Strength retrieval of artificially cemented bauxite residue using machine learning: an alternative design approach based on response surface methodology

Abstract

The aim of the present study is to propose an alternative artificial neural network model based on response surface methodology over conventional approach to estimate the unconfined compressive strength of artificially cemented bauxite residue. The artificial neural network model uses molding moisture content (w), curing time (t) and porosity/volumetric lime (η/L_v′) as input parameters and unconfined compressive strength as the output parameter. Bayesian regularization as training function with sigmoid and pure linear at hidden and output layers is used for modeling the artificial neural network. The proposed response surface methodology designed ANN model is comparable with the conventional designed ANN model and can be used effectively with significantly less number of data set. Sensitivity analysis, to make out the significant input factors based on connection-weight approach, is also discussed. Further, neural interpretation diagram is incorporated to study the effects of individual input parameters over the response. Finally, a predictive equation is presented based on response surface methodology designed artificial neural network model for the range of parameters studied.

Appendix A

1.1 Calculation of volumetric lime content (L _v) and porosity of artificially cemented bauxite residue

In the present study, the following procedure has been followed:

• a. The amount of lime for each mixture was calculated based on the mass of dry bauxite residue.

• b. The dry density of the specimens was calculated as the dry mass of the bauxite residue and lime divided by the total volume of the sample.

1.2 Volumetric lime content (L _V)

Defined as volume of lime with respect to total volume/volume of specimen and expressed in percentage.

Suppose,

We take 100 gm of bauxite residue and L gm of lime.

Then, total mass (m_T) = (100 + L) gm

So,

$$ \begin{aligned} L_{\text{v}} \left( \% \right) & = \frac{\text{Volume of lime}}{\text{Total volume}} = \frac{{\frac{{{\text{Mass of lime}} \left( {m_{L} } \right)}}{{{\text{Density of lime}} (\gamma_{L} )}} }}{{\frac{{{\text{Total mass}} \left( {m_{\text{T}} } \right)}}{{{\text{Density of specimen}}(\gamma_{\text{d}} )}}}} = \frac{{\gamma_{\text{d}} \times m_{L} }}{{G_{\text{L}} \times \gamma_{\text{w}} \times m_{\text{T}} }} \\ & = \frac{{\gamma_{\text{d}} \times L}}{{G_{\text{L}} \times \gamma_{\text{w}} \times \left( {100 + L} \right)}} = \frac{{\gamma_{\text{d}} \times \left( {\frac{L}{100}} \right)}}{{G_{\text{L}} \times \gamma_{\text{w}} \times \left( {1 + \frac{L}{100}} \right)}} = \left[ {\frac{{\frac{{\gamma_{\text{d}} }}{{\left( {1 + \frac{L}{100}} \right)}} \times \left( {\frac{L}{100}} \right)}}{{G_{\text{L}} \times \gamma_{\text{w}} }}} \right] \times 100 \\ \end{aligned} $$

(8)

where: γ_w = density of the specimen (defined as the weight of the water per its unit volume). γ_d = dry density of the specimen (defined as the weight of solid in a given volume). L = lime content (defined as the mass of lime with respect to total mass). G_L = specific gravity of lime (defined as the ratio of the density of lime solids to the density of water). G_RM = specific gravity of bauxite residue (defined as the ratio of the density of bauxite residue solids to the density of water).

Similarly,

Porosity (η) is defined as volume of voids with respect to total volume/volume of specimen and expressed in percentage.So,

$$ \begin{aligned} \eta & = \frac{\text{Volume of voids}}{\text{Total volume}} = \frac{{{\text{Total volume}} {-}{\text{Volume of solids}}}}{\text{Total volume}} = 1 - \frac{\text{Volume of solids}}{\text{Total volume}} \\ & = \frac{\text{Volume of bauxite residue }}{\text{Total volume}} + \frac{\text{Volume of lime}}{\text{Total volume}} \\ \end{aligned} $$

Now,

$$ \frac{\text{Volume of lime}}{\text{Total volume}} = L_{\text{v}} = {\text{Volumetric lime content}} = \left[ {\frac{{\frac{{\gamma_{\text{d}} }}{{\left( {1 + \frac{L}{100}} \right)}} \times \left( {\frac{L}{100}} \right)}}{{G_{\text{L}} \times \gamma_{\text{w}} }}} \right] \times 100 $$

Similarly, from Eq. (9), we can calculate,

$$ \begin{aligned} \frac{\text{Volume of bauxite residue }}{\text{Total volume}} & = \frac{\text{Volume of bauxite residue }}{\text{Total volume}} \\ & = {\text{Volumetric bauxite residue content}} = \left[ {\frac{{\frac{{\gamma_{\text{d}} }}{{\left( {1 + \frac{L}{100}} \right)}} }}{{G_{\text{RM}} \times \gamma_{\text{w}} }}} \right] \times 100 \\ \end{aligned} $$

Finally we get,

$$ \eta = 100 - 100 \left\{ {\left[ {\frac{{\gamma_{\text{d}} }}{{1 + \left( {\frac{L}{100}} \right)}}} \right]\left[ {\frac{1}{{G_{\text{RM }} \gamma_{\text{w}} }} + \frac{{\left( {\frac{L}{100}} \right)}}{{G_{\text{L }} \gamma_{\text{w}} }}} \right]} \right\} $$

(9)