# Rheological Study of an Extruded Fish Diet with the Addition of Hydrolyzed Protein Flour
## 2 Materials and Methods
### Rheological Measurement Using an In-Line Slit Die Rheometer  
A Thermo Scientific (HaakeTM, MiniLab II, Dreieich, Germany) brand slit dies to type vertical line rheometer was used to measure the rheological properties of mixtures, including hydrolyzed protein flour, with an outlet opening of 1.5 \(\times\) 20 mm, with ports for a melt temperature sensor and three pressure sensors (700, 500, and 300 bar) (see Figure 1), coupled to a compact single screw extruder (Haake Polylab OS Rheemex 19/25 OS, Dreieich, Germany), which consists of a 475 mm long (L) and 19 mm diameter (D) worm with a 25: 1 L/D ratio, a maximum working temperature of 450 \({}^{\circ}\)C, a maximum speed of 250 min\({}^{-1}\), and a maximum torque of 160 Nm, coupled to a Haake RheeDrive 4 system, with a motor power of 4 kW. The feeding was carried out using the Haake Metering Feeder OS. The temperature and the range of working speeds were adjusted between 160 and 250 min\({}^{-1}\), obtaining four samples (160, 190, 220, and 250), which were weighed, and the values were recorded in the PolySoft OS Capillary Rheology software for MS Windows. The determination of the volumetric flow rate was carried out by quantifying the density of the melt. According to the ASTM D1238-2004c standard, the measurement was carried out in a Tinius Olsen model MP 1200 (Germany) plastometer, composed of a vertical cylinder, a die with a 2.095 mm hole diameter and 8 mm long. The test temperature was  
Figure 1: In-line slit die rheometer (**A**). General view (**B**). Longitudinal cut. Source: Adapted from Thermo Scientific. \begin{table}
\begin{tabular}{c c} \hline \hline
**Raw Materials** & **Inclusion (\%)** \\ \hline Cassava starch & 55.0 \\ Hydrolyzed protein flour & 15.0–35.0 \\ Fish flour & 25.0–5.0 \\ Vitamin nucleus & 2.0 \\ Calcium carbonate & 0.8 \\ Bentonite & 1.7 \\ Sodium chloride & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Raw materials are required for the preparation of the diet. adjusted to 115 \({}^{\circ}\)C with 3.6 kg of weight on the piston, with a preheating time of 30 s. The travel meter was adjusted to 6.35 mm \(\pm\) 0.25 mm, the plastometer was loaded with an 8 g sample, the sample was made to flow through the die, and the weight was recorded for five examples for each material, obtaining the values of volumetric melt index, mass melt index, and melt density. According to the following equations, the parameters evaluated were shear rate, shear stress, and apparent viscosity [17]. \[\gamma_{app}=\frac{6\cdot Q}{W\cdot H^{2}} \tag{1}\]  
where \(s\gamma_{app}-1\) is the apparent shear rate, \(Q\) is the volumetric flow rate (m\({}^{3}\)/s), \(W\) is the width of the slit (m), and \(H\) is the height of the slit (m) (Equation (1)). \[\tau=\frac{\Delta P\cdot H}{2\cdot L} \tag{2}\]  
where \(\tau\) is the shear stress (Pa), \(\Delta P\) is the pressure variation in the capillary (Pa), \(L\) is the length of the cleft (m), and \(H\) is the height of the slit (m) (Equation (2)). \[\gamma_{true}=\frac{1}{3}\cdot\gamma_{app}\cdot\left(2+\frac{\partial Log \gamma_{app}}{\partial Log\tau}\right) \tag{3}\]  
\(\gamma_{true}\) is the cutting speed. The correction of the shear rate was performed using the Weissenberg-Rabinowitsch equation (Equation (3)). \[\eta_{app}=\frac{\tau}{\gamma_{true}} \tag{4}\]  
\(\eta_{app}\) is the apparent viscosity (Pa\(\cdot\)s) quantified as the relationship between shear stress and true shear velocity (Equation (4)). The rheological behavior of the samples was represented by the Power Law (\(lp\)), Bingham (\(bg\)), and Casson (\(cs\)) models (Equations (5)-(7)). \[\tau=K_{lp}\gamma^{n} \tag{5}\]  
\[\tau=\tau_{0 bg}+K_{bg}\gamma \tag{6}\]  
\[\tau^{0.5}=\tau_{0 cs}^{0.5}+K_{cs}\gamma^{0.5} \tag{7}\]  
where \(K_{lp}\), \(K_{cs}\), and \(y\)\(K_{bg}\) are the consistency indices of the samples; they correspond to the minimum force to initiate the flow through the capillary \(\tau_{0}\)[18].