A Novel Equation Correlating the Rheological Properties of Some Commercial Tomato Ketchups
I. Stanciu*, A. Messaâdi, O. DíezSales, S.S. AlJameel, E. Mliki, J.V. Herráez, N. Ouerfelli
Abstract
This article is focused on evaluating the rheological properties of tomato ketchup. The purpose of the paper was to show the importance of knowledge about the rheological properties of ketchup and to find new more adequate equations of the temperatureviscosity dependence. All measurements were performed under the same temperature conditions at approximately 25 °C and shear velocities between 5.46 and 700.41 s^{1}. Ketchup is a nonNewtonian material, so the apparent viscosity was measured.
We found out that the apparent viscosity decreased exponentially with increasing temperature, so the Arrhenius equation is valid. Ketchup fluidity increases exponentially with temperature, which leads us to propose an original logarithmic expression correlating the shear stress and the square root of shear rate with only two adjustable parameters with some physical significance. The model is compared with some previous ones showing that the calculated rheological characteristics can be used to design the technological equipment or containers for the distribution of the product to the endusers. A comparison between parameter values of different fluids can contribute to the knowledge of the flow behavior is also important for the development of new recipes and the direct qualitative evaluation of the products.
Keywords: Tomato ketchup, Modeling, Pseudoplastic, Viscosity, Powerlaw, Rheology, Correlation.
Introduction
Tomatoes are consumed worldwide, and represent an important part of the human diet and a rich source of bcarotene, folates (vitamin B9), potassium, vitamin C, vitamin E, flavonoids, and lycopene (Fakirabad & Abedi, 2019). Tomatoes can be consumed fresh, but important quantities are processed to obtain different products, including tomato juice, paste, puree, ketchup, and sauces (Bourne, 2002; Sahin & Ozdemir, 2004; Fortuna et. al., 2004; Śmigielska et. al., 2013).
To meet consumers' quality requirements, some quality parameters of tomato saucebased products are to be considered very important, such as rheological and appearance indicators (i.e. consistency and color). It was reported that tomatobased products characterized by high viscosity and red color are preferred by consumers (Bourne, 2002; Sahin & Ozdemir, 2004; Fortuna et. al., 2004; Śmigielska et. al., 2013).
Different ingredients (e.g. additives) are allowed to be included in tomato sauce and have been reported to affect multiple properties of tomato sauce. Longchain polymers, such as hydrocolloids (polysaccharides and proteins), are extensively used by the food industry in different systems as soups, beverages, desserts, ketchups, sauces, or salad dressings, due to their functionality as thickening (Bourne, 2002; Sahin & Ozdemir, 2004; Fortuna et. al., 2004; Śmigielska et. al., 2013).
This study aims to explore and model the effect of different ingredients on commercial textured tomato ketchup with modified regular corn starch in terms of rheological properties.
The knowledge of the physical qualities of food materials is necessary in the processes controlled in their manufacture, operation, and handling. In order to determine the quality of food materials, it is important to know the physical properties such as mechanical, thermal (Kubík Ľ et. al., 2014; Kubík Ľ et. al., 2017; Dizayee & Maaroof, 2019), thermo physical (Božiková et. al., 2010; Glicerina et. al., 2013; Micić et. al., 2014; Hailu et al., 2018), and rheological (Glicerina et. al., 2013; Hlaváč et. al., 2012; Bikić et. al., 2012; Bukurov et. al., 2012; Diósi et. al., 2014).
Tomatoes can often be eaten fresh, but there are several varieties that are processed into various products, such as tomato sauce, soup, pasta, puree, juice, ketchup, and salsa (Tan et.al. 2015). Some researchers have investigated the benefits of tomatoes in reducing the risk of heart disease, improving bone health, and lowering the risk of cancer (Tan et.al., 2015; Alexandrovich et al., 2018). Through the industrial processing of tomatoes, they lead to a great variety of products. The most relevant products are as follows: concentrated tomato products, either in the form of puree or paste, depending on the percentage of natural soluble solids; pizza sauce, made from shells and seeds; tomato powder, in the form of dehydrated concentrated tomatoes; tomatoes peeled, whole or cut into cubes; ketchup, tomato sauce seasoned with vinegar, sugar, salt and a few spices, etc. (Ruiz et. al., 2009). Studies of changes in tomato powder during storage at different storage temperatures have been evaluated for five months by (Liu et. al., 2010). Some researchers have found that higher storage temperatures have a significant effect on the qualities of tomato puree, while lower temperatures had a smaller effect. Tomato puree has many benefits, including ease of packing, transport, and mixing. In addition, tomato puree can be used as an ingredient in many foods, mainly soups, sauces, and ketchup (Liu et. al., 2010).
The rheological properties of tomato products are considered as one of the most important quality attributes because they influence the processing parameters of the products, in particular, the flow properties during transport, as well as the acceptability of consumers (Torbica et. al., 2016). Investigation of the rheological properties and microstructure of tomato puree using continuous highpressure homogenization was performed (Tan et. al., 2015). The influence of the rheological and structural characterization of tomato paste on the quality of ketchup has been investigated by (Bayod et. al., 2008). According to (Sharoba et. al., 2005), ketchup is a nonNewtonian, semisolid, timeindependent liquid with a defined shear rate. The effect of temperature on fluid viscosity at a specified shear rate could be described by the Arrhenius equation, where the apparent viscosity decreases as an exponential function with temperature (Sharoba et. al., 2005). According to (Bayod et. al., 2008), the viscosity of tomato ketchup is a major quality component for consumer acceptance. Some parameters affect the flow behavior of tomato ketchup, including the quality of the raw material (eg tomato paste) and the processing conditions. In order to obtain a constant and desirable quality in the final product (i.e. ketchup), a highquality paste is required and continuous control and adjustment of the variables for its processing (Bayod et. al., 2008). Tomato ketchup could be presented as a concentrated dispersion of insoluble matter in aqueous media, and its complex structure determines that it exhibits a nonNewtonian, thin shear, and timedependent rheological behavior (Torbica et. al., 2016). Ketchup can be included in one of the most consumed spices, obtained from either fresh or concentrated tomatoes, such as tomato puree and tomato paste (Mert, 2012). Concentrated tomato paste is usually stored and used as an intermediate with water and other ingredients to be reconstituted into end products such as ketchups and sauces (Zhang et. al., 2014).
The effect of temperature and concentration on the rheological properties of ketchup processed cheese mixtures was analyzed using a constant and dynamic oscillatory shear by (Yilmaz et. al., 2011). The temperature dependence of the apparent viscosity at a specified shear rate (50 s^{1}) could be described by the Arrhenius model (Yilmaz et. al., 2011). Some researchers have analyzed linear creep and recovery of ketchup processed cheese mixtures using mechanical simulation models based on temperature and concentration (Karaman et. al., 2012). The authors found that the effect of temperature on whistle phase parameters (for example, viscosity) could be successfully described by the Arrhenius relation. The authors also determined the activation energies for ketchup analyzed and described the effect of temperature and concentration on the activation energy (Karaman et. al., 2012; Sahin et. al., 2004; Bird, 1960). Some have used highpressure microfluidization to improve the physical properties and lycopene content of ketchup products (Mert, 2012). Torbica et al. (Torbica et. al., 2016) evaluated the nutritional, rheological, and sensory properties of tomato ketchup with a high content of natural fibers obtained from grape fruit.
In the present work, we propose an original logarithmic expression correlating the shear stress and the square root of shear rate with only two adjustable parameters with some physical significance. The model is compared with some previous ones (Scott et. al., 1939; Reiner et. al., 1933; Ostwald , 1929; Herschel et. al., 1926; Tang et. al., 2004; Nagarani et. al., 2012; Mernone et. al., 2002; Ponalagusamy et. al., 2011; Chaturani et. al., 1982; Chaturani et. al., 1986; Hlaváč et. al., 2019; Vercet et. al., 2002; Herrmann et. al., 2013; Figura et. al.., 2007; Belda et. al., 2004; Heyes et. al., 1994).
Experimental part, Procedure (Materials and Methods of research)
The studied sample is a mixture, with the same mass composition, of about five different commercial textured tomato ketchup thickened with modified regular corn starch.
The mixing is realized using a blenderhomogenizer (IKA Dispersers ULTRATURRAX ) at about 25 °C (1500 rpm and 15 min) for homogenization and to obtain a creamy and smooth texture.
The mixture was stored for 7 days to reach the equilibrium. Following this period, viscosity measurements were made. Rheological measurements were carried out with a Brookfield Digital DVII + rotary viscometer. Taking into account the experimental characteristics involved (cylindrical spindles of different standard sizes were used); shear rates of between 5.46 and 700.41 s^{1} were employed to determine viscosity. The measurements of apparent viscosity were initially made with the nonagitated dispersions in increasing order of shear rate and a time intervals of 10 s, which enabled the corresponding up curve rheograms to be plotted. The temperature was maintained at 25 ± 0.5 °C throughout the measurement process by immersing the flasks in a thermostatic bath.
To reduce the temperature gradient between the Rheometer and the laboratory room for making the Rheometer thermostat block working at ease and giving more precision for the working fixed temperature, we maintained the temperature of the laboratory room also at 25°C using a split airconditioner 24000 btu (Samsung Digital Inverter).
Results and Discussion
The results of the rheological properties of the studied tomato ketchup are summarized in Table 1. As shown in Table 1, the shear rate increases with the shear stress by analyzing the commercial ketchup. The relationship between rheological parameters for a nonNewtonian fluid is described principally by the following equation:
(1)
The values are in general an average of triplicate measurements; that is why some digits are written. We notice that our data behavior is roughly similar to those of the ketchup thickened with modified regular corn starch (symbolized by RSK) obtained in the literature under approximately similar conditions (Śmigielska et. al., 2013; Scott et. al., 1939).
The increase of shear stress (τ) and the decrease of apparent viscosity (η) versus the shear rate () (Table 1, Figure 1), show that the tomato ketchup is a nonNewtonian complex fluid and it follows the behavior of the pseudoplastic or shearthinning fluid (Reiner et. al., 1933; Ostwald, 1929).
In addition, the 3Dplot of the three rheological parameters (Figure 2), shear rates ( ), shear stress (τ), and apparent viscosity (η) gives an overview of monotonous mutual dependence. This global vision of the trends of scatter points and different types of curvature allows us to propose an adequate mathematical expression taking into account the different behaviors at very low and very high share rate ()values, such as the power law and the parabolic branch in ()direction, respectively.
We notice that when the shear stress (τ) and the shear rate () tend simultaneously to zero (Eq. 1), the ratio (τ/) representing the apparent viscosity (η) has an indeterminate form in zero for which the lifting of the indetermination gives a nonzero limit value of the apparent viscosity of 13.00 Pa·s (Table 1). This operation required a nonlinear regression in a 9^{th} degree polynomial with a correlation coefficient R greater than 0.99999, for which we have changed manually and slightly the obtained free value of the initial apparent viscosity until we observe the maximum of the correlation coefficient R. We have tested the limiting value of the derivative (∂τ/∂) at zero of () for confirming the obtained nonnull limiting apparent viscosity.
Power low dependence and theoretical background
The most used model for complex fluids it the famous power law dedicated principally to the Ostwaldde Waele model (Bourne, 2002; Sahin et. al., 2004; Fortuna et. al., 2002), which is simply expressed as follows:
(2)
where K is the flow consistency index (SI units Pa·s^{n}), is the shear rate or the velocity gradient perpendicular to the plane of shear (SI unit s^{−1}), n is the flow behavior index (dimensionless), and τ is the shear stress (Pa).
We remember that the powerlaw fluids can be subdivided principally into three different types of nonNewtonian fluids based on the value of their flow (n) behavior index such as: pseudoplastic or shearthinning (n<1), Newtonian fluid or shearthickening fluids (n =1), and dilatants or shearthickening (n<1),
The second used model (Eq. 3) is due to Herschel–Bulkley (Herschel et. al., 1926; Tang, et. al., 2004) in which the strain experienced by the fluid is related to the stress in a complicated, nonlinear way.
(3)
where is shear rate, K is the flow consistency index (SI units Pa·s^{n}).
The third used model (Eq. 4) is due to the Casson model (Nagarani et. al., 2012; Mernone et. al., 2002; Ponalagusamy et. al., 2011; Chaturani et. al., 1982; Chaturani et. al., 1986; Hlaváč et. al., 2019) characterized by yield stress (τ_{0}) and it has been widely used for modeling certain biological fluids.
(4)
where τ_{0} is yield stress and η_{C} the Casson plastic viscosity. We note that the Eq. 4 is valid for (τ ≥ τ_{0}) and we must consider = 0 when τ ≤ τ_{0}.
Modeling Methodology
Due to the great varieties of rheological fluids and its very different behaviors for which some existing models succeed more than others due to some specific features for each studied complex fluid, we suggest an original empirical equation with logarithm and power mathematical form (Eq. 5). In fact, regarding the power behavior of the shear stress (τ) of the tomato ketchup against the shear rate () with a flow behavior index (n) less than unit due to the pseudoplastic (shear thinning) behavior, we propose an original empirical two parametersequation esteeming the abovementioned characteristics with a global form expressed as follows:
(5)
where f_{0} and ε are two free adjustable parameters, and x and Y represent the shear rate ( ) and the shear stress (τ) respectively.
In fact, at high shear rate values () the plot of the logarithm form exhibits a parabolic branch and the variation can describe eventual yield stress characterizing the studied pseudoplastic sauce. On the other hand, at very low shear rate values (), the asymptotic expansion of the expression (Eq. 5) gives a dominant term (x^{0.5}) expressed as follows:
(6)
Then the common factorization leads to the following simplified equation:
(7)
For which this obtained expression shows principally a powerdependence (n_{0}=0.5) and the polynomial in square brackets adjusts (or corrects) somewhat the true powerdependence (with an increment δn) of the studied rheological system, and gives then a global power (n=0.5±δn) correcting well like the (Eqs. 24), the variation of the shear stress (τ) vs. the shear rate values () around zero for the studied sauce.
Moreover, similarly to the existing power laws (Eqs. 24), which exhibit a divergence of the derivative of (τ) vs. () characterized by quasivertical asymptote at very low shear rate values () values, the suggested model satisfies also this property where the asymptote expansion can be expressed around ≈0^{+} as follows:
(8)
where the (x^{0.5})term in the denominator is responsible for the observed divergence.
In addition, taking into account the dimensional analysis on physical quantities described in Eq. 5, we have tried to upgrade the proposed equation to a semiempirical model by giving explicit sense and physical meaning of each adjustable parameter. So, Eq. 5 can be reexpressed as follows:
(9)
Where τ_{0} represents somewhat yield stress or critical shear stress, and represents a critical shear rate and can be close to the equivalent reciprocal flow consistency index related to the powerlaw (Eq. 2), for which we can define an extra parameter such as a critical apparent viscosity (η_{0}) expressed as follows:
(10)
In the experimental section, we have chosen to test the suggested model on a blended and homogenized mixture of five different commercial textured tomato ketchups thickened with modified regular corn starch in order to obtain average values of obtained optimal free adjustable parameters, which can be considered as a certain reference for future discussion and interpretation on new parameters values of individually studied ketchup or why not for any other pseudoplastic plastic systems.
So, for future application and assessment of the present suggested model on other individual tomato ketchup systems (Vercet et. al., 2002; Herrmann et. al., 2013; Figura et. al.., 2007; Belda et. al., 2004; Heyes et. al., 1994), the different obtained parameters’ values can give a specific discussion, criterion on each feature of the studied system especially when the other ones are fixed. We can cite, for example, the comparison of parameters’ values between studied sauces derived from different botanical origins used different thickeners such as corn, potato, maize, waxy maize starches, etc. Moreover, the effect on the parameters values of the added ingredients, pH, texture, etc. can be an interesting subject of discussion and interpretation of the amount of obtained parameters.
Model application and comparison
Table 2 represents the results of the fitting using the nonlinear regression method for the suggested models and the threeabovementioned ones. In the same way, Figure 3 shows the variation of the experimental and the four different calculated shear stresses (τ) versus shear rate ().
Regarding the values of the standard deviation (SD) and standard error (SE), we can conclude that our model is reliable and well describes the variation of the shear stresses (τ) versus shear rate ( ).
In the same context, Table 3 summarized the obtained parameters values of different models. We note that our results are close to those of the ketchup thickened with modified regular corn starch (symbolized by RSK) obtained in the literature under the same conditions [4, 36].
Mathematical analysis of the model
Starting from our suggested model (Eq. 9) containing a halfpower dependence into the argument of logarithm and expressed by the square root of shear rate ( ), we have thought to test another power (n) other than 0.5. So, we have rewritten the suggested model to a new power form expressed by Eq. 11.
(11)
In a new fitting in nonlinear regression, when we take both the three parameters (τ_{0},,n) as free adjustable parameters in the fitting with nonlinear regression, we observe that there are infinite possible optimal solutions and the software’s iterations move to the sense of the decrease of nvalues and the increase of (τ_{0}, _{0}) for each round of iteration.
Nevertheless, when the parameter is fixed with high values (from 10^{3} to 10^{15} or more), we observe that the discrepancy is satisfactorily reduced and the power (n) tends to a limiting value about n_{lim} ≈ 0.3295 (Figure 4). We can then use this option to obtaining a very satisfactory agreement but with a very high couple values of (τ_{0},_{0}), which are devoid of any physical significance. The statistical parameters values become about (SD≈1.578, SE≈0.298), which are best compared with those of Table 3 for fixed n = 0.5. Nevertheless, we note that these values of SD and SE are identical to those of the Ostwaldde Waele model (Table 3, Eq. 2).
Considering the very high values of (_{0}) compared to those of experimental shear rate () given in Table 3, we can consider that the ratio (/) is very low and the expression of Eq. 11 can be written in an asymptotic expansion expressed as follows:
(12)
And the common factorization leads to the following simplified equation:
(13)
This clearly joins the expression of the Ostwaldde Waele model (Eq. 2). That is why we found the same value of (n=0.3295) in Table 3 and we can see easily the equality between the Kvalue of Eq. 2 and ( )value of Eq. 13 which is equal to 19.50 (Table 3).
Regarding these complications on the flow behavior index (n) for different models, we saw good to think about the real flow behavior index ν(), which depends on each value of () reflecting then the shear behavior for each shear rate value. Then, we can call it the current shear rate (ν), which can be calculated through the derived function defined by Eq. 14 and deduced from the transformed derivative of the Ostwaldde Waele powerlaw expression (Eq. 2).
(14)
If we suppose that the rheological behavior follows our suggested model (Eq. 9), the current shear rate (ν) can be calculated using the following expression:
(15)
Mathematically, we can see that it is about a descending function starting from 0.5, which is confirmed by the asymptotic expansion of Eq. 7 and tending slowly to zero at infinity.
Experimentally, the plot of the curve representing ln(τ) as a function of ln() for the studied commercial textured tomato ketchup is shown by (Figure 5). We observe a quasilinear dependence at high values of shear stress () showing normal power dependence very close to the Herschel–Bulkley model (Eq. 3), while at low values of shear stress () the rheological behavior is somewhat complicated by some change of curvatures.
Nevertheless, for the experimental data representation, the complicated curvature in the low shear rate region and the lack of precision and experimental data number necessitates using fittings to a high polynomial degree. So, remedying to this inconvenience, we can also simply determinate otherwise the values of the current shear rate (ν) by fitting the plot of the curve (Figure 6) representing the natural logarithm of the experimental apparent viscosity ln(η) against the natural logarithm of the shear rate ln() in polynomial nonlinear regression. Due to the double dependence (Eq. 1) of (η) on (τ) and (), the apparent viscosity becomes an endogen variable and gives more simplified and low curvature (Figure 6) at low values of shear stress ( ) permitting then an optimal fitting with less degree polynomial.
Otherwise, the current shear rate (ν) by the following equation:
(16)
So, the derivation of the natural logarithm of the experimental apparent viscosity ln(η) against the natural logarithm of the shear rate ln() represented by Figure 6 gives interesting results on the variation of the experimental and calculated (Eq. 9) current shear rate (ν) with the shear rate (). Figure 7 shows that experimental values of ν decrease from 1 confirming then the abovementioned nonzero limit value of the apparent viscosity (13 Pa·s) while the calculated ones decrease from 0.5 which is predicted by the asymptotic expansion of Eq. 7. The spectacular finding is that the experimental current shear rate exhibits a curious minimum of about 0.2731 at a low shear rate (≈30s^{1}) and then increases to reaching a pseudoplateau with a limiting value of about 0.3553 and it continues to increase very slightly with a negligible variation. We note that this new finding of the variations of the current shear rate (ν) can open new interesting theoretical researches to explain what happens as structural changes or rheological phenomena in molecular layers, shear plans, etc.
We conclude that our suggested model (Eq. 9) has the advantage to give a variable current shear rate (ν) against the shear rate () while the Ostwaldde Waele (Eq. 2) and Herschel–Bulkley (Eq. 3) models give a global constant shear rate (n). For this purpose, we think about the value of the experimental global shear rate (n_{g,exp}), as well as the calculated global one related to the suggested model (n_{g,cal}). So, we must calculate the average value of the ()function over the interval of [0,700.41] using the integral expressed as follows:
(17)
where _{min} =0, _{max}=700.41, and ν(γ) should be expressed by a wellchosen polynomial with an optimal degree or directly calculated by the software. For our case, we found the experimental global shear rate (n_{g,exp}) equal to 0.3441 and the calculated global one (n_{g,cal}) equal to 0.3126. Consequently, we see that the two obtained values are near that of Ostwaldde Waele (n=0.3295), which shows a good agreement with the experimental data (Figure 3). Note, regarding the variation of the new concept of current shear rate (ν) against the shear rate (), the experimental global shear rate (n_{g,exp}) is strongly affected by the studied shear rate range.
Conclusion
We have suggested a new equation correlating the shear stress (τ) against the shear rate () for the studied commercial textured tomato ketchup thickened with modified regular corn starch. The proposed expression has two mathematical characteristics, such as (a): the logarithm function has a nonexponential parabolic branch, i.e., for the large values of the shear rate (), which fits well with the variation of the experimental values of shear stress (τ). (b): the squarerooted term of the shear rate () into the logarithmic argument has the property of the power trend in the region of very low values of the shear rate (), (i.e. the mathematical function has an asymptotic expansion in powerlaw against the shear rate (), which is a good framework for the behavior of the shear stress (τ) for the case of ketchup in very feeble values of shear rate (). In addition, we have given physical significance for the adjusted parameters of the proposed empirical equation.
We conclude that this model is in good agreement with experimental data and has approximately the same quality or better than the previous models of the literature.
Regarding that, the ketchup has a pseudoplastic (thinning) rheological behavior; we notice that this model can be tested for other nonNewtonian fluids having the same rheological properties (Vercet et. al., 2002; Herrmann et. al., 2013; Figura et. al.., 2007; Belda et. al., 2004; Heyes et. al., 1994).
Disclosure statement
No potential conflict of interest was reported by the authors.
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Table 1. Experimental values of the shear stress (τ) and the apparent viscosity (η) for different shear rates ( ) of the analyzed commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure.

τ_{exp} 
η_{exp} 
s^{1} 
Pa 
Pa·s 
0.0000 
0.0000 
13.000^{a} 
5.4582 
34.433 
6.3085 
13.648 
50.194 
3.6778 
20.922 
56.671 
2.7087 
37.292 
65.901 
1.7672 
52.761 
71.302 
1.3514 
67.309 
76.682 
1.1392 
83.138 
82.070 
0.98715 
100.06 
87.456 
0.87406 
120.07 
93.607 
0.77960 
146.44 
100.64 
0.68720 
171.92 
105.16 
0.61168 
200.12 
110.54 
0.55240 
229.22 
115.94 
0.50578 
261.97 
120.57 
0.46025 
300.17 
125.96 
0.41963 
333.83 
130.68 
0.39146 
372.04 
135.22 
0.36347 
400.23 
139.07 
0.34748 
430.25 
142.92 
0.33218 
468.45 
146.78 
0.31333 
500.29 
150.64 
0.30110 
507.56 
151.19 
0.29786 
536.67 
154.49 
0.28787 
573.97 
159.10 
0.27719 
600.35 
162.18 
0.27014 
650.38 
166.21 
0.25555 
700.41 
170.93 
0.24404 
a: extrapolated value using nonlinear regression of 9^{th}degree polynomial with R> 0.99999.
Table 2. Comparison between experimental and calculated values of the shear stress (τ) or different shear rates ( ) of the analyzed commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure. τ_{cal} (Eq. 9), τ_{Ostwald} (Eq. 2), τ_{Herschel} (Eq. 3), τ_{Casson} (Eq. 4), and the corresponding standard deviation (SD) and standard error (SE).

τ_{exp} 
τ_{cal} 
τ_{Ostwald} 
τ_{Herschel} 
τ_{Casson} 
s^{1} 
Pa 

0.0000 
0.000 
0.000 
0.000 
2.0220 
46.933 
5.4582 
34.433 
27.594 
34.0228 
37.833 
55.042 
13.648 
50.194 
41.170 
46.0174 
48.148 
60.053 
20.922 
56.671 
49.222 
52.9733 
54.332 
63.416 
37.292 
65.901 
62.097 
64.0868 
64.470 
69.492 
52.761 
71.302 
70.989 
71.8497 
71.718 
74.264 
67.309 
76.682 
77.770 
77.8527 
77.407 
78.259 
83.138 
82.070 
84.016 
83.4639 
82.786 
82.241 
100.06 
87.456 
89.776 
88.7179 
87.875 
86.192 
120.07 
93.607 
95.699 
94.2106 
93.244 
90.558 
146.44 
100.64 
102.435 
100.580 
99.531 
95.929 
171.92 
105.16 
108.095 
106.040 
104.97 
100.81 
200.12 
110.54 
113.629 
111.482 
110.43 
105.92 
229.22 
115.94 
118.718 
116.582 
115.59 
110.96 
261.97 
120.57 
123.854 
121.827 
120.93 
116.38 
300.17 
125.96 
129.217 
127.416 
126.66 
122.44 
333.83 
130.68 
133.495 
131.957 
131.35 
127.58 
372.04 
135.22 
137.936 
136.754 
136.32 
133.23 
400.23 
139.07 
140.973 
140.085 
139.80 
137.29 
430.25 
142.92 
144.016 
143.464 
143.33 
141.52 
468.45 
146.78 
147.637 
147.542 
147.61 
146.79 
500.29 
150.64 
150.469 
150.774 
151.02 
151.08 
507.56 
151.19 
151.094 
151.492 
151.78 
152.05 
536.67 
154.49 
153.522 
154.302 
154.75 
155.90 
573.97 
159.10 
156.473 
157.756 
158.42 
160.74 
600.35 
162.18 
158.462 
160.110 
160.92 
164.12 
650.38 
166.21 
162.035 
164.389 
165.49 
170.41 
700.41 
170.93 
165.378 
168.453 
169.85 
176.58 

SD / Pa 
2.945 
1.578 
1.146 
10.232 

SE / Pa 
0.577 
0.298 
0.221 
1.934 
Table 3. Values of rheological model parameters used to describe flow curves of analyzed commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure. Ostwaldde Waele (Eq. 2), Herschel–Bulkley (Eq. 3), Casson (Eq. 4), suggested model (Eq. 9), and corresponding standard deviation (SD) and standard error (SE).
Model 
τ_{0} 
K 
n 
γ_{0} 
η_{0} 
SD 
SE 
Pa 
Pa·s^{n} 
 
s^{1} 
Pa·s 
Pa 
Pa 

Ostwaldde Waele 
 
19.449 ±0.384 
0.3295 ±0.0034 
 
 
1.578 
0.298 
Herschel–Bulkley 
13.103 ±0.491 
12.968 ±0.123 
0.3804 ±0.0008 
 
 
1.146 
0.221 
Casson 
46.933 ±1.468 

0.5 ±0.0 ^{a} 
793.189^{ b} ±61.406 
0.05917 ±0.00273 
10.232 
1.934 
Suggested model 
122.229 ±7.406 
 
0.5 ±0.0^{ a} 
85.089 ±16.972 
1.4365^{b} ±0.3736 
2.945 
0.577 
3.544×10^{7} ±4.67×10^{6} 
 
0.3295 ±0.0036 
10^{19} ±0.0^{ a} 
3.54×10^{12b} ±4.7×10^{13} 
1.578 
0.298 
a: Fixed parameter in the fit; b: extra parameter calculated from Eq. 1.
Figure 1. Variation of the apparent viscosity (η) versus shear rate ), in double logarithm scale, for five blended commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure.
Figure 2. 3Dplot of the three coupled rheological parameters of Eq. 1 describing the studied commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure.
Figure 3. Variation of the shear stress (τ) versus shear rate () for five blended commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure. (●): Experimental; (○): suggested model (Eq. 9); (▲): Ostwaldde Waele (Eq. 2); (∆): Herschel–Bulkley (Eq. 3), and (■): Casson (Eq. 4),
Figure 4. Variation of the pseudo flow behavior index (n) of Eq. 11 with the decimal logarithm of critical shear rate log(γ_{0}) defined in Eq. 9, for the studied commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure.
Figure 5. Variation of the natural logarithm of the experimental shear stress ln(τ) against the natural logarithm of the shear rate ln(γ) for the studied commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure.
Figure 6. Variation of the natural logarithm of the experimental apparent viscosity ln(η) against the natural logarithm of the shear rate ln() for the studied commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure. (●): Experimental; (○): Calculated by the suggested model (Eq. 9);
Figure 7. Variation of current shear stress (ν) against the shear rae () for the studied commercial textured tomato ketchup thickened with modified regular corn starch at room temperature and atmospheric pressure. (●): Experimental; (○): Calculated by fitting of ln(η) vs. ln(γ) of the suggested model (Eq. 9).