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Understanding Wheat Grain Steaming Daniel J. Horrobin,† Kerry A. Landman,*,† and Lyndon Ryder‡ Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia, and Research and Development Centre, The Uncle Tobys Company Limited, Rutherglen, VIC 3685, Australia
Wheat grain cooking involves heating and moisture uptake. Stapley et al. (Int. J. Food Sci. Technol. 1997, 32, 355) documented the differences between moisture uptake in boiled and steamed grains. The relationship and coupling between the heat- and mass-transfer processes is investigated here. By determining the moisture-activity diagram arising from the Stapley et al. steaming data (Chem. Eng. Sci. 1999, 54, 965), it is argued for the first time that heat transfer exerts the key controlling influence on the moisture mass transfer in steaming. A heat- and mass-transfer model for grain steaming is developed that extends and improves on the previous model (Chem. Eng. Sci. 1999, 54, 965). This model allows for the coupling of the two processes and the introduction of a dimensionless parameter, as well as providing neater solutions. In steaming, the observed temperature elevation in the grain suppresses the driving force for mass transfer. This study contributes to a better understanding of the cooking operation and will lead to improvements in batch cooking operations in breakfast cereal manufacturing. Introduction Cereal grain cooking involves heating and moisture uptake. The fact that heat conduction within the grain is rapid compared to moisture uptake has been widely documented.1,2 This implies that the temperature of the grain can be considered to be uniform across the whole grain over the time scales of interest. In this paper we investigate the relationship and intimate coupling between the heat-transfer and the mass-transfer processes. Stapley et al.3 determined the total moisture and the moisture distribution within grains that had been cooked for different times. They examined two different cooking processes, boiling in pure liquid water and steaming in pure gaseous water, and observed two key differences: (i) the mass-transfer rate is much higher in boiling than in steaming, and (ii) substantial moisture gradients are apparent within boiled grains, but not in steamed grains. The moisture gradients in boiled grains suggest that molecular diffusion within the grain does limit the mass-transfer rate under these conditions. However, the spatially uniform (but time-dependent) moisture distributions in steamed grains, along with the low moisture uptake rate, indicate that some other factor limits mass transfer under these conditions. Stapley et al.4 argued that this additional factor cannot be related to the diffusion of water in the fluid outside the grains, because this fluid, in both boiling and steaming, is pure water. There is therefore no need for water molecules to diffuse through another chemical species from the bulk to the grain surface, and a significant external mass-transfer resistance should not arise. Another suggested factor is a mass-transfer resistance associated with the grain’s surface layers. * To whom correspondence should be addressed. Address: Kerry A. Landman, Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia. Tel.: 613-8344 6762. Fax: 613-8344 4599. E-mail: k.landman@ ms.unimelb.edu.au. † The University of Melbourne. ‡ The Uncle Tobys Company Limited.
This idea has some merit, as wheat grains are not homogeneous entities, the starchy endosperm being enclosed by the bran layers or pericarp. The outermost pericarp layer is the epidermis, which is covered by a waxy cuticle that, in nature, serves to prevent excessive moisture loss.5 Another possible explanation, suggested by Stapley et al.,4 is that the first heat and moisture to reach the grain’s surface “cooks” this material and creates a barrier for further water movement that does not exist in the grain in its natural state. This notion goes against an idea, common in grain cooking models, that cooking (meaning moisture uptake and the associated gelatinization and subsequent changes undergone by starch) actually assists further moisture ingress. A diffusivity that is a strongly increasing function of moisture content has been measured directly by Stapley et al.6 in NMR experiments. This appears to be necessary to account for the steep moisture fronts observed in boiling, that penetrate into the grain in the early stages of cooking.3 Both of these conjectures immediately raise the problem of why this surface mass-transfer resistance should dominate the internal mass-transfer resistance in steaming, but not in boiling. This leads us to return to the idea suggested by Stapley et al.4 that it is heat transfer in steaming that indirectly controls the moisture uptake rate. Stapley et al.4 cooked whole Riband wheat grains by contacting them with saturated steam at either 1 or 2 bara. Both gravimetric moisture measurements and grain temperature elevation measurements were carried out. In a gravimetric test, the final moisture was determined from the change in a grain’s mass during a cook; it was assumed that all grains initially had the same 10 % wet-basis moisture content. By cooking grains for different times, a moisture uptake curve (moisture content versus cook time) could be constructed. It was assumed that this curve closely approximates the change in moisture experienced by a grain during a single cook. Grain temperature measurements were made by inserting a thermocouple into a hole drilled in a grain. The temperature could be logged as a function
10.1021/ie0208725 CCC: $25.00 © 2003 American Chemical Society Published on Web 06/10/2003
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In this paper, we argue that heat transfer exerts a key controlling influence on moisture uptake. In fact, a proportional relationship between dry-basis moisture and activity seems to be a poor description of the moisture-activity curve in the activity range relevant to grain steaming. Specifically, the gradient of moisture with respect to activity is too small, so that the equilibrium moisture is too insensitive to the grain temperature. We compare the cook data of Stapley et al.4 with known isotherm data for wheat starch. This comparison suggests that the moisture-activity relationship is sensitive enough, in the region concerned, for heattransfer control with a small temperature elevation. A coupled heat- and mass-transfer model is developed for steaming using concentration driving forces (instead of partial-pressure driving forces4) and an equilibrium moisture content that varies with grain temperature. The solutions obtained by this approach are neater than Stapley et al.4 solutions. Our solutions are fitted to the cook data, providing estimates of the heat- and masstransfer coefficients. The Equilibrium Relationship
Figure 1. Stapley steaming data.
of time from a single cook. A second temperature sensor recorded the steam temperature inside the cooking vessel as a function of time, and the difference between the two readings gave the grain’s temperature elevation. The moisture and temperature elevation results for cooks at 1 and 2 bara are plotted in Figure 1 as functions of cook time. The temperature data were logged at a much more frequent intervals than Figure 1b suggests; we report here only the temperature measurements corresponding to the cook times for which moisture measurements were made, rather than the complete temperature data set. The main feature to note at this stage is that the temperature elevations are small. At both pressures, the initial temperature elevation is about 2-3 °C, and the value falls as the cook proceeds (and the rate of moisture uptake decreases). If heat transfer is to control the mass-transfer process, the equilibrium moisture level in the grain must therefore be very sensitively dependent on temperature. The relationship between the equilibrium moisture and the temperature elevation can be constructed knowing the relationship between the moisture and the water activity in the grain at equilibrium (discussed in later sections). Stapley et al.4 assumed that the drybasis moisture was proportionally related to the activity. If the equilibrium partial pressure of grain moisture was allowed to vary with the temperature, the resulting dimensionless group was calculated as 0.2, and the authors concluded that “the depressant effect on mass transfer rates due to the rise in grain temperature is relatively small”. Furthermore, such a coupled model for heat and mass transfer was not adopted, as it appeared to contain too many degrees of freedom to obtain meaningful parameter values. Hence, a simpler uncoupled model was analyzed instead to obtain parameter values.
General Form. We consider the equilibrium between a grain and gaseous surroundings in which the mole fraction of water is y. If the grain is in contact with pure steam, as is typically the case in cooking, y is equal to 1. However, because we also refer to results from lowtemperature sorption experiments, where the surrounding gas is not pure water, we retain y as an additional variable. The equilibrium relationship tells us the moisture in equilibrium with the gas, meq, when the whole system is at temperature T and pressure P. The relationship can be written in the following form, commonly used for describing the equilibrium between a nonideal nongaseous mixture and an ideal gas phase
a(meq,T,P ) )
yP Psat(T )
(1)
Here, Psat(T ) is the saturation temperature of pure water at temperature T, and a is an activity function. The activity depends on the composition of the nongas phase (here represented by the moisture), and in general, it might also depend on the temperature and pressure. In some circumstances, the temperature or pressure dependence is important, whereas in other cases, it can be neglected. We shall encounter both situations below. The activity function can be represented by a plot of the equilibrium moisture meq versus activity, calculated according to the right-hand side of eq 1, at constant temperature and pressure. Such curves are referred to as isotherms or as moisture-activity curves, an example of which is shown in Figure 3 below. The temperature or pressure dependence of the activity function is manifested as a potential movement of the curve as T or P is varied. Relationship between Activity and Temperature Elevation. To analyze grain steaming, we need to be able to relate the equilibrium moisture to the temperature elevation, which means relating the activity to the temperature elevation. If the atmosphere surrounding a grain is pure steam, then the water mole fraction y is 1. Furthermore, if saturated steam is used and the temperature elevation is 0, Psat(T) will equal
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Figure 2. Activity versus temperature elevation.
P, and the activity will equal 1. If there is a small temperature elevation, then Psat(T ) will slightly exceed P, and the activity will be slightly less than 1. However, it is important to note that, for temperature elevations that are at most 2-3 °C, as observed by Stapley et al.,4 the activity is always close to 1. The relationship between Psat and T, and hence between a and temperature elevation ∆T, is given (slightly approximately) by the Clausius-Clapeyron equation
∂ ln Psat Wλ ) ∂T RT 2
(2)
Here, W is the molar mass of water, and λ is the specific vaporization enthalpy for pure water at temperature T. Because the temperature elevation is small, we are justified in using an approximate integrated form of eq 2. Specifically, we have a choice between the following approximate expressions relating a and ∆T
1 Psat Wλ∆T ) ≈1+ a P RT 2
(3)
P Wλ∆T ≈1Psat RT 2
(4)
sat
and
a)
sat
where λ is treated as a constant and its value is taken at the saturation temperature Tsat. Stapley et al. effectively used eq 3 in developing their model for fully coupled heat and mass transfer, which leads to an implicit expression for the dry-basis moisture md as a function of cook time. However, greater simplicity is achieved by using eq 4, as this is a linear relationship between activity and temperature elevation. As we shall see, by using eq 4, we obtain an explicit expression for md as a function of cook time. In terms of the accuracy of the approximation, there is little to choose between eqs 3 and 4. In Figure 2, we have plotted the ratio a ) P/Psat against ∆T for P ) 1 bara and P ) 2 bara. The solid lines correspond to an accurate determination of the ratio, based on exact integration of eq 2 taking λ to be a piecewise linear function of temperature (using steam table values7). The approximate nonlinear relation in eq 3 slightly overpredicts a, whereas the alternative linear relation in eq 4 slightly underpredicts a. The nonlinear expression in eq 3 might give slightly better accuracy than the linear
expression in eq 4, but not enough to warrant the additional complexity that results. Inserting appropriate values into eq 4, we find that the temperature elevations measured by Stapley et al.4 correspond to the narrow activity range 0.91 j a j 0.98. In principle, if a cook is continued for long enough, the temperature elevation should eventually fall to 0, and the activity should approach 1. However, the activity will never be less than its value at the start of a cook, and this is still high, exceeding 0.9. Moisture-Activity Curve for Starchy Materials. We now examine the general form of the moistureactivity relationship for starchy materials. Although considerable effort has been expended in measuring this relationship for grains and ungelatinized starch at low temperatures, as far as the authors are aware, no measurements exist for whole wheat grains at temperatures exceeding 100 °C. Stapley8 believed the closest available data to be the results of van den Berg,9 for wheat starch that had been gelatinized and then freezedried. van den Berg carried out sorption experiments where starch was exposed to a moist atmosphere (not pure steam) for a sufficiently long time for apparent steady state to be established. The moisture level in the atmosphere was initially increased in steps (the sorption phase), waiting for steady state to be reached after each step change, and then decreased again in steps (the desorption phase). The van den Berg9 data, shown in Figure 3, represent measurements carried out under isothermal conditions at 20 °C. (van den Berg10 noted that the system does not appear to reach true equilibrium within the experimental time scale, as evidenced by the fact that the sorption and desorption curves exhibit some hysteresis.) In Figure 3a, the curve described by the points is very steep in the high-activity region. This is not a reflection of the nonideality of the system. The same would be true for an ideal mixture, since it is the dry-basis moisture (a mass ratio) that is plotted against activity. For an ideal mixture, the activity would equal the mole fraction, x, of water in the mixture, and there would be a singularity in the equilibrium relationship at x ) 1, indicating that only pure liquid water could coexist in equilibrium with pure steam. It is a little difficult to discern whether there is a singularity at the right-hand side of Figure 3a, or whether the data points indicate a finite value when the activity equals 1. If we were to plot the wet-basis moisture (a mass fraction) for the ideal mixture, the curve would not have a singularity. At x ) 1, the wet-basis moisture would also equal 1. The data in Figure 3a are reproduced in Figure 3b, but this time, with the moisture expressed in wet-basis form. The curve is still quite steep when the activity is 1, but this feature is not as exaggerated as in Figure 3a. Figure 3b appears to indicate that the wet-basis moisture approachs a value smaller than 1 as the activity approaches 1. This implies that, at least at low temperatures, gelatinized starch reaches equilibrium with pure steam without the starch becoming infinitely diluted with liquid water. If this is correct, then Figure 3a does not have a singularity when a ) 1, but a finite intercept. Following Stapley et al.,4 we use the dry-basis moisture to describe the grain’s water content. This is because (i) equilibrium data, such as van den Berg’s results, are usually presented in this form, and (ii) the algebra involved in the mathematical model for the
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Figure 4. Two possible scenarios for the cooking process. The operating line is the actual path followed by the grain during a cook.
Figure 3. Moisture-activity relationship for wheat starch from van den Berg.9 R marks the point of reverse, between sorption and desorption. (a) Dry-basis moisture versus activity. (b) Wetbasis moisture versus activity.
steaming process is slightly simpler. van den Berg’s data suggest that, in the high-activity range relevant to grain steaming, the equilibrium dry-basis moisture is sensitively dependent on the activity and, hence, on the grain’s temperature elevation. Whether this dependence is sensitive enough for heat transfer to control moisture uptake remains to be seen, but it is already clear that a proportional relationship between dry-basis moisture and activity is a poor approximation to the equilibrium relationship at high activities. Data of Stapley et al.4 Treated as Equilibrium Data Stapley et al.4 constructed a coupled heat- and masstransfer model for grain steaming, which was used to make an assessment about whether the small temper-
ature elevations observed were sufficient to influence the moisture uptake rate. If this were the case, then the actual moisture level in the grain at a given time should be close to the equilibrium moisture level at the current grain temperature. We can get an idea of whether heat-transfer control is a feasible proposition, without the need for detailed modeling, by plotting the cook data obtained by Stapley et al.4 against isotherm data on a moisture-activity diagram. The measured temperature elevations can be converted into activity values by applying eq 4 to yield an x coordinate for each cook data point; the measured moisture levels supply the y coordinates directly. When plotted in this way, the cook data define an “operating line”, which can be compared with an equilibrium line defined by data from low-temperature sorption experiments. We envisage the operating line/equilibrium line relationship revealed in this plot to lie between two extremes, as shown schematically in Figure 4. The first extreme corresponds to the situation where molecular diffusion controls the rate of moisture uptake. Here, the actual moisture level in the grain is always rather less than the equilibrium moisture level at the current grain temperature. The operating line (cook data) is therefore relatively distant from, and below, the equilibrium line (sorption data), as shown in Figure 4a. The other extreme corresponds to the heat-transfer-control scenario. Here, the equilibrium moisture level is substantially depressed as the temperature elevation increases above 0. The equilibrium line is therefore relatively steep and lies close to the operating line, as shown in Figure 4b. In this case, the actual moisture level in the grain is always close to the equilibrium moisture level at the current grain temperature. The data of Stapley et al.4 are plotted in moistureactivity form in Figure 5, alongside van den Berg’s sorption data. As expected, the cook data points cover a small activity range close to 1, and the curves they describe are very steep, indicating that the equilibrium moisture must be sensitively dependent on temperature elevation for heat-transfer control to be valid. Figure 6a shows a detail of Figure 5 in the high-activity region. There are two main differences between the sorption data and the cook data. First, the cook data points that correspond to short cook times (and therefore high temperature elevations and lower activities) generally lie below the sorption curves. The cook data points on the moisture-activity plot can be explained in terms of a temperature shift
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Figure 5. Stapley cook data plotted on a moisture-activity diagram.
and are consistent with that observed by Becker and Sallans11 for wheat at 25 and 50 °C. Second, the cook data points suggest a larger gradient of moisture with respect to activity than the sorption data points, the difference being a factor of about 2.5 for the 2 bara cooks. On temperature grounds, we would expect these data points to be shifted to higher activities or lower moistures compared with the 1 bara cook results. In fact, Figure 6a illustrates that they lie slightly on the other side of the 1 bara data points. At longer cook times, this feature could possibly be explained by chemical processes, such as starch degradation, that take place during cooking (see Conclusions). However, even for the shortest cooks, where little starch degradation can yet have occurred, the 2 bara data points lie quite close to the 1 bara data points. The location of the 2 bara data on the moisture-activity plot is unable to be justified at this stage, and there are no experimental observations with which to compare. Simple Representations of the Moisture-Activity Relationship in the High-Activity Range. As noted in the previous section, the data of Stapley et al.4 require a good description of the moisture-activity relationship in the activity range 0.91 j a j 0.98. The simplest adequate algebraic form is a linear one, written as
md∞ - mdeq ) c1(1 - a)
(5)
This relationship is defined by two constants: md∞, which is the intercept with the axis a ) 1 (and is therefore the final equilibrium moisture), and the slope c1. A linear relationship of this type describes the asymptotic behavior, as a approaches 1, of the Guggenheim-Anderson-de Boer (GAB) isotherm, which is often used to fit low-temperature sorption data for starchy materials.
Figure 6. Closer examination of the moisture-activity relationship described by the cook data: (a) moisture versus activity a, (b) moisture versus (1-a)-1.
Judging from Figure 6a, a linear relationship would appear to describe the data points derived from the 2 bara cooks very well, but the data points derived from the 1 bara cooks not so well. We therefore anticipate that we will obtain a good fit to the moisture-time and temperature-time data for the 2 bara cooks from a steaming model in which heat transfer controls the mass-transfer rate and the moisture-activity relationship is linear. On the other hand, we anticipate that we would not obtain such a good fit to the data for the 1 bara cooks.
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Another relatively simple algebraic form that might give a good fit to the moisture-activity data is the hyperbolic relationship
c2 mdeq - m/d ) 1-a
parameters such as md∞ and m/d. For the linear moisture-activity relationship, we have
md∞ - mdeq(T) ) (6)
This relationship is again defined by two constants, m/d and c2, where m/d is a hypothetical moisture that would correspond to an activity that is either very much less than 0 or very much greater than 1. If m/d can be taken to be 0, then we have a single-parameter form that might be better at representing the moisture-activity relationship in the high-activity region than the proportional form adopted by Stapley et al. Equation 6 represents the asymptotic behavior, as a approaches 1, of the Brunauer-Emmett-Teller (BET) isotherm, which can be regarded as a simpler precursor of the GAB isotherm. The most important feature to note is that there is a singularity when a ) 1, so the final equilibrium moisture is infinite. In other words, eq 6 asserts that only pure liquid water can exist in equilibrium with pure steam and that, in the final steady state, all nonwater components of the grain would become infinitely diluted. Figure 6b is similar to Figure 6a, but now, the drybasis moisture has been plotted against (1 - a)-1. In this case, the data points derived from the 1 bara cooks define a reasonably straight line, whereas the data points derived from the 2 bara cooks exhibit some curvature. We therefore conclude that we would obtain a good fit to the moisture-time and temperature-time results for the 1 bara cooks from a steaming model in which heat transfer is controlling and the moistureactivity relationship has the hyperbolic form of eq 6. On the other hand, we would expect the fit to the data for the 2 bara cooks to be less good. The intercept of the best-fit straight line through the 1 bara data points is close to 0, so it might be reasonable to take m/d to be 0. However, removing a degree of freedom in this way might incur a cost in terms of the quality of fit obtained from a steaming model. We therefore have two possible algebraic representations of the moisture-activity curve to investigate. Both appear as though they might provide an adequate description of the equilibrium relationship for the high activities encountered in steaming. However, there is a significant difference between the two in terms of the final equilibrium moisture content (finite in one case and infinite in the other) that makes consideration of both worthwhile. Moisture-Temperature Relationship. Whichever form we choose for the moisture-activity relationship, we combine this with eq 4 to obtain a relationship between the equilibrium moisture and the temperature elevation, which then forms part of a steaming model. In general, we can write this relationship as
mdeq(T ) ) F(R∆T,[dimensionless constants]) (7) where R is a parameter that makes the temperature elevation ∆T dimensionless and the “dimensionless constants” that appear as the second argument to F are
c1Wλ RT 2sat
∆T
(8)
2 ). For the and so the constant R is equal to c1Wλ/(RT sat hyperbolic moisture-activity relationship, we have
mdeq(T) -
m/d
c2RT 2sat 1 ) Wλ ∆T
(9)
2 so in this case, the constant R is equal to Wλ/(c2RT sat ).
Modeling In this section, we develop a coupled heat- and masstransfer model for grain steaming. Although the model is nominally coupled, it encompasses, as special cases, uncoupled models where either molecular diffusion only or heat transfer only limit the rate of moisture uptake. Mass Balance. When writing the mass balance equation, Stapley et al.4 did not specify where the masstransfer resistance lies, choosing to relate the molecular flux to a pressure-difference driving force using a form of overall gas-side mass-transfer coefficient. In the accompanying discussion, they argued that there could not be a significant gas-phase mass-transfer resistance and, therefore, that the mass-transfer resistance lies entirely within the grain. They raised the possibility that the most significant mass-transfer resistance could arise from the bran layers, rather than the bulk endosperm. Here, we explicitly assume that there is no gas-phase mass-transfer resistance and choose to express the molecular flux in terms of a moisture-difference driving force. Explicitly, we too do not specify exactly where the mass-transfer resistance arises within the grain. However, if molecular diffusion is the main limitation on the rate of moisture uptake, we should argue that the most significant mass-transfer resistance does indeed occur in the bran layers. This is to be consistent with the observed uniformity of the moisture distribution in the bulk endosperm during steaming, which provides the justification for a “lumped-parameter” ODE model. On the other hand, if heat transfer is the main factor controlling the rate of moisture uptake, it is not so essential to invoke a bran-layer mass-transfer resistance. The expression below could be regarded as a crude way of accounting for the minor contribution to the overall limitation on the moisture uptake rate from diffusion in the bulk endosperm. The rate of moisture uptake is equal to the rate of change of total mass M of the grain. Stapley et al.4 related this to partial pressure differences. Instead, as is more commonly done, we relate it to a concentrationdifference driving force using a conventional masstransfer coefficient k (units m s-1)
dM ) WkA[ceq(T ) - c] dt
(10)
Here, A is the area available for mass transfer, c is the concentration of water at the inner boundary of the region that provides the most significant mass-transfer resistance, and ceq is the concentration at the outer
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boundary. Because there is no gas-phase mass-transfer resistance and equilibrium is assumed to be achieved rapidly when there is direct contact between phases, ceq is the concentration of water in equilibrium with pure steam at pressure P and the current grain temperature T. Using M ) Md(md + 1), cW/Fd ) md, and Fd ) Md/V, where Md is the dry mass of the grain, md is the drybasis moisture, and Fd is the mass of the nonwater components of the grain per unit total volume V, eq 10 can be rewritten in terms of the dry-basis moisture content as
md′(t) ) kφ[mdeq(T ) - md]
(11)
where we have defined φ as the area-to-volume ratio A/V. This equation is probably only a crude description of the moisture transfer that takes place within the cooking grain. Stapley et al. argued that, in boiling, where the bulk endosperm provides the major masstransfer resistance, the process is non-Fickian, being limited by the time required for starch molecules to undergo structural rearrangement. This possibility casts doubt on the validity of the fundamental eq 10, maybe even if the mass-transfer resistance in steaming is concentrated at the edge of the grain. (The bran layers or a thin layer of “cooked” starch that might form early in the cook, would comprise material that is similar in its polymeric nature to the bulk endosperm.) Furthermore, even if eq 10 is a good description of moisture transfer within the grain, we take the mass-transfer coefficient k to be constant, which is doubtful given the observation that the diffusivity of small molecular species in polymeric substances is strongly concentration dependent. Similarly, we assume that the density Fd is constant, although as the moisture increases, we expect that this quantity will tend to decrease. This could be a particularly bad assumption if the moisture reaches very high levels, because when md∞ approaches ∞, Fd tends to 0. When we come to consider a model for which md∞ approaches ∞ in a later section, we therefore abandon this mass balance equation and consider only the case where the rate of moisture uptake is entirely controlled by heat-transfer effects. Despite these provisos, eq 11 might still be adequate if the rate of moisture uptake is predominantly controlled by heat transfer but molecular diffusion is not fast enough to be completely ignored. In this situation, the advantages of eq 11 in terms of simplicity arguably outweigh its potential drawbacks. Heat Balance. Following Stapley et al.,4 we assume that the grain is at quasi-steady state as far as heat transfer is concerned, and so, the rate at which heat is released by the incoming water is equal to the rate of sensible heat transfer back to the surroundings
λ
dM ) hA(T - Tsat) dt
(12)
Here, h is the heat-transfer coefficient, and Tsat is the temperature of the bulk steam, which is assumed to be the saturation temperature at pressure P. Strictly speaking, λ (units J kg-1) should be the difference between the specific enthalpy of gaseous water at temperature T and the specific enthalpy of absorbed water in the grain at the same temperature. However, this enthalpy difference should be dominated by the fact
that it is a difference between gas- and condensed-phase enthalpies. It should therefore be well approximated by the specific enthalpy of vaporization of water at Tsat. Again, in terms of md, φ, and temperature elevation ∆T ) T - Tsat, this equation becomes
md′(t) )
hφ ∆T λFd
(13)
Equilibrium Relationship (Case in which md∞ Is Finite). Here, we use the linear equilibrium moisturetemperature elevation relationship in eq 8
md∞ - mdeq(T ) ) R∆T, R )
c1Wλ RT 2sat
(14)
In a later section, we investigate the hyperbolic equilibrium moisture-temperature elevation relationship in eq 9. Manipulation. By eliminating ∆T between eqs 13 and 14 and then eliminating mdeq using eq 11, these three equations can be combined as
md′(t) )
(
)
RλFd 1 + kφ hφ
-1
(md∞ - md)
(15)
where the constant [1/(kφ) + RλFd/(hφ)]-1 acts as an overall transfer coefficient. It has units of s-1, and we introduce a time constant τ as
τ)
RλFd 1 + kφ hφ
(16)
Equation 15 is easily integrated, resulting in
md∞ - md ) exp(-t/τ) md∞ - md0
(17)
This equation is the same form as the result given by Stapley et al.4 for their one-way-coupled model, where the moisture uptake is controlled by molecular diffusion within the grain. However, within our framework, the same equation applies for all possible scenarios between, at one extreme, moisture uptake governed solely by molecular diffusion and, at the other extreme, moisture uptake governed solely by heat transfer and equilibrium thermodynamics. We define a dimensionless parameter Π that determines where we lie in this spectrum as
Π)
kRλFd h
(18)
The influence of this parameter is apparent from the definition of τ or alternatively as follows. Once we have eliminated ∆T between eqs 13 and 14, if, instead of eliminating mdeq using eq 11, we eliminate md′(t), we obtain
md∞ - mdeq(T ) mdeq(T ) - md
)Π
(19)
Now, if h is large (fast heat transfer) or k is small (slow molecular diffusion), R is small (mdeq insensitive to ∆T ), or λ is small (little heat released by incoming water), then Π , 1, and the moisture uptake will tend to be limited by molecular diffusion. The current equilibrium moisture mdeq, which we interpret as the actual mois-
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ture in the grain at the outer boundary of the region of major mass-transfer resistance, will be closer to the final moisture md∞ than the bulk moisture md. Conversely, if h is small or k, R, or λ is large, then Π . 1, and heat transfer and equilibrium thermodynamics tend to limit the moisture uptake, and mdeq will be closer to md than to md∞. The effect of the density Fd is more subtle and seems to merely reflect the way we choose to measure the rate of moisture uptake. The rate of change of the mass of water in the grain is equal to the rate of change of the total grain mass dM/dt, and this is equal to Mdmd′(t). Therefore, if the dry mass Md is small, implying that Fd is small, then the rate at which the amount of water in the grain is changing in real terms will be small, even though md′(t) might not be. If dM/dt is small, the temperature elevation will be small, and molecular diffusion will become the controlling factor determining the rate of moisture uptake. Once the solution for the moisture has been obtained, it is a simple matter to substitute back into the heat balance to find the solution for the temperature elevation. The result is of the form
∆T ) ∆T0 exp(-t/τ)
(20)
where the initial temperature elevation ∆T0 is given by
∆T0 )
md∞ - md0 R(Π
-1
+ 1)
)
λMd md∞ - md0 hA τ
(21)
There is therefore a linear relationship between the drybasis moisture and the temperature elevation at time t
md∞ - md ∆T ) md∞ - md0 ∆T0
(22)
Stapley et al.4 derived an equation of the same form for their one-way-coupled model, where moisture uptake is controlled solely by molecular diffusion. However, in our case, eq 22 applies for all situations between molecular diffusion control at one extreme and heat transfer/equilibrium thermodynamics control at the other. Comparison with Experimental Data (Case in which md∞ Is Finite) In this section, we estimate the heat- and masstransfer coefficients from values of the fit parameters and other parameter values obtained from independent sources (shown in Table 1). The values of W, R, Tsat, and λ are known with good accuracy and have been obtained from steam tables.7 However, we are using the enthalpy of vaporization of water, λ, as an approximation to the enthalpy change when water is absorbed from pure steam into the grain. The other independent parameter estimates that we need relate to the size of a wheat grain, and these are considerably more uncertain. For the dry mass Md, Stapley et al.4 used a value of 35 mg, which we retain. For purely geometrical properties, Stapley et al. assumed that a wheat grain could be approximated as a sphere of diameter 3 mm. The volume and surface area of such a sphere are about 15 mm3 and 30 mm2, respectively. These estimates are perhaps a little on the low side, and Table 1 lists some alternative values based
Table 1. Independent Parameter Values parameter
value K-1)
gas constant (J molar mass of water (kg kmol-1) saturation temperature of water (K) enthalpy of vaporization of water (J kg-1)
Wa Tsat λa
Mdb Vc Ac φd
c
definition (units) kmol-1
Ra
dry grain mass (kg) grain volume (m3) grain surface area (m2) surface area-to-volume ratio (m-1)
8314.5 18.015 372.8 at 1.0 bara 393.4 at 2.0 bara 2.258 × 106 at 1.0 bara and 372.8 K 2.202 × 106 at 2.0 bara and 393.4 K 35 × 10-6 20 × 10-9 50 × 10-6 2500
a Taken from steam tables.7 b Value used by Stapley et al.4 Our estimates. d Calculated as A/V.
on our own measurements of wheat grain dimensions. Our suggested surface area value of about 50 mm2 is considerably larger than the value used by Stapley et al. In the process of calculating the heat-transfer coefficient, we divide by this surface area. The fact that our heat-transfer coefficients are about half the size of those calculated by Stapley et al. is almost entirely due to this difference in the surface area estimates. When fitting the temperature elevation data, Stapley et al.4 chose to express the data in temperature elevation versus moisture content form and then to fit an equation analogous to eq 22. They expressed the data in this way by extracting from the temperature elevation versus time data the temperature elevations corresponding to the cook times for which the final moistures were measured. We have chosen to fit the temperature elevation data in the original form, where time is the independent variable, and we fit eq 20. We do not have all of the original thermocouple readings from Stapley et al., but we do have the temperature elevations corresponding to the cook times employed for the moisture measurements, and it is these data that we use for fitting. We fit the moisture and temperature elevation data using eqs 17 and 20 by adjusting the four parameters md0, md∞, ∆T0, and τ. The parameter τ appears in both equations and therefore should be constrained to be the same for both fits. Stapley et al.4 also had a common parameter in the two equations they used for data fitting, but they did not enforce the same value for both data sets. They did, however, comment that there was good consistency between the two estimates. In our case, we do constrain τ to be the same for both the moisture-time fit and our temperature elevationtime fit, and we determine the overall best fit that minimizes the following quantity Nm
∑ i)1
NT
(o) (m) 2 [mdi - mdi ]
+ Nm
(o)2 mdi ∑ i)1
(m) 2 [∆T(o) ∑ i - ∆Ti ] i)1 NT
(23)
∆T(o)2 ∑ i i)1
(o) Here, mdi is the observed moisture associated with ith (m) data point in the moisture set, and mdi is the moisture calculated from the model for the time associated with (m) the ith data point. ∆T(o) have similar meani and ∆Ti ings, but relate to the temperature elevation data set.
Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4117 Table 2. Model 1 (Linear Equilibrium Relationship): Best-Fit Parameter Values steam pressure (bara) parameter md0 md∞ ∆T0 τ
units
1
2
(K) (s)
0.157 ( 0.016 0.529 ( 0.026 2.32 ( 0.08 4760 ( 340
0.180 ( 0.006 0.684 ( 0.016 2.60 ( 0.03 5550 ( 240
Table 3. Model 1 (Linear Equilibrium Relationship): Derived Parameter Values P (bara) parameter R(Π-1+1)a hb
units (K-1)
(W m-2 K-1)
1
2
0.160 53
0.194 54 II
Π R kc
(K-1) (10-7 m s-1)
0.1
1
10
0.1
1
10
0.015 0.92
0.080 1.7
0.15 9.2
0.018 0.79
0.097 1.4
0.18 7.9
a R(Π-1 + 1) is calculated as (m b d∞ - md0)/∆T0. h is calculated as [λMd/A][R(Π-1 + 1)/τ]. c k is calculated as (Π + 1)/(φτ).
Figure 7. Fit of model 1 (linear equilibrium relationship) to the Stapley cook data.
Nm and NT are the numbers of points in the moisture and temperature elevation data sets, respectively. The reason for weighting the sums of square differences by the denominators shown is to try to obtain an overall fit that is not biased toward either data set. The weights attempt to take into account the facts that there are different numbers of points in each data set and that the orders of magnitude of the moistures and the temperature elevations are different. Figure 7 shows the combined fit of eqs 17 and 20 to the moisture and temperature elevation data of Stapley et al.4 The moisture data are shown in Figure 7a and the temperature elevation data in Figure 7b. Although our fitting procedure is slightly more restrictive than that employed by Stapley et al., the resulting qualities of fit seem to be very similar. In general, the model is capable of fitting the data reasonably well. The exception to this is the temperature elevation data for the 1 bara cook, where the fit is not especially good. This is not because the fit is compromised by having to choose τ so that the 1 bara moisture data are also fitted adequately. When eq 17 is fitted to the 1 bara temperature elevation data only, ignoring the moisture data, a very similar fit is obtained. The reason behind the poor fit is that the linear isotherm approximation in eq 14 is inadequate for the 1 bara cooks (as anticipated previously). The best-fit parameter values that we obtained are given in Table 2. We are fitting the same experimental data as Stapley et al. using equations of the same form, and so, not surprisingly, the fit parameter values that we obtain are similar to those obtained by Stapley et al. (after some manipulation to obtain values that can be compared). This is particularly true for the initial moisture (md0) and initial temperature (∆T0) values,
because the experimental data define these values quite precisely. There are larger differences between our md∞ and τ values and those of Stapley et al., reflecting the fact that the experimental data define these parameters less precisely. Heat-Transfer Coefficient. From md0, md∞, and ∆T0, we can calculate the quantity R(Π-1 + 1), which is equal to (md∞ - md0)/∆T0. This quantity is listed in Table 3 for the 1 and 2 bara cooks, and from these results, we can calculate the heat-transfer coefficient h as (λMd/A)R(Π-1 + 1)/τ, which is also listed in the table. The values turn out to be almost exactly the same for the 1 and 2 bara cooks, being about 53-54 W m-2 K-1. Note that the heat-transfer coefficient is well-defined even when we do not know whether the moisture uptake is diffusion- or heat-transfer-controlled. Our heat-transfer coefficient values are smaller than those obtained by Stapley et al., which were around 100 W m-2 K-1. As explained above, the main reason for this difference is the fact that we use a larger value for the surface area A. Stapley et al. compared their heat-transfer coefficients with values obtained from a correlation due to Ranz and Marshall12,13 for heat transfer from a sphere enhanced by natural convection. This correlation is of the form
Nu ) 2.0 + 0.6Pr1/3Gr1/4
(24)
where Nu, Pr, and Gr are the Nusselt, Prandtl, and Grashof numbers, respectively. The Nusselt and Grashof numbers have the following definitions
Nu )
F2gβg∆Tgd3 hd , Gr ) κg µ2
(25)
g
where the quantities with a subscript g are properties of saturated steam, which are listed in Table 4, along with the values appropriate for steam at 1 and 2 bara. g itself is the acceleration due to gravity, equal to 9.81 m s-1. The Prandtl number is defined as µgcpg/κg, where cpg is the specific heat capacity of saturated steam, but values of the Prandtl number are given in steam tables,
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Table 4. Further Properties of Saturated Steam Used in Applying the Correlation of Ranz and Marshall14 for Heat Transfer from a Sphere with Natural Convectiona P (bara) parameter κg Fg µg Prg βgb
definition (units) thermal conductivity (W m-1 K-1) density (kg m-3) viscosity (Pa s) Prandtl number thermal expansivity (K-1)
1
2 10-3
26.8 × 10-3
0.590 12.0 × 10-6 0.97 2.68 × 10-3
1.129 12.8 × 10-6 1.00 2.54 × 10-3
24.8 ×
a All properties taken from steam tables.7 reciprocal of the saturation temperature.
b
Calculated as the
so we do not need to know the value of the specific heat capacity as a separate quantity. Substituting the appropriate values into the correlation, we obtain the following relationships between h, d, and ∆T
at 1 bara at 2 bara
0.0496 ∆T 1/4 + 1.32 d d
( ) ∆T 0.0536 + 1.89( ) h) d d h)
1/4
(26a) (26b)
Stapley et al. suggested that d be taken as 3 mm, and using this value, we obtain the same constant terms on the right-hand sides as Stapley et al., but we find that the coefficients of the ∆T-dependent terms are much smaller. We can attribute this difference only to an arithmetical error on the part of the previous authors. Also, as we have already mentioned, an equivalent diameter of 3 mm is perhaps a little small. It is difficult to know exactly what equivalent diameter is appropriate for a nonspherical object, but we choose a value of 3.5 mm, which lies somewhere between the equivalent spherical diameters obtained using the volume and surface area values that we suggested earlier. With this d value, the expressions for the heat-transfer coefficient become
at 1 bara
h ) 14.2 + 5.43∆T1/4
(27a)
at 2 bara
h ) 15.3 + 7.77∆T1/4
(27b)
For small values of the temperature elevation ∆T similar to those encountered in steaming, the heattransfer coefficient is therefore only moderately dependent on ∆T (as opposed to strongly dependent on ∆T, as the equations presented by Stapley et al. suggest). The model we have derived assumes that h is constant, so if h, in fact, varies, then the value that we extract from the fit parameters should be regarded as some sort of average heat-transfer coefficient over the range of cook times employed in the experiments. In the experiments carried out by Stapley et al., the temperature elevation fell from an initial value of about 2.5 °C to about 1.0 °C at the longest cook times, with these rough values being appropriate for both the 1 and 2 bara cooks. Applying these temperature elevations to eq 37, the correlation of Ranz and Marshall indicates that 19.6 j h j 21.0 for cooks at 1 bara and 23.1 j h j 25.1 for cooks at 2 bara. This suggests that there is, in fact, little variation in h during cooking, as a result of natural convection effects, up to the longest cook times covered by the experiments.
The heat-transfer coefficients calculated from the fit parameters are 2-3 times larger than those obtained from the correlation. (Stapley et al. obtained larger heattransfer coefficients from both sources for different reasons, but found that the values obtained from the fit parameters were slightly smaller than those obtained from the correlation.) The grains were sitting on a mesh above a boiling pool of water, with steam continually being vented from the cooker, so we would not consider it surprising if the heat-transfer coefficient were larger than could be explained in terms of diffusive heat transfer and natural convection alone. It seems quite likely that the movement of steam around the grains, due to factors other than the temperature difference between the grains and the steam, could enhance the convective heat transfer to a significant degree. The fact that a plausible heat-transfer coefficient emerges from the data fitting procedure is not at all a searching test of the model. In effect, md′(t) and ∆T are both measured in the experiments. The heat-transfer coefficient can be calculated directly from these quantities using eq 13, which can be viewed as little more than a statement defining the heat-transfer coefficient. All that is required to evaluate h are some reasonable estimates for A, Md, and λ. Any relationship between md and t that provides a satisfactory fit to the moisture uptake data allows md′(t) to be easily determined and will produce a similar heat-transfer coefficient value. This moisture uptake “model” need not be based on any physical principles for h to be obtained. Mass-Transfer Coefficient. We cannot, from the experimental data alone, determine the values of R and Π individually, and so we cannot establish whether we are working in the molecular-diffusion-controlled regime, the heat-transfer-controlled regime, or somewhere in between. The fundamental reason for this is that the equations have the same form, regardless of which factor is controlling. (Similarly, Stapley et al.4 found that, when they tried to incorporate a more general linear isotherm into their model, defined by two constants rather than the single constant required for the unrealistic proportional isotherm, the error bounds on the fitted parameters became very large.) However, we can examine different values of Π and calculate the corresponding values of R and the mass-transfer coefficient k. In Table 3, we have listed for each cook pressure the R and k values corresponding to Π ) 0.1, 1, and 10. Π ) 0.1 represents a cook in which the moisture uptake is predominantly controlled by molecular diffusion; Π ) 10, a cook in which the moisture uptake is predominantly controlled by heat transfer; and Π ) 1, the intermediate case where both factors are important. Given that the moisture uptake rate in steaming is about 10 times less than that in boiling, if we assume that the mass-transfer coefficient in the same in both cases (which is likely because there is no mass-transfer limitation in the fluid surrounding the grain), we might expect Π ) 10 to be about the right value. Our mass-transfer coefficient values are quite different from those of Stapley et al., but this is almost entirely because we define the mass-transfer coefficient in a different way. Our mass-transfer coefficient is a “grain” mass-transfer coefficient, whereas the masstransfer coefficient of Stapley et al. is a “gas” masstransfer coefficient, even though the mass-transfer resistance must be entirely within the grain. If the
Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4119
activity can be considered to be a linear function of moisture with constant slope a′, then the relationship between the two is
kSt )
RTFd k a′PW
(28)
where kSt is the mass-transfer coefficient defined by Stapley et al. kSt is inversely proportional to the pressure P, and this largely determines how kSt changes as the cook pressure is increased from 1 to 2 bara, as T and k only change slightly. This explains the “surprising” observation of Stapley et al. that the mass-transfer coefficient was only about half the size at 2 bara that it was at 1 bara. Suppose that Π is the same at both 1 and 2 bara. Then, our k values are still slightly smaller at 2 bara than they are at 1 bara. This is surprising, because the mass-transfer coefficient can be viewed as a diffusivity divided by a characteristic length. Because the relevant diffusivity is that of absorbed water within the grain, we would expect it to be insensitive to pressure, and perhaps to increase slightly as the temperature increases from 373 to 393 K. k should therefore, if anything, increase slightly from a 1 bara cook to a 2 bara cook. However, the decrease in the k values that we find is small and could easily be explained by uncertainty in the fit parameters. However, there is no fundamental reason that Π should be the same at both 1 and 2 bara. If k increases between 1 and 2 bara, Π should also tend to increase. In fact, the direction in which Π should change is difficult to determine, because the increase in temperature might also affect other parameters that make up this quantity. To conclude this section, we note that our k values are in the right range for diffusion of a small molecular species within a polymeric material. Diffusivities for such systems can be very concentration dependent, but Cussler14 gives some values for relatively small organic molecules such as benzene and cyclohexane in various polymers that are in the range 10-10-10-13 m2 s-1. We might expect the diffusivity of a very small molecule such as water to be at, or perhaps slightly above, the upper limit of this range. The characteristic length for diffusion in our system should be somewhat less than the dimension of a grain, i.e., on order less than 10-3 m. A mass-transfer coefficient of 10-7 m s-1 therefore seems about right. Comparison with Experimental Data (Case in which md∞ f ∞) In the previous section, we assumed that the final moisture when the activity is equal to 1, md∞, has a finite value. This certainly appeared to be the case in van den Verg’s sorption experiments. However, even if md∞ is finite for gelatinized starch at low temperatures, it is not clear that the same should be true for starch that is in the process of being cooked. The starch appears to undergo substantial changes even after gelatinization is complete, related to a reduction in the molecular weight of the polymer. Such changes might allow more water into the grain. Given the uncertainty regarding whether the eventual dry-basis moisture in cooking is finite or not, it is interesting to investigate a model in which md∞ approaches ∞.
Figure 8. Fit of model 2a (hyperbolic equilibrium relationship) to the Stapley cook data.
In eq 9, there is an equilibrium relationship that has the property that md∞ approaches ∞ as the temperature elevation approaches 0. We combine this with the heat balance equation and now assume that mass transfer is fast enough to be completely ignored. (As explained earlier, our mass balance equation is probably a poor approximation when the moisture, and hence Fd, changes substantially during the cook. It is, however, possible to write a slightly more complicated mass balance that takes this into account.) This means that we replace mdeq in eq 9 by md. Equations 9 and 13 can then be solved simultaneously to give
∆md ) ∆md0x1 + t/τ ∆T )
∆T0
x1 + t/τ
(29)
where ∆md is the difference md - m/d. ∆md0 and ∆T0 are the initial values of ∆md and the temperature elevation ∆T, respectively, which are related according to the equation R ) (∆md0∆T0)-1. The time constant τ is now defined as [λMd/(2hA)](∆md0/∆T0). When fitting eq 29 to the experimental data, we again have four adjustable constants in total: ∆md0 and m/d from the moisture equation, ∆T0 from the temperature equation, and τ common to both equations. The fits of eq 29 to the moisture and temperature data are shown in Figure 8. The moisture fits are very similar to those obtained from the exponential model (eq 17). The temperature equation is found to fit the temperature data very well at 1 bara but not quite so well at 2 bara. This is the reverse of the previous situation, where the 2 bara data could be fitted well and the 1 bara data
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Table 5. Parameter Values for Model 2 (Hyperbolic Equilibrium Relationship)
Table 6. Parameter Values for Model #3 (hyperbolic equilibrium relationship with m/d ) 0)
steam pressure (bara) parameter ∆md0a m/da ∆T0a τa md0b hc Rd
steam pressure (bara)
units
1
2
parameter
units
1
2
0.244 ( 0.019 -0.060 ( 0.022 2.85 ( 0.08 1270 ( 130 0.184 52 1.4
md0 ∆T0 τ h R
(W m-2 K-1) (K-1)
0.111 ( 0.007 0.041 ( 0.012 2.99 ( 0.08 529 ( 44 0.152 55 3.0
(K) (s) (W m-2 K-1) (K-1)
0.133 ( 0.005 2.88 ( 0.08 598 ( 52 61 2.6
0.197 ( 0.007 2.99 ( 0.07 1050 ( 92 48 1.7
(K) (s)
a Fit parameters. b Calculated as m/ + ∆m . c Calculated as d0 d [λMd/(2A)][∆md0/(τ∆T0)]. d Calculated as (∆md0∆T0)-1.
relationship. The fits are therefore not so good, as we have lost a degree of freedom, but they are far from poor. With this modified md∞ f ∞ isotherm, we could reintroduce the mass balance equation, and by fitting the resulting (implicit) moisture and temperature equations, deduce a value for Π. We choose not to do this here, as it would not add anything to our arguments. The quantity Π that we would calculate would effectively be merely a measure of the quality of fit to the cook data, in moisture-activity form, that could be obtained from the functional form selected for the isotherm. If Π is very large, then a good fit can be obtained, and if Π is very small (as found by Stapley et al.), then the fit obtained is poor (the proportional relationship). Conclusions
Figure 9. Fit of model 2b (hyperbolic equilibrium relationship with m/d ) 0) to the Stapley cook data.
not so well. The fit parameter values associated with these fits are given in Table 5. Not surprisingly, given that the data define the initial moisture quite precisely, the ∆md0 and m/d values translate into md0 values (listed in the table) that are very close to those obtained by fitting eq 17 . Similarly, the ∆T0 values are close to those obtained by fitting eq 20. The ∆md0 and τ values for the 2 bara cooks are both a little over twice as big as the values for the 1 bara cooks. When calculating h, this difference cancels out, so that the h values are quite similar at 1 and 2 bara. As we found when we fitted the previous model, h is in the range 50-60 W m-2 K-1, and again, this agreement is not surprising, given that, as we have already noted, h is defined by the experimental data in a way that does not rely on any modeling assumptions. Figure 9 and Table 6 show the fits obtained from this model when m/d is taken to be 0. In this case, there is only a single adjustable parameter (R) in the equilibrium
Careful consideration has been made of the concepts underpinning the grain steaming model in light of observed differences between the wheat boiling and steaming processes. We provide several arguments to support the theory that heat transfer controls the masstransfer process in steaming. This theory hinges on the equilibrium moisture level in the grain being very sensitively dependent on temperature. Available experimental data confirm that such a sensitive dependence is possible. Stapley et al.4 were unable to reach these conclusions and resolve the importance of heat transfer in mass transfer because of the limitations of the particular mathematical model they developed. The Stapley et al.4 data are plotted on a moistureactivity diagram and shown to be consistent with independent experimental observations. In most cases, only rough calculations are needed to check for this consistency. An example is the comparison between the cook data, plotted on the moisture-activity diagram, and low-temperature sorption data. We are unable to justify the location of the 2 bara data in relation to the 1 bara data. However, no experimental observations of isotherms corresponding to different pressures are available for comparison; it would be interesting to obtain some measured isotherms for wheat/starch at pressures greater than atmospheric. We have constructed a mathematical framework to represent our conceptual picture of the heat and mass transfer during steaming. This model extends and improves the earlier one.4 As expected, our equations and results are similar to those obtained by Stapley et al.4 for their one-way-coupled model, where moisture uptake is controlled solely by molecular diffusion. In particular, our equations apply for all situations between molecular diffusion control at one extreme and heat transfer/equilibrium thermodynamics control at the other, as controlled by our dimensionless parameter Π. In addition, our modeling allows us to address two issues of concern to Stapley et al.: (i) the reason they
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Nomenclature a ) activity A ) area available for mass transfer, m2 c ) (with super- and/or subscripts) concentration of water in grain, kmol m-3 c1 ) constant in linear equilibrium moisture-activity relationship c2 ) constant in hyperbolic equilibrium moisture-activity relationship h ) heat-transfer coefficient, W m-2 K-1 k ) mass-transfer coefficient, m s-1 m ) moisture, expressed on either a wet or dry basis M ) total mass of grain, kg md ) dry-basis moisture Md ) dry mass of grain, kg P ) pressure, Pa R ) gas constant, J kmol-1 K-1 t ) cook time, s T ) temperature, K V ) total grain volume, m3 W ) molar mass of water, kg kmol-1 x ) mole fraction y ) mole fraction of water Greek Letters Figure 10. Possible effect of chemical processes such as starch breakdown on the apparent moisture-activity curve in steaming.
were unable to fit an equilibrium isotherm containing more than one adjustable constant and (ii) the reason that their mass-transfer coefficient was inversely proportional to pressure. An issue we have so far avoided addressing is the possible effect that irreversible structural and chemical changes that the grain experiences during cooking might have on the steaming process as we have described it. We could potentially accommodate such as effect within our conceptual framework by considering an isotherm that moves in response to permanent changes, in addition to temperature and pressure. If we denote as X the parameter that represents the extent of chemical processes that irreversibly alter the grain, we would write the activity as a(T,P,md,X). If permanent changes are such that they increase the grain’s capacity to hold water, then the actual moisture-activity path followed during cooking would be steeper than that if chemical changes were arrested, keeping X constant. This situation is depicted in Figure 10. This work has increased our understanding of the importance of heat-transfer processes in wheat grain cooking. Further elucidation of the connection between moisture uptake and changes in flavor and texture will lead to improvements in batch cooking operations in breakfast cereal manufacturing. Acknowledgment The authors gratefully acknowledge the support of the Australian Research Council, through the award of a SPIRT grant. The assistance of Dr. Andrew Stapley (Loughborough University, U.K.), for providing data relating to the experiments carried out by him and coworkers, is also gratefully acknowledged.
λ ) specific enthalpy of vaporization of water, J kg-1 φ ) area-to-volume ratio A/V, m-1 Fd ) dry mass of grain divided by total volume of grain, kg m-3 ∆T ) temperature elevation of grain above surrounding steam, K General Subscripts eq ) equilibrium property sat ) saturation temperature or pressure for water
Literature Cited (1) Fortes, M., Okos, M. R.; Barrett, J. R. Heat and mass transfer analysis of intra kernel wheat drying and rewetting. J. Agris. Eng. Res. 1981, 26, 109. (2) McGuinness, M. J.; Please, C. P.; Fowkes, N.; McGowan, P.; Ryder, L.; Forte, D. Modelling the wetting and cooking of a single cereal grain. IMA J. Math. Appl. Business Ind. 2000, 11, 4. (3) Stapley, A. G. F.; Hyde, T. M.; Gladden, L. F.; Fryer, P. J. NMR imaging of the wheat grain cooking process. Int. J. Food Sci. Technol. 1997, 32, 355. (4) Stapley, A. G. F.; Landman, K. A.; Please, C. P.; Fryer, P. J. Modelling the steaming of whole wheat grains. Chem. Eng. Sci. 1999, 54, 965. (5) Evers, A. D.; Bechtel, D. E. Microscopic structure of the wheat grain. In Wheat: Chemistry and Technology, 3rd ed.; Pomeranz, Y., Ed.; American Association of Cereal Chemists: St. Paul, MN, 1988; Vol. 1, pp 47-95. (6) Stapley, A. G. F.; Fryer, P. J.; Gladden, L. F. Diffusion and reaction in whole wheat grains during boiling. AIChE J. 1998, 44 (8), 1777. (7) Rogers, G. F. C.; Mayhew, Y. R. Thermodynamic and Transport Properties of Fluids, 4th ed.; Blackwell Publishers: Oxford, U.K., 1988. (8) Stapley, A. G. F. Diffusion & Reaction in Wheat Grains. Ph.D. Dissertation, University of Cambridge, Cambridge, U.K., 1995. (9) van den Berg, C. Vapour Sorption Equilibria and Other Water-Starch Interactions. Doctoral Dissertation, Wageningen Agricultural University, Wageningen, The Netherlands, 1981 (cited in van den Berg10 and Stapley8).
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(10) van den Berg, C. Food-water relations: Progress and integration, comments and thoughts. In Water Relationships in Foods; Levine, H., Slade, L.; Plenum Press: New York, 1991. (11) Becker, H. A.; Sallans, H. R. A study of the desorption isotherms of wheat at 25 °C and 50 °C. Cereal Chem. 1956, 33 (2), 79. (12) Ranz, W. E.; Marshall, W. R. Evaporation from drops. Part I. Chem. Eng. Prog. 1952, 48, 141. (13) Ranz, W. E.; Marshall, W. R. Evaporation from drops. Part II. Chem. Eng. Prog. 1952, 48, 173.
(14) Cussler, E. L. Diffusion Mass Transfer in Fluid Systems; Cambridge University Press: Cambridge, U.K., 1997; pp 125129.
Received for review November 4, 2002 Revised manuscript received April 28, 2003 Accepted May 1, 2003 IE0208725
