Experimental investigation and modelling of the mechanical properties of palm oil fuel ash concrete using Scheffe’s method

This study explores the enhancement of mechanical properties in concrete blended with palm oil fuel ash (POFA) through Scheffe's optimization. The utilization of POFA as supplementary cementitious material in concrete has gained attention for its potential environmental benefits. Utilizing a (5,2) simplex-lattice design, a systematic approach is employed for optimizing mixture proportions based on response parameters. The laboratory tests to evaluate concrete's mechanical behavior were conducted using the computed mixture ratios from the design experimental points after 28 days of hydration. The results showed maximum flexural strength at 8.84 N/mm2 and compressive strength at 31.16 N/mm2, achieved with a mix of 0.65:0.54:2.3:3.96:0.35 for cement, water, coarse aggregate, fine aggregate, and POFA. Additionally, maximum splitting tensile strength reached 8.84 N/mm2 with a mix of 0.62:0.55:2.09:3.86:0.38 for the same components. Conversely, the minimum flexural, splitting tensile and compressive strength within the experimental factor space was 4.25, 2.08 and 19.82 N/mm2 respectively. The results obtained indicated a satisfactory mechanical strength performance at POFA replacement of 35 percent in the concrete mixture. The developed mathematical model was statistically validated using analysis of variance (ANOVA) at a 95% confidence interval which showed satisfactory prediction performance. The findings from this study provide valuable insights into optimizing POFA-blended concrete for enhanced mechanical performance, offering potential sustainable solutions for the construction industry.

The assessment of mixture response properties involves employing a mathematical (polynomial) function of degree m-order and q, where q represents the count of mixture components.This yields a (q, m) polynomial in the overall structure outlined in Eq. ( 3) and the parameter Y represents the response function 33 .
where 1 ≤ i ≤ q, 1 ≤ i ≤ j ≤ q, 1 ≤ i ≤ j ≤ k ≤ q and b i is the model coefficient Scheffe's Simplex lattice design Scheffe's Simplex lattice design is a structured experimental approach used to optimize mixture compositions for various properties.It involves arranging experimental points in a systematic manner on a simplex lattice, where each point represents a specific mixture proportion.This design allows for the exploration of multiple factors simultaneously and accounts for their interactions 34,35 .The resulting data is used to create mathematical models that predict how changes in component proportions affect the desired outcomes 36 .A simplex is a geometrical shape with one more vertex than variable factor spaces (q), projected from n-dimensional space to n-1-dimensional coordinates.For q = 1, it's a two-vertex line; q = 2 forms a triangle, and q = 3 creates a tetrahedron 37 .Scheffe 28 expanded and generalized the simplex lattice design, pioneering it in mixture design.His work established the notion of each component as a vertex within a regular simplex-lattice with q-1 factor space.Scheffe's simplex lattice patterns have become a widely recognized term for lattice designs, setting the stage for optimizing mixture compositions.Scheffe's Simplex lattice design is particularly useful in industries like materials science, where the properties of mixtures (e.g., in concrete or polymers) need to be optimized efficiently.It simplifies the experimentation process, reduces resource consumption, and provides insights into the complex relationships among mixture components 38 .
A simplex design is a form of mixture design where design points are systematically positioned on a simplex lattice.The coordinate system employed for each ingredient's value x i (i = 1, 2, …, q) is referred to as the simplex coordinate system.Here, q signifies the count of ingredients for each experimental run 45 .The simplex coordinate system is defined as x i = 0, 1 m , 2 m , . . . 1 .The design space encompasses all rational combinations of factor values, where m represents the lattice degree (or dimensional space).When considering an entire factor space in the design, a (q, m) simplex lattice emerges, characterized by uniformly distributed and saturated points 39 .Proportions assigned to each factor comprise m + 1 evenly spaced levels within the simplex coordinate system, with all feasible combinations stemming from these component concentration values.In the scenario of a quadratic lattice (q, 2), where the response surface is approximated by second-degree polynomials (m = 2), the simplex cooinate x i = 0, 1 2 , . . ., 1 .is utilized, signifying 3 spaced levels.Similarly, in a full cubic mixture design model with m = 3, the simplex lattice coordinate x i = 0, 1 3 , 2 3 , . . ., 1 indicates 4 spaced levels 40 .Every combination of these variables is applied in an experiment run, denoted as space points A i , A ij , or A ijk for (i ≠ j ≠ k = 1, 2, 3, …, q).The summation of component volume fractions equates to one, and this constraint defines a regular tetrahedron in the simplex factor space.This arrangement is illustrated in Fig. 1.
(1) www.nature.com/scientificreports/Scheffe's factor space and components interaction Mixture experiment designs are arrangements for response surface experiments where each component adheres to a scale value between 0 and 1, with the sum of components equating to one.In a mixture containing q components, the shape and layout of experimental points are defined by q−1 component.These constraints establish a (q−1) dimensional design space known as a simplex.Essentially, the design space dimension is consistently one less than the component count (Borkowski et al., 2014).Consequently, the factor space manifests as a regular (q−1) dimensional simplex 41 .
In a (q−1) dimensional simplex, when q = 2, two connecting points create a straight-line simplex lattice.For q = 3, a triangular simplex lattice forms with equal sides; similarly, q = 4 yields a tetrahedron simplex lattice.(q−1) dimensions shape the boundary where q components interact within a mixture.The number of design space points N, represented as A i or Aij or A ijk for (i ≠ j ≠ k = 1, 2, 3, …, q), is determined by Eq. 3.4 in a simplex lattice design.This result guides the required number of experimental runs 42 .
Thus, in the general canonical form of mixture models is mathematically expressed in Eqs.(5-7); for Linear, Second order/Quadratic and Cubic function respectively.
Substituting the values of i and j in Eq. ( 5) the quadratic equation for (0 ≤ i ≤ j ≤ 4) transforms to Eq. (7).
Here, b i represents the linear mixing proportion attributed to the pure blend when x i = 1 and x j = 0, where i = j = k .The expected response is denoted as E y .b ij .signifies the quadratic non-linear blending between pairs of components, featuring parameters that could demonstrate either synergistic or antagonistic blending effects.Similarly, b ijk stands for the complete cun-linear blending coefficients among components, where parameters may indicate either synergistic or antagonistic blending interactions 43,44 .

Derivation of Scheffe's quadratic response function
Further expansion of Eq. 6 through the substitution of 0 ≤ i ≤ j ≤ 5 into the values of i and j transforms to Eq. ( 8).
Multiplying Eq. ( 1) by b o we obtain a mathematical expression presented in Eq. (9)   Multiplying in succession Eq. ( 2) by X 1 , X 2 , X 3 , X 4 , and X 5 we get the relationship in Eq. (10)   By inserting Eqs.(9 and 10) into Eq.( 8), we derived the comprehensive quadratic polynomial model structure for a mixture consisting of five components, resulting in the expression given in Eq. (11).
Vol.:(0123456789) www.nature.com/scientificreports/ We denote the mathematical relationship between the pure blends coefficients (β i ) and the tenary blends (β ij ) as shown in Eqs.(12, 13)   Then we derive the second-degree polynomial for the response function as presented in Eq. (14).
Here, x i signifies the coded variables known as pseudo components within the mixture design, whereas β i rep- resents the response model coefficients in Scheffe's optimization equation.These coefficients can be denoted as β i for pure or binary blends, and as β ij for ternary blends or the amalgamation of mixture components.Their definition is formulated using the mathematical expression presented in Eq. ( 15) 45 . Where

Actual components and pseudo components
Pseudo-components are designated as fictitious or encoded variables adopted to streamline design creation and model adjustment.This approach curtails the linkages between component boundaries in constrained designs, resulting in reduced correlations among coefficients.This is accomplished by converting actual components, Z, into pseudo-components, X. Essentially, pseudo-components recalibrate the restricted data space to align with zero as the minimum permissible value (lower bound) for each component in mixture designs, similar to Scheffe's model 46 .This mathematical association is illustrated in Eq. (16).Z signifies the real components which is the actual fraction of ingredients added to the mixture for a given run of experiment, whereas X stands for the surrogate components, with A as the constant, represented by a 5 × 5 matrix.The matrix A components are derived from the initial five mix ratios, forming a q × q dimensional matrix based on this preliminary experimental mixture 47 .

Mix ratio development
The generation of initial trial mixes was initiated through a combination of expert judgment, practical experience, economic considerations, and insights from pertinent literature.This approach aimed to initiate the calculation of interaction points using Eq. ( 16) The aim is to find an optimal blend of mixture components that enhances desired properties.Scheffe's methodology enables the identification of interactions and correlations between components, leading to a refined mix ratio that achieves enhanced performance in various applications such as concrete, materials, and other fields.By iteratively adjusting the mixture proportions and evaluating the predicted responses, the optimal mix ratio can be identified that maximizes or minimizes the desired properties 48 .
X 1 = fraction of ordinary Portland cement; X 2 = fraction of water cement ratio; X 3 = fraction of fine aggregate.X 4 = fraction of coarse aggregate; X 5 = fraction of POFA.The matrix notation form of Eq. ( 16) which was utilized for computation of the components' ratio for the concrete mixture.The initial five points correspond to the vertices of the simplex factor space, while the remaining ten points situated within the simplex, known as interaction points, are determined by applying Eq. ( 16) through the following substitution process: Therefore, for A 12 for A 13    The computation matrix table for the mixture proportion formulation is presented in Table 1.The mixture compositions for the experimental control points are also computed.These compositions were specifically designed to validate the generated Scheffe's regression model.
for                The computation matrix table for the mixture proportion formulation is presented in Table 2.          www.nature.com/scientificreports/

Test of adequacy of the model
The effectiveness of the developed Scheffe's model is evaluated using statistical techniques, specifically through analysis of variance (ANOVA).These tests are applied to ascertain any disparities between the actual outcomes and the model-predicted results for the control experiment 49 .The statistical assessment is performed separately for both second and third order models, with a 95% confidence level.It focuses on the achieved compressive and flexural strengths results.To derive the predicted values (Y-predicted) for the control test points, the model equation is employed by substituting the respective values of coded units X 1 , X 2 , X 3 , X 4 , and X 5 50,51 .•Null Hypothesis There is no noteworthy distinction observed between the strength outcomes predicted by the model and those obtained from laboratory tests.
•Alternative Hypothesis A considerable contrast exists between the strength outcomes projected by the model and those derived from laboratory testing.

Materials
Palm oil fuel ash POFA samples shall be collected in a sack from different location within Cross River South and Central Palm Mill.Global Positioning System (GPS) coordinates and site photographs shall also be taken.The collected POFA samples shall be taken to the laboratory where it is oven dried.Sieving of the POFA samples may be necessary to remove all unwanted materials contained in the sample in agreement with BS 8615-1 (2019) and ASTM C618 standards 51 .

Coarse aggregates
The coarse aggregates shall be 15-22 mm grain size, obtained from Saturn quarry at Akamkpa Local Government Area of the State.

Fine aggregates
Fine aggregates were obtained from the Calabar river and classified within the sharp sand envelope in accordance with BS EN12620 specifications.

Ordinary Portland cement
Material shall be obtained from United Cement Company of Nigeria (UNICEM), Akamkpa Local Government Area with 32.5 grade and normal consistency of 30% in accordance with Nigerian Industrial Standard (NIS) 444-1 requirement 52 .

Water
Drinkable or potable water, a crucial element, influences the mechanical and rheological characteristics of concrete, meeting the specifications outlined in ASTM C1602-12.

Study area
The area of study under review was tailored around the availability of the concept material in view which is central and southern Cross River State shown in Fig. 2. Materials for the study will be obtained from Calabar Municipal, Ikom and Biase Local Government Areas.The laboratory work was done in the Concrete Laboratory of the Cross River University of Technology (CRUTECH), Calabar.

Experimental investigation and setup
The concrete mix involves partially substituting cement in the matrix with POFA.Utilizing Scheffe's statistical technique, the mix formulation was determined using the mathematical relationship between real and pseudocomponents.The target strength of 25 N/mm 2 was aimed for, with a cement content of 290 kg/m 3 , coarse aggregate content of 1198.65 kg/m 3 , and fine aggregate content of 766.35 kg/m 3 for the design 53 .The concrete ingredients were mixed thoroughly with water to ensure uniformity before compaction and placement into the mold for mechanical strength assessments.Fresh concrete was also tested to evaluate setting time and workability.Following this, the concrete samples were cured in a tank at room temperature for 28 days after which they were subjected to structural property investigations.This study aims to explore the structural attributes of POFA concrete 54 .

Grain size distribution
Sieve analysis is a fundamental technique used to determine the grain size distribution of aggregates in concrete.This method involves passing a sample of aggregates through a series of sieves with progressively smaller openings in accordance with BS EN 933-1:2012 specification.The amount of material retained on each sieve is measured and used to create a particle size distribution curve.This curve provides valuable insights into the composition and grading of aggregates, which directly influence the properties of concrete, such as workability, strength, and durability.By analyzing the distribution of different particle sizes, engineers and researchers can optimize concrete mix designs, ensuring the right balance of coarse and fine aggregates for desired performance characteristics 55 .

Slump tests
The slump test is a commonly employed method to assess the workability or cohesion of newly mixed concrete.This technique offers insights into the concrete's capacity to flow and adapt without separation or excessive liquid release.Furthermore, it offers valuable insights into how effortlessly the concrete can be positioned, condensed, and refined during the construction process.Fresh concrete's workability is assessed through slump tests.These tests adhere to the guidelines stipulated in BS EN 12350-2:2009 56 .

Compressive strength
The compressive strength experiment is a fundamental test conducted on concrete specimens to assess their ability to withstand axial loads without failure.This procedure will be conducted on solidified concrete cubes with dimensions of 150mm x 150mm x 150mm, following the guidelines outlined in BS EN 12390-3:2019 for concrete compressive strength testing.After casting, the concrete cubes will be allowed to solidify for 24 h, followed by a 28-day curing period before testing.Subsequently, each cured sample will be placed between two plates in a Universal Test Machine (UTM), which will apply a load uniformly across the two opposing faces.The load will aim to flatten the sample, causing compression along the load's direction and expansion perpendicular to it.The loading will be incrementally increased at a rate of 140 kg/cm 2 per minute until failure occurs.The compressive strength of each cube will then be calculated as the load at failure divided by the sample's surface area, expressed mathematically as shown in Eq. ( 17) 57,58 .
where F is the applied load in (N) and A the cross-sectional area in (mm 2 )

Split tensile test
The splitting tensile strength experiment is a test conducted on cylindrical concrete specimens to evaluate their resistance to cracking and failure under tensile stresses.The cylindrical specimens are prepared and cured for a specific duration, commonly 28 days.After curing, the specimens are placed horizontally in a testing machine, and a diametrical load is applied to the cylindrical surface following the specifications defined in BS EN 12390-6:2019.The blended concrete mixture is filled into cylindrical molds measuring 100mm in diameter and 300mm in height.Following a day's period, the specimens are extracted from the molds and transferred to a water bath maintained at a consistent temperature range of 23-25 °C.This step is undertaken to determine the tensile strength of the mixture after it has been cured for 28 days, utilizing a crushing apparatus.The splitting tensile strength of the specimen is calculated using the formula presented in Eq. ( 18) 59,60 where, T = Splitting tensile strength; P = Maximum applied load; L = Length, D = Diameter.

Flexural strength
The flexural strength experiment involves subjecting concrete beams or prisms to a bending load to evaluate their ability to withstand applied forces.This test measures the concrete's ability to resist bending stresses and is crucial for assessing its performance in structural applications.Concrete specimens, typically shaped like beams or prisms, are cast and cured according to standard procedures.After the curing period, these specimens are placed on a testing apparatus where a gradually increasing load is applied at the center of the specimen.As the load increases, the specimen bends until it eventually cracks and fails.The cement-POFA blended concrete were filled in a beam mold of 100 mm*100 mm*400 mm dimension, after 24 h the hardened samples are immersed in a curing tank and cured for 28 days in accordance with BS EN 12390-5:2019 requirement.The flexural strength response expressed in units of force per unit area, such as N/mm 2 is determined using the mathematical expression in Eq. ( 19) 61 .
Here, P is the applied load at fracture; L is the beam's length; B is the width and D is the beam's thickness.

Results discussion and analysis
The results, discussion, and analysis of the Scheffe's optimization of the mechanical properties of palm oil fuel ash (POFA) concrete reveal significant insights into the behavior and performance of the optimized concrete mixture.Through the utilization of Scheffe's statistical approach, the study aimed to enhance the mechanical properties of concrete by incorporating POFA as a partial replacement for cement.The investigation began with the development of mixture ratios using mathematical models based on Scheffe's approach.The desired characteristic strength of 25 N/mm 2 was chosen, and the corresponding proportions of cement, coarse aggregate, POFA and fine aggregate were determined.The concrete mixture was prepared by thoroughly blending these ingredients with water to ensure homogeneity 62,63 .

Characterization of test materials
Laboratory tests were conducted on the mixture components to assess their overall engineering behaviour as civil construction materials.The test aggregates and admixtures underwent sieve analysis and specific gravity tests to evaluate their gradation and particle size distribution.Figure 3 displays the sieve analysis test outcome represented on a semi-log graph, depicting the variation of soil grain sizes using a cumulative frequency distribution curve.The results indicate that for the coarse aggregate, the percentage passing through the sieve sizes 10-2 mm ranges from 75.3 to 11.5%.Similarly, for the fine aggregates and POFA, the percentage passing through sieve sizes 2 mm-75 μm ranges from 91.4 to 0.15% and 98.67-24.55%,respectively.The coefficients of gradation are provided in Table 3, showing well-graded sand and gravel particles that meet the specified requirements by BS 882, ensuring improved concrete durability performance 64,65 .

Chemical properties of the test cement and POFA
hydration reaction mechanism involves aluminum and silicon oxides derived from the pozzolanic material (POFA) blending with lime to form hydration products such as calcium silicate hydrate (C-S-H), resulting in a progressively harder mass over time 66,67 .

Slump test results
Laboratory tests were conducted to assess the workability properties of the freshly mixed POFA-cement blend concrete.The aim was to determine the place ability and workability properties of the fresh concrete mixture with varying ratios of POFA-cement combinations from Scheffe's mixture design.The experimental results signify that the slump test value decreases with increasing POFA fractions in the concrete mixture, requiring more water to enhance workability.This may be due to the presence of alumino-silica content in the admixtures and increased surface area.The obtained experimental result is presented in a graphical plot shown in Fig. 4. The obtained results show that Y 1 experimental point achieved a maximum slump of 125mm with ratio of 0.92:0.44:2.18:3.85:0.03for cement, water, coarse agg., fine agg., and POFA respectively.In contrast, Y 5 experimental point achieved a minimum slump of 41mm with mix ratio of 0.5:0.6:1.95:3.72:0.5 for cement, water, coarse agg., fine agg., and POFA respectively.The findings indicate a linear decrease in slump response as the admixture percentages in the mix increase.In the context of this study, the slump test results for palm oil fuel ash (POFA) concrete offer valuable insights into the concrete's ability to flow, deform, and retain its homogeneity without excessive segregation or bleeding 68

Compressive strength Results
Concrete compressive strength results are a vital indicator of the material's ability to withstand compressive loads without failure.It is a fundamental property used to assess the structural integrity and performance of concrete in various engineering applications.To optimize the compressive strength of POFA blended concrete, laboratory responses for compressive strength were collected for fifteen different mixture designs, each with three replicates after 28 days of curing.These values were used to develop the Scheffe's model.Among the results, Y 4 had the highest compressive strength at 31.16 N/mm 2 , while Y 5 had the lowest at 19.82 N/mm 2 (Table 5).Control points were also tested for model validation, with C 35 showing the highest compressive strength at 28.18 N/mm 2 , and C 45 having the lowest at 21.82 N/mm 2 (Table 6).Overall, the incorporation of POFA admixtures in the concrete matrix led to improved mechanical strength, making it suitable for sustainable structural applications 69,70 .

Flexural strength response
Concrete flexural strength results provide valuable insights into the material's ability to resist bending or deflection when subjected to external loads.It is a critical mechanical property that reflects the concrete's performance in applications where it needs to span over supports or resist bending stresses, such as in beams, slabs, and other structural elements.Flexural strength testing involves applying a load perpendicular to the surface of a concrete beam or sample until it reaches failure 61 .The flexural strength laboratory responses were obtained for concrete beam samples of dimension 100*100*400 mm cured for 28 days, involving fifteen different mixture design points, each with three replicates.These values were used to develop Scheffe's second-order regression model for optimizing concrete with POFA blended concrete's flexural strength properties.Table 7 shows Y 4 had the

Spitting tensile strength response
Concrete splitting tensile strength results provide important information about the material's ability to resist tensile stresses perpendicular to the applied load.It is a critical mechanical property used to assess the performance and durability of concrete in applications where tensile forces are present, such as in pavements and slabs.The splitting tensile strength results were obtained for cylindrical concrete samples measuring 150 mm in diameter and 300 mm in height.The samples were cured for 28 days, and the testing involved fifteen Scheffe's experimental design points, each with three replicates.The experimental response is shown in Table 9 which presents the highest strength value of 5.23 N/mm 2 for Y 35 , while Y 1 had the lowest at 2.08 N/mm 2 .Moreover, Table 10 shows the control point responses for validating the regression model, with C 5 having the highest strength result of 5.06 N/mm 2 and C 1 with the lowest at 3.85 N/mm 2 .These findings indicate that the concrete's strength improves as the percentage of POFA replacement of cement in the matrix increases 45,72 .www.nature.com/scientificreports/

Compressive strength
The model equation is derived by inputting the derived experimental response values into Eq.14, which represents the relationship between the obtained response and the model coefficients, resulting in the Scheffe's coefficients.These coefficient values are then substituted into Eq.( 15), leading to the model equation presented in Eq. ( 20) and Table 11.

Flexural strength model equation
The regression model equation is obtained by substituting the laboratory responses for the concrete's flexural strength into Eq.( 15), resulting in Eq. ( 21) and the coefficients is shown in   15), which leads to the formation of Eq. ( 22).The regression coefficients obtained from this process are presented in Table 13.

Test of adequacy and validation of developed models
Regression model validation is a crucial step in the analysis of data and the development of predictive models.
In the context of a regression model, validation refers to the process of assessing the model's accuracy and reliability in predicting outcomes for new or unseen data.This helps in gauging the model's generalization capability,   www.nature.com/scientificreports/which is essential for its practical applicability 73 .The control points of the experiment were employed to assess the accuracy of the developed Scheffe's model using statistical assessment method such as analysis of variance (ANOVA).Graphical representations in Fig. 5 display a comparison between the actual experimental results and the values obtained from the generated regression second-order model for the compressive, flexural and splitting tensile strength properties 74 .

Statistical evaluation of model performance using ANOVA
Statistical evaluation of model performance using ANOVA is a powerful technique commonly used to assess the accuracy and significance of regression models.ANOVA helps to determine whether the variation in the response variable can be adequately explained by the model's predictors.It essentially compares the variability of the observed data to the variability of the data predicted by the regression model.The main objective is to evaluate whether the model's predictors (independent variables) have a significant impact on the response variable (dependent variable) 75 .A significant ANOVA result indicates that the regression model is valid and provides valuable insights into the relationship between the predictors and the response variable.However, a non-significant ANOVA result suggests that the model might not be appropriate, and further investigation or model refinement may be necessary.In summary, ANOVA is a fundamental tool for statistical evaluation in regression analysis, helping researchers and analysts determine the adequacy and significance of their models, ultimately leading to better decision-making and insights from the data 76 .
Analysis of variance (ANOVA) was carried out using Microsoft Excel statistical software which was deployed to evaluate the prediction performance of the developed second-order regression models for the compressive, flexural and splitting tensile strength optimization.This evaluation is based on the comparison of laboratory and model-predicted values at a 95% confidence interval, as presented in Tables 14, 15 and 16.The condition for assessment is as follows: if F > F crit, we reject the null hypothesis.After performing the statistical computation, for the compressive strength we find that F = 0.165 and F crit = 4.196.As F crit > F, we accept the null hypothesis with a p-value of 0.688, which is greater than the alpha value of 0.05 77,78 .
Similarly, for the flexural strength we obtained F = 0.538, F crit = 4.196 and p-value of 0.47.Also, we observe the criteria that F crit > F. finally, for the splitting tensile strength results, F = 0.000188, F crit = 4.196 and p-value of 0.989 was calculated which indicated that F crit > F. The statistical examination outcome showed that the  www.nature.com/scientificreports/difference between the actual laboratory-derived results and the model results was not significant which signified that the models developed possessed good prediction accuracy 79,80 .

Conclusion
The study focused on exploring the mechanical behaviour of green concrete containing palm oil fuel ash (POFA) through an optimization process using Scheffe's theory.The conclusions drawn from the research findings and results are as follows: 1.The study employs Scheffe's quadratic polynomial to predict mechanical properties in POFA-cement concrete.
The model estimates strength values based on mix ratios or determines ratios for desired strengths.2. The evaluation of the rheological properties of the freshly mixed POFA-cement blended concrete shows that the incorporation of admixtures led to a reduction in the slump response of the fresh blended concrete specimens.3. The maximum compressive and flexural strength at 28 days were 31.16N/mm 2 and 8.84 N/mm 2 , achieved with a mix of 0.65:0.54:2.3:3.96:0.35for cement, water, coarse aggregate, Fine aggregate and POFA respectively.Additionally, splitting tensile strength reached 8.84 N/mm 2 with a mix of 0.62:0.55:2.09:3.86:0.38. 4. The lowest compressive and flexural strength, at 19.82 N/mm 2 and 4.25 N/mm 2 , was achieved with a mix of 0.5:0.6:0.95:3.72:0.5 for cement, water, coarse aggregate, Fine aggregate and POFA respectively.Similarly, splitting tensile strength was minimal at 2.08 N/mm 2 with a mix of 0.97:0.44:2.18:3.85:0.03. 5.The model's accuracy was assessed through ANOVA, revealing a strong correlation between the model's predictions and the laboratory-derived control results.The strengths predicted by the model closely align with the corresponding experimentally observed outcomes, indicating good agreement between them.6.Furthermore, using desired percentages of agricultural waste and their derivatives such as POFA in concrete production leads to cost savings, promoting sustainability, and playing a significant role in waste management.

Research limitations
1. Limited Scope of Variables: One limitation of this research is the focus on specific variables for optimization, such as mix proportions and curing conditions.2. Short-Term Evaluation: The mechanical properties of concrete were evaluated after 28 days of curing.Longerterm studies tracking the performance over several years or decades would provide a more comprehensive understanding of how POFA concrete behaves over time.3. Laboratory Setting: The research was conducted in a controlled laboratory environment.Real-world construction sites may have variations in conditions that could affect the performance of POFA concrete differently.Field studies and on-site assessments could complement these findings.4. Material Variability: The properties of POFA can vary based on its source and processing methods.Future research should explore how these variations impact the effectiveness of Scheffe's optimization in different contexts.

Figure 2 .
Figure 2. (a) Nigerian map showing Cross River State.(b) Cross River State showing various L.G.A.

Figure 5 .
Figure 5.Comparison of the developed model and experimental results.

Table 1 .
Second order mixture formulation matrix table.

Table 2 .
Second order mixture formulation matrix table for control points.

Table 4
presents the results of the chemical properties assessment of the test admixtures achieved using x-ray fluorescence (XRF).The findings show that POFA predominantly consists of SiO 2 (62.8%),Al 2 O 3 (4.6%),andFe 2 O 3 (3.02%),totaling70.52% by composition indicating good pozzolanic properties in line with ASTM C618, 98 specifications.The test Portland cement and POFA under study from the obtained results, exhibit significant levels of calcium oxide (CaO) at 10.23% and 7.6% respectively.This abundance of CaO boosts complete hydration of cement, improving the mechanical strength and durability characteristics, as depicted in Table4.The cement

Table 4 .
Elemental composition of test samples using X-ray fluorescence (XRF).

Table 5 .
71mpressive strength results.maximumstrengthvalueat8.84 N/mm 2 , while Y 5 and Y 45 had the minimum at 4.25 and 4.89 N/mm 2 respectively.Table8presents control point responses for validating the regression model, where C 5 had the highest flexural strength at 7.91 N/mm 2 , and C 45 had the lowest at 4.96 N/mm 2 .These results demonstrate the improved strength performance of the concrete as POFA percentage replacement of cement in the matrix increases to about 35%71.

Table 6 .
Compressive strength results for control points.

Table 8 .
Flexural strength experimental response for control points.

Table 9 .
Splitting tensile strength experimental response.Splitting tensile strength model equationThe Scheffe's quadratic model equation is derived by incorporating the experimental results for the concrete's splitting tensile strength into Eq.(

Table 10 .
Splitting tensile strength experimental response for control points.

Table 12 .
Regression model coefficient for flexural strength.

Table 13 .
Regression model coefficient for flexural strength.

Table 14 .
ANOVA results for compressive strength.

Table 15 .
ANOVA results for flexural strength.

Table 16 .
ANOVA results for splitting tensile strength.