# Rheological Study of an Extruded Fish Diet with the Addition of Hydrolyzed Protein Flour
## 3 Results
### Rheological Behavior Models
The equation presents the assumption that the apparent viscosity (\(n\)) has a dependence on the Power Law with the shear rate, an Arrhenius dependence on temperature (T), and an exponential dependence on moisture content (MC) \(\gamma\)[3], assuming that the flow is fully developed through the capillary and is in laminar regime, there is no viscous dissipation or slippage of the material in the wall, there is no loss of pressure at the end of the capillary, and the fluid behaves according to the Power Law [7; 23]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{\(n\)} & \multicolumn{2}{c}{\(K_{lp}\) (Pa\(\cdot\)s\({}^{n}\))} \\ \cline{2-5}
**Fountain** & **F** & **P** & **F** & **P** \\ \hline Moisture & 159.39 & 0.000 & 317.33 & 0.000 \\ Temperature & 251.89 & 0.000 & 506.18 & 0.000 \\ HPF & 1470.92 & 0.000 & 2355.22 & 0.000 \\ Moisture \(\times\) Temperature & 1.46 & 0.242 & 19.53 & 0.000 \\ Moisture \(\times\) HPF & 7.34 & 0.000 & 105.57 & 0.000 \\ Temperature \(\times\) HPF & 4.54 & 0.006 & 175.12 & 0.000 \\ Moisture \(\times\) Temperature \(\times\) HPF & 1.42 & 0.233 & 8.24 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Analysis of variance model of Power Law. The models described for the different percentages of inclusion of hydrolyzed protein flour (15, 25, and 35%) show values with a positive sign for the term (E/R), indicating that the increase in viscosity is related to the decrease in the temperature of the product in the range of temperatures studied [3]. The increase in the consistency index \(K_{lp}\) is related to the proportional increase of hydrolyzed protein flour, behavior possibly related to more significant interaction between the macromolecules presents in the mixture [32]. The equations show an R\({}^{2}\) between 0.89 and 0.95, which allows the model to fit the data and the possibility of using the equation as a data design and estimation tool under similar test conditions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample** & **Flow Behavior Index (_n_)** & **Consistency Index (\(K_{lp}\), Pa \(\cdot\)s\({}^{\text{n}}\))** \\ \hline T1 & 0.6108 \(\pm\) 0.0009 h & 982 \(\pm\) 10 e \\ T2 & 0.72 \(\pm\) 0.04 e & 868 \(\pm\) 124 f \\ T3 & 0.743 \(\pm\) 0.009 d & 561 \(\pm\) 29 g \\ T4 & 0.67 \(\pm\) 0.03 f & 535 \(\pm\) 23 g \\ T5 & 0.74 \(\pm\) 0.03 d & 408 \(\pm\) 57 h \\ T6 & 0.82 \(\pm\) 0.02 b & 322 \(\pm\) 27 i \\ T7 & 0.79 \(\pm\) 0.02 c & 333 \(\pm\) 20 i \\ T8 & 0.849 \(\pm\) 0.005 b & 176 \(\pm\) 2 j \\ T9 & 0.976 \(\pm\) 0.012 a & 159 \(\pm\) 9 j \\ T10 & 0.62 \(\pm\) 0.02 g & 1474 \(\pm\) 132 e \\ T11 & 0.68 \(\pm\) 0.02 f & 992 \(\pm\) 81 e \\ T12 & 0.73 \(\pm\) 0.03 d & 585 \(\pm\) 99 g \\ T13 & 0.673 \(\pm\) 0.009 f & 1087 \(\pm\) 49 e \\ T14 & 0.71 \(\pm\) 0.03 e & 677 \(\pm\) 108 g \\ T15 & 0.800 \(\pm\) 0.013 b & 334 \(\pm\) 22 i \\ T16 & 0.76 \(\pm\) 0.02 c & 666 \(\pm\) 66 g \\ T17 & 0.81 \(\pm\) 0.03 b & 431 \(\pm\) 69 h \\ T18 & 0.885 \(\pm\) 0.006 a & 207 \(\pm\) 5 j \\ T19 & 0.372 \(\pm\) 0.012 m & 6362 \(\pm\) 371 a \\ T20 & 0.381 \(\pm\) 0.007 l & 4472 \(\pm\) 142 b \\ T21 & 0.441 \(\pm\) 0.004 k & 2299 \(\pm\) 43 d \\ T22 & 0.393 \(\pm\) 0.003 l & 4125 \(\pm\) 69 b \\ T23 & 0.42 \(\pm\) 0.02 k & 2983 \(\pm\) 302 c \\ T24 & 0.458 \(\pm\) 0.013 k & 2188 \(\pm\) 147 d \\ T25 & 0.47 \(\pm\) 0.02 j & 2453 \(\pm\) 290 c \\ T26 & 0.513 \(\pm\) 0.013 j & 1542 \(\pm\) 103 e \\ T27 & 0.554 \(\pm\) 0.005 l & 1014 \(\pm\) 31 e \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Mean values and standard deviation of flow behavior index (_n_) and consistency index (\(K_{lp}\)) for the studied treatments.