Biotechnol. Prog. 1991, 7, 540-543
540
Model for Transient Moisture Profiles of a Drying Apple Slab Using the Data Obtained with Magnetic Resonance Imaging Michael J. McCarthy' and Ernest Perez Departments of Food Science & Technology and of Agricultural Engineering, University of California, Davis, California 95616-8598
Mustafa Ozilgen Department of Food Engineering, Middle East Technical University, 06531 Ankara, Turkey
T h e use of magnetic resonance imaging (MRI) spectroscopy as a real time probe t o noninvasively obtain transient moisture profiles during food dehydration is demonstrated. A laboratory-scale wind tunnel drier was constructed. A 2 X 5 X 5 cm3 apple slab was inserted in the probative zone and one-dimensional drying was effected a t quasiisothermal conditions. Spatial resolution of 0.60 mm was obtained. T h e transient moisture profiles were simulated with a simple mass transfer model. Effective diffucm2/s. sivity of water in the apple was calculated as 8.3 X
Introduction Knowledge of a food's moisture-temperature distribution as a function of time during drying is essential to predict the extent of deterioration in quality factors such as flavor, color and vitamins. A great deal of literature has been published on mathematical modelling of simultaneous heat and mass transfer in foodstuffs to meet this objective. The need to support theoretical drying models with experimental data has been well documented ( 1 , 2 ) . The definitive test of the theoretical heat and mass transfer models is the comparison of computer-generated temperature and moisture profiles based on the theoretical models with experimental temperature and moisture profiles. Traditional methods of obtaining moisture profiles in drying research include the use of microtometry coupled with gravimetric analysis (3-7) and y-ray densitometry (2,B). The disadvantages of the former method are that it is an invasive and destructive technique with inherently poor spatial resolution. The latter method is noninvasive, which eliminates uncertainties due to sample variations and inability to duplicate experimental conditions. Magnetic resonance imaging (MRI) is a spectroscopic method which allows us to construct images of the internal structure of materials. MRI, like y-ray densitometry, is noninvasive and is capable of providing a significant improvement in spatial resolution over the above methods of obtaining moisture profiles. In addition to obtaining submillimeter resolution, data can be obtained in two or three dimensions, which allows for a check of basic assumptions such as uniform initial moisture distribution and one-dimensional mass transfer. MRI is also useful in obtaining data which can be used to define permeabilities (9), diffusion coefficients (IO),structural information such as the porosity, and other adjustable parameters. These parameters can then be used in the same theoretical models that are being tested. Hence, MRI can assist in the modeling process as well as the validation of the theoretical models. An application is demonstrated in this study, where the transient moisture profile data is used to obtain a diffusion coefficient.
* To whom correspondence
should be addressed.
8756-7938/91/3007-0540$02.50/0
When a water-containing sample is placed between the poles of a magnet, the magnetic moments of the hydrogen nuclei contained within the sample become polarized. This produces a net magnetization vector aligned parallel to the magnetic field (the z direction in Cartesian coordinates). The magnetization can be measured by displacing the system from the established equilibrium. This is done by applying a second magnetic field perpendicular to the main field in the form of a pulse of radio frequency radiation for a precise length of time. The pulse of radio frequency energy required must equal the frequency of precession of the hydrogen nuclei magnetic moments and is proportional to the strength of the applied magnetic field Bo. The frequency, w , is given by the relationship 0 = yBo (1) where y is a scalar constant of proportionality and a property of the nucleus known as the magnetogyric ratio. This is the fundamental nuclear magnetic resonance (NMR) relationship known as the Larmor relationship. The component of the magnetization rotating in the x-y plane at the above frequency induces a signal in the receiver coil of the spectrometer. The real-time signal amplitude is Fourier transformed and the resulting area under the signal versus frequency plot is proportional to the number of hydrogen nuclei in the sample (for appropriate experimental conditions). The induced signal usually decays exponentially with time due to several relaxation processes. The relaxation processes of interest in this study and the time constants associated with the processes are spin-lattice relaxation, T I ,and spin-spin relaxation, Tz. Spin-lattice relaxation describes the exchange of energy between the excited nuclear spin and the thermal motions associated with the molecules (11). Spin-spin relaxation is due to interactions between similar nuclear moments that occur without exchange of energy to surrounding molecules (12). The NMR experiment is extended to MRI by incorporating linear magnetic field gradients to give the signal a spatial dependence. The external magnetic field is
0 1991 American Chemical Society and American Institute of Chemical Engineers
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Bbtechnol. Rag.., 1991, Vol. 7, No. 6
modified by applying a linear gradient:
B,(x) = Bo+ G,x
2 x 4.9 x 4.9 CUBIC CM. APPLE SLAB
(2) where G , is the uniform magnetic field in the x direction (= dB/dx = constant), x is distance, and Bo is the homogeneous magnetic field. The frequency of absorption now depends upon position:
4 ~= r(Bo ) + G,x)
(3)
In MRI, two or three gradients are employed to spatially encode data in one, two, or three dimensions. By controlling the magnetic field gradients, we know the resonant frequency as a function of x , y, and/or z. In a onedimensional spin projection experiment, a gradient and a frequency-controlled pulse are used to select a thin plane from which data is spatially encoded by a second gradient, the uread gradient", in a direction perpendicular to the first gradient. Gradients and pulses can be combined in an infinite variety of pulse sequences. In medical imaging, the sequences are designed to take advantage of the variation in TI,Tz,and nuclei density within various tissues to obtain a high degree of contrast between different tissues. The signal amplitude a t a given frequency interval (or spatial interval AX) obtained from the spin-echo imaging sequence used in this study is proportional to the nuclei density and related to the T2 in the following manner (11): Su
p
exp(-TE/T2)
/
2 mm THICK IMAGING PLANE
(4)
It is apparent that if TE (an experimental delay time) is on the order of T2, small changes in TZcan have a significant effect on the signal amplitude. This has important ramifications in quantitative MRI studies. A direct relationship between signal intensity and moisture density cannot be assumed a priori in MRI experiments to obtain moisture profiles. The interested reader is referred to the texts by Morris (11)and Gadian (12)for more detailed discussions of NMR imaging.
SIGNAL ,AMPLITUDE
Figure 1. Schematic illustrating the volumetric element from which data was spatially encoded in the one-dimensional spin projection experiments. Profile at right is actual data from a fruit sample with uneven moisture distribution. 0.11
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Experimental Procedures Materials. Red Delicious apples (>340.0g) were stored a t 7 "C in a refrigerator and then kept a t room temperature 24 h prior to imaging studies. An apple sample was then sliced to 5 cm X 4.9 cm X 2.0 cm wide and inserted into a plastic sample holder accommodating those dimensions. The top and bottom faces (2.0 X 4.9 cmz) of the sample were exposed and two plastic tubes were inserted through the central plane of two of the vertical faces (4.9 X 4.9 cmz) to maintain the sample in a fixed position (see Figure 1). A vial filled with CaSOd-doped water was placed near the sample within the probative zone to provide a reference signal to which signal intensity from the apple sample was scaled. Equipment. The laboratory drier consisted of a centrifugal fan which blows air through a long horizontal cylindrical duct. A sample holder fits into a shaft which slides into the cylindrical duct. An air flow divider at the end of the shaft divides the air stream into two sections such that air flows over the top and bottom faces of the sample. The dry bulb temperature near the exit of the heater was measured by a thermocouple. Air velocity was measured with a hot-wire anemometer and the relative humidity of the ambient environment was measured with a capacitance-measuring RH probe. The cylindrical duct fit through the 15.2-cm-diameter bore of the imaging coil and the shaft is adjusted to position the sample in the
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Figure 2. Transient moisture profiles of the drying apple slab. Solid lines are the simulations; experimental data are shown as symbols. probative zone. To use the drier as an MRI probe, the system is constructed of nonmagnetic parts. A General Electric CSI-I1 magnetic resonance imaging spectrometer with a 2-T magnet was used to obtain onedimensional projections. A 15.2-cm internal diameter imaging coil is used to record lH signals a t 85.5 MHz. Methodology. The drying equipment was installed through the bore of the MRI magnet and a spin-echo type pulse sequence was used to acquire one-dimensional spin projections every 30 min. Signal intensity vs frequency plots were converted to moisture density vs distance plots by (1)dividing the frequency axis by the gradient strength and (2) multiplying the signal intensity axis by the appropriate scaling factor. The scaling factor is calculated by dividing the experimentally obtained water density of fresh apple tissue by the average signal intensity obtained in the initial one-dimensional spin projection of fresh apple tissue. Over the range of moistures studied, no dependence
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Biotechnol. Prog., 1991, Vol. 7, No. 6
Table 1. Diffusion Coefficient Data for Amles
Roman et al. (19) Alzamora (20) Singh and Lund (16)
D , cm2/s 11 x 10-5 6.4 x 10-5 2.6 X lo4 3.6 X lo6 1.6 x 10-5 3.67 X 10" to 1.73 X 1@
Lomauro et al. (21)
4.055 x
McCarthy et al. (present study)
8.3 x 10-5
reference Labuza and Simon (17) Rotstein et al. (18)
of the relaxation times on moisture content was observed; thus only a proportionality constant was needed to obtain moisture content from MRI signal intensities (13). The sample was positioned to align the linear imaging gradient, G,, normal to the drying surface. Drying conditions are chosen as follows: 24 "Cor roomtemperature air to effect quasi-isothermal drying, 3.0 m/s air velocity to minimize external resistances, and ambient humidity of 32% RH. The echo time was set a t 15 ms, the predelay time was set to 8 s to assure complete relaxation of magnetization between acquisitions, and the number of scans was 16 for adequate signal-to-noise ratio. Echo time, within the imaging pulse sequence, was varied to determine if spinspin relaxation time changed with moisture content. In the limited range of moisture contents used in this study, TZwas not found to be a function of moisture content. If TZhad varied with moisture content, the NMR signal would need to be corrected for relaxation effects in order to calculate a correct density (14, 15). The projection contains 128 intervals; hence, 128 data points are collected per scan. The imaging volume selected and origin of the signal when using one-dimensional spin projections is illustrated in Figure 1.
commenta McIntosh, var, T = 66 "C T=66OC T = 30 "C, Granny Smith, var T = 76 "C, Granny Smith, var T=7l0C intermediate moisture apple (MC = 0-1.3 g of HzO/ g of dry solid) T = 15-45 "C D,E = 27.5 exp[12.97MC - (4216MC + 5267)/(T + 273)) ow = 75, freeze-dried apple from adsorption curves and a fit to analytical solution Red Delicious apples, T = 24 OC
aC/dX = 0 a t t 1 Ofor X = L C = C, a t t = 0 for all X
(10) (11) Equation 9 implies that the interface dries immediately and attains the equilibrium moisture concentration as soon as the drying process starts. Equation 10 indicates that the maximum of the symmetrical concentration profile is on the center line of the slab. A uniform initial moisture content is assumed in the apple tissue in eq 11. Equation 8 is rewritten by replacing the differential terms with the forward difference formula
Parameters A X and At indicate one-dimensional location and time increments, respectively. Parameter tC, indicates grams of water per unit volume in the apple tissue for location of n increments from the apple-air interface. Equation 12 may be rearranged:
Drying Model The equation of continuity describing one-dimensional moisture transport in the apple slab is
a -(N,) =0 at ax The parameter N,, water flux in x direction, is defined as
+
N , = X,(N,
dC + N,) - D dX
The terms X,(N, + N,) and -D dC/dX simulate the convective moisture transport and the Fickian diffusion, respectively. Transport of the non-water constituents, N,, may be neglected, and mass fraction of water, X,, may be substituted as C X,=- c+ c,
(7)
Equation 5 is rearranged after substituting eqs 6 and 7 and locating the origin of the coordinate system, Le., X = 0, on the apple-air interface as
The boundary conditions and the initial condition required for solution of eq 8 are C=C, attZOforX=O
(9)
Numerical values of t+~tCn can be computed from eq 13 when the numerical values of parameters tCn,&+I, tCn-l, and the constants D, C, AX, and At are known.
Results and Discussion Moisture profiles in an apple sample with a uniform initial moisture distribution were used for modeling. The initial moisture profiles in this sample were symmetrical; therefore, they were simulated between the air-apple interface and the center line of the slab. The bulk density of the apple tissue a t zero moisture content, C,, was reported to be 0.3284 g of dry solid/cm3 (16). The dry basis equilibrium moisture content was experimentally determined to be 0.163 g/g of dry solid and equilibrium water density, C, was calculated to be 0.054 g/cm3. Numerical values of the location increment AX and time increment At were assigned as 0.10 cm and 2 s, respectively. The average initial moisture density of the slab was experimentally determined as 0.627 g/cm3; thus, the numerical value of parameter J, was known for all the locations a t t = 0 (initial condition, given in eq 11). Numerical values of parameter &, were also known for all times on the interface, i.e., n = 0 (boundary condition given in eq 9). Numerical values of t+AtCn were computed for all the locations and times with eq 13 after a value was assigned for D, using the initial and the boundary conditions and the numerical values of parameters C,, C,,
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At, and AX. The best agreement was found between the experimental data and the simulations when diffusivity D was 8.3 X 10-5 cm2/s, The simulations with this value of the diffusivity were shown and compared with the experimental data in Figure 2. The numerical value of diffusivity, D, was compared with the results of the previous studies in Table I. Similarity of the numerical value of the diffusivity with those of the other studies strengthens the point that MRI is a powerful noninvasive technique for following the moisture migration in biological tissues.
MRI was used in this study to obtain transient moisture profiles with spatial resolution of 0.60 mm. The use of this noninvasive method for obtaining an effective diffusivity useful in modeling and process control is demonstrated. No relaxation dependence on moisture content was observed in the high end of the range of moisture content (from 6.02 to 3.081 g/g of dry solid); thus, the signal intensity obtained from MRI can be directly related to moisture density for the purpose of obtaining moisture profiles.
Notation
Bo Bz
c
ce
co
8"
cs D Gi L MRI n Nz
Ns NMR S
T1 T2 TE t At
X AX X W P
Y
static magnetic field strength magnetic field strength with spatial dependence moisture content, g of water/cm3 equilibrium moisture content, g of water/cm3 initial moisture content, g of water/cm3 numerical value of parameter C at a distance of nxAX cm from the interface at time t solids content in the apple, g of solids/cm3 effective diffusivity, cm2/s dBi/di = linear magnetic field gradient in direction i half thickness of the slab, cm magnetic resonance imaging location index water flux, g/(cm2 min) flux of the solids, g/(cm2 min) nuclear magnetic resonance signal intensity, arbitrary units spin-lattice relaxation time spin-spin relaxation time echo time time, min time increment distance from surface into the solid distance increment water content on dry basis, g of water/g of dry apple spin density, number of spinslunit volume magnetogyric ratio
w
Larmor frequency at which radio frequency from transmitter and frequency of precession are in resonance
Acknowledgment This work was partially supported by a grant from the University of California, Davis NMR facility. Mustafa Ozilgen was partially supported by a grant (CRG 890445) from NATO Office of Scientific Affairs.
Literature Cited (1) Chirife, J. In Advances in drying; Mujumdar, A. S., Ed.; Hemisphere Publishing Co.: New York, 1981. (2) Chiang, W. C.; Peterson, J. N. In Drying '86; Mujumdar, A. S., Ed.; Hemisphere Publishing Co.: New York, 1986. (3) Litchfield, J. B. Analysis of mass transfer for the drying of
extruded durum semolina. Ph.D. Thesis, Purdue University, West Lafayette, IN, 1986. (4) Igbeka, J. C. J. Food Technol. 1982,17, 28-36. (5) Mittal, G. S.;Blaisdell, J. L. J.Food Process. Preserv. 1982, 6, 111-126. (6) Villota, R.; Karel, M. J.Food Process. Preserv. 1980,4,111134. (7) Vaccarezza, L. M.; Lombardi, J. L.; Chirife, J. J.Food Technol. 1974,9,317-327. (8) Cunningham, R. M.; Kelly, J. J. In Drying 'BO: Developments in Drying; Mujumdar, A. S., Ed.; Hemisphere Publishing Co.: New York, 1980;Vol. 1, pp 40-47. (9) Banavar, J. R.; Schwartz, L. M. Phys. Rev. Lett. 1987,58, 1411-1414. (10) Blackband, S.;Mansfield, P. J.Phys. C 1986,19,L49-L52. (11) Morris, P. G. Nuclear Magnetic Resonance Imaging in
Medicine and Biology; Clarendon Press, Oxford, England,
1986. (12) Gadian, D. G. Nuclear magnetic resonance and its
application to living systems, Clarendon Press, Oxford, England, 1982. (13) Perez, E.; Kauten, R.; McCarthy, M. J. In Drying '89;Mujumdar, A. s.,Roques, M. A., Eds.; Hemisphere Publishing Co.: New York, 1990. (14)Haacke, E. M.; Liang, Z.-P.; Tkach, J. A. T2-Deconvolution in MR Imaging and NMR Spectroscopy. J. Magn. Reson. 1988,76,440-457. (15) McCarthy, M. J. Interpretation of the Magnetic Resonance Imaging Signal from a Foam. AZCHE J. 1990,36,287-290. (16) Singh, R. K.;Lund, D. B. J. Food Process. Preserv. 1984, 8,191-210. (17) Labuza, T. P.; Simon, I. B. Food Technol. 1970,24,712716. (18) Rotstein, E.; Laura, P. A.; Cemborain, M. E. J. Food Sci. 1974,39,627-631. (19) Roman, G. E.; Rotstein, E.; Ubicain, J. J. Food Sci. 1979, 44,193-197. (20) Alzamora, S. M. Thesis, Facultad de Ciencias Exactas y
Natureles, Universidad de Buenos Aires, Buenos Aires, Argentina, 1970. (21) Lomauro, C. J.; Bakshi, A. S.; Labuza, T. P. J. Food Sci. 1985,50,379-400.
Accepted September 10, 1991. Registry No. H&, 7732-18-5.
