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high-power microwave heating
Journal of Food Engineering 40 (1999) 81±88
Predicting temperature range in food slabs undergoing short-term/high-power microwave heating Gregory J. Fleischman 1 US Food and Drug Administration, National Center for Food Safety and Technology, 6502 S. Archer Road, Summit Argo, IL 60501, USA Received 6 March 1998; received in revised form 15 September 1998; accepted 7 December 1998
Abstract A closed-form solution to the heat equation, derived for one-dimensional microwave-heated slabs, is presented. The microwave energy distribution used in the heat equation is obtained from MaxwellÕs equations. Therefore, the potential exists of standing waves of microwave energy within the food. This phenomenon contributes to the nonuniform temperature pattern characteristic of microwave heating but is often neglected in theoretical predictions of temperatures. The solution allows temperature to be calculated as a function of thermal, geometric and dielectric properties of the food being heated and is valid for all microwave power levels and durations of heating. Previous work allowed such a solution only if very long durations and very low power levels of microwave heating were used. That the solution is closed-form allows its implementation in spreadsheet format. Simpli®cations to the closedform solution that retain the standing wave phenomenon are shown. Their accuracy, however, is limited to approximately 40 s of heating. Results from the full closed-form solution, using parameters measured in beef, show that the range of temperatures experienced by a microwave-heated beef slab is a sensitive function of slab width and time, with ranges as great as 50°C. It is shown that thermal insulation at the slab faces slightly increases the temperature range. Another feature of the theoretical solution is the extreme sensitivity of temperature range to slab width in the 1±3 cm width interval, with decreasing sensitivity as slab widths increase beyond 4 cm. Sensitivity increases, however, as time increases, regardless of slab width. Lastly, a time-invariant temperature range minimum is seen in the 3±4 cm slab width interval. Ó 1999 Elsevier Science Ltd. All rights reserved.
Notation b Cp E0 h H k Nm P0 r jRj t T jT j Ti Tref DTf z
a slab width (m) heat capacity (J/kg K) impinging electromagnetic ®eld (V/m) surface heat transfer coecient (W/m2 K) Biot number, hb/2k (dimensionless) thermal conductivity (W/m K) normalization integral (dimensionless) volumetric power generation created by E0 (W/m3 ) argument of the re¯ection coecient or phase angle (rad) magnitude of the re¯ection coecient (dimensionless) time (s, or dimensionless) temperature, (°C, or dimensionless) magnitude of the transmission coecient (dimensionless) food initial temperature (°C) reference temperature, taken as ambient temperature (°C) the minimum-to-maximum temperature range in a microwave heated slab at a speci®c time (°C) perpendicular distance from the center of the slab, (m, or dimensionless)
1 Tel.: +1-708-728-4122; fax: +1-708-728-4177; e-mail: Gjf@vm. cfsan.fda.gov
b d 0 0 00 k, km n q w, wm x
attenuation factor (1/2a is the penetration depth) (rad/m) phase factor (rad/m) constant appearing in dimensionless heat equation (see Table 2) (dimensionless) permittivity of free space (8.854 ´ 10ÿ12 F/m) relative dielectric constant (dimensionless) relative dielectric loss (dimensionless) eigenvalue or mth eigenvalue (dimensionless) constant, de®ned in Eq. (3) in the text, obtained from food dielectric parameters (dimensionless) density (kg/m3 ) eigenfunction or mth eigenfunction (dimensionless) radian frequency of microwave radiation, based on 2450 MHz operating frequency (rad/s)
1. Introduction In a previous paper (Fleischman, 1996), a mathematical model was developed to analyze the standing wave phenomenon within microwave-heated food slabs and to determine how it in¯uences temperature distribution for heating situations involving long-term/low-
0260-8774/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 9 9 ) 0 0 0 4 0 - 0
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G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
power microwave heating. A part of that study was the development of a closed-form solution for temperature distribution, thus eliminating the need for sophisticated numerical routines that obtain the temperature distribution directly from the de®ning partial dierential equations. Direct inspection of the closed-form solution made evident the role of the various thermal and dielectric parameters in modifying the temperature distribution. In order to obtain the closed-form solution, however, only steady-state heating was examined. Steady-state heating occurs when the heat generated by microwave energy is balanced with the heat lost from the slab. This was achieved by limiting the analysis to either long-term heating or low power heating. Steadystate temperatures calculated with thermal and dielectric parameters associated with agar gel and beef showed wide variation throughout slabs of diering widths. A major ®nding was the strong sensitivity of the theoretical global minimum-to-maximum temperature range to changes in slab width. Temperature range could increase by 7°C in either agar gel or beef with slight changes of slab width on the order of 0.5 cm. This sensitivity, prominent in slab widths under 9 cm, decreased as slab width increased and became negligible at 10 cm and beyond. These results could have interesting implications for the heating of slabs from safety and quality points-ofview. However, actual applications of microwave heating in the food industry involve transient heating, not steady-state heating. Transient heating is caused by the power levels used in commercial practice, on the order of 1000 W. At these levels, heat transfer by conduction or convection is overwhelmed by the rate of heat generation, resulting in time-dependent temperatures. Food thermal processing with microwaves puts an upper limit on these temperatures, eectively preventing steadystate if food is to survive in recognizable form. It was possible to experimentally obtain steady-state heating in a food-like substance, agar gel, but only with a power level of 40 W continuous. Therefore the term high power is used to distinguish power levels greater than 100 W continuous from lower powers required for steady-state heating. Thus, to determine if these results are possible in transient heating, the long-term/low-power microwave heating requirement had to be removed. This paper describes the derivation of the ®rst closedform solution to the transient heat equation, incorporating an energy distribution obtained from MaxwellÕs equations and valid for any power level or heating duration. Other approaches to the solution of the transient heat equation derived for microwave heating either employ sophisticated numerical approaches to solve the combined heat and MaxwellÕs equations, or solve the heat equation in closed-form by replacing MaxwellÕs equations with LambertÕs law (Buer, 1993). On one hand, numerical approaches, because they deal with the
de®ning dierential equations directly, can obscure results such as those described above. On the other, LambertÕs law, an approximation to MaxwellÕs equations that is valid when microwave power decreases exponentially with distance, has been shown not to be generally applicable (Ayappa, Davis, Crapiste, Davis & Gordon, 1991). Furthermore, it precludes the formation of standing waves that contribute to the nonuniform temperature distribution characteristic of microwave heating and that were responsible for the interesting results of steady-state heating. The assumptions used in this analysis, as with those in the long-term/low-power analysis, are temperatureindependent thermal and dielectric properties, no moisture loss at the slab surfaces and impinging TEM microwaves. This work impacts directly the food service area where slabs of food in pans are routinely heated in a microwave environment. 2. Materials and methods Slab geometry is invoked to restrict heat and microwave energy transfer to one dimension. Coordinates are axed to the slab such that the z axis is parallel to the width, with the zero point at the middle of the slab (see Fig. 1). Microwaves impinge on the slab across the two broad, opposing faces with no thermal or microwave energy transfer across the other faces. Also, the oppos-
Fig. 1. The physical system consists of a slab of food having uniform microwave impingement across the broad, opposing faces. The width of the slab is b. The z-axis lays along the width of the slab with the zero point at the slab center. Ambient and initial temperatures are 25°F.
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
ing, impinging microwaves are assumed to have parallel polarity. Because microwave energy is a vector quantity, the energy at a point within the food is found from the vector addition of the standing wave created by each impinging wave train. The assumption of parallel polarity allows the components to be added algebraically, creating a worst-case analysis by allowing the full eect of the energy in each standing wave to be manifested at each point within the food. The parameters used for calculation of temperature pro®les are those of raw beef. The values are shown in Table 1 along with typical values used for slab width, impinging microwave power and the surface heat transfer coecient. The volumetric heat generation term, to be used in the heat equation, is of the form P z P0 f z;
1
where P0 is a constant, de®ned by 1 P0 x0 E02 ; 2 and f(z) is the dimensionless energy distribution
2
n
2
jRj eÿ2a e2a 2jRj cos 2b ÿ r
:
3
Eq. (3) is obtained from MaxwellÕs equations and follows the development given previously by Fleischman (1996). Note that the symbols for the dielectric parameters and the phase and attenuation factors used in the present paper are changed from those used in previous papers (e.g., Ayappa et al., 1991; Fleischman, 1996). This is to make the notation consistent among other Table 1 Parameter values used for calculations
a
Parameter
Value for beef
Units
b Cp h k P0 r jRj jT j Ti Tref a b 0 00 q
0.025 3200a 10 0.466a 2 ´ 106 ÿ3.09b 0.703b 0.301b 25 25 44.2b 288b 30.5c 9.6c 1030
m J/kg K W/m2 K W/m K W/m3 rad Dimensionless Dimensionless °C °C rad/m rad/m Dimensionless Dimensionless kg/m3
See Heldman and Singh (1981). This parameter is obtained from 0 and 00 (see Ayappa et al., 1991 for details). c See To, Mudgett, Wang, Goldblith and Decareau (1974). b
®elds of study that also employ these parameters. Total consistency, however, necessitated a change of signs, from positive to negative, of the imaginary part of the complex representation of the permittivity constant. As a result, some signs of terms in various equations are also changed. If care is taken, however, comparison of equations with previous papers should not be a problem. The parameters in Eq. (2) are those which are independent of the food being heated whereas Eq. (3) contains all food-dependent parameters. The form of the dimensionless, nonhomogeneous heat equation used is oT t; z o2 T t; z d2 f z: 4 ot oz2 The dimensionless boundary conditions for the heat equation are: oT t; z HT t; ÿ1; oz zÿ1 oT t; z ÿ HT t; 1; oz z1 T 0; z 0:
f z n cosh2az cos 2bz; 200 jT j2
83
5
The three columns of Table 2 show the dimensionless variables and parameters appearing in Eqs. (4) and (5), the grouping of dimensioned variables that each represent, and their typical values or ranges of values. Dimensionless parameters H and d2 arose as a result of nondimensionalization. The value of d2 is generally small, on the order of 0.01, at microwave power levels usually encountered in food processing. This opens the possibility of the use of perturbation analysis in obtaining a simpli®ed solution to Eq. (4). The integral transform method is used to solve Eq. (4). The method extends standard separation of variables to nonhomogeneous partial dierential equations and boundary conditions. The method breaks the heat equation into two equations: one equation is a ®rst-order, nonhomogeneous ordinary dierential equation in Table 2 Dimensionless quantities and their dimensioned equivalents and values Dimensionless variable/parameter
Dimensioned equivalent
Typical range or value (range with dimensions)
H t
hb/2k tP0 /qCp Tref
T
(T ÿ Ti )/Tref
z
2z/b
a b d2
ab/2 bb/2 4k0 Ta /P0 b2
0.27 0±45 (0±30 min) 0±7 (25±100°C) ÿ1 to 1 (ÿb/2 to b/2) 0.551 3.584 0.04
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G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
time that contains the driving term f(z) while the other is a standard eigenvalue problem. The transform and its inversion formula are de®ned as: Z 1 T t; km T t; zw km ; zdz; Transform : ÿ1
Inverse : Z Nm
T t; z
1 X w km ; z m1
1
ÿ1
w km ; z2 dz:
Nm
T t; km ; 6
The overbar on a variable designates a transformed quantity. For brevity, the dependent variables will be written henceforth without their associated independent variables. Thus, T designates T(t,z), T designates T t; km ; f designates f(z), f designates f km , and wm designates w km ; z. Note that Eq. (6) is simply a recasting of an eigenfunction expansion of the function T. The inverse equation of Eq. (6) is the expansion itself, where wm is the mth eigenfunction and T =Nm is the mth coecient of the expansion. If the transform equation of Eq. (6) is divided through by Nm , it becomes the de®ning equation for the mth coecient. The link between the eigenfunctions and T(t,z) occurs when the transform equation in Eq. (6) is applied to all terms of Eqs. (4) and (5) after multiplying through by w(km ,z) and integrating across the problem domain of [ÿ1,1]. All of the terms are directly transformed, as de®ned by the transform equation of Eq. (6), except for the second-order derivative of Eq. (4). The goal of the transform is to replace this derivative with an algebraic equivalent in the transformed equation. The details of this are given by Ozisik (1980). In general, though, the derivative is transformed with the establishment of an eigenvalue problem, the eigenfunctions of which are those to be used in the inverse equation of Eq. (6). The result of the transformation is Z 1 2 o2 T oT wm dz ÿk2m T : 7 2 oz2 ÿ1 oz Thus, Eq. (4) now becomes a nonhomogeneous, ®rstorder ordinary dierential equation for T as a function of time: oT 8 d2 k2m T f : ot The eigenvalue problem used to obtain Eq. (7) is given by: o2 w m k2m wm 0; oz2 owm H wm ; oz zÿ1 owm ÿ H wm : oz z1
9
When solved, the eigenfunctions, and thus the kernel of the transform, are given by wm cos km z:
10
The eigenvalues, km , are the roots of km tan km H : Solving for T from Eq. (8) is straightforward: f 2 2 2 2 T 2 2 1 ÿ eÿd km t T i eÿd km t : d km
11
12
Substituting Eq. (12) into the inversion equation of Eq. (6) yields the solution: ! 1 X f wm ÿd2 k2m t ÿd2 k2m t 1ÿe : 13 T ie T N d2 k2m m1 m To explore the general behavior of the full solution, Eq. (13) is used to calculate temperatures. This was accomplished using the spreadsheet, Excel 97 (Microsoft, Redmond, WA). Two Visual Basic macros were written for Excel to calculate temperature, T, at various points across the slab width and at various times. One macro summed the individual terms of Eq. (13). The summation was stopped when a chosen accuracy level was achieved. Each term of the summation required evaluation of its eigenvalue, km . This was accomplished with a call to the second macro that used a modi®ed version of the Newton±Raphson-based root ®nding routine given by Press, Flannery, Teukolsky and Vetterling (1989). The modi®cation allowed the routine to take advantage of the bracketing of eigenvalues of Eq. (11) in intervals of m ÿ 1p; m ÿ 1=2p for m 1; 2; 3; . . . All of the equations needed to carry out the calculation of T are shown in Appendix A. The results of the calculations were veri®ed as to their correctness by comparison with results from a numerical ®nite element program, PDEase (Macsyma, Arlington, MA). To examine the eect of thermally insulated slab faces, the value of H is set to zero in Eq. (5). The solution for insulated slab faces has the same form and eigenfunctions as Eq. (13) with the exception that Rthe summation includes an additional term, 1 t=2 ÿ1 f z dz Ti , corresponding to the eigenvalue km 0. Also the eigenvalues themselves are of the form km mp. One of the advantages of a closed-form solution is that temperatures may be calculated without the use of specialty software (e.g., ®nite element analysis). However, implementing Eq. (13) in Excel 97 was nevertheless formidable and required an extensive knowledge of the software and some programming ability. Its potential use as a tool for the analysis of actual microwave heating in commercial situations could be increased if a simpler version of the solution were available. The small value of d2 provides a means to such a solution. The
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
terms involving d2 in Eq. (13) were expanded in a Taylor series that, with minor rearrangement, yielded: f 00 z 2 IV f z f VI z 14 d6 t 4 O d8 t5 : d4 t 3 6 24 The term O(d8 t5 ) represents further terms in the series having values smaller than d8 t5 . T ÿ Ti f zt d2 t2
3. Results and discussion Several approaches could have been taken to nondimensionalize the heat equation, leading to Eq. (4). The speci®c nondimensionalization chosen highlighted the two processes in¯uencing the temperature distribution within a microwave heated food±microwave volumetric heat generation and heat conduction. Reference quantities (see Table 2) were chosen such that the derivatives of Eq. (4) were all O(100 ). This nondimensionalization led to a parameter, d2 , that is a ratio of the rate of conductive heat transfer to the rate of volumetric heat generation by microwaves at temperatures near the reference temperature, Tref . The importance of d2 is that for typical values of the parameters it comprises (see Table 1), its value is of O(10ÿ2 ). Thus the second term of Eq. (4), representing conductive heat transfer, is initially small compared with the third term, representing the rate of microwave heat generation. At longer times, however, the increasing temperature gradient within the food increases the rate of heat conduction to the point where it is no longer negligible compared with microwave heat generation. The closed-form nature of Eq. (13) allows functional forms to be derived for short-and long-term behavior. For short heating durations, the time value will be small, reducing Eq. (13) to T tf z Ti :
85
For very long durations of heating the following equation was obtained from Eq. (13): n cosh2a ÿ cosh2az cos 2b ÿ cos 2bz ÿ T 2 2a2 d 2b2 sinh 2a sin 2b : 16 Ha Hb To obtain Eq. (16), the functional form of f(z), given by Eq. (3), was substituted into Eq. (13). Time was allowed to approach in®nity, eliminating the exponential terms. This result was also obtained by Fleischman (1996) by ®rst modifying the de®ning equations of microwave heating (Eqs. (4) and (5)) for long term heating, then solving. In Eq. (16), the parameters of both microwave heating and conductive and convective heat transfer are present in the parameters of H and d2 . What is missing, is time dependency. Thus, at long times, the rate of microwave energy generation within the food is balanced by the rate of heat movement out of the food creating dynamic steady-state behavior. Although Eq. (13) represents the full solution to the heat equation with a source term derived from MaxwellÕs equations, using it to calculate temperatures was not straightforward. A simpler solution was desired. It was obtained by taking advantage of the existence of the small d2 term to produce the asymptotic expansion of Eq. (14). A comparison between the full and simpli®ed solution using parameters in Table 1 is shown in Fig. 2. The curves were obtained for the center point of the slab. It is here that Eq. (14) is the least accurate. The short time approximation can be seen to be valid only during the ®rst 10 s of heating. The fourth-term asymptotic expansion has a longer interval of accuracy, approximately 40 s. However, this is not satisfactory for
15
This resulted from the expansion and subsequent simpli®cation of the exponential terms in Eq. (13), retaining only the ®rst term of each expansion. A similar result was obtained by Dolande and Datta (1993) through the use of order-of-magnitude estimates to simplify the original heat equation. Eq. (15) shows that at short times into microwave heating, the temperature distribution is dominated by the microwave energy distribution, which appears in the form of f(z). Conductive and convective heat transfer in this time interval is overwhelmed by the rate of microwave energy generation within the food. Indeed, no parameters relating to either mode of heat transfer appear in Eq. (15). The equation also shows that temperature is initially a linear function of time while the distribution of temperatures matches the distribution of microwave energy within the slab.
Fig. 2. A plot of the temperature versus time for three solutions of the heat equation: the full solution, Eq. (13); the short duration (small t) approximation, Eq. (15); and the four term asymptotic expansion, Eq. (14). Parameter values used are those in Table 1.
86
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
application to commercial processes with processing times from 3 to 20 min. Adding terms to the asymptotic expansion extends the interval of accuracy. However, adding more terms to the expansion defeats the purpose of simpli®cation by substituting one expansion for another. Henceforth only the full solution will be considered. Eq. (13), the full closed-form solution of Eq. (4), relates the temperature at speci®c locations within the slab to the duration of the heating time and the thermal and dielectric parameters that govern the transmission and distribution of thermal and microwave energy. For the variables and parameters shown in Table 1, Fig. 3 shows sample temperature pro®les calculated with Eq. (13) at 20, 40 and 80 s. Included are curves calculated with Eq. (13) modi®ed for insulated slab faces. In the early stages of heating, the curves corresponding to the insulated slab faces closely match those of the open slab faces. Eventually, however, they begin to diverge, with the temperatures in the slab with insulated faces showing greater variation. It thus appears that insulation increases the temperature nonuniformity, at least initially. One indicator of temperature nonuniformity, the minimum-to-maximum temperature range, was used previously (Fleischman, 1996) to show the relationship between the nonuniformity and slab width. It was found that temperature range was a very sensitive function of slab width for widths less than 9 cm. Slight changes of slab width, on the order of 0.5 cm, increased the pre-
Fig. 3. Temperature as a function of distance from the slab center at various heating times, calculated with Eq. (13) and, for insulation, Eq. (13) modi®ed as described in the text. The horizontal line represents the initial uniform temperature. The broken curves represent open face slabs. The solid curves represent slabs insulated against heat loss. The three sets of curves shown represent, starting at the bottom, 20, 40 and 80 s of heating. Parameter values used are those in Table 1.
dicted temperature range by 7°C in either agar gel or beef. The study also showed that the temperature range was lower with insulation than without insulation for slabs widths below 9 cm. To determine if this sensitivity occurs in the short-term/high-power region examined in this study, the temperature range was calculated for heating durations upto 2 min using Eq. (13). The results are shown for 0.5 and 2 min into heating in the graph in Fig. 4. It is seen that temperature range is a function of both slab width and heating duration. The relationship of the temperature range on slab width is that of damped-sinusoid having greatest amplitude at slab widths of 1±3 cm, with a maximum range of 50°C. The steepness of the slopes of the curves shows that in various regions the temperature range is very sensitive to the slab width. The interval of greatest sensitivity is in the same 1±3 cm thickness range. Sensitivity generally decreases with increasing thicknesses, but increases with time. An unusual feature of the graph is a minimum temperature range in the region between 3 and 4 cm. This minimum is maintained at increasing times, exempli®ed by the two chosen times in the graph. Ignoring for the moment that food is being heated, at times far beyond that seen in food processing, conductive and convective heat transfer begins to compete with the rate of microwave heat generation within the heating substance. This is when temperature range and its sensitivity to slab width decreases to values obtained by Fleischman (1996). Also included in Fig. 4 is the eect of thermal insulation at the slab faces experiencing microwave impingement. It is seen that insulation gives somewhat higher values of the temperature range which is opposite of the result of long-term heating where insulation decreased temperature range. With heat generation dominant at the early moments in microwave heating, insulation prevents convective heat transfer at the slab surfaces that helps to moderate, however slightly, the unevenness of the temperature.
Fig. 4. Temperature range as a function of slab width, at 0.5 min and 2.0 min of heating time, calculated with Eq. (13) and, for insulation, Eq. (13) modi®ed as described in the text. Parameter values used are those in Table 1.
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
4. Conclusions The integral transform technique was used to solve, in closed-form, the heat equation derived for microwave food processing subject to the assumptions of temperature-independent parameters, TEM microwave impingement, slab geometry and no moisture loss. The solution revealed an hitherto unknown phenomenon ± that of signi®cant temperature range sensitivity to slab width. This result has implications for food service operations where pans of food are heated by microwaves. Having slab geometry, food in pans may experience signi®cant changes in temperature uniformity with only slight changes in thickness. Furthermore, an optimal thickness may exist where a pan of food undergoing microwave processing will heat with minimal nonuniformity. Small deviations from this optimum, on the order of a centimeter, could result, however, in very large increases of nonuniformity with temperature ranges within the food increasing by 10°C. The addition of thermal insulating in¯uences, such as a pan cover, would add to the nonuniformity. All of these results are food-dependent, based on the dielectric constant and dielectric loss. Using parameter values measured in raw beef, it was seen that the standing wave phenomenon, the key contributor to these results, is prominent. The standing wave phenomenon diminishes in foods that more easily transform microwave energy to heat. Thus these foods will have less of the nonuniformity associated with microwave heating and the results of the present analysis would apply less to them. Another result of this work is the important role played by closed-form analysis in revealing interesting behaviors. Closed-form analysis produces a solution that is a functional relationship among variables and parameters that may not be readily apparent in the dierential equations whence the solution was derived. The functional relationship may be inspected and manipulated to reveal interesting behaviors. The drawback of closed-form analysis is the possibility of oversimpli®cation of the dierential equations to make a closedform solution tractable, resulting in the elimination, not revelation, of interesting behaviors. Numerical analyses, operating as they do directly on the dierential equations, produce tables of numbers that represent the solution in a nonfunctional form. Thus, any interesting behaviors not evident in the dierential equations could be overlooked. The advantage of numerical analyses, however, is their ability to deal with sophisticated differential equations. It is therefore important to recognize the strength of each approach in the overall analysis of physical situations. In this paper, the integral-transform technique provided a means to investigate in closed-form the problem of heat transfer with generation. Numerical analysis could have dealt with the problem more easily. How-
87
ever, the resulting relationship between temperature nonuniformity and slab width would not have been apparent as it was in the closed-form analysis. Acknowledgements This paper was supported (or partially supported) by a Cooperative Agreement, No. FD-000431, from the US Food and Drug Administration (and the National Center for Food Safety and Technology). Its contents are solely the opinions of the author, and do not necessarily represent the ocial views of the U.S. Food And Drug Administration. Appendix A For microwave heated slabs having temperature-independent thermal and dielectric properties, experiencing an exposure across its broad faces of TEM microwaves of parallel polarity, and having no moisture loss at the faces, the temperature as a function of distance into the slab and duration of heating is given by: ! 1 X f wm ÿd2 k2m t ÿd2 k2m t 1ÿe ; 17 T ie T N d2 k2m m1 m where wm cos km z;
18
km tan km H ;
19
1 sin km cos km 1; km
Nm f
Z
1
ÿ1
Z n
20
f zwm km ; z dz 1 ÿ1
2 cosh2az 2 cos 2bz cos km z dz
4a sinh 2a cos km 2km cosh 2a sin km 4a2 k2m sin 2b ÿ km sin 2b km 2b ÿ km 2b km
n
and Ti
Z
1
ÿ1 Z 1 ÿ1
21
Ti wm km ; z dz Ti cos km z dz:
22
For the analysis presented in this paper, Ti is a constant with respect to z. For the speci®c cases examined, Ti is zero. The value of n in Eq. (21) is calculated via Eq. (3):
88
n
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
200 jT j
2
2
jRj eÿ2a e2a 2jRj cos 2b ÿ r
;
23
The construction of a spreadsheet to calculate T from Eq. (17) takes several steps. To start with, all of the dimensionless parameters, a, b, jT j, r, jRj, n, d, and H, and the reference quantities for the variables T, z, and t, are calculated from fundamental parameters 0 , 00 , b, E0 , Cp , k, h, q. Parameters a and b are obtained from 0 and 00 through the complex relationship: x p 24 0 ÿ i00 ; c p where i ÿ1 and c is the speed of light. Microsoft Excel can deftly handle this relationship through its built-in complex operators. However, a and b can be obtained algebraically by squaring both sides of Eq. (23), expanding the resulting binomial on the left-hand side, equating real and imaginary components and solving simultaneously for a and b. Values for jRj, r and jT j are obtained by: 1 ÿ p 0 ÿ i00 p ; 25 jRj 1 0 ÿ i00 a ib i
1 ÿ p 0 ÿ i00 p ; r arg R arg 1 0 ÿ i00
26
2 p : jT j 0 00 1 ÿ i
27
Here, again, ExcelÕs built-in complex operators are very useful. Otherwise, by alternating the polar with cartesian forms of the complex quantities on the right-hand sides of Eqs. (24)±(26) the equations may be reduced to obtain a simpli®ed form whence the values in question can be easily calculated. Note, however, that calculating the argument of a complex number is not just a simple division of the imaginary by the real components. The sign of each component must separately be taken into account to correctly place the argument in the correct quadrant of the complex plane. Finally, the quantity P0 , used to establish the reference quantities in Table 2, is calculated via Eq. (2): 1 P0 x0 E02 ; 2
28
Usually, however, P0 is more easily inferred than E0 is measured. Thus P0 is usually an input or a ®tted parameter, and Eq. (28) is used to calculate E0 . The one-dimensional nature of the solution allows the remainder of the spreadsheet to be constructed with a column having values of dimensionless distance into the slab and a row having the dimensionless heating times. Each cell that corresponds to a particular distance and time contains a call to a macro that calculates the dimensionless temperature via Eq. (17). The macro sums terms in the in®nite series until a relative accuracy requirement, set by the user, is met. Each term of the sum requires the evaluation of an eigenvalue per Eq. (19). Consecutive eigenvalues are the roots of Eq. (19) in the consecutive intervals where the function H =k intersects the periodic tan k. Any suitable root-®nding algorithm, such as Newton±Raphson, can be used. Bracketing of the root is important to reduce computation time and can be easily accomplished by recognizing that the function H =k is positive for k > 0 and will intersect each curve of the tangent function in the interval where it is positive as well, i.e., p m ÿ 1 < km < p m ÿ 1=2 for m 1; 2; 3; . . . Once temperature is known, the reference quantities in Table 2 can then be used to calculate the dimensional values of the temperature, distance and time. References Ayappa, K. G., Davis, H. T., Crapiste, G., Davis, E. A., & Gordon, J. (1991). Microwave heating: An evaluation of power formulations. Chemical Engineering Science, 46, 1005±1016. Buer, C. R. (1993). Microwave Cooking and Processing: Engineering Fundamentals for the Food Scientist, New York: Van Nostrand Reinhold. Dolande, J., & Datta, A. (1993). Temperature pro®les in microwave heating of solids: A systematic study. Journal of Microwave Power and Electromagnetic Energy, 28, 58±67. Fleischman, G. J. (1996). Predicting temperature range in food slabs undergoing long-term/low-power microwave heating. Journal of Food Engineering, 27, 337±351. Heldman, D. R., & Singh, R. P. (1981). Food Process Engineering, Westport, Connecticut: AVI Publishing. Ozisik, M. N. (1980). Heat Conduction, New York: Wiley. Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. (1989). Numerical Recipes: The Art of Scienti®c Computing (FORTRAN Version), New York: Cambridge University Press. To, E. C., Mudgett, R. E., Wang, D. I. C., Goldblith, S. A., & Decareau, R. V. (1974). Dielectric properties of food materials. Journal of Microwave Power, 9, 303±315.
Predicting temperature range in food slabs undergoing short-term/high-power microwave heating Gregory J. Fleischman 1 US Food and Drug Administration, National Center for Food Safety and Technology, 6502 S. Archer Road, Summit Argo, IL 60501, USA Received 6 March 1998; received in revised form 15 September 1998; accepted 7 December 1998
Abstract A closed-form solution to the heat equation, derived for one-dimensional microwave-heated slabs, is presented. The microwave energy distribution used in the heat equation is obtained from MaxwellÕs equations. Therefore, the potential exists of standing waves of microwave energy within the food. This phenomenon contributes to the nonuniform temperature pattern characteristic of microwave heating but is often neglected in theoretical predictions of temperatures. The solution allows temperature to be calculated as a function of thermal, geometric and dielectric properties of the food being heated and is valid for all microwave power levels and durations of heating. Previous work allowed such a solution only if very long durations and very low power levels of microwave heating were used. That the solution is closed-form allows its implementation in spreadsheet format. Simpli®cations to the closedform solution that retain the standing wave phenomenon are shown. Their accuracy, however, is limited to approximately 40 s of heating. Results from the full closed-form solution, using parameters measured in beef, show that the range of temperatures experienced by a microwave-heated beef slab is a sensitive function of slab width and time, with ranges as great as 50°C. It is shown that thermal insulation at the slab faces slightly increases the temperature range. Another feature of the theoretical solution is the extreme sensitivity of temperature range to slab width in the 1±3 cm width interval, with decreasing sensitivity as slab widths increase beyond 4 cm. Sensitivity increases, however, as time increases, regardless of slab width. Lastly, a time-invariant temperature range minimum is seen in the 3±4 cm slab width interval. Ó 1999 Elsevier Science Ltd. All rights reserved.
Notation b Cp E0 h H k Nm P0 r jRj t T jT j Ti Tref DTf z
a slab width (m) heat capacity (J/kg K) impinging electromagnetic ®eld (V/m) surface heat transfer coecient (W/m2 K) Biot number, hb/2k (dimensionless) thermal conductivity (W/m K) normalization integral (dimensionless) volumetric power generation created by E0 (W/m3 ) argument of the re¯ection coecient or phase angle (rad) magnitude of the re¯ection coecient (dimensionless) time (s, or dimensionless) temperature, (°C, or dimensionless) magnitude of the transmission coecient (dimensionless) food initial temperature (°C) reference temperature, taken as ambient temperature (°C) the minimum-to-maximum temperature range in a microwave heated slab at a speci®c time (°C) perpendicular distance from the center of the slab, (m, or dimensionless)
1 Tel.: +1-708-728-4122; fax: +1-708-728-4177; e-mail: Gjf@vm. cfsan.fda.gov
b d 0 0 00 k, km n q w, wm x
attenuation factor (1/2a is the penetration depth) (rad/m) phase factor (rad/m) constant appearing in dimensionless heat equation (see Table 2) (dimensionless) permittivity of free space (8.854 ´ 10ÿ12 F/m) relative dielectric constant (dimensionless) relative dielectric loss (dimensionless) eigenvalue or mth eigenvalue (dimensionless) constant, de®ned in Eq. (3) in the text, obtained from food dielectric parameters (dimensionless) density (kg/m3 ) eigenfunction or mth eigenfunction (dimensionless) radian frequency of microwave radiation, based on 2450 MHz operating frequency (rad/s)
1. Introduction In a previous paper (Fleischman, 1996), a mathematical model was developed to analyze the standing wave phenomenon within microwave-heated food slabs and to determine how it in¯uences temperature distribution for heating situations involving long-term/low-
0260-8774/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 9 9 ) 0 0 0 4 0 - 0
82
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
power microwave heating. A part of that study was the development of a closed-form solution for temperature distribution, thus eliminating the need for sophisticated numerical routines that obtain the temperature distribution directly from the de®ning partial dierential equations. Direct inspection of the closed-form solution made evident the role of the various thermal and dielectric parameters in modifying the temperature distribution. In order to obtain the closed-form solution, however, only steady-state heating was examined. Steady-state heating occurs when the heat generated by microwave energy is balanced with the heat lost from the slab. This was achieved by limiting the analysis to either long-term heating or low power heating. Steadystate temperatures calculated with thermal and dielectric parameters associated with agar gel and beef showed wide variation throughout slabs of diering widths. A major ®nding was the strong sensitivity of the theoretical global minimum-to-maximum temperature range to changes in slab width. Temperature range could increase by 7°C in either agar gel or beef with slight changes of slab width on the order of 0.5 cm. This sensitivity, prominent in slab widths under 9 cm, decreased as slab width increased and became negligible at 10 cm and beyond. These results could have interesting implications for the heating of slabs from safety and quality points-ofview. However, actual applications of microwave heating in the food industry involve transient heating, not steady-state heating. Transient heating is caused by the power levels used in commercial practice, on the order of 1000 W. At these levels, heat transfer by conduction or convection is overwhelmed by the rate of heat generation, resulting in time-dependent temperatures. Food thermal processing with microwaves puts an upper limit on these temperatures, eectively preventing steadystate if food is to survive in recognizable form. It was possible to experimentally obtain steady-state heating in a food-like substance, agar gel, but only with a power level of 40 W continuous. Therefore the term high power is used to distinguish power levels greater than 100 W continuous from lower powers required for steady-state heating. Thus, to determine if these results are possible in transient heating, the long-term/low-power microwave heating requirement had to be removed. This paper describes the derivation of the ®rst closedform solution to the transient heat equation, incorporating an energy distribution obtained from MaxwellÕs equations and valid for any power level or heating duration. Other approaches to the solution of the transient heat equation derived for microwave heating either employ sophisticated numerical approaches to solve the combined heat and MaxwellÕs equations, or solve the heat equation in closed-form by replacing MaxwellÕs equations with LambertÕs law (Buer, 1993). On one hand, numerical approaches, because they deal with the
de®ning dierential equations directly, can obscure results such as those described above. On the other, LambertÕs law, an approximation to MaxwellÕs equations that is valid when microwave power decreases exponentially with distance, has been shown not to be generally applicable (Ayappa, Davis, Crapiste, Davis & Gordon, 1991). Furthermore, it precludes the formation of standing waves that contribute to the nonuniform temperature distribution characteristic of microwave heating and that were responsible for the interesting results of steady-state heating. The assumptions used in this analysis, as with those in the long-term/low-power analysis, are temperatureindependent thermal and dielectric properties, no moisture loss at the slab surfaces and impinging TEM microwaves. This work impacts directly the food service area where slabs of food in pans are routinely heated in a microwave environment. 2. Materials and methods Slab geometry is invoked to restrict heat and microwave energy transfer to one dimension. Coordinates are axed to the slab such that the z axis is parallel to the width, with the zero point at the middle of the slab (see Fig. 1). Microwaves impinge on the slab across the two broad, opposing faces with no thermal or microwave energy transfer across the other faces. Also, the oppos-
Fig. 1. The physical system consists of a slab of food having uniform microwave impingement across the broad, opposing faces. The width of the slab is b. The z-axis lays along the width of the slab with the zero point at the slab center. Ambient and initial temperatures are 25°F.
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
ing, impinging microwaves are assumed to have parallel polarity. Because microwave energy is a vector quantity, the energy at a point within the food is found from the vector addition of the standing wave created by each impinging wave train. The assumption of parallel polarity allows the components to be added algebraically, creating a worst-case analysis by allowing the full eect of the energy in each standing wave to be manifested at each point within the food. The parameters used for calculation of temperature pro®les are those of raw beef. The values are shown in Table 1 along with typical values used for slab width, impinging microwave power and the surface heat transfer coecient. The volumetric heat generation term, to be used in the heat equation, is of the form P z P0 f z;
1
where P0 is a constant, de®ned by 1 P0 x0 E02 ; 2 and f(z) is the dimensionless energy distribution
2
n
2
jRj eÿ2a e2a 2jRj cos 2b ÿ r
:
3
Eq. (3) is obtained from MaxwellÕs equations and follows the development given previously by Fleischman (1996). Note that the symbols for the dielectric parameters and the phase and attenuation factors used in the present paper are changed from those used in previous papers (e.g., Ayappa et al., 1991; Fleischman, 1996). This is to make the notation consistent among other Table 1 Parameter values used for calculations
a
Parameter
Value for beef
Units
b Cp h k P0 r jRj jT j Ti Tref a b 0 00 q
0.025 3200a 10 0.466a 2 ´ 106 ÿ3.09b 0.703b 0.301b 25 25 44.2b 288b 30.5c 9.6c 1030
m J/kg K W/m2 K W/m K W/m3 rad Dimensionless Dimensionless °C °C rad/m rad/m Dimensionless Dimensionless kg/m3
See Heldman and Singh (1981). This parameter is obtained from 0 and 00 (see Ayappa et al., 1991 for details). c See To, Mudgett, Wang, Goldblith and Decareau (1974). b
®elds of study that also employ these parameters. Total consistency, however, necessitated a change of signs, from positive to negative, of the imaginary part of the complex representation of the permittivity constant. As a result, some signs of terms in various equations are also changed. If care is taken, however, comparison of equations with previous papers should not be a problem. The parameters in Eq. (2) are those which are independent of the food being heated whereas Eq. (3) contains all food-dependent parameters. The form of the dimensionless, nonhomogeneous heat equation used is oT t; z o2 T t; z d2 f z: 4 ot oz2 The dimensionless boundary conditions for the heat equation are: oT t; z HT t; ÿ1; oz zÿ1 oT t; z ÿ HT t; 1; oz z1 T 0; z 0:
f z n cosh2az cos 2bz; 200 jT j2
83
5
The three columns of Table 2 show the dimensionless variables and parameters appearing in Eqs. (4) and (5), the grouping of dimensioned variables that each represent, and their typical values or ranges of values. Dimensionless parameters H and d2 arose as a result of nondimensionalization. The value of d2 is generally small, on the order of 0.01, at microwave power levels usually encountered in food processing. This opens the possibility of the use of perturbation analysis in obtaining a simpli®ed solution to Eq. (4). The integral transform method is used to solve Eq. (4). The method extends standard separation of variables to nonhomogeneous partial dierential equations and boundary conditions. The method breaks the heat equation into two equations: one equation is a ®rst-order, nonhomogeneous ordinary dierential equation in Table 2 Dimensionless quantities and their dimensioned equivalents and values Dimensionless variable/parameter
Dimensioned equivalent
Typical range or value (range with dimensions)
H t
hb/2k tP0 /qCp Tref
T
(T ÿ Ti )/Tref
z
2z/b
a b d2
ab/2 bb/2 4k0 Ta /P0 b2
0.27 0±45 (0±30 min) 0±7 (25±100°C) ÿ1 to 1 (ÿb/2 to b/2) 0.551 3.584 0.04
84
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
time that contains the driving term f(z) while the other is a standard eigenvalue problem. The transform and its inversion formula are de®ned as: Z 1 T t; km T t; zw km ; zdz; Transform : ÿ1
Inverse : Z Nm
T t; z
1 X w km ; z m1
1
ÿ1
w km ; z2 dz:
Nm
T t; km ; 6
The overbar on a variable designates a transformed quantity. For brevity, the dependent variables will be written henceforth without their associated independent variables. Thus, T designates T(t,z), T designates T t; km ; f designates f(z), f designates f km , and wm designates w km ; z. Note that Eq. (6) is simply a recasting of an eigenfunction expansion of the function T. The inverse equation of Eq. (6) is the expansion itself, where wm is the mth eigenfunction and T =Nm is the mth coecient of the expansion. If the transform equation of Eq. (6) is divided through by Nm , it becomes the de®ning equation for the mth coecient. The link between the eigenfunctions and T(t,z) occurs when the transform equation in Eq. (6) is applied to all terms of Eqs. (4) and (5) after multiplying through by w(km ,z) and integrating across the problem domain of [ÿ1,1]. All of the terms are directly transformed, as de®ned by the transform equation of Eq. (6), except for the second-order derivative of Eq. (4). The goal of the transform is to replace this derivative with an algebraic equivalent in the transformed equation. The details of this are given by Ozisik (1980). In general, though, the derivative is transformed with the establishment of an eigenvalue problem, the eigenfunctions of which are those to be used in the inverse equation of Eq. (6). The result of the transformation is Z 1 2 o2 T oT wm dz ÿk2m T : 7 2 oz2 ÿ1 oz Thus, Eq. (4) now becomes a nonhomogeneous, ®rstorder ordinary dierential equation for T as a function of time: oT 8 d2 k2m T f : ot The eigenvalue problem used to obtain Eq. (7) is given by: o2 w m k2m wm 0; oz2 owm H wm ; oz zÿ1 owm ÿ H wm : oz z1
9
When solved, the eigenfunctions, and thus the kernel of the transform, are given by wm cos km z:
10
The eigenvalues, km , are the roots of km tan km H : Solving for T from Eq. (8) is straightforward: f 2 2 2 2 T 2 2 1 ÿ eÿd km t T i eÿd km t : d km
11
12
Substituting Eq. (12) into the inversion equation of Eq. (6) yields the solution: ! 1 X f wm ÿd2 k2m t ÿd2 k2m t 1ÿe : 13 T ie T N d2 k2m m1 m To explore the general behavior of the full solution, Eq. (13) is used to calculate temperatures. This was accomplished using the spreadsheet, Excel 97 (Microsoft, Redmond, WA). Two Visual Basic macros were written for Excel to calculate temperature, T, at various points across the slab width and at various times. One macro summed the individual terms of Eq. (13). The summation was stopped when a chosen accuracy level was achieved. Each term of the summation required evaluation of its eigenvalue, km . This was accomplished with a call to the second macro that used a modi®ed version of the Newton±Raphson-based root ®nding routine given by Press, Flannery, Teukolsky and Vetterling (1989). The modi®cation allowed the routine to take advantage of the bracketing of eigenvalues of Eq. (11) in intervals of m ÿ 1p; m ÿ 1=2p for m 1; 2; 3; . . . All of the equations needed to carry out the calculation of T are shown in Appendix A. The results of the calculations were veri®ed as to their correctness by comparison with results from a numerical ®nite element program, PDEase (Macsyma, Arlington, MA). To examine the eect of thermally insulated slab faces, the value of H is set to zero in Eq. (5). The solution for insulated slab faces has the same form and eigenfunctions as Eq. (13) with the exception that Rthe summation includes an additional term, 1 t=2 ÿ1 f z dz Ti , corresponding to the eigenvalue km 0. Also the eigenvalues themselves are of the form km mp. One of the advantages of a closed-form solution is that temperatures may be calculated without the use of specialty software (e.g., ®nite element analysis). However, implementing Eq. (13) in Excel 97 was nevertheless formidable and required an extensive knowledge of the software and some programming ability. Its potential use as a tool for the analysis of actual microwave heating in commercial situations could be increased if a simpler version of the solution were available. The small value of d2 provides a means to such a solution. The
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
terms involving d2 in Eq. (13) were expanded in a Taylor series that, with minor rearrangement, yielded: f 00 z 2 IV f z f VI z 14 d6 t 4 O d8 t5 : d4 t 3 6 24 The term O(d8 t5 ) represents further terms in the series having values smaller than d8 t5 . T ÿ Ti f zt d2 t2
3. Results and discussion Several approaches could have been taken to nondimensionalize the heat equation, leading to Eq. (4). The speci®c nondimensionalization chosen highlighted the two processes in¯uencing the temperature distribution within a microwave heated food±microwave volumetric heat generation and heat conduction. Reference quantities (see Table 2) were chosen such that the derivatives of Eq. (4) were all O(100 ). This nondimensionalization led to a parameter, d2 , that is a ratio of the rate of conductive heat transfer to the rate of volumetric heat generation by microwaves at temperatures near the reference temperature, Tref . The importance of d2 is that for typical values of the parameters it comprises (see Table 1), its value is of O(10ÿ2 ). Thus the second term of Eq. (4), representing conductive heat transfer, is initially small compared with the third term, representing the rate of microwave heat generation. At longer times, however, the increasing temperature gradient within the food increases the rate of heat conduction to the point where it is no longer negligible compared with microwave heat generation. The closed-form nature of Eq. (13) allows functional forms to be derived for short-and long-term behavior. For short heating durations, the time value will be small, reducing Eq. (13) to T tf z Ti :
85
For very long durations of heating the following equation was obtained from Eq. (13): n cosh2a ÿ cosh2az cos 2b ÿ cos 2bz ÿ T 2 2a2 d 2b2 sinh 2a sin 2b : 16 Ha Hb To obtain Eq. (16), the functional form of f(z), given by Eq. (3), was substituted into Eq. (13). Time was allowed to approach in®nity, eliminating the exponential terms. This result was also obtained by Fleischman (1996) by ®rst modifying the de®ning equations of microwave heating (Eqs. (4) and (5)) for long term heating, then solving. In Eq. (16), the parameters of both microwave heating and conductive and convective heat transfer are present in the parameters of H and d2 . What is missing, is time dependency. Thus, at long times, the rate of microwave energy generation within the food is balanced by the rate of heat movement out of the food creating dynamic steady-state behavior. Although Eq. (13) represents the full solution to the heat equation with a source term derived from MaxwellÕs equations, using it to calculate temperatures was not straightforward. A simpler solution was desired. It was obtained by taking advantage of the existence of the small d2 term to produce the asymptotic expansion of Eq. (14). A comparison between the full and simpli®ed solution using parameters in Table 1 is shown in Fig. 2. The curves were obtained for the center point of the slab. It is here that Eq. (14) is the least accurate. The short time approximation can be seen to be valid only during the ®rst 10 s of heating. The fourth-term asymptotic expansion has a longer interval of accuracy, approximately 40 s. However, this is not satisfactory for
15
This resulted from the expansion and subsequent simpli®cation of the exponential terms in Eq. (13), retaining only the ®rst term of each expansion. A similar result was obtained by Dolande and Datta (1993) through the use of order-of-magnitude estimates to simplify the original heat equation. Eq. (15) shows that at short times into microwave heating, the temperature distribution is dominated by the microwave energy distribution, which appears in the form of f(z). Conductive and convective heat transfer in this time interval is overwhelmed by the rate of microwave energy generation within the food. Indeed, no parameters relating to either mode of heat transfer appear in Eq. (15). The equation also shows that temperature is initially a linear function of time while the distribution of temperatures matches the distribution of microwave energy within the slab.
Fig. 2. A plot of the temperature versus time for three solutions of the heat equation: the full solution, Eq. (13); the short duration (small t) approximation, Eq. (15); and the four term asymptotic expansion, Eq. (14). Parameter values used are those in Table 1.
86
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
application to commercial processes with processing times from 3 to 20 min. Adding terms to the asymptotic expansion extends the interval of accuracy. However, adding more terms to the expansion defeats the purpose of simpli®cation by substituting one expansion for another. Henceforth only the full solution will be considered. Eq. (13), the full closed-form solution of Eq. (4), relates the temperature at speci®c locations within the slab to the duration of the heating time and the thermal and dielectric parameters that govern the transmission and distribution of thermal and microwave energy. For the variables and parameters shown in Table 1, Fig. 3 shows sample temperature pro®les calculated with Eq. (13) at 20, 40 and 80 s. Included are curves calculated with Eq. (13) modi®ed for insulated slab faces. In the early stages of heating, the curves corresponding to the insulated slab faces closely match those of the open slab faces. Eventually, however, they begin to diverge, with the temperatures in the slab with insulated faces showing greater variation. It thus appears that insulation increases the temperature nonuniformity, at least initially. One indicator of temperature nonuniformity, the minimum-to-maximum temperature range, was used previously (Fleischman, 1996) to show the relationship between the nonuniformity and slab width. It was found that temperature range was a very sensitive function of slab width for widths less than 9 cm. Slight changes of slab width, on the order of 0.5 cm, increased the pre-
Fig. 3. Temperature as a function of distance from the slab center at various heating times, calculated with Eq. (13) and, for insulation, Eq. (13) modi®ed as described in the text. The horizontal line represents the initial uniform temperature. The broken curves represent open face slabs. The solid curves represent slabs insulated against heat loss. The three sets of curves shown represent, starting at the bottom, 20, 40 and 80 s of heating. Parameter values used are those in Table 1.
dicted temperature range by 7°C in either agar gel or beef. The study also showed that the temperature range was lower with insulation than without insulation for slabs widths below 9 cm. To determine if this sensitivity occurs in the short-term/high-power region examined in this study, the temperature range was calculated for heating durations upto 2 min using Eq. (13). The results are shown for 0.5 and 2 min into heating in the graph in Fig. 4. It is seen that temperature range is a function of both slab width and heating duration. The relationship of the temperature range on slab width is that of damped-sinusoid having greatest amplitude at slab widths of 1±3 cm, with a maximum range of 50°C. The steepness of the slopes of the curves shows that in various regions the temperature range is very sensitive to the slab width. The interval of greatest sensitivity is in the same 1±3 cm thickness range. Sensitivity generally decreases with increasing thicknesses, but increases with time. An unusual feature of the graph is a minimum temperature range in the region between 3 and 4 cm. This minimum is maintained at increasing times, exempli®ed by the two chosen times in the graph. Ignoring for the moment that food is being heated, at times far beyond that seen in food processing, conductive and convective heat transfer begins to compete with the rate of microwave heat generation within the heating substance. This is when temperature range and its sensitivity to slab width decreases to values obtained by Fleischman (1996). Also included in Fig. 4 is the eect of thermal insulation at the slab faces experiencing microwave impingement. It is seen that insulation gives somewhat higher values of the temperature range which is opposite of the result of long-term heating where insulation decreased temperature range. With heat generation dominant at the early moments in microwave heating, insulation prevents convective heat transfer at the slab surfaces that helps to moderate, however slightly, the unevenness of the temperature.
Fig. 4. Temperature range as a function of slab width, at 0.5 min and 2.0 min of heating time, calculated with Eq. (13) and, for insulation, Eq. (13) modi®ed as described in the text. Parameter values used are those in Table 1.
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
4. Conclusions The integral transform technique was used to solve, in closed-form, the heat equation derived for microwave food processing subject to the assumptions of temperature-independent parameters, TEM microwave impingement, slab geometry and no moisture loss. The solution revealed an hitherto unknown phenomenon ± that of signi®cant temperature range sensitivity to slab width. This result has implications for food service operations where pans of food are heated by microwaves. Having slab geometry, food in pans may experience signi®cant changes in temperature uniformity with only slight changes in thickness. Furthermore, an optimal thickness may exist where a pan of food undergoing microwave processing will heat with minimal nonuniformity. Small deviations from this optimum, on the order of a centimeter, could result, however, in very large increases of nonuniformity with temperature ranges within the food increasing by 10°C. The addition of thermal insulating in¯uences, such as a pan cover, would add to the nonuniformity. All of these results are food-dependent, based on the dielectric constant and dielectric loss. Using parameter values measured in raw beef, it was seen that the standing wave phenomenon, the key contributor to these results, is prominent. The standing wave phenomenon diminishes in foods that more easily transform microwave energy to heat. Thus these foods will have less of the nonuniformity associated with microwave heating and the results of the present analysis would apply less to them. Another result of this work is the important role played by closed-form analysis in revealing interesting behaviors. Closed-form analysis produces a solution that is a functional relationship among variables and parameters that may not be readily apparent in the dierential equations whence the solution was derived. The functional relationship may be inspected and manipulated to reveal interesting behaviors. The drawback of closed-form analysis is the possibility of oversimpli®cation of the dierential equations to make a closedform solution tractable, resulting in the elimination, not revelation, of interesting behaviors. Numerical analyses, operating as they do directly on the dierential equations, produce tables of numbers that represent the solution in a nonfunctional form. Thus, any interesting behaviors not evident in the dierential equations could be overlooked. The advantage of numerical analyses, however, is their ability to deal with sophisticated differential equations. It is therefore important to recognize the strength of each approach in the overall analysis of physical situations. In this paper, the integral-transform technique provided a means to investigate in closed-form the problem of heat transfer with generation. Numerical analysis could have dealt with the problem more easily. How-
87
ever, the resulting relationship between temperature nonuniformity and slab width would not have been apparent as it was in the closed-form analysis. Acknowledgements This paper was supported (or partially supported) by a Cooperative Agreement, No. FD-000431, from the US Food and Drug Administration (and the National Center for Food Safety and Technology). Its contents are solely the opinions of the author, and do not necessarily represent the ocial views of the U.S. Food And Drug Administration. Appendix A For microwave heated slabs having temperature-independent thermal and dielectric properties, experiencing an exposure across its broad faces of TEM microwaves of parallel polarity, and having no moisture loss at the faces, the temperature as a function of distance into the slab and duration of heating is given by: ! 1 X f wm ÿd2 k2m t ÿd2 k2m t 1ÿe ; 17 T ie T N d2 k2m m1 m where wm cos km z;
18
km tan km H ;
19
1 sin km cos km 1; km
Nm f
Z
1
ÿ1
Z n
20
f zwm km ; z dz 1 ÿ1
2 cosh2az 2 cos 2bz cos km z dz
4a sinh 2a cos km 2km cosh 2a sin km 4a2 k2m sin 2b ÿ km sin 2b km 2b ÿ km 2b km
n
and Ti
Z
1
ÿ1 Z 1 ÿ1
21
Ti wm km ; z dz Ti cos km z dz:
22
For the analysis presented in this paper, Ti is a constant with respect to z. For the speci®c cases examined, Ti is zero. The value of n in Eq. (21) is calculated via Eq. (3):
88
n
G.J. Fleischman / Journal of Food Engineering 40 (1999) 81±88
200 jT j
2
2
jRj eÿ2a e2a 2jRj cos 2b ÿ r
;
23
The construction of a spreadsheet to calculate T from Eq. (17) takes several steps. To start with, all of the dimensionless parameters, a, b, jT j, r, jRj, n, d, and H, and the reference quantities for the variables T, z, and t, are calculated from fundamental parameters 0 , 00 , b, E0 , Cp , k, h, q. Parameters a and b are obtained from 0 and 00 through the complex relationship: x p 24 0 ÿ i00 ; c p where i ÿ1 and c is the speed of light. Microsoft Excel can deftly handle this relationship through its built-in complex operators. However, a and b can be obtained algebraically by squaring both sides of Eq. (23), expanding the resulting binomial on the left-hand side, equating real and imaginary components and solving simultaneously for a and b. Values for jRj, r and jT j are obtained by: 1 ÿ p 0 ÿ i00 p ; 25 jRj 1 0 ÿ i00 a ib i
1 ÿ p 0 ÿ i00 p ; r arg R arg 1 0 ÿ i00
26
2 p : jT j 0 00 1 ÿ i
27
Here, again, ExcelÕs built-in complex operators are very useful. Otherwise, by alternating the polar with cartesian forms of the complex quantities on the right-hand sides of Eqs. (24)±(26) the equations may be reduced to obtain a simpli®ed form whence the values in question can be easily calculated. Note, however, that calculating the argument of a complex number is not just a simple division of the imaginary by the real components. The sign of each component must separately be taken into account to correctly place the argument in the correct quadrant of the complex plane. Finally, the quantity P0 , used to establish the reference quantities in Table 2, is calculated via Eq. (2): 1 P0 x0 E02 ; 2
28
Usually, however, P0 is more easily inferred than E0 is measured. Thus P0 is usually an input or a ®tted parameter, and Eq. (28) is used to calculate E0 . The one-dimensional nature of the solution allows the remainder of the spreadsheet to be constructed with a column having values of dimensionless distance into the slab and a row having the dimensionless heating times. Each cell that corresponds to a particular distance and time contains a call to a macro that calculates the dimensionless temperature via Eq. (17). The macro sums terms in the in®nite series until a relative accuracy requirement, set by the user, is met. Each term of the sum requires the evaluation of an eigenvalue per Eq. (19). Consecutive eigenvalues are the roots of Eq. (19) in the consecutive intervals where the function H =k intersects the periodic tan k. Any suitable root-®nding algorithm, such as Newton±Raphson, can be used. Bracketing of the root is important to reduce computation time and can be easily accomplished by recognizing that the function H =k is positive for k > 0 and will intersect each curve of the tangent function in the interval where it is positive as well, i.e., p m ÿ 1 < km < p m ÿ 1=2 for m 1; 2; 3; . . . Once temperature is known, the reference quantities in Table 2 can then be used to calculate the dimensional values of the temperature, distance and time. References Ayappa, K. G., Davis, H. T., Crapiste, G., Davis, E. A., & Gordon, J. (1991). Microwave heating: An evaluation of power formulations. Chemical Engineering Science, 46, 1005±1016. Buer, C. R. (1993). Microwave Cooking and Processing: Engineering Fundamentals for the Food Scientist, New York: Van Nostrand Reinhold. Dolande, J., & Datta, A. (1993). Temperature pro®les in microwave heating of solids: A systematic study. Journal of Microwave Power and Electromagnetic Energy, 28, 58±67. Fleischman, G. J. (1996). Predicting temperature range in food slabs undergoing long-term/low-power microwave heating. Journal of Food Engineering, 27, 337±351. Heldman, D. R., & Singh, R. P. (1981). Food Process Engineering, Westport, Connecticut: AVI Publishing. Ozisik, M. N. (1980). Heat Conduction, New York: Wiley. Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. (1989). Numerical Recipes: The Art of Scienti®c Computing (FORTRAN Version), New York: Cambridge University Press. To, E. C., Mudgett, R. E., Wang, D. I. C., Goldblith, S. A., & Decareau, R. V. (1974). Dielectric properties of food materials. Journal of Microwave Power, 9, 303±315.