Application of the Zhurkov model
The Zhurkov model was used by Cai and Diosady (1993) to describe the effects of shear stress on the rate constants of starch gelatinization during extrusion. The shear stress term in Eq. (6) was calculated from apparent viscosity (η) and shear rate () using
 (7)
The rheological model developed by Harper and others (1971) for cooked cereal dough was used by Cai and Diosady (1993) to calculate the apparent viscosity of the extruded wheat starch:
 (8)
where W donates moisture content. For a constant screw configuration, is proportional to screw speed (Yacu 1985):
 (9)
where G, geometrical coefficient, is a constant for a given screw configuration; N is screw speed.
Substituting Eqs. (8) and (9) in Eq. (7) yields
 (10)
Making Eq. (10) is simplified to
 (11)
The value of can be calculated for a given extrusion operation where material temperature, moisture content, and screw speed are known.
Substituting Eq. (11) in Eq. (6) gives
 (12)
While determining the model parameters in Eq. (12), term VG^{0.51} is regarded as one parameter. The results given by Cai and Diosady (1993) were E_{0}= 45636.9 J/kgmol, and VG^{0.51}= 1.784 × 10^{−3} m^{3}/kgmol. Here, the unit of activation energy is joule per kilogrammole (a unit used in earlier literature = 1000 J/g mol, now the unit is joule per mole). This showed that the authors should take the general gas constant R as 8314 J/kgmol. After estimating residence time t from Figure 3 in the original article, we calculate starch gelatinization degree (D_{g}) using Eq. (8) in the article with R = 8314 J/kgmol and the parameter values given by the authors. The resulting D_{g} is 1. Even taking R = 83.14 J/kgmol, still D_{g}= 1. While taking R = 0.8314 J/kgmol, D_{g}= 0. Only when taking R = 8.314 J/kgmol, is the resulting D_{g} on the order 0.1, which is a reasonable value for gelatinization degree. The general gas constant R should be 8.314 J/g_{}mol. So, we inferred that the authors may have made a mistake on the unit of R. If this is the case, the resulting E and VG^{0.51} should be 45.6369 kJ/mol and 1.784 × 10^{−3} m^{3}/mol, respectively. We calculated from temperature, moisture content, and screw speed, to be 35134 to 162226 Pa. Based on the above inference, the activation energy reduction and its percentage of E_{0} (45.6369 J/mol) were calculated to be 0.063 to 0.289 kJ/mol and 0.14% to 0.64%, respectively.
Ilo and Berghofer (1998, 2003) found that VG^{0.15} is 0.06568 m^{3}/mol for thiamin destruction and 0.01164, 0.004989, and 0.010809 m^{3}/mol for the loss of lysine, arginine, and cystine, respectively.
We estimated the material temperatures in the report of Ilo and Berghofer (2003) using the regression model developed by Ilo and others (1996), who used the same extruder and processing conditions as Ilo and Berghofer (2003) did. Then, was evaluated using . The shearinduced activation energy reduction for lysine, arginine, and cystine loss was calculated, 1.89 to 3.18, 0.81 to 1.36, and 1.75 to 2.95 kJ/mol, respectively. As a percentages of E_{0} (124.71, 67.60, and 74.33 kJ/mol for lysine, arginine, and cystine, respectively), the values are 1.5% to 2.5%, 1.2% to 2.0%, and 2.4% to 3.9%, respectively. The material temperature and its corresponding feed moisture and screw speed were not detailed in the report of Ilo and Berghofer (1998). The central levels of N, T, and M (73 rpm, 170 °C, and 13%) were used to calculate . The activation energy reduction calculated using Eq. (13) and its percentage of E_{0} (99.41 kJ/mol) are 16.09 kJ/mol and 16.19%, respectively.
Van den Hout and others (1998) used the Zhurkov model to express the effect of shear force in an extruder on the inactivation of TI; however, they calculated ΔE with a different assumption. They proposed that the activation energy for the inactivation of a TI molecule is reduced by an amount equal to the work done in moving a segment of the molecule with a crosssectional area S over a distance x with respect to the other parts of the molecule:
 (14)
where N_{a} is the Avogadro constant, S =π d^{2}_{0}/4, x can be approximated by τ d_{0}/ M_{e}, M_{e} and d_{0} are the elastic modulus and diameter of the molecule, respectively. Equation (14) becomes
 (15)
M_{e} and d_{0} were estimated as 10^{6} Nm^{−2} and 3.5 × 10^{−9} m, respectively. The viscosity model of Morgan (1979) was used to calculate the shear stress. Shear rates in the screw channel, clearance, and die were calculated separately. Then, ΔEs in the 3 regions were obtained and ranged from 0.1 kJ/mol in the screw channel to 0.7 kJ/mol in the extruder die.
The earlier work of van den Hout (1997) demonstrated that the inactivation of TIs when heated in shearless environments showed a 2phase inactivation behavior:
 (16)
where TIA_{t} and TIA_{0} are TI activity after heating time t and without heating, respectively. k_{1} and k_{2} are inactivation constants in the 1st and 2nd phase, and expressed as the Arrhenius relation with their corresponding preexponent factors k_{1,0}= 0.142 s^{−1} and k_{2,0}= 0.0148 s^{−1}, and activation energies E_{1}= 200 kJ/mol and E_{2}= 158 kJ/mol at reference temperature T_{r}= 110 °C. α is a fitting parameter = 0.583.
When shear effect on TI inactivation is considered, E_{1} and E_{2} are reduced by ΔE calculated using Eq. (15). TIA was calculated numerically from the feed port of the extruder to die exit using Eq. (16) with the reduced E_{1} and E_{2}. The predicted trypsin inhibitors activity (TIA) levels were comparable with the measured values of the extrudates.
While modeling shearinduced inactivation of αamylase in a plain shear field, van der Veen and others (2004) assumed that the shearinduced reduction of the activation energy for bond breakage is equal to the work done by the external applied force. Although the exact relation between the force applied on the bond and the external applied shear force is unknown, a positive relation between the external applied shear stress and the bond energy reduction may exist. They proposed a simple power law relation to describe the reduction of activation energy due to the external shear stress:
 (17)
where ζ and m are constants to be determined experimentally.
Substituting Eq. (17) into Eq. (1) and integrating gives:
 (18)
where C denotes the enzyme activity at time t, C_{0} the initial enzyme activity. Equation (18) was used to fit all experimental data at once, using ζ, k′_{∞}, and m as adjustable parameters and was numerically integrated over the complete experimental τt profile of each shearing treatment. The estimated parameters were k′_{∞}= 6.05 × 10^{−5} s^{−1}, ζ= 25.1 m^{2} mol^{−1} Pa^{1−n}, and m = 0.57. The range of τ was not presented in the article. Taking τ= 18375 Pa (half of the τ_{max} average), we calculated ΔE using Eq. (17)= 6.77 kJ/mol. The activation energy for thermal inactivation E_{0} is on the order of 240 kJ/mol (De Cordt and others 1994).
The E_{0} for starch gelatinization (45.64 J/mol, Cai and Diosady 1993) is higher than the strength of hydrogen bindings (10 to 40 kJ/mol, Semenova and Dickinson 2010) slightly. This suggests that other interactions stronger than hydrogen bindings may be involved in starch gelatinization. The E_{0}s for the loss of 3 amino acids (67.60 to 124.71 kJ/mol, Ilo and Berghofer 2003), thiamin destruction (99.41 kJ/mol, Ilo and Berghofer 1998), inactivation of TIs (158 to 220 kJ/mol, van den Hout and others 1998), and αamylase (240 kJ/mol, De Cordt and others 1994) are all lower than the strengths of CC bonds (360 kJ/mol in C_{2}H_{6}, Israelachvili 1992) and CO bonds (340 kJ/mol in CH_{3}OH, Israelachvili 1992) that are very common in food molecules. This indicates that these processes may involve interactions weaker than the covalent bonds.
The activation energy reduction and the rate constant increase induced by thermomechanical processes are summarized in Table 1. Comparing the estimated ΔE values showed that the reduction of the activation energy for starch gelatinization is about one order of magnitude smaller than that for amino acid loss, thiamin destruction, and inactivation of TI and αamylase. This suggests that shear stress has lower efficiency in catalyzing starch gelatinization than in catalyzing other reactions. In other words, thermal effects are the major contributor to the starch gelatinization in the experimental range investigated. The size of starch granules is much larger than that of amino acid, thiamin, and TI. Theoretically speaking, the larger starch granules should be more easily deformed by shear force and demonstrate more activation energy reduction than the smaller. A possibility is that shear leads to viscous dissipation and results in an additional thermal effect on the loss of smaller molecules.
Table 1–. The activation energy reduction and rate constant increase induced by thermomechanical processes.Thermomechanical process  Kinetic topic  Activation energy and its reduction  k/k_{T}*  Source 


Extrusion  Starch gelatinization  E = 45.6369 kJ/mol, ΔE = 0.063 to 0.289 kJ/mol ^{a}  1.02 to 1.10  Cai and Diosady (1993) 
Extrusion  Thiamine destruction  E = 99.41 kJ/mol, ΔE = 16.09 kJ/mol ^{b}  76.88  Ilo and Berghofer (1998) 
Extrusion  Amino acid loss  Lys: E = 124.71 kJ/mol, ΔE = 1.89 to 3.18 kJ/mol ^{b}  lys: 1.67 to 2.42  Ilo and Berghofer (2003) 
 Arg: E = 67.60 kJ/mol, ΔE = 0.81 to 1.36 kJ/mol ^{b}  Arg: 1.25 to 1.46  
 Cys: E = 74.33 kJ/mol, ΔE = 1.75 to 2.95 kJ/mol ^{b}  Cys: 1.61 to 2.27  
Extrusion  Water solubility index increase  E_{0}= 94.99 kJ/mol ΔE = 77.96 kJ/mol ^{c}  8.13 × 10^{8} to 3.14 × 10^{10}  Lei and others (2005) 
Extrusion  Inactivation of trypsin inhibitor  E_{1}= 200 kJ/mol, E_{2}= 158 kJ/mol  1.03 to 1.25  van den Hout and others (1998) 
 ΔE = 0.1 to 0.7 kJ/mol ^{d}   
Shear treatment  Inactivation of αamylase  E = 240 kJ/mol ^{e}, ΔE = 6.77 kJ/mol ^{f}  T = 90 °C: 9.41  van der Veen and others (2004) 
  T = 110 °C: 8.37  
Different from the commonly used linear relationship ΔE = Vτ, Eq. (17) shows that the reduction of the activation energy for αamylase inactivation was less than linear with increasing shear stress (m < 1). A possible explanation given by van der Veen and others (2004) is that the αamylase is somewhat flexible and allows a certain alignment to the shear field. Alignment to the shear field reduces the hydrodynamic diameter of the enzyme, as a result of which the enzyme becomes less sensitive to the shear forces.
Application of the Basedow model
Shearinduced rate constants can be modeled by the Basedow model, while thermal induced by the Arrhenius relation. The problem is how to combine these 2 effects when both shear and thermal effects contribute to rate constants? One approach is that of Zhurkov and Korsukov (1974) by modifying the Arrhenius equation. Wang and others (1992) proposed another approach. They assumed that the contributions of thermal and shear energy to starch conversions are additive based on laws of thermodynamics. The overall rate constant k is equal to the summation of shearinduced rate constant k_{S}, thermally induced rate constant k_{T}, and shear and thermal interaction term.
 (19)
When k_{T} is sufficiently small to be neglected, Eq. (19) is reduced to Eq. (4) and was used by Wang and others to model shearinduced starch conversion in extruder and capillary rheometer (Wang and others 1992; Zheng and Wang 1994; Gogoi and others 1995; Wang and Zheng 1995). In their reports, Wang and others took the molar volume of anhydroglucose units in the starch–water mixture as the activation volume V in Basedow model, V = 16200/(P_{s}ρ_{s}+ (100 − P_{s})ρ_{w}), where P_{s} and P_{w} are the weight percentage of starch and water, respectively; ρ_{s} and ρ_{w} are the density of starch and water, respectively.
The activation energy for starch conversion by shear in extruder (5.10 to 19.19 J/mol, Zheng and Wang 1994; 167.4 J/mol, Gogoi and others 1995) and capillary rheometer (272.1 to 1023.8 J/mol, Zheng and Wang 1994) is about 2 or 3 orders of magnitude lower than that by heating only at extrusion temperatures and moisture contents (141.2 to 185.0 kJ/mol, Wang and others 1989). This suggests that shear energy is orders of magnitude more efficient in initiating the conversion of starch to cooked starch when compared with thermal energy. An interesting finding is that the activation energy for starch gelatinization in absence of shear from Cai and Diosady (1993) (45.6369 kJ/mol) is much lower than that for starch conversion also in absence of shear from Wang and others (1989) (141.2 to 185.0 kJ/mol). The starch gelatinization degree was determined based on formation of a blue iodine complex by released during gelatinization, while starch conversion degree was determined based on the enthalpy change of starch using a differential scanning calorimeter. Maybe, the difference in determining methods leads to the discrepancy in activation energy.
The results of Zheng and Wang (1994) indicated that shear activation energy for starch conversion decreases with temperature increase, and their relation can be expressed byE_{S}= A exp (BT). For starch extruded in a rheometer at 30% moisture content, coefficients A and B are 1.740 × 10^{6} kJ/mol and −0.05068 k^{−1}, at 35% moisture content, 4.443 × 10^{6} kJ/mol and −0.0555 k^{−1}. The exponential dependence of E_{S} on temperature indicates that thermal effect still plays an important role in the shear conversion of starch. In addition, multiplepass extrusion in a capillary rheometer showed that shear activation energy increases with shear rate. An unexpected finding of Zheng and Wang (1994) is that the shear activation energy from extruder treatment (8.0 to 19.2 J/mol at 80 to 110 °C and 30% moisture content) is much lower than that from rheometer treatment (464.0 to 1116.1 J/mol at 21 to 50 °C and same moisture content). Although the temperature for the rheometer treatment is lower, it is difficult to attribute the much higher activation energy from rheometer treatment to the lower temperature only.
Application of a simplified Basebow model
If the range of shear stress in the Basedow model is small, the exponential function for rate constants is simplified to a linear function of shear stress:
 (20)
where k′ is a constant. Davidson and others (1984) used this linear relation to model starch degradation during extrusion. Assuming n = 1, they derived the following equation from Eq. (1)
 (21)
where C =[η]/[η]_{0}, d is a constant. [η], [η]_{0} are the intrinsic viscosity of starch after and before extrusion, respectively.
Diosady and others (1985) assumed that only fully cooked starch was susceptible to shear degradation and modified Eq. (21) by introducing gelatinization degree of the extruded starch to give
 (22)
While determining shear stress, Diosady and others (1984, 1985) calculated apparent viscosity using the model of Harper and others (1971) and shear rate using NπD/H (D: extruder screw diameter, H: screw channel depth). Results showed that, for starch with 20% to 25% moisture content, k′= 1.7 × 10^{−6} m^{2}/Ns and d =−0.05 (Davidson and others 1984), with 25% to 30% moisture content, k′= 0.39 × 10^{−6} m^{2}/Ns, and d =−0.08 (Diosady and others 1985). This indicated that lower moisture content leads to a pronounced response of degradation rate constant to shear stress.
It is known from Eq. (11) that the term τG^{−0.51} in Eq. (23) is equal to . Its value can be calculated from the material temperature, moisture content, and screw speed for a given extrusion run. The values of k′G^{0.51} and d were obtained by fitting starch degradation data to Eq. (23), being 6.01 × 10^{−7} m^{2}/Ns and −0.559, respectively.
Assuming n = 1, the following starch gelatinization kinetic model is obtained by substituting Eq. (6) in Eq. (1):
 (24)
This model can predict the degradation of extruded starch based on material temperature, feed moisture, extruder screw speed, and starch gelatinization degree.
Equation (20) was used in an alterative way by Cha and others (2003) to describe shear effects on thiamin retention in extruded wheat flour. They obtained the following empirical model by regressing the shear history on thiamin retention:
 (26)
where b_{1} and b_{2} are constants, R_{Φ} is the thiamin retention when only shear history Φ () is considered, R_{Φ} = C_{Ω,Φ}/C_{Ω,Φ=} _{0}, C_{Ω,Φ} is the thiamin concentration after extrusion, C_{Ω,Φ=} _{0} is the thiamin concentration after a treatment where the thermal history Ω is equal to that in extrusion and Φ= 0. Then
 (27)
Differentiating both sides with respect to time, we get
 (28)
At relatively low reaction conversions, C_{Ω,Φ}≈ C_{Ω,Φ=} _{0}, then Eq. (28) can be written as
 (29)
Clearly, Eq. (29) suggests that thiamin retention as a function of shear history follows a 1storder kinetic model, and its rate constant is . Equation (20) shows that the rate constant is k′τ. When η is constant, also noticing , Eq. (29) is in line with Eq. (20).
Bueche and Zhurkovlike models
In the Basedow and Zhurkov models, the shear energy input is expressed as the product of shear stress and activation volume. Another way to quantify the shear effects in a thermomechanical treatment, especially in extrusion cooking, is specific mechanical energy (SME). SME is defined as shear energy per unit weight of treated material. It reflects the thermomechanical history of the material during the entire mechanical treatment. To overcome the constraint by using only product temperature, Lei and others (2005) constructed an SME model (Eq. (30)) by adapting the Arrhenius equation, and SMEArrhenius models (Eqs. (31) and (32)) by incorporating SME into the Arrhenius equation to describe the increase in water solubility index of rice flour after extrusion.
 (30)
 (31)
 (32)
where E_{SME} is SME constant, E_{T} is temperature constant, E_{SMET} is SME and temperature constant. Regression analysis showed that, compared with the Arrhenius relationship, all these 3 models gave improved correlation coefficients.
Equation (30) has a similar form of the Basedow model, except that τV is replaced by SME in this case. So, Eq. (30) can be called Basedowlike model. Equation (32) can be rewritten as:
 (33)
where E_{0}= E_{T}R. When E_{SME} is negative, term (−RTE_{SME})/SME can be regarded as the ΔE in the Zhurkov model. So, Eq. (32) can be called as Zhurkovlike model. In this case, ΔE is a function of temperature, SME and E_{SME}. After copying the Figure15 of Lei and others (2005) into Photoshop 5.0, we estimated the extrusion temperatures of the run 1 and its corresponding rate constants with the aid of the ruler in the operating window. The obtained data were regressed using Eq. (33) with taking (RT·E_{SME})/SME as one parameter. The resulting E_{0} and ΔE are 94.99 and 77.96 kJ/mol, respectively. The calculation of k/k_{T} (see Table 1) showed that mechanical effect dominates the water solubility index increase. However, Lei and others (2005) demonstrated that the correlation coefficients with the Arrhenius equation, even lower than those of Eqs. (30) to (32), were still considerable (0.87). This discrepancy may suggest that the Zhurkov theory is not valid for the very complicated event such as water solubility increase, which involves starch degradation and gelatinization, and even protein degradation.
Maximum stressbased degradation model
As for shearinduced starch degradation, Davison and others (1984) and Diosady and others (1985) related the extent of degradation to the product of shear stress and time. This means that starch degradation is time dependent. Contrary to these studies, the fundamental investigations of Grandbois and others (1999) and Beyer (2000) suggested that breakdown of a covalent bond in an amylose molecule takes place almost instantaneously in the case that the force applied exceeds the bond strength. Inspired by this finding, van den Einde and others (2004a) proposed that shear degradation of starch during a thermomechanical treatment will occur only when the shear stress exceeds a critical value τ_{cri}. The degradation is maximum stress dependent, other than time dependent, and follows an exponential relation with the maximum stress.
 (34)
where X_{M} is relative intrinsic viscosity of the molten fraction of the treated starch due to mechanical breakdown (=[η]/[η]_{0}), c_{1} and c_{2} are constants, τ_{max} is the maximal shear stress occurring during a treatment. van den Einde and others (2004b) indicated that, at 43% moisture content, c_{1}= 1.46, c_{2}=−1.63 × 10^{−5}, τ_{cri}= 2.32 × 10^{4} N/m^{2}; at 30% moisture content, c_{1}= 1.15, c_{2}=−2.18 × 10^{−5}, τ_{cri}= 7.74 × 10^{3} N/m^{2}.
The above model gave a good prediction of the relative intrinsic viscosity of extruded waxy corn starch with assuming that shear and elongation in an extruder have the same effect on the molecular breakdown and that the combined effect of simultaneous shear and elongation be equal to the effect of the larger one (van den Einde and others 2005). In another work, van den Einde and others (2004c) proposed a molecular scale model that relates the molecular breakdown of waxy corn starch with the maximum shear stress on the polymer and molecular structural characteristics. The application of this model to their experimental data yielded a fractal dimension of 1.9 of amylopectin and a CO bond strength of 272 kJ/mol, which were in good agreement with literature data.
van den Einde and others (2004b) indicated that the critical stress necessary for the degradation of waxy corn starch with 30% moisture content is 2.32 × 10^{4} N/m^{2}. This value is very close to the shear stress threshold (2.5 × 10^{4} N/m^{2}) above which αamylase is rapidly deactivated in a plain shear field (van der Veen and others 2004). The minimum stress necessary for the conversion of wheat starch with 30% moisture content at 100 °C is a little lower (1.0 × 10^{4} N/m^{2}, Zheng and Wang 1994). With temperature increase (Zheng and Wang 1994) or moisture decrease (van den Einde and others 2004b), the minimum stress increases. The critical shear stress beyond which the viability of Chaetoceros muelleri in solution drop sharply is 4 orders of magnitude lower (1 to 1.3 N/m^{2}, Michels and others 2010). Probably, having a larger size than starch and αamylase makes C. muelleri prone to be deformed and lose viability.