Mass Transfer in Food Preservation Processes
III. MASS TRANSFER BETWEEN PHASES A Interface Between Phases
Phases in contact are usually assumed to be in equilibrium at the plane of contact, that is, the interface. This assumption means that the controlling factor in transfer of components from one phase to another is the rate of transport to and from the interface. The analysis of transfer between phases therefore requires that primary consideration be given to transport in each phase.
The following pages deal with transport of diffusing components in gases, liquids, and porous solids. Water is the substance usually transferred in processes for food preservation, but the same general principles apply to flavor components, preservatives, and other chemical species.
B. Diffusion Within a Homogeneous Phase
Within a homogeneous phase (gas, liquid, or solid) the mass transfer of a molecular species occurs primarily by diffusion. The driving force for diffusion is the difference in the chemical potential of the diffusant, when the chemical potential is constant the system is at equilibrium. At constant temperature the condition of equilibrium is given by constancy of activity of the diffusant and in most cases is approximated by equality of concentrations. Figure 4.7 represents the concentration gradient of a diffusing chemical species during approach to equilibrium. The mathematical formulation is given by the well-known Fick’s first law shown in Fig. 4.8. Fick’s law is analogous to the equation for steady-state heat transfer equation presented in Chapter 3.
Diffusion may be considered as the result of a very large number of small steps in random directions and the diffusion coefficient (D) has been related to the length of individual steps, and the time for each step by the classic Einstein- Smoluchowski equation (Fig. 4.9).
As in the case of heat transfer, it is possible to derive a differential equation for unsteady transfer by conducting a balance on a differential element in the space in which the transfer occurs. In the case of heat transfer the balance considered is energy balance, in diffusion mass balance. In fact Fick derived both of his laws on the basis of analogy to heat transfer relation.
The equation for unsteady-state diffusion is the Fick’s second law shown in Fig. 4.10. The solutions to this differential equation for various boundary conditions (geometry, initial concentrations as a function of spatial coordinates) has well-known solutions analogous to those available for unsteady heat transfer (Treybal, 1968; Sherwood et al., 1975; Crank, 1956). Figure 4.11 shows a general solution for an infinite slab and approximations which are useful for specified time periods (short times and long times), Figure 4.12 shows graphically the conformance and deviation of the approximations from the general solutions. In addition to slab geometry, solutions have been developed for other geometric shapes including cylinders and spheres. Computer software is available for obtaining these solutions, and there are examples in literature of their application to food engineering problems (Barnard and Quintero, 1998). A useful approach is the graphical presentation of solutions of the equation analogous to those for heat transfer, such as the well-known Guernie-Lurie charts, the use of which for heat transfer calculations was explained in Chapter 3.
The analogy with heat transfer presentations can be recognized by consideration of the following dimensionless numbers used in these graphs. The unaccomplished change is often given the symbol Y and represents (T2 Ta)/
(Ti2 Ta) for heat transfer and (c2 ca) / (ci2 ca) for mass transfer. The subscripts
a and i stand for initial and ambient conditions, respectively. In these charts
FIGURE4.7 Schematic representation of the approach to equilibrium in diffusion process. General condition for equilibrium with respect to molecular species A. The
chemical potential of A = mAis constant.
the symbol X representing the abscissa stands for the quantity at/l2in heat transfer and in mass transfer for Dt/l2. a is the thermal diffusivity, D the (mass) diffusivity, t is time and l is a characteristic length for the geometric shape considered.
FIGURE4.9 Statistical representation of diffusion.
As in the case of heat transfer the graphs show solutions for specified locations within the object and for specified surface resistance ratios. Similar charts may be constructed for specific applications in mass transfer, such as infusion of compounds into foods. Infusion is used in curing, which is a set of processes in which microbial activity is inhibited by infused compounds such as salts and sugars, which lower water availability to microorganisms and may also have direct bacteriostatic effects.
Figure 4.13 shows accomplished concentration change, i.e., fraction of the total change attained if equilibrium is reached. If we use previously cited
FIGURE4.11 Some solutions to the differential equation of Fick’s second law.
definition of Y, the accomplished change is equal to (12 Y). The curves in Figure 4.5 – 13 refer to conditions in the center of these geometric shapes. Similar curves may be obtained for other locations.
Another set of solutions may be developed for the average concentration in given geometric shapes. Figure 4.14 shows a set of such curves for spheres, slabs, and cylinders. The use of mathematical models for process design calculations in various leaching and diffusion processes important in preservation is reported in numerous reviews and research papers (Schwartzberg and Chao, 1983; Califano and Cavelo, 1983; Liu 1992; Wand and Sastry, 1993; Luna and Chavez, 1992). We can illustrate the applicability of above solutions to food technology by considering the process for curing Parma Ham (Prosciutto). This rather expensive delicacy is produced by a very well controlled operation both in terms of raw materials and process conditions. The process involves trimming 12 to 14 kg hams to a weight of about 8 to 10 kg and covering them with solid sodium chloride. This results in the existence of a saturated salt solution on the surface of the hams, which are placed in refrigerated rooms at a relative humidity of 85% at 1 to 48C for an initial equilibration period of approximately days, after which the remaining visible salt crust is removed and the ham are stored for an additional 15 to 18 days at the same temperature, but at somewhat lower temperature. The purpose of this two-step salt infusion period is to achieve a local salt concentration adequate to prevent microbial growth. Following this equilibration
FIGURE 4.13 Accomplished concentration change (12 Y) as a function the
diffusion number (X) for some geometric shapes. Curves are for the center of each shape.
the surface is cleaned of remaining salt, and the hams are placed for several months in a room at 388C to develop the desired characteristic flavor.
The composition of the cured Parma ham is water 62% proteins 27%, salt 6%, and fat 4%. The periods of initial equilibration are adjusted on the basis of extensive experimental data relating weight of the ham to the time needed for salt penetration.
The problem we shall consider is the estimation of the equilibration period on the basis the known salt/water diffusion coefficient, an assumed geometric shape of the ham, and the graphical solutions of Fick’s second law.
We make the following “rough” approximations:
Geometric shape: Cylinder 18 cm in diameter and 36 cm in length. This gives the characteristic length l ¼ 9 cm or 0.09 m. Assuming a density of ham of 1070 kg/m3(Lewis, 1987), the weight of the cylinder is approximately 9.8 kg.
Diffusion coefficient: We estimate D of sodium chloride in the meat as 4£ 10210(Saravacos and Maroulis, 2001).
Accomplished concentration change (12 Y ): We approximate the surface concentration of salt (saturated salt solution) as 240 kg/m3and assume that a post- equilibration concentration in the center of the ham of 50 kg/m3is adequate to assure microbial safety. Compare this with the equilibrium composition for the cured ham reported above, where salt content is 6%. We obtain a value of 12 Y of 50/240 ¼ 0.208.
FIGURE4.14 Accomplished average concentration change (12 Y) as a function of the diffusion number (X) for the following spheres, infinite slabs, and infinite cylinders.
To evaluate the diffusion behavior, we consult an accomplished concentration change chart for the center of a cylinder (Fig. 4.15). Note that at low values of 12 Y, the difference between cylinders with diameter equal to length and infinite cylinders is not significant. We obtain a value of X ¼ Dt/12 by consulting the chart as equal to 0.11. We can now estimate equilibration time t: t ¼ (X12)/D. Therefore t ¼ 0.11£ 0.092/ (4£ 10210) ¼ 6.2£ 1026s ¼ 26 days.
We note that the estimated time required is in remarkably good agreement with the 22 to 25 days used according to literature description of the process. Given the large number of approximations we made, this close agreement is probably fortuitous. Nevertheless, estimations of this type are useful as guides to process design but must be confirmed by experiment or in cases where very good data on geometry and on physical properties are available by computer-aided solutions of Fick’s Second Law.
C. Nonideal Diffusion Behavior
The diffusion equations described above constitute a description of so-called Fickian, or ideal behavior. The prerequisites for such behavior are
1. Diffusion constant independent of concentration of diffusant.
2. Absence of geometric changes in the matrix (absence of shrinkage or swelling).
FIGURE 4.15 Accomplished concentration change (12 Y) at the center of two cylinders: infinite cylinder and cylinder with height equal to diameter.
3. Temperature dependence of D may be described by the Arrhenius equation.
In many materials, in particular in glassy polymers, and in foods which in many cases are composed of glassy polymers, significant deviations from Fickian behavior are observed (Crank and Park, 1968). The primary causes of such behavior of foods are swelling or shrinkage. Water is the universal plasticizer in foods, and since plasticization increases molecular mobility of polymer chains and often results in swelling, it is in the diffusion of water that nonideal behavior is most common in foods. The effect of polymer mobility on diffusion is expressed by the Deborah Number, De shown in Fig. 4.16.
In Fig 4.17 are shown schematically the effects of De on diffusion behavior as a function of diffusant (in this case water) concentration and temperature. When the rate of relaxation of the matrix (due to polymer segment mobility) is very much slower than the rate of diffusion, De .. 1.
The matrix geometry does not change significantly during the diffusion and the regime of diffusion is ideal or Fickian. This occurs when the diffusant does not plasticize the matrix, or when the concentration and temperature condition limit such plasticization and the polymer remains in the glassy state. When the value of De is very much lower than 1, mechanical relaxation dominates the process (Peppas and Brannon-Peppas, 1994).
A special case observed in diffusion is the so-called Case II diffusion in which the initial sorption of the diffusant shows a linear dependence on time (Fig. 4.18). It will be recalled that the expected Fickian behavior gives a linear dependence on the square root of time (Fig. 4.11). There have been many attempts to model Case II behavior, and the most widely accepted model
postulates a coupling of the diffusion process with viscoelastic effects. The viscoelastic properties of the polymers change as the diffusant plasticizes the matrix. Food polymers are known to behave in this manner. A recent study using nuclear-magnetic-resonance imaging reported the diffusion behavior of water in starch and found Case II behavior in diffusion into amylose pellets (Hopkinson et al., 1997).
FIGURE4.17 Case II diffusion in swellable polymers. (From Peppas and Brennon- Peppas, 1996.)
FIGURE 4.18 The initial portion of sorption curves in two types of diffusion.
D. Film Coefficients for Mass Transfer
In analyzing mass transfer between phases, it is convenient to express rates in terms of coefficients incorporating various resistances to mass transfer and relating fluxes to gradients of concentration or pressure.
For instance, consider a liquid phase in contact with a gaseous phase, as shown in Fig. 4.19. Assume that a component is transferred from liquid to gas, In the bulk of the liquid, the concentration of the component is uniform and equal to C. The gradient of concentration is assumed to occur in a film adjacent to the gas-liquid interface. At the interface, the concentration (Ci) is in
equilibrium with the partial pressure ( p1) of the component at the interface. The
partial pressure in the bulk of the gas phase is also considered uniform and equal to p, with all the gradient in partial pressure occurring in a gaseous film adjacent to the interface. Given steady-state conditions, the following equation gives the rate of transfer in the liquid:
J ¼ kLðC 2 C1Þ ð22Þ
where kLis a film transfer coefficient in the liquid in units consistent with those
FIGURE4.19 Schematic representation of mass transfer from a liquid to a gas phase.
of J and C. If for instance J is in mol/s cm2and C is in mol/cm3, then kLis in
mol
ð cm2ÞðsÞðmol / cm3Þ¼ cm / s
Note that Eq. (23) relates kL to the diffusion coefficient
kL¼
D DXliq film
ð23Þ where DXliq film is the thickness of the liquid film. Of course, under usual
conditions DXliq film is not known and kLcannot be simply related to D.
The rate of transfer in the gas phase is given by
J ¼ kGð pj2 pÞ ð24Þ
where kGis the film transfer coefficient in consistent units. For instance, it J is in
mol/s21cm2and p is in atmospheres, then kGis in mol/s21cm22atm21. Note that
kGmay be related to D by Eq. (25) if the following assumptions hold: (1) D Xgas flim
is known, and (2) partial pressure of the component transferred in the gas is very much less than the total pressure.
kG¼
D RTðDXgas filmÞ
ð25Þ where D Xgas filmis the thickness of the gas film. Under conditions of steady state,
the flux from the bulk of the liquid to be interface equals the flux from the interface to the bulk of the gas phase, and we can write
kLðC 2 ClÞ ¼ kGðPi2 PÞ ð26Þ
In the usual situation, the actual concentrations and partial pressures at the interface are not measured and the need arises to relate flux to concentrations and partial pressures in the bulk phase. If a function relating gas partial pressures in equilibrium with concentration in liquid is known, as symbolized in Eqs. (27) and (28) then overall transfer coefficients can be defined as shown in Eqs. (29) and (30)
C0¼ f ð pÞ ð27Þ p0¼ f ðCÞ ð28Þ where C0is the concentration is equilibrium with partial pressure p and p0is the partial pressure in equilibrium with concentration C.
where KLis the overall transfer coefficient defined in terms of concentrations in
liquid.
J ¼ KGð p02 pÞ ð30Þ
where KGis the overall transfer coefficient defined in terms of partial pressures.
It follows that in steady state,
KLðC 2 C0Þ ¼ KGð p02 pÞ ð31Þ
If Henry’s law can be applied to the equilibrium situation, Eq. (27) may be rewritten:
C ¼ p/s0 ð32Þ
where S0is the Henry’s law constant in suitable units. Equation (31) can then be written:
KLð p 2 p0Þ ¼ S0KGð p02 pÞ ð33Þ
and it follows that
KL¼ S0KG ð34Þ
E. Transport in Porous Solids
In many of the operations performed on foods for purposes of preservation, water is removed from porous solids. Details of transport mechanisms in dehydration are discussed in another chapter and here the subject is introduced only in its most general aspects.
Transport in solids may occur by diffusion in the solid itself, in which case Fick’s law can be used.
In porous solids, however, various mechanisms may be involved including 1. Diffusion in the solid itself
2. Diffusion in gas-filled pores
3. Surface diffusion along the walls of pores and capillaries 4. Capillary flow resulting from gradients in surface pressure
5. Convective flow in capillaries resulting from differences in total pressure
As discussed in a subsequent chapter, transport in porous solids can be related to various properties, such as porosity, internal surface area, and tortuosity of pores.
F. Simultaneous Heat and Mass Transfer
Often, when mass is transferred, heat also must be transferred. When a component is transformed from a liquid to a gas phase, latent heat of vaporization must be supplied. The same amount of heat must be removed when mass transfer occurs in the opposite direction. In every instance the temperature of the interface and transfer rate adjust themselves so that at steady state the rate of heat transfer and rate of mass transfer is balanced.
In some operations, heat transfer is of minor importance is controlling the overall rate of transport from one phase to another. This situation is true of operations in which latent heat affects are small. In liquid-liquid extraction and in leaching, for instance, the overall transfer rate may be considered as limited only by resistances to mass transfer. In other operations, including evaporization and crystallization, heat transfer is the controlling mechanism. Both heat and mass are transferred in large amounts, but the rates at which both occur can be determined by considering only the rate of heat exchange with an external sources.
In still other operations, particularly humidification, drying, and freeze drying, both heat transfer and mass transfer must be considered and balanced in determining overall rates. The methods used for that purpose are discussed in the chapters which follow.
SYMBOLS a activity C, c concentration C0 equilibrium concentration D diffusion coefficient De Deborah number d diameter
d hopping length (Defined in Fig. 4.9.) F number of intensive variables h height
J fux
K overall coefficient of mass transfer k constant
kL,kG mass transfer coefficients in the liquid and gas phases
respectively l characteristic length
N number of moles in gas phase p partial pressure
P8 vapor pressure R gas constant
r radius S, S0 constants T temperature t time V volume v molar volume W weight fraction X diffusion number ¼ Dt/12 XLmole fraction in liquid
x distance
Y unaccomplished concentration change y mole fraction in gas phase
D Hv latent heat of vaporization
D XLiquid Film Liquid film thickness
D XGas Film
a
Gas film thickness Thermal diffusivity
u Characteristic diffusion time (Fig. 4.15) G number of components
g activity coefficient DHv latent heat of vaporization DHv volume change
l mechanical relaxation time f number of phases h viscosity m chemical potential Subscripts a ambient G gas phase i initial L liquid phase u all components REFERENCES
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