# Rheological Study of an Extruded Fish Diet with the Addition of Hydrolyzed Protein Flour
## 3 Results  
### Rheological Behavior Models  
The density values in the molten state at different percentages of hydrolyzed protein flour (HPF) are presented in Table 4. The results (\(p<0.05\)) did not show significant differences. Therefore, the mean melt density of 1.176 g/cm\({}^{3}\) was used to calculate the volumetric flow rate and the corresponding determination of the shear rate (\(\gamma\)) in the evaluated treatments. The density obtained was lower than that obtained by researchers [19], who reported a density of 1.253 g/cm\({}^{3}\) in starch mixtures and isolated soy protein in different percentages. Table 5 shows the values of humidity, extrusion temperature, and percentage of hydrolyzed protein flour of the 27 treatments studied and the values of adjustment of their rheological behavior to the models of Law of Power, Casson, and Bingham. The coefficient of determination (R\({}^{2}\)) and the standard error (E.M) were determined for each model. Table 5 presents three models of rheological behavior according to different processing conditions. The standard error is observed for each model, where the lowest values correspond to the Power Law model, mainly due to lower hydrolyzed flour content (15%). Therefore, this is the model that best fits the behavior of the experimental data. The coefficients of determination and standard error for the Casson and Bingham models that incorporate the term \(\tau_{\text{or}}\) have the slightest adjustment concerning the experimental values. The Power Law model predominates in the analysis of viscosity behavior in extrusion processes [17]. Similar studies with mixtures of soybeans and corn for animal feed obtained a better fit with the Power Law model [18]. Other studies evaluated yield stress (\(\tau_{\text{or}}\)) [20; 21], described in the Herschel-Bulkley model, obtaining as a result that this parameter is related to a structuring of the material, possibly by the presence of semicrystalline structures. However, crystals could be unstructured during the shear treatment before the viscosity measurement, generating controversy regarding the results. Some authors recommend complementing the analysis with X-ray diffraction to observe changes in crystallinity [22]. \begin{table}
\begin{tabular}{c c} \hline
**Treatment** & **Density in the Molten State (g/cm\({}^{3}\)) \(\ast\)** \\ \hline HPF 15\% & 1.176 \(\pm\) 0.003 \\ HPF 25\% & 1.174 \(\pm\) 0.002 \\ HPF 35\% & 1.179 \(\pm\) 0.002 \\ \hline \end{tabular} \(\ast\) The number of replicas was 3.  
\end{table} 
Table 4: Density values in the fade state. \begin{table}
\begin{tabular}{c c} \hline
**Factors** & **Levels** \\ \hline Temperature (\({}^{\circ}\)C) & 115 \\  & 125 \\  & 135 \\ \hline  & 20 \\ Moisture (\%) & 22 \\  & 24 \\ \hline  & 15 \\ Hydrolyzed protein flour (\%) & 25 \\  & 35 \\ \hline \end{tabular}
\end{table} 
Table 3: Factorial design 3\({}^{3}\) applied to rheological evaluation. In any of the cases reported above, the information was finally interpreted by the Power Law model, which for the case study was the one that presented the best fit, verified by the values of R\({}^{2}\) and E.M. Similarly, [PERSON]'s rheological model, which explains the behavior of extruded polymeric matrices, assumes for its use that the flow must be fully developed and that the fluid behaves according to the Power Law [7; 23]. Under the conditions of this experiment, the mixture presented a pseudoplastic behavior with a decrease in apparent viscosity as the shear rate increased, as indicated by the value of \(n\) in the Power Law model. Various authors [10; 24; 25; 26] reported the use of the Power Law model to describe flow behavior using in-line and off-line rheology and experimental conditions similar to the present experiment. Similar studies investigated the rheological behavior of a formulation for animal feed [18]. They studied the rheological behavior of mixtures of native and pregelatinized wheat starch.