Numerical and experimental application of the automated slump test for yield stress evaluation of mineralogical and polymeric materials

Abstract

This work presents reliable empirical correlations of slump and spread measurements with yield stress of different materials based on the automated slump test methodology (capable of monitoring the transient regime of the slump test and minimizing operational errors — better control and reproducibility of tests). Measures of the final samples, such as slump and spread, are then correlated to yield stress, assessed through a rotational rheometer. The reliability of this methodology is corroborated by numerical simulations of the slump test using Ansys Fluent 14.5. A series of sensitivity tests was made and numerical results showed the importance of including the mold and its lifting velocity in the simulations. Such experimental control and good agreement of numerical-experimental results allowed the definition of well-validated empirical models for yield stress estimation. Correlations with an elevated coefficient of determination (${{\varvec{R}}}^{2}>0.90$) were obtained using a new dimensionless parameter relating slump and spread measurements.

Graphical abstract

Appendix

Figures 22, 23, 24, and 25 show the sensitivity test results regarding the mesh size, CFL condition, regularization parameter, and residuals, respectively. The results indicate that the slump $\mathrm{S}$ has low sensitivity to the mesh size, CFL condition, and regularization parameters both on the transient and steady phases and the convergence criteria are the main uncertainty source for the simulation. Based on the chosen numerical parameters ($\Delta x=1.5\mathrm{ mm}$, $CFL=0.25$, ${\dot{\gamma }}_{c}=0.01 {\mathrm{s}}^{-1},$ and $res=7.5\times {10}^{-8}$), the sensitivity tests provide the following uncertainties for each numerical parameter regarding the final slump: $0.078\mathrm{\%}$ for mesh size; $0.0042\mathrm{\%}$ for CFL condition; $0.00017\mathrm{\%}$ for regularization parameter; and $0.27\mathrm{\%}$ for convergence criteria. The composition of uncertainties leads to a total uncertainty of $0.28\mathrm{\%}$ regarding the final slump.

Fig. 22

Mesh sensitivity test using three different mesh sizes ($\Delta {x}_{1}=1.5\mathrm{ mm}$, $\Delta {x}_{2}=2.25\mathrm{ mm},$ and $\Delta {x}_{3}=3.375\mathrm{ mm}$). The uncertainty bar for $\Delta {x}_{1}$ is showed in the detail window

Fig. 23

Time sensitivity test using three different CFL conditions (CFL_1 = 0.0625, CFL_2 = 0.25, and CFL_3 = 0.90). The uncertainty bar for CFL_1 is showed in the detail window

Fig. 24

Regularization parameter sensitivity test using three different critical shear rates (${\dot{\gamma }}_{{c}_{1}}=0.01 {\mathrm{s}}^{-1}$, ${\dot{\gamma }}_{{c}_{2}}=0.1 {\mathrm{s}}^{-1}$, and ${\dot{\gamma }}_{{c}_{3}}=0.5 {\mathrm{s}}^{-1}$). The uncertainty bar for ${\dot{\gamma }}_{{c}_{1}}$ is showed in the detail window

Fig. 25

Convergence criteria sensitivity test using three different criteria based on residuals (${res}_{1}=7.5\times {10}^{-8}$, ${res}_{2}=1.0\times {10}^{-7}$, and ${res}_{3}=5.0\times {10}^{-7}$). The uncertainty bar for ${res}_{1}$ is showed in the detail window

The results indicate that the simulation is highly dependent on convergence criteria. Even the larger criterion is stricter than the criterion recommended by ANSYS Fluent (Fluent 2012) and used as a rule of thumb by CFD simulations ($res={10}^{-6}$). Thus, the stricter convergence criterion was chosen to simulate all the numerical cases.

The slump behavior does not present significant dependence on the regularization parameter, as shown in Fig. 24. In order to correctly choose the regularization parameter, it was chosen based on the spread velocity along time, as shown in Fig. 26. Higher regularization parameters yield to significative spread velocities at final instants, while a value of $0.01 {\mathrm{s}}^{-1}$ shows to be sufficient to model the spread stoppage.

Fig. 26

Spread velocity ${V}_{D}$ in function of time $t$. Detail window shows residual spread velocity due to the bi-viscous regularization model

Additionally, a time-step sensitivity test of the gradual slump test was carried out using the sample $C0.17$ (${\tau }_{c}=57.12 Pa$, ${K}_{n}=15.35 Pa\cdot {s}^{n}$, $n=0.42$, and $\rho =1000 \mathrm{kg}/\mathrm{m}3$). Figure 27 presents slump over time using three different CFL conditions ($CF{L}_{1}=0.03125$, $CF{L}_{2}=0.0625$, and $CF{L}_{3}=0.125$) and shows no relevant implications over the numerical slump curves over time.

Fig. 27

Time sensitivity for gradual slump test using three different CFL conditions (${CFL}_{1}=0.03125$, ${\mathrm{CFL}}_{2}=0.0625$, and ${CFL}_{3}=0.125$)