). The rate of drying of solid materials. I&. Engng Chem., 13(5), 427-32. Luikov, A.?‘. (1966). Heat and Mass Trunsjer in Capillary Porous Bodies. Pergamon Press, New York. Madarro, A., Pinaga, E, Carbonell, J.V. & Peiia, JL. (198 1). Des~drataci~n de frutas y hortalizas con aire ambiente.1. Ensayos exploratorios con zanahorias. Rev. A~o~~~rn. TecnoE. ALiment., 21(4), 525-33. Maneval, J.E. & Whitaker, S. (1988). Effects of saturation heterogeneities on the interfacial mass transfer relation. In Doing ‘88, ed. A.S. Mujumdar. Elsevier, Amsterdam, pp. 238-45. McCarthy, M.J. & Perez, E. (1990). Measurements of effective moisture diffusivities using magnetic resonance imaging. In Engineering and Food, Vol. 1, eds W.E.L Spiess & H. Schubert. Elsevier Applied Science, London, pp. 473-8 1. Mulet, A., Berna, A., Borras, M. & Piiiaga, F. (1987). Effect of air flow rate on carrot drying. Drying Technology, 5(2), 245-58. Mulet, A., Berna, A. & Rosello, C. (1989a). Drying of carrots I. Drying models. Drying Technology, 7(3), 537-57.
Mulet, A., Berna, A., Rosello, C. & Piiiaga, F. (19896). Drying of carrots II. Evaluation of drying models. Drying Technology, 7( 4), 64 l-6 1. Pasternak, I.S. & Gauvin, W.H. (1960). Turbulent heat and mass transfer from stationary particles. Can. J. Chem. Engng, 38,35-42. Ratti, C. & Crapiste, G.H. (1992). A generalized drying curve for shrinking food materials. In Drying ‘92, ed. A.S. Mujumdar. Elsevier, Amsterdam, pp. 864-73. Rossello, C., Canellas, J., Simal, S. & Berna, A. (1992). A simple mathematical model to predict the drying rate of potatoes. J. Agric. Food Chem., 40( 12), 2374-8.
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A. h4ulet
Sherwood,T.K. ( 1929).The drying of solids. I&. &zgag C&m., 2 l( 1 ), 12- 16. Sherwood, T.K. (1931). Applications of theoretical diffusion equations to the drying of solids. Trans. AIChE, 27,190-202. Shrader, G.W. & Lichtfield, J.B. (1992). Moisture profiles in a model food gel during drying: measurement using magnetic resonance imaging and evaluation of the Fickian model. Dryins Technology, 10(2), 295-332. Sk;elkd, A.H.P. ( 1974). D i& sional Mass Transfer. John Wiley and Sons, New Tsotsas; E. (1992). Measurement and modelling of intraparticle drying kinetics: a review. In Drying ‘92,ed. A.S. Mujumdar. Elsevier, Amsterdam, p. 174 1. Voilley, A. & Vidal, D. ( 1990). Difision Properties of Foods. Food Science Publishers, Hornchurch, Essex. Project COST 90-bis. Wang, N. & Brennan, LG. (1992). Effect of water binding on the drying behaviour of potato. In Dryitzg ‘92, ed. AS. Mujumdar. Elsevier, Amsterdam, pp. 135-9. Whitaker, S. (1977). Simultaneous heat, mass and moments transfer in porous media: A theory of drying. In Advances in Heat T~a~r, Vol. 13, ed. T.F. Irvine. Academic Press, New York, pp. 119-203.
of Food
Engineering 22 ( 1994) 329-348 8 1994 Elsevier Science Limited
Printed in Great Britain. All rights reserved 0260-8774/94/$7.00 ELSEVIER
Drying Modelling and Water Ditisivity Potatoes Antonio
in Carrots and
Mulet
Food Technology Department, Universidad PolitCcnica de Valencia, Camino de Vera s/n, 0707 1 Valencia, Spain
ABSTRACT Modelling is a useful way to validate mechanisms of drying and to establish properties, and thus can be used to determine the reliabilityof calculated values of the eflective di~ivi~ (De). Di~ive models are considered for the desc~.pt~on of convective drying of p~~~cul~te vegetables (carrots and potatoes). Four models with different degrees of complexity are d&cussed. The degree of complexity depends on consideration of shrinkage, heat tranqer and particle temperature variation during drying, and the D, dependence on moisture and temperature simultaneously. Expe~.ments have been carried out to testthe models and the assumed bounder conditions. Expe~ment~l results show that the eflect of shrinkage cannot be neglected in establishing reliable valuesfor 0,. The effect of the external resistance, which is affected by the velocityof the air used in drying, also appears to be an important factor for testing De values. Two methods are proposed to carry out this test, both of which could be readily applied if ihe dying curve is available. Expe~ments on drying carrots and potatoes are used to illustratethe procedures.
NOTATION
adw
De G
r T t W Y
Constants Effective diffusivity (m2 /s) Air flow rate (kg/m2h) Half-thickness (m) Temperature (“C, K) Time(s) Moisture content (kg water/kg dry solid) Dimensionless moisture ( W- rY,)/( WC- We) 329
A. Mulet
330
Subscripts c
1” P
Critical Equilibrium Local Particle
INTRODUCTION Modelling is essentially a way of representing processes or phenomena to explain the observed data and to predict behaviour under different conditions. Two kinds of models are generally considered: one tries to interpret phenomena from physical laws, and the other represents these phenomena to account for the experimental data and to generalize regardless of the physical reality. Often the bounda~es between the two kinds of model are ill defined, mostly because physical laws are, in themselves, models. In fact, physical phenomena are regulated by laws that we do not know, and ‘physicai laws’ deal with the abstractions of our models. The essence of scientific activity is to establish more realistic models, or, in other words, to attain a better description of reality. Complexity is the key characteristic of reality, and sapling is the key process in modelling. Thus modelling is science with a great deal of art and sometimes luck. A considerable amount of work has been carried out in recent years (Jayaraman & Das Gupta, 1992) on the drying of fruits and vegetables. To simplify and delimit the subject to some extent, we will concentrate mostly on drying of particulate foods by convective drying. For this particular kind of drying, models based on physical laws should explain two drying periods; the first showing a constant rate of drying, and the second with a falling rate. The first period does not always appear, and this poses a problem in defining the starting point for the second one. In this case, the initial characteristics of the material are assumed as the starting point. The first drying period, or constant rate period, persists until the transpo~ mechanisms inside the particle are able to feed the surface with enough water to maintain the constant mass flow from the surface. Usually, the first period is associated with the existence of free water at the surface, which is a wet surface. Nevertheless, it has been shown that with only partia~y wet surfaces the first drying period is detected in some cases (Maneval & Whitaker, 1988). During this period a moisture
gradient builds up in the particle and this should be considered in determining the falling rate period starting point. Strong ar~me~ts took place in the past about the transport mechanisms describing water migration during the second drying period, or the falling rate period (Tsotsas, 1992). Nowadays, although we recognize the complexity of the physical phenomena and the fact that no simple model can account far real behaviour, two types of models are commonly used. One is based on the diffusional transport of water (the diffusion model), and the other is based mostly on capillary transport (the receding front model). The diffusion model was formulated by Lewis (1922 ) and further developed by Sherwood (1929). The transport on capiIlary porous media was mathemati~~y foliated by Luikov f 1966). A third kind of model is based on the complete ~onse~ation equations ~W~taker, 3977), and gives rna~erna~~~y complex formu~a~ons with the added problems linked to the lack of ~owledge of model parameters. In addition to these rn~h~s~~ models, more empirical des~~p~~~s of the dreg prucess are often used. These empirical models do not pretend to describe how the process takes place but rather give an outline of what happens. This is done through the use of drying curves, which may or may not be standard~ed. Despite the fact that these modeis do not provide a physical desc~ptiun of the phenamena taking place, they are often useful for design purposes in excessively complex cases. The ~haracte~stic drying curve is useful for this purpose even with shrinking food materials (Ratti & Crapiste, 1992). The use of mechanistic models for design purposes is preferred wherever possible (Tsotsas, 1992). In this case, the drying kinetics of a single particle is the best choice for scaling-up. The scaling-up procedure is done using estab~shed engineering procedures. Among the mechanistic models the d~sion~ ones seem to be the most popular, They are ease to formulate and normally provide reasonable resuhs. The only difficulty is that along the way a number of simphfying assump~ons are admitted. Therefore we should talk about effective di~~i~ty (l>,)” This di~sivity includes the effects of the known hypetheses as well as the unknown phenomena not being modeled, Under these cir~u~tances, the question of the vale of the values computed is raised as well as that of what is realIy happe~g inside the particle, Many so-called ‘m~ch~sti~’ models have such safe assumptions that they are in effect empirical models. The mudel shoutd be viewed as a descriptive tool of the phenomena taking place. Its use as an analytical tool makes it valuable for understanding meehanisms in some cases, although it can be of very limited usefi&ress in other cases.
A. Mulet
332
The next step will be to validate the models through experimentation. The experiments needed depend on the kind of model; thus varying degrees of detail will be necessary. Until recently, the measurements commonly used were the weight of the samples or the particle temperature. The water distribution inside the particle was not measured becaue of the absence of techniques; it was computed with the use of a model. The total water content was obtained by weighing the sample, and the accuracy of the model was evaluated by integrating the profile of moisture and comparing it with the sample weight. This type of procedure is disconcerting because we gain insight into the behaviour of the solid only through the integration of the model to match the water content. Of course, this matching can be done with different models and assumptions; thus, testing models can be a difficult task. Consequently, the need arises for more powerful experimental tools which can show water profiles inside particles. Two techniques are now very promising - NMR imaging (McCarthy & Perez, 1990; Schrader & Lichtfield, 1992) and neutron transmission (Ketelaars et al., 1992). These could be an additional and interesting way of testing hypotheses in modelling. In addition, the NMR technique could be used for determining water binding (Wang & Brennan, 1992). These newer techniques will bring a better understanding of the phenomena. One of the main conclusions arising from the use of these techniques is that, in general, the diffusional model does not appear to describe the process inside the particle accurately. Nevertheless, the use of effective diffusivity of water in drying will be a useful way of representing the phenomena involved in the description of the overall drying rate and thus is useful for design. The diffusion-like process taking place in drying is then described through water diffusivity, a transport property which will be examined more closely below.
COMPLEXITY
OF THE MODELS
The complexity needed in a model, or in other words the level of detail, depends on the target to be reached. The end use of the model will establish the degree of complexity. In general, a model could be built to describe microscopic particle or macroscopic level behaviour. The lower the scale of description the greater the complexity, because a more detailed description of the phenomena is involved. This means also that, at the microscopic level, more theoretical models are considered ,and at the other extreme, at the
Drying modelling and waterdi@sivity in vegetables
333
macroscopic level, lumped parameters and empirical approaches are commonly used. The ma~ematical form of the model fo~ulation also influences the different approaches used to identify the numerical value of the parameters and thus the effective diffusivity. Microscopic level descriptions are mainly useful for understanding and describing the drying mechanisms. Particle level could be reached by proper integration of microscopic models. The properties measured are then averages within the particle. The kind of water more or less bound to the material changes during the drying operation. At the beginning, the free water is taken out and the remaining water is then bound more strongly to the solid matrix. Of course, there is no si~c~t change in the kind of water being evaporated. Thus, when observing the effective d~si~ty it is difficult to establish different zones within the falling rate period. Often, only one zone is reported in the falling rate period and then only one value of the effective diffusivity identified, regardless of the amount of water still present in the solid, or the character of the remaining water. As pointed out above, the use of more precise measurements, such as those provided by NMR imaging, could throw light on this problem (Wang & Brennan, 1992). At the macroscopic level, a description of the drying env~o~ent is needed. Flow distribution, air velocity, air mois~re, bed characte~stics, etc., should be taken into account. A general way of attaining this level of description could be by integration of particle level models together with the description of the drier characteristics. This approach is preferred by some workers (Tsotas, 1992) in the scaling-up process. It constitutes a more general approach than direct modelling at macroscopic level. From these considerations, it seems appropriate to consider the particle level description as reasonable for most applications. At this level, the effective diffusivity should be carefully evaluated and special attention paid to bounda~ conditions of the model. At particle level, the macroscopic phenomena taking place are well known. The main factors affecting the drying rate are related to the external and internal resistance to mass and heat transfer. If the boundary conditions are not in accordance with the physical reality, e.g. if we admit negligible external resistance when this is not the case, the results are of little value. Most foods shrink when drying, thus moving boundaries need to be taken into account. This makes the model difficult to solve and often only approximate solutions are possible because of anisotropic shrinkage.
334
A. Mulet
The external resistance is linked to the flow around the particle. Thus, shrinkage will affect this resistance considerably. In some instances, the effect of this phenomenon on the ratio of external to internal resistances can be very important. In this case, in some circumstances, the value of the effective diffusivity in the solid could be misleading because it is affected by the external resistance. Shrinkage in food drying appears to be an important aspect to consider in a model. If shrinkage is neglected, as is often the case in the literature, the simple di~sional model still gives a good description of the experimental data. Ahhough it is useful for describing the drying rates, the effective diffusivity reflects this phenomenon. Consequently, it is difficult to view this di~usivity as linked only to water transport. Thus this effective diffusi~ty is not strictly speaking a transport property. In simplified diffusional models the effective diffusivity is identified as the value that gives the better fit to experimental data. Within the better fit, many phenomena known to a greater or lesser extent are included, so the models lose some physical meaning, It appears that effective diffusivity could lose its meaning if the model is not carefully written and solved. It seems that this transport property is not very selective of the phenomena it is supposed to represent, the main reason being that it cannot be directly measured. Although it is very useful to consider shrinkage, for the reasons already mentioned, and also because it makes the values of D, more meaningful, it could be that the added complexity does not compensate for the advantages of simplified models for most of the common applications. It is useful to consider models with different degrees of complexity to evaluate the effort required to obtain the additional complexity more effectively. Four different models will be considered (Table 1) to show the effects of shrinkage, and the drying of carrot cubes (Mulet et al., 1989a,b) and potatoes (Rossello; et al., 1992) will be examined. The main characteristics of the models are shown in Table 1. In all four models it is assumed that external resistance to heat and mass transfer is negligible; thus the external surface of the solid is at equilibrium. It is also assumed that the geometry (shape) remains unchanged. To describe the volume variation of the carrots with change in moisture content, a relationship from literature was used (Madarro et al., 198 1). The first model (A) is the simplest and can be solved by variable separation. Models B, C and D, with moving boundaries, need numerical tec~iques (e.g. finite differences) to solve the partial di~erential equation representing the mass balance. The heat balance ordinary differen-
Drying model~i~g and water di~~vi~
Characteristics
Model Model Model Model
A B C D
in vegetables
335
TABLE 1 of Four Drying Models
Solid temperature
Difusivity
Volume
Constant (air) Constant (air)
Constant Constant
Constant V=fi w 1
D=.flqJ
V=fi w 1 V=fiWi
T,=fM T,=f(4
D=f( T,,. W,)
tial equation is solved in models C and D by a numerical integration method (Runge-Kutta), A full description of the model definition and solution has been published (Mulet et al., 1989a,b) for the models A, B and D. Model C is a simplified version of model D, where the effective diffusivity is considered to vary only with the particle tempera~re during the experiment. In drying model D the diffusivity is considered to vary as suggested by Crank (1975), according to D-exp[a+(b/T)+
cW]
0)
By introducing the heat transfer in the modelling (models C and D), the temperature influence on L) could be established from a single experiment, as the particle temperature varies from the initial value to nearly the drying air tempera~re. A method for testing the models will be by predicting the drying curves at various air temperatures. The agreement between the results given by the model and those obtained experimentally will constitute a way of testing the model. Experiments were carried out at various temperatures with carrot cubes of 1 cm and an air flow rate of around 8000 kg/m* h. The results are shown in Fig. 1 for models A and B. The effective diffusivities obtained for the first diffusional period by using three of the models are: Model A:D,=5.08
x lOI_“exp( -2964/T)
Model B:D, = 3.84 x lo’-‘exp(
(2)
- 3185/T )
Model D:n, = exp[ - 0.97 - (3460/T ) + 0.059WJ
(4)
336
A. Mulet
When analysing the experiments with models A and B it is observed (Fig. 1) that in both cases the effective diffusivity (D,) varies linearly with l/K Also, the temperature influence (slope) is similar in both cases. The main difference lies in the nume~ca~ value of f),. From these data it can be seen that it is important to take shrinkage into account when studying experimental data to obtain values of 0, for shrinking materials. Those values are different even if the temperature influence is well described. Model D is difficult to compare with models A and B, as it also takes into consideration the local moisture content of the particle. Nevertheless, by considering only the tem.perature influence on D,, a higher influence of the local temperature than in models A and B (higher activation energy) can be observed (eqns (2)-( 4)). A feature of the models presented, which has not yet been outlined, is the amount of work necessary to compute II,. The emphasis is nowadays placed mainly on obtaining good experimental results and defining properly the models and boundary conditions. The computational time required is becoming a less important feature as a result of the increased speed of computers. Whereas in models A and B an experiment needs to be performed to obtain each of the values of D,(T), like those shown in Fig. 1, in model D only one experiment is needed. This is because model D considers, as already mentioned, the temperature change of the particle during drying. This lower amount of expe~mental work is somewhat offset by greater computational complexity. To test the models further, the first diffusional period was simulated
1.0 1
0.8
-
0.6
-
Oa4
:
e^ 3
4 x Q ” 0.2
-
l/T x l@ (K)
Fig. 1.
Influence of air temperature on the effective di~sivity. Drying of l-cm carrot cubes at an air flow rate of G = 8000 kg/m2 h.
Drying modelling and water diffusivity in vegetables
337
by introducing the expression for D, in the models. The results for all three models were acceptable when compared with the experimental ones. The root mean square deviations (~SD) between experiments and predictions are shown in Table 2. Although model B appears to give on average the best agreement, it should be noted that ~dividual values of D, were computed by fitting experimental data obtained at each drying air temperature and then used for calculating the FWSD. This was done only at 50°C in model D; for all the other temperatures reported the predictions were made without using any information on drying experiments carried out at those drying air temperatures. This is evidence that heat transfer and the effect of the temperature on D, are reasonably well integrated in model D. The higher RMSD at 30 and 7O”C, although similar to that of model A, could be explained by the fact the these temperatures are never attained within the particle that dried at 50°C. An assumption that should also be examined as a possible source of the higher deviation is the use of a constant value of the heat transfer coefficient. From the above results, it can be concluded that model D is more precise for analysing a particular experiment; also it needs little experimental work, and it is possible to extrapolate with little risk to surrounding temperatures. Model B is simpler but requires more experimental work to describe a wide range of temperatures adequately. The target to be attained will help in choosing the model. Other points to be taken into account are the simplicity of the experiments and the computations difficulties in model D. When defin~g an effective di~sivi~ by considering it as a water transport property, it is implicidy assumed that its value is not dependent on particle shape and size. It is interesting to test how the three models considered behave when considering different particle sizes. This test is
Agreement
TABLE 2 (RMSD x 10’) Between Experimental and Computed Particle Size r= 5 X 10-j m
Moisture
Content,
Temperature “C
Model A Model B Model D
30
40
50
60
70
Average
6.2 2.6 9.0
4.4 26 2.5
6.4 2.5 1.3
11.5 2.7 1.5
10.6 2.0 5.4
7.8 2.5 3.9
338
A. Mulet
carried out easily with models A and B. This is not the case for model D because r>, is moisture dependent and varies within the particle. The De values were computed for various particle sizes, and are shown in Table 3. It can be observed that with model B reasonably constant values of D, are obtained, whereas with model A large variations are observed. For cubes smaller than 10 mm {r= 5 mm), the effect of size increased in model B; this could be explained by the fact that smaller particles lose their shape more easily than larger ones. By using model D, where the function I), was identified by employing only one experiment, it can be seen (Table 4) that the RMSD values are similar for all the sizes, and also are similar to those obtained previously for the testing temperature (50°C). These results obtained with model D are fairly good, not only because of the fit to the experimental data but also because the II, function was obtained with an experiment performed at another temperature (30°C higher) and particle size,
TABLE 3
Effective Diffusivities in Carrots at 60°C (G= 8000 kg/m” h) r
D, (IOh
x 10,’
M
3hj
Motel A
Motel E
4-O 5-O
4%
2-7
6.0
6-O 7.5
6.6 7%
3.2 3-3 3.6
63 (1.3)
3.2 (O-4)
Average
TABLE 4
Agreement
r
x 10.’ 4.0 5.0 6.0 7.5
Average
of Experimental
Data and Predictions with Model D; Experiments and G= 8000 kg/m’ with Carrots
at 60°C
RMSD x 10’ 2-7 2.6 2.6 1.5 2.3
Drying modelling and water di~ivi~
in vegetables
339
Again, model D gives more general and accurate results, although model B is similar in some ways. For design purposes, model B presents the advantage of being easier to handle. Where moisture profiles inside the particle have to be established, model D should be chosen. The effective d~usivi~ is strongly influenced by shrinkage, and this effect cannot be neglected unless great care is taken. Both models seem to deal with this adequately. Nevertheless, the more complex the model for diffusivity, the more realistic it should be. Data on the drying of carrots showed that the influence of temperature on drying was well described even with a s~pli~ed model. It has also been found that model B gives results which in some cases are comparable with those furnished by the more complex model D. At this point, it is interesting to try an intermediate model, C, where the diffusivity was assumed to be moisture independent, although it varied with the particle temperature during the drying process. To test another raw material, potatoes were chosen, and experiments at 30,40,50,60,70,80 and 90°C were carried out, with an air flux of 10 000 kg/m2 h. With models B and C, again, the effect of the air temperature is similar when plotting the values of D, for all the experiments. A straight line is obtained in an Arrhenius plot, the slope being similar for both models. The numerical values for D, obtained with models B and C and the corresponding RMSD values are shown in Table 5. From the RMSD values it can be observed that model C gave a better fit when the values of rs?, were identified for each experiment. These values are similar to those obtained when only the experiment at the higher temperature is used (figures in parentheses). As was previously mentioned, with model C a temperature dependence of De is identified. To check the reliabi~tv of the procedure, this dependence was evaluated in all cases for the model: lnL),=6+P/(T+273.16)
(5)
The results are shown in Table 6. It can be observed that the effect of temperature represented by p is surprisingly constant, thus indicating that the influence of this variable is fairly well established with the model considered. 8 is also constant. These results show that the use of model C could be convenient for establis~ng the temperature effects by using only one experiment. Another point to be raised when examining the D, values is the ratio of internal to external resistance to mass transfer. The internal resistance depends on temperature, the material being dried and the kind of water being extracted, which can be detected in some cases by different drying
340
A. Mulet
TABLE 5 Effective Diffusivities and Agreement with Experimental Data According and C; drying of 1-cm Potato Cubes at G= 10 000 kg/m* h
T-r%)
Model B
Fidel
to Models B
C
D, x IO7 (m’/h)
RMSD x 10’
D, x 1O7(ml/h)
RMSD x lo”
3.98 5.60 6.79 9.19 10.0 15.4 16.3
1.2 1.3 1.8 15 1.8 2.5 l-6
511 7.73 9.91 14.23 15.84 21.15. 2955
45 (10.9) 4.9 (185) 4.3 (10.5) 7.5 (24.0) 5.4 (4.9) 76 (197) 2@2 (2@2)
30 40 50 60 70 80 90
TABLE 6 Values for the Parameters 8 and /3 in Model C (eqn (5)) 0
T (“C) 30 40 50 60 70 80 90 Average (SD)
P
- 4.093 -4011 - 4.073 - 4005 -4,173 -4.144 - 4.054
-3151.0 -3151-o -3151.0 -3151-o -3151.0 -3151.0 -3151.5
- 4.079 (0,063)
- 3151.0 (0,2)
periods. The external resistance depends on the boundary layer that develops around the particle. All the factors affecting the boundary layer will influence this resistance, these factors mainly being air velocity, shape and size of the particle and fluid properties. The external resistance can be evaluated approximately through existing correlations, although it is very difficult to find a correlation to cover all the possible situations encountered, and so reliability decreases. This is why the influence of external resistance in drying is often determined by indirect methods. This evaluation is crucial because otherwise the value of D, is meaningless. Although this seems evident, in the experiments reported in the literature it is very often simply assumed to be negligible or is ignored. Then, according to the model structure and bounda~ conditions, the values of De are reliable to a greater or lesser extent.
Drying mode&g
and waterdi#usivityin vegetables
341
Up to this point, the question is still how to check D, data from the literature because they are highly dependent on the model and experimental conditions. Two major points appear to need careful checking: one related to shrinkage, which we have briefly discussed above, and another related to the external resistance influence, which merits further consideration. It is common to find data on D, that should be similar to other data, but in fact differ by more than experimental error. To date, screening of D, data has not been carried out, although an effort has been made recently to gather data on 0, (Voilley & Vidal, 1990). To gather valuable data, checking should be considered for the aspects related to the model complexity and experimental procedures. Otherwise, the data are representative of only a particular experimental set-up. TESTING DATA In the literature, drying curves from which values for De are derived are often shown. As we have mentioned above, these values often need to be checked to assess their validity regarding the experimental set-up for at least external resistance. One approach used in the past (Mulet et aZ., 1987; Rossello et al., 1992) was to increase the air velocity until the value D, remained constant. The kind of variation shown in Fig. 2 observed for carrots was also found for potatoes. Of course, this testing is valid for a type of material, a shape and a size, assuming constant air flow characteristics. This procedure has several disadvantages. The most important is the fact that it is very time consuming, and much experimentation is needed to determine the air velocity at which external resistance becomes negligible, In addition, as mentioned above, this is dependent on size and shape, which means that for a different shape or size it is necessary to repeat at least part of the experiments. To determine the ranges in which each resistance is predominant, correlations from the literature were used to establish the external mass transfer coefficient. For this purpose, the flux density was computed both expe~menta~y and with the mass transfer models. The results for the drying of potatoes are shown in Fig. 3. It can be observed that the data at lower flow rates correspond well to the model with external resistance prevailing (Skelland, 1974), and at higher ones to the diffusion model. There is a large transition zone which is difficult to establish and predict. The method outlined above could be used to check an experimental situation, but is useless for testing the data in the literature. In this latter case, what is available, at best, is a drying curve from which D, is derived.
A. Mulet
342
2
4
6
G x l@
8
fkg~m’hf
Fig. 2. Infiuence of air flow rate (G) on the effective diffusivity D,. Drying of l-cm carrot cubes at 30°C.
2 "8
6.0 -
‘M 5 5
4.0 I
# -a--,
2’
A.._*_-__
*__-__-_-‘_a_*_
2.0 -
0.0 0
t 20000
30000
G Fig. 3. Mass flux densities:
experimental, externaf and internal Potato, l-cm cube, at 50°C.
(kg&? resistance
h) predictions.
Drying modelling and waterdi~ivi~
in vegetables
343
Consequently, a new way of checking D, data should be found (Berna et al., 1992). For the sake of illustration and simplicity we will consider model A. In this case (considering only one term of the series development), the dimensionless moisture content of the particle is given by Y(t) = exp[ - (3Jr2/4)(L),/L2ft] If the value of D, is obtained by plotting ln Y(t) against t, a straight line should be found, the slope (a)being L),. These results are similar to those obtained by other numerical methods (Karathanos, 1990). However, if L), is not constant, no linear reiations~p will appear. By plotting the slope (a, dlnY/dt), which is proportional to O,, vs t, it can be observed that there is really no constant range as model A should predict. Instead a smooth variation is observed, as in Fig. 4. This smooth variation constitutes a first check ~dicat~g a dependence on other variables, the most likely being the moisture content, according to the literature. This fact was pointed out by Sherwood (1931), who observed that ‘the diffusion constant decrease with the moisture concentration’. By plotting against this variable, a similar variation is observed, thus pointing to the moisture content as a variable to be taken into account to clarify variations in D,. Because the slope (a),mentioned previously, will represent an effective diffusivity only in the case where internal resistance prevails, if this is not the case the influence of the external resistance is also present and does not really represent 0,. Consequently, it is expected that the slope of the curves will be different according to the prevailing resistance. To check this point, experiments at different air flow rates were carried out -1
0 G=10500
-6
0
2000
4000
6000
8000 t
Fig. 4.
10000
-I 12000
(s)
Test of the constancy for De.Drying of carrot cubes at 40°C and C= 10 500 kg/ m2 h.
A. Mulet
344
-5 -
l
G=4000
*
G=6000
*
G=8000
l
-6. 0.0
,
G=lOOOO . , 0.2
.
+
.
GrlOSOO , . 0.4
++ .
,
.
.
0.6
,
++ .
0.8 Y
Fig. 5. Drying of carrot cubes at various air flows.
and the results are plotted in Fig. 5. It can be observed that, effectively, there is a different slope. This fact could be used to establish whether the external resistance influences the value of 0, obtained. It is observed that if external resistance is important the influence of Y is negligible or slightly positive, for values of Y larger than 0.3. To test previous results on carrot drying according to this method, and to understand this change in slope better, the results of several experiments run at different air velocities (Fig. 2) were considered. In Fig. 2 it can be observed that when the slope a variation is positive for the higher values of !P the values of D, vary with air velocity. It can also be observed that, for negative variation, the prevailing resistance is the internal one. Agreement exists between these results and those previously found. According to these results, because the slope a is calculated in a single experiment, it can readily be seen whether the experiment was carried out with the external resistance being negligible or not, and then identify whether the value of D, was affected by air velocity in each case. If it is affected, the figure obtained is related to the experimental conditions, and is not very useful for other situations. The slope of the curves shown in Fig. 5 indicates that for the particular case examined internal resistance always prevails at the end of the drying period. The steep change in slope that occurs at about Y = 0.1 could be explained by the existence of more strongly bound water. In fact, Wang and Brennan (1992) showed, using NMR techniques, that in the case of potatoes, for values of Y lower than 0.2, two species of more strongly
Drying model&
and water difhsivity in vegetables
345
bound water start to prevail, over the one (less strongly bound) that prevailed from the beginning of the drying operation. Another approach to testing the influence of external resistance on drying has been developed. It is also based on the use of empirical correlations to compute the mass flux from the particles being dried. After stating the basic mass flux equations and ad~tting some s~plif~ng assumptions, using the Pasternak and Gauvin correlation (1960) for the calculation of the external mass transfer coefficient, and admitting the volume change given by Madarro et al. ( 198 1), the following expression was found (Berna, er al., 1992): dlnY/dt=(a+
bY))0.5/Y
(7)
This indicates that the plot of (W (dlnY/dt)) vs Y should be a straight line if external resistance prevails. By examining the results shown in Fig. 5 and replotting them according to eqn (7) (Fig. 6), it can be observed that a linear variation appears for the whole range of dimensionless moisture content for the lower air flow rates. The same procedure was applied to the drying of potatoes, and similar results were obtained. It was observed that a linear relationship also appears for low flow rates and a parabola-like shape for high flow rates.
CONCLUSIONS Various approaches are possible for modelling the drying operation with varying degrees of comple~ty. At present, di~siou~ models seem to be among the most commonly used in the convective drying of food particulates for engineering applications. The complexity of drying foodstuffs arises from the diversity of materials and shrinkage. Thus, it is difficult to make generalizations. The models are also mathematically cumbersome if shrinkage is considered. These aspects give rise to a diversity of models with degrees of complexity which vary according to the end use of the model. Of course, the values of the effective diffusivities for the same product and experimental conditions will vary according to the model. The boundary conditions are where most of the simplifying assumptions are made. Two of the boundary conditions commonly differentiate the models. One is related to shrinkage (moving or static boundaries) and the other to the equilibrium (attained or not) at the outer surface. This latter assumption, related to the external resistance being negligible, is often made but seldom tested.
346
0.0
0.2
0.4
0.6
0.8
1.0
F
Fig. 6. Influence of external resistance on drying. Drying of l-cm carrot cubes at 40°C.
It has been seen that the influence of temperature on the value of D,, during the first diffusional period, is the same regardless of the consideration of shrinkage by the model. What makes the difference is the value of r>,, which differs according to the model. Although this has been tested only in potatoes and carrots, for other vegetable materials the same phenomena are expected. In any case, shrinkage should be accounted forto attain reliable values of D,, otherwise they are dependent on size and experimental conditions, and thus are meaningless as transport properties. With models which take into account heat transfer and particle temperature variation during drying, the temperature influence on D, can be identified by using only one experiment. The errors in the prediction of the drying curves at different air temperatures are similar to those obtained identifying D, at each particular temperature by using model B (shrinkage). Ident~ying a De vs Y relations~p does not seem to offer great advantage in en~eering applications. Three ways have been developed to test data to assess whether the effective diffusivity is influenced by the drying air velocity. One consists of carrying out experiments with increasing air velocity until a constant value for D, is obtained. The others are based on examining the drying curves. One of the latter is based on plotting dln Y / dt, a magnitude which is linked to D, vs Y. If the slope is very low or positive at high moisture contents, it can be assumed that the external resistance is important. In the second of the latter methods, a different shape is also found according to the prevailing resistance. The use of these methods could help in screening De data from the literature if drying curves are available.
Drying modelling and water difisivity in vegetables
347
This work was ~ancia~y supported by IMP~A/~ICYT (RE 3473). The co~aboration of A. Berna, C. Rossell6, S. Simal and J. Garcia is acknowledged. REFERENCES Berna, A., Garcia, J. & Mulet, A. (1992). AINIA Report. Valencia, AINIA. Crank, J. (1975). The Mathematics ofthe Di@sion, 2nd edn. Oxford University Press, London, 4 14 pp. Jayaraman, KS. & Das Gupta, D.K. (1992). Dehydration of fruits and vegetables. Recent developments in principles and techniques. Drying Technology, lO( l), l-50. Karathanos, V.T., Villalobos, G. & Saravacos, G.D. (1990). Comparison of two methods of estimation of the effective moisture diffusivity from drying data. 1. FoodScience, 55(l), 218-23. Ketelaars, A.A.J., Lauf,O., Coumans, J.W. & Kerhof, P.J.A.M. (1992). Determination of diffusion coefficients from experimental moisture concentration ~;yhlil;in clay. In Drying ‘92, ed. A.S. Mujumdar. Elsevier, Amsterdam, pp. Lewis, W.K. (1921). The rate of drying of solid materials. I&. Engng Chem., 13(5), 427-32. Luikov, A.?‘. (1966). Heat and Mass Trunsjer in Capillary Porous Bodies. Pergamon Press, New York. Madarro, A., Pinaga, E, Carbonell, J.V. & Peiia, JL. (198 1). Des~drataci~n de frutas y hortalizas con aire ambiente.1. Ensayos exploratorios con zanahorias. Rev. A~o~~~rn. TecnoE. ALiment., 21(4), 525-33. Maneval, J.E. & Whitaker, S. (1988). Effects of saturation heterogeneities on the interfacial mass transfer relation. In Doing ‘88, ed. A.S. Mujumdar. Elsevier, Amsterdam, pp. 238-45. McCarthy, M.J. & Perez, E. (1990). Measurements of effective moisture diffusivities using magnetic resonance imaging. In Engineering and Food, Vol. 1, eds W.E.L Spiess & H. Schubert. Elsevier Applied Science, London, pp. 473-8 1. Mulet, A., Berna, A., Borras, M. & Piiiaga, F. (1987). Effect of air flow rate on carrot drying. Drying Technology, 5(2), 245-58. Mulet, A., Berna, A. & Rosello, C. (1989a). Drying of carrots I. Drying models. Drying Technology, 7(3), 537-57.
Mulet, A., Berna, A., Rosello, C. & Piiiaga, F. (19896). Drying of carrots II. Evaluation of drying models. Drying Technology, 7( 4), 64 l-6 1. Pasternak, I.S. & Gauvin, W.H. (1960). Turbulent heat and mass transfer from stationary particles. Can. J. Chem. Engng, 38,35-42. Ratti, C. & Crapiste, G.H. (1992). A generalized drying curve for shrinking food materials. In Drying ‘92, ed. A.S. Mujumdar. Elsevier, Amsterdam, pp. 864-73. Rossello, C., Canellas, J., Simal, S. & Berna, A. (1992). A simple mathematical model to predict the drying rate of potatoes. J. Agric. Food Chem., 40( 12), 2374-8.
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Sherwood,T.K. ( 1929).The drying of solids. I&. &zgag C&m., 2 l( 1 ), 12- 16. Sherwood, T.K. (1931). Applications of theoretical diffusion equations to the drying of solids. Trans. AIChE, 27,190-202. Shrader, G.W. & Lichtfield, J.B. (1992). Moisture profiles in a model food gel during drying: measurement using magnetic resonance imaging and evaluation of the Fickian model. Dryins Technology, 10(2), 295-332. Sk;elkd, A.H.P. ( 1974). D i& sional Mass Transfer. John Wiley and Sons, New Tsotsas; E. (1992). Measurement and modelling of intraparticle drying kinetics: a review. In Drying ‘92,ed. A.S. Mujumdar. Elsevier, Amsterdam, p. 174 1. Voilley, A. & Vidal, D. ( 1990). Difision Properties of Foods. Food Science Publishers, Hornchurch, Essex. Project COST 90-bis. Wang, N. & Brennan, LG. (1992). Effect of water binding on the drying behaviour of potato. In Dryitzg ‘92, ed. AS. Mujumdar. Elsevier, Amsterdam, pp. 135-9. Whitaker, S. (1977). Simultaneous heat, mass and moments transfer in porous media: A theory of drying. In Advances in Heat T~a~r, Vol. 13, ed. T.F. Irvine. Academic Press, New York, pp. 119-203.