Usual factors that lead to deformation in food materials are moisture change (examples include drying and rehydration) and internal pressure generation (puffing and bread bak- ing). Between the two, deformation due to moisture change is a complex phenomena and is highly dependent on the state of the food material. The physics of deformation due to gas transport is relatively easy as the effect of gas pressure can be easily expressed as a source term in the solid momentum balance (more later).
Processes with Moisture Change as the Driving Mechanism
Most wet food materials are initially in a soft rubbery state. For such materials, it is usually observed that total volume change at equilibrium is equal to volume of moisture lost or gained20. In other words, as long as the material is in rubbery state and drying
rate is not too high to cause surface cracks, the solid matrix remains saturated and the gas phase does not enter the pores. In such a case, the pore pressure is simply the pressure
of liquid water, and Equation 5.7 can be written as:
r: N0D rpw (5.11)
In a series of papers, moisture transport has been investigated in detail by Scherer29,30
for soft and deforming polymer gels, which behave in a similar fashion. Scherer argued that for a uniform pore size medium with inert liquids in its pores, effective stress at equilibrium (or during a slow drying process) is equal to pore pressure:
N0 D pw (5.12)
As a soft material dries out, two important phenomena happen, the pores shrink and the bulk modulus of the material increases (turning a soft, rubbery food into a rigid, glassy state). For uniform moisture distribution, the volume change is equal to the volume of water lost. The material will stop shrinking when the liquid-vapor meniscus moves inside the matrix and, with the increased bulk modulus, the solid stresses can balance the compressive capillary pressure, pc. Until that point, the solid skeleton is too soft to
allow the meniscus to move inside and create compressive pressure. Assuming the solid skeleton to be elastic, the normal effective stress (shear stress will be zero at equilibrium as there are no pressure gradient or external shear load) can be related to volume change:
d N0D KdV =V (5.13)
Inserting the stress-strain relation from Equation 5.13 into differential form of Equa- tion 5.12 and integrating from initial stress-free volume, V0 to final volume at which
shrinkage stops, V, we obtain: Z V V0 K V dV D p c (5.14)
critical volume, Vcan be established from Equation 5.14. However, due to highly het- erogeneous and hygroscopic nature of food material, we can only say that since K and pc are functions of moisture content, M and temperature, T , critical volume, V, will
also be a function of temperature and moisture at equilibrium.
VD V.M; T / (5.15)
Also, for a general food material with range of pore sizes, the capillaries will empty at different values of shrinkage. Thus, in food materials, we may observe a gradual decrease (rather than a sharp change which is expected for uniform pore size material) in the slope of volume vs. moisture content plot to zero, as shown in Figure 5.1 (dotted line instead of the solid line). Fortunately, volume vs. moisture content data is available for many food materials. This allows us to treat volume change due to moisture loss as a free strain analogous to thermal expansion.
Moisture Content Vo lume Critical Volume, V* Mainly Liquid Transport Mainly Vapor Transport Liquid + Vapor Transport Gradual Transition Ideal Transition
Figure 5.1: Volume change versus moisture content curve for a typical food material
Small Deformation
For small deformation, volume changes due to temperature and moisture change, i.e., the moisture and thermal strains (" and" , respectively) are subtracted from the total
strain to get the mechanical strain,"m:
"mD " "M "T (5.16)
Now, with the effect of liquid (moisture) pressure accounted for as a free strain, the mechanical strain, "m can be related to the stress due to mechanical load only, N00, i.e
the effective stress, N0minus the pressure of water, pw:
N0 pwD N00 D D:"m (5.17)
The solid momentum balance, Equation 5.11 can also be written in terms of N00
r: N00 D 0 (5.18)
Depending on the time scales of the process and deformation, the food material can be treated as elastic or viscoelastic and the corresponding stress-strain relationship can be used along with the solid momentum equation.
Large Deformation
For large deformation analysis, a multiplicative split31 in deformation gradient tensor,
F, can be used to separate volume changes due to moisture and temperature changes
from volume change due to mechanical effects. As shown in Figure 5.2, the material is first assumed to under go stress-free deformations due to moisture and temperature changes and, then, mechanical stresses act on this stress-free deformed material. The deformation tensor, F, can be split as:
FD FTFMFel (5.19)
The dilatation (volume-changing) stress is related to elastic jacobian, Jel D det.Fel/, which is obtained as the ratio of total volume change and volume change due to moisture and temperature effects (details in Section 5.4.1). Thermal jacobian, JT D det.FT/ is often small for food materials and is usually ignored. Moisture jacobian, FM can easily
To , Mo , σ'=0 T , Mo , σ'=0 T , M , σ'=0 T , M , σ' FT FM Fel F
Figure 5.2: Steps indicating multiplicative split in the deformation tensor, separating moisture, temperature and mechanical effects
Processes with Gas Pressure as the Driving Mechanism
For some processes such as microwave heating or bread-baking, large internal pressure generation (due to water vapor in microwave heating and carbon dioxide in baking) can cause swelling/puffing of the material. In such cases, the gas pressure gradient term of Equation 5.8 (first term on RHS) may dominate. Swelling due to gas pressure in such cases can be much larger than shrinkage due to moisture loss, and, therefore, stresses and strains due to the latter can be ignored. In the absence of thermal strains, the total strain is approximately equal to the mechanical strain:
"m " (5.20)
Also, as the stress due to moisture transport is neglected, the solid momentum balance 5.8 becomes
with effective stress, N0related to strain,".
Of course, if deformation due to both phenomena (moisture change and gas pressure) need to be accounted for, the governing equation and the constitutive law will take the form:
r: N00 D rpg; N00 D D:"m (5.22)