Systemic Response Analysis of Final Setting Time of Concrete Based on Initial Setting Time and Cement Replacement with Bagasse Ash

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2.1. Model Formulation

Experimental data obtained from research work ^[16] were used for this work. Computational analysis of the data ^[16]shown in Table 1, gave rise to Table 2 which indicate that;

(1)

Introducing the values of S, S_e, K and N into equation (1) reduces it to;

(2)

(3)

(4)

Where

(ϑ) = Initial setting time of concrete (hrs)

(ɤ) = Bagasse ash input (%)

(ζ ) = Final setting time of concrete (hrs)

S = 0.0286, S_e = 1.0001, K = 0.0796 and N = 0.0275. S, S_e, K and N are empirical constants (determined using C-NIKBRAN ^[17]

3. Boundary and Initial Condition

Consider a concrete forming mixture of required consistency (in a cylindrical container) interacting with quantities of bagasse ash. The container’s atmosphere was not contaminated i.e (free of unwanted gases, dusts and other micro organisms). Range of bagasse ash input: 5- 40%. Range of initial setting time considered: 1.5 – 14.2 hrs. The range of final setting time resulting from replacement of cement with bagasse ash: 39.6 – 110.06 hrs. The mass of wastes used as well as other process conditions are as stated in the experimental technique ^[16].

The boundary conditions are: free movement of oxygen across the cylindrical container. At the bottom of the particles, a zero gradient for the gas scalar are assumed and also for the gas phase at the top of the concrete forming mixture. The treated concrete forming mixture was stationary. The sides of the waste particles are taken to be symmetries.

4. Results and Discussion

4.1. Model Validation

The validity of the model was found to be rooted on equation (2) where both sides of the equation are correspondingly approximately equal. Table 2 also agrees with equation (2) following the values of 0.0286 ζ - 1.0001 and 0.0796 ϑ + 0.0275ɤ precisely evaluated from the experimental results in Table 1.

Equation (3) is the derived model.

Table 2. Variation of 0.0286 ζ - 1.0001 with 0.0796 ϑ + 0.0275ɤ

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In addition, the derived model was validated by comparing the final setting time predicted by the model and that obtained from the experiment ^[16]. This was done using the 4^th Degree Model Validity Test Techniques (4^th DMVTT); computational, graphical, statistical and deviational analysis ^[18]

Figure 1. Coefficient of determination between final setting time and bagasse ash input as obtained from experiment [16]

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Figure 2. Coefficient of determination between final setting time and bagasse ash input as predicted by derived model

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Figure 3. Coefficient of determination between final setting time and initial setting time as obtained from experiment [16]

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Figure 4. Coefficient of determination between final setting time and initial setting time as predicted by derived model

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Figure 5. Comparison of final setting times (relative to bagasse ash input)as obtained from experiment [16] and derived model

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Figure 6. Comparison of final setting times (relative to initial setting times) as obtained from experiment [16] and derived model

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Figure 7. Comparison of final setting times (relative to bagasse ash input)as obtained from experiment [16], derived model and regression model

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Figure 8. Comparison of final setting times (relative to initial setting time) as obtained from experiment [16], derived model and regression model

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4.2. Statistical Analysis
4.2.1. Standard Error (STEYX)

Standard errors incurred in predicting final setting time for each value of the bagasse ash input and initial setting time considered as obtained from experiment, derived model & regression model were 4.0687, 3.8533, & 2.0163 x 10^-5% and 8.403, 3.960 & 7.9206% respectively. The standard error was evaluated using Microsoft Excel version 2003.

4.2.2. Correlation (CORREL)

The correlation coefficient between final setting time and bagasse ash input & initial setting time were evaluated from the results of derived model and experiment, considering the coefficient of determination R² from Figs. 1-4, was calculated using Microsoft Excel version 2003.

(5)

The evaluated correlations are shown in Table 3. These evaluated results indicate that the derived model predictions are significantly reliable and hence valid considering its proximate agreement with results from actual experiment.

Table 3. Comparison of the correlations evaluated from derived model predicted and ExD results based on bagasse ash input

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Table 4. Comparison of the correlations evaluated from derived model predicted and ExD results based on initial setting time

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4.3. Graphical Analysis

Comparative graphical analysis of Figure 5 and Figure 6 shows very close alignment of the curves from the experimental (ExD) and model-predicted (MoD) final setting time. Furthermore, the degree of alignment of these curves is indicative of the proximate agreement between both experimental and model-predicted final setting times.

4.4. Comparison of Derived Model with Standard Model

The validity of the derived model was further verified through application of the regression model (Reg) (Least Square Method using Excel version 2003) in predicting the trend of the experimental results. Comparative analysis of Figure 7 and Figure 8 shows very close alignment of curves and significantly similar trend of data point’s distribution for experimental (ExD), derived model (MoD) and regression model-predicted (ReG) results of final setting time. Also, the evaluated correlations (from Figure 7 and Figure 8) between final setting time and bagasse ash input & initial setting time for results obtained from regression model gave 1.0000 & 0.9686 respectively. These values are in proximate agreement with both experimental and derived model-predicted results. The standard errors incurred in predicting final setting time for each value of the bagasse ash input and initial setting time considered as obtained from regression model were 2.0163 x 10^-5 and 7.9206% respectively.

4.5. Computational Analysis

Comparative computational analysis of the experimental and model-predicted final setting time was carried out to ascertain the degree of validity of the derived model. This was done by comparing results of evaluated final setting time per unit input of bagasse ash and initial setting time of as obtained from experimental and derived model within bagasse ash input and initial setting time ranges: 5 – 40% and 1.5 – 14.2 respectively.

4.5.1. Final Setting Time per unit Initial Setting Time

Final setting time per unit initial setting time ζ / ϑ was calculated from the equation;

(6)

Re-written as

(7)

Equation (7) is detailed as

(8)

Where

Δζ = Change in the final setting time ζ₂, ζ₁ at two initial setting times ϑ₂ϑ₁.

Considering the points (5, 39.6) & (40, 110.06), (5, 43.9828) & (40, 113.0623) and (5, 43.0983) & (40, 110.4814) as shown in Figure 8, and designating them as (ζ₁, ϑ₁) & (ζ₂ ϑ₂) for experimental, derived model and regression model predicted results respectively, and then substituting them into equation (8), gives the slopes: 8.403, 3.960 & 7.9206% respectively as their corresponding final setting time per unit initial setting time.

4.6. Final Setting Time per unit Bagasse Ash Input

Similarly, substituting into equation (8) points (1.5, 39.6) & (14.2, 110.06), (1.5, 43.9828) & (14.2, 113.0623) and (1.5,43.0983) & (14.2, 110.4814) from Figure 7, as (ζ₁, ɤ₁) & (ζ₂, ɤ₂) for experimental, derived model and regression model predicted results respectively also gives the slopes: 4.0687, 3.8533, & 2.0163 x 10^-5% respectively as their corresponding final setting time per unit bagasse ash input. The proximity between values in each result set indicates significantly high validity level for the derived model.

4.7. Deviational Analysis

A comparative analysis of the final setting times obtained from experiment ^[16] and derived model shows low deviations on the part of the model-predicted values relative to values obtained from the experiment. This is attributed to the fact that the surface properties of the bagasse ash and the physiochemical interactions between bagasse ash and the concrete forming mixture which were believed to have played vital roles during the process ^[10] were not considered during the model formulation. This necessitated the introduction of correction factor to bring the model-predicted final setting time to those of the corresponding experimental values.

Deviation Dev, of model-predicted final setting time from the corresponding experimental result is given by

(9)

Where Es and Ps are transmissivities evaluated from experiment and derived model respectively.

Figure 9. Variation of final setting time with initial setting time and associated deviation (of model-predicted values)

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Deviational analysis of Figure 9 indicates that the precise maximum deviation of model-predicted final setting time from the experimental results is less than 12%. This translated into over 88% operational confidence and response level for the derived model as well as over 0.88 response coefficients of produced concrete final setting time to the collective operational contributions of bagasse ash and initial setting time.

Consideration of equation (9) and critical analysis of Figure 1- Figure 4 and Figure 9 show that the least and highest magnitudes of deviation of the model-predicted final setting time (from the corresponding experimental values) are -2.53 and + 11.07%. This corresponds to final setting times: 86.16 and 43.9828 hrs; bagasse ash inputs: 30 and 5g as well as initial setting times: 8 and 1.5 hrs respectively.

Equations (9) and (10) indicate that correction factor is the negative of the deviation.

The correction factor took care of the negligence of operational contributions of surface properties of the bagasse ash and the physiochemical interactions between bagasse ash and the concrete forming mixture which actually played vital roles during the process^[10]. The model predicted results deviated from those of the experiment because these contributions were not considered during the model formulation. Introduction of the corresponding values of Cf from equation (10) into the model gives exactly the corresponding experimental final setting time.

Correction factor, Cf to the model-predicted results is given by

(10)

Figure 10. Variation of final setting time with bagasse ash input and associated correction factor to model-predicted values

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