Empirical Compression Model of Ultra-High-Performance Concrete Considering the Effect of Cement Hydration on Particle Packing Characteristics

The mix design of UHPC has always been based on a large number of experiments; in order to reduce the number of repeated experiments, in this study, silica fume (SF), fly ash (FA), and limestone powder (LP) were used as the raw materials to conduct 15 groups of experiments to determine the particle size distribution (PSD) properties of UHPC. A model of multi-component hydration based on the SF, FA, and LP pozzolanic reactions was devised to quantify the rate and total heat release during the hydration process. Additionally, a microscopic pore development model, which was based on the accumulation of hydration products, was established to measure the effect of these products on the particle-packing properties. Utilizing this model, a UHPC strength prediction technique was formulated to precisely forecast the compressive strength based on a restricted experimental data set. The applicability of this prediction method was verified using 15 sets of existing experimental data along with the data collected from 4 research articles. The results show that the prediction method can predict the strength values of different mix proportions with an accuracy rate of over 80%.


Introduction
Ultra-high-performance concrete (UHPC), a form of concrete designed to maximize the particle size distribution (PSD) of cementitious materials, was founded on the notion of reactive powder concrete (RPC). To increase uniformity and reduce defects, ultrafine reactive powder is added and normally coarse aggregates are not included. To bolster toughness and reduce the water to binder and porosity ratios, steel fibers are also incorporated [1,2]. UHPC has very high strength, toughness, and durability compared with ordinary concrete [3]. A recent research project on UHPC, the hydraulic tunnel project in China [4,5], is considered to be of great importance. This project features many of the usual traits of UHPC, such as its remarkable depth of burial, intense in situ pressure, and intricate geological conditions. As an effective way to resolve relevant problems, the concrete of the hydraulic tunnel was released from water, and UHPC was developed for the tunnel lining structure. A few studies [4,5] have been conducted to investigate the comprehensive value of using UHPC for improving the service life of structures.
Considering that there are few theoretical bases for UHPC design, further experimental studies are essential for achieving UHPC [6]. A few mix design methods have been investigated to simplify the UHPC design workload [7][8][9][10]. Four types of mix design methods have been proposed in the literature, which are based on rheological properties, statistical designs, neural network models, and close packing models [11][12][13][14][15]. No theoretical model exists for the design of mixed materials based on rheological properties, thus necessitating the constant alteration of these properties. A considerable quantity of Two kinds of sand were used: coarse quartz sand with a size of 0-4.75 mm and fine quartz sand with the size of 0-2 mm. The density of these two kinds of sand was 2.64 g/cm 3 . To enhance the structure's mechanical properties, straight steel fibers that were 0.2 mm in diameter and 13 mm in length were employed. To adjust the workability of the UHPC, a poly-carboxylic acid-based superplasticizer (SP) was employed with a dosage of 30 kg/m 3 for the mix design. Table 2 presents 15 amalgamations of cement-based substances, with the water to binder and sand to binder ratios being 0.20 and 1.1, respectively. These substances are cementitious and comprise cement, SF, FA, and LP.

Preparation of UHPC and Curing
All mixtures were created and examined at a temperature of 20 ± 2 • C. The lab conditions necessitated us to perform the mixing of dried aggregates and powder materials.
The mixing was performed in three steps: (1) Put all dried powder materials and sand in a mixer for 60 s to mix.
(2) Add water and SP to 80%, remix the sample for 360 s, and then stop the mixer for 30 s.
The GB/T 31387-2015 test for mechanical properties necessitates a cubic specimen with the dimensions of 100 mm × 100 mm × 100 mm for compressive strength; a loading rate ranging from 1.2 MPa/s to 1.4 MPa/s; and a prism specimen with the dimensions of 100 mm × 100 mm × 400 mm for flexural strength, with a loading rate ranging from 0.08 MPa/s to 0.1 MPa/s.

Experimental Results
To examine the impact of diverse mineral blends on the compressive strength of UHPC concrete, SF, FA, and LP were chosen for the mix formation.
The results are presented in Table 3. The compressive strength of UHPC increases with the increase in the concentration of mineral amalgamations after 7 and 28 days of curing. As the SF content in the system rises to 10% and 40%, the compressive strength of the UHPC increases by 7% and 38%, respectively; this increase is mainly attributed to SF' s dense microstructure and rapid pozzolanic reaction rate [38]. After seven days of curing, the compressive strength of UHPC with SF being replaced by cement is greater than that of UHPC with FA or LP. Nevertheless, when FA replaces cement, the average 7-day compressive strength of UHPC decreases by 4% compared with UHPC with SF, implying that FA's pozzolanic reaction is sluggish and incomplete. After 28 days, the pozzolanic reaction intensifies, thus diminishing the disparity between UHPC's compressive strength when FA or SF is substituted for cement [39]. However, LP, being an inert admixture, only plays a filling role in the cement paste [40]. No discernible effect of UHPC's density on its potency is visible, and its compressive strength is inferior to that of SF and FA.
When the content of SF is 10%, its 28-day compressive strength meets the requirement of 100 MPa (Table 3). From an economic perspective, the experiments that were conducted with multi-powder material instead of cement used a 10% SF content along with a minimum compressive strength of 100 MPa, as per the GB/T 31387-2015 standard. Thus, FA or LP was used instead of cement while keeping the SF content at 10%. As shown in Table 3, there is a gradual rise in compressive strength over the 7-day and 28-day periods with the rise in FA content. As the FA content rises from 10% to 20%, the compressive strength of the system increases; however, when the FA content surpasses 20%, it decreases. A slight rise in compressive strength is also seen with the increase in LP content.
where the percentage of particle size D (%) is denoted by P(D i ); the maximum particle size (µm) is D max ; the minimum particle size is D min ; and the distribution modulus, q, is determined by the ratio of large to small particles in the system. The larger the particles, the larger the modulus. Figure  The least square method (LSM) was used to adjust the mass ratio of the in the dry mixture. The residual sum of squares (RSS) was used to furthe divergence between the cumulative distribution curve of the mixture and th Particle size (µm) The least square method (LSM) was used to adjust the mass ratio of the raw materials in the dry mixture. The residual sum of squares (RSS) was used to further elucidate the divergence between the cumulative distribution curve of the mixture and the model target curve, which is derived from various amounts of raw materials. The RSS was calculated as follows: where RSS is the sum of the squares of residuals and Pmix and Ptra are the actual accumulation curve and target curve, respectively. The raw materials' particle sizes, D max and D min , were 9500 µm and 0.503 µm, respectively, at their peak and minimum. The distribution modulus (q) was 0.23 to establish the objective function of the MAA model ( Figure 1). The MAA model's RSS, which is a dimensionless metric, quantitatively reflects the close packing degree of UHPC particles. A lower RSS implies a higher packing density; thus, RSS reflects the initial accumulation state of concrete with varying mix designs. The MAA model's calculated RSS value curve was then mirrored (Figure 2) to compare the compressive strength trend as predicted by the model to the actual test trend. The least square method (LSM) was used to adjust the mass ratio of the raw materials in the dry mixture. The residual sum of squares (RSS) was used to further elucidate the divergence between the cumulative distribution curve of the mixture and the model target curve, which is derived from various amounts of raw materials. The RSS was calculated as follows: where RSS is the sum of the squares of residuals and Pmix and Ptra are the actual accumulation curve and target curve, respectively. The raw materials' particle sizes, Dmax and Dmin, were 9500 µm and 0.503 µm, respectively, at their peak and minimum. The distribution modulus (q) was 0.23 to establish the objective function of the MAA model ( Figure 1). The MAA model's RSS, which is a dimensionless metric, quantitatively reflects the close packing degree of UHPC particles. A lower RSS implies a higher packing density; thus, RSS reflects the initial accumulation state of concrete with varying mix designs. The MAA model's calculated RSS value curve was then mirrored (Figure 2) to compare the compressive strength trend as predicted by the model to the actual test trend.
(a) 7-day compressive strength (b) 28-day compressive strength The results suggest that the initial accumulation state can predict future strength trend laterally and that the initial accumulation can qualitatively characterize the compressive strength of concrete. Therefore, this study only needed to consider the influence of hydration products based on the initial accumulation, which could accurately predict the compressive strength.

Multi-Component Hydration Model Considering the Influence of Different Powders
An examination of the influence of the raw materials on the compressive strength of UHPC was conducted, which was based on physical accumulation while disregarding the effect of alterations in the water to binder ratio on concrete performance. Thus, incorporating the chemical hydration reaction process into the accumulation model was essential. The hydration of cement and the effect of product volume alterations on accumulation are the primary discussion of the subsequent two sections. Figure 3 illustrates the entire process, including both the physical accumulation and the chemical hydration reaction. UHPC was conducted, which was based on physical accumulation while disregarding the effect of alterations in the water to binder ratio on concrete performance. Thus, incorpo rating the chemical hydration reaction process into the accumulation model was essential The hydration of cement and the effect of product volume alterations on accumulation ar the primary discussion of the subsequent two sections. Figure 3 illustrates the entire pro cess, including both the physical accumulation and the chemical hydration reaction.

Hydration of Cement in Concrete
The four major minerals of Portland cement, alite (C 3 S), belite (C 2 S), aluminate (mainly C 3 A), and ferrite (C 4 AF), are powdered cement clinker and gypsum ( Figure 4). However, when cement clinker is burned, aluminate and ferrite (collectively known as the gap phase) fill the gaps between calcium silicates [35,36].  The proportions of mineral components in various types of cement vary. Therefore, a hydrothermal model suitable for any given kind of cement must accurately describe the heat dissipation process of cement based on its mineral content. Various pozzolans can be employed to reduce or substitute Portland cement, thereby inhibiting the production of heat. Therefore, different raw materials are regarded as one unit, and the hydration transfer process of different components must be defined based on their interactions. A hydrothermal model based on such multi-component concepts represents various types of cement based on their clinker mineral composition [40].
An original multi-component cement hydration model was developed by Koichi Maekawa [40], which calculates the hydrothermally generated rate of cement by using the sum of the individual hydrothermal heat rate of each component based on the percentage of each component; meanwhile, the exothermic reaction of each mineral is described sep- The proportions of mineral components in various types of cement vary. Therefore, a hydrothermal model suitable for any given kind of cement must accurately describe the heat dissipation process of cement based on its mineral content. Various pozzolans can be employed to reduce or substitute Portland cement, thereby inhibiting the production of heat. Therefore, different raw materials are regarded as one unit, and the hydration transfer process of different components must be defined based on their interactions. A hydrothermal model based on such multi-component concepts represents various types of cement based on their clinker mineral composition [40].
An original multi-component cement hydration model was developed by Koichi Maekawa [40], which calculates the hydrothermally generated rate of cement by using the sum of the individual hydrothermal heat rate of each component based on the percentage of each component; meanwhile, the exothermic reaction of each mineral is described separately. The total heat consumption of cement that contains the mixed powders (H C ) is the sum of the heat consumption of the component reactions [35].
where P i is the weight composition ratio and H C3AET and H C4AFET are the rates of heat generation during the formation of ettringite based on the aluminate and ferrite phases, respectively. After the reaction of calcium hydroxide formation has stopped due to the disappearance of unreacted gypsum, hydration heat is generated in C 3 A and C 4 AF (expressed as H C3A and H C4AF , respectively). H i is the heat release rate of mineral i per unit weight, as defined in Equation (5): where the activation energy of component i, which is denoted by E i , is accompanied by the gas constant R, and the reference heat-generation rate of component i at a constant temperature T 0 is H i,T0 . The coefficient of change when no other effect is present, γ i , is defined as the delaying effect of the chemical admixture and FA in the initial exothermic hydration process; β i is the decrease in the heat generation rate due to the reduction in the availability of free water. λ i is the coefficient of the heating rate change caused by the absence of Ca(OH) 2 in the powder mixture's liquid phase. A coefficient of heat generation rate alteration, µ i , is the interdependence between alite and belite in Portland cement.
(−E i /R) is defined as the thermal activity.

Effect of Fly Ash on Hydration
FAs are typically regarded as single reaction units that blend cement. However, these components are considered in this model as a single component, and the mixing of cement partially suppresses the generation of heat. The reaction of the powder mixture is determined based on the hydration of cement, which is linked to the amount of Ca(OH) 2 [36].
where r and s are the material constants that are common to all minerals. The comparison of the experimental and analytical results show that R = 5.0 and S = 2.4. The coefficient of β i varies from 0 to 1, and s i is a function of the normalized Blaine value, which represents the change in the reference heating rate due to powder fineness. w f ree is the free water ratio, and η i is the internal reaction layer of the thickness component i. It is essential for the model to calculate the amount of Ca(OH) 2 consumed by cement hydration, as other powder mixtures require it. The rate of reduction after mixing is expressed as the ratio of the remaining Ca(OH) 2 to the amount required for the active reaction of FA.
where F CH is the amount of Ca(OH) 2 , that is produced by the hydration of C 3 S and C 2 S but not consumed by the C 4 AF reaction, and R FACH is the amount of Ca(OH) 2 required for the reaction with FA when a sufficient amount of Ca(OH) 2 is available. Because of limited test equipment, this study did not carry out an isothermal exothermic test, but instead used the data obtained from isothermal exothermic tests in other scholars' articles to verify the model. In the study by He [33], isothermal calorimetry tests were conducted on UHPC samples with different levels of FA content to measure the heat dissipation rate during the hydration process. A comparison was drawn between the outcomes of the isothermal experiment and the numerical simulations in the present research, which utilized the original multi-component hydration model. Figure 5 illustrates the contrast between the experimental and numerical simulation outcomes of the UHPC samples with varying FA content in terms of heat dissipation rate. The experimental results demonstrate that the inclusion of FA can significantly reduce the heat dissipation rate of the UHPC samples. The original multi-component hydration model's simulation results demonstrate a strong correlation with the experimental data, thereby demonstrating its broad applicability to UHPC with varying levels of FA content.

Effect of Silica Fume on Hydration
As an active admixture, the exothermic process of SF is similar to that of FA; thus, the heat release rate formula of SF is as follows:

Effect of Silica Fume on Hydration
As an active admixture, the exothermic process of SF is similar to that of FA; thus, the heat release rate formula of SF is as follows: A comparison of SF concrete's isothermal heat dissipation test results ( Figure 6) with its numerical simulation results was performed to guarantee the broad applicability of the multi-component hydration model's original design.

Effect of Limestone on Hydration
In the multi-component hydration model presented in this study, LP was included as a powder material. LP, although not actively engaged in the hydration process, has a critical role in optimizing the model's performance. The packing density and porosity are both increased and reduced by its micro-filling effect on other active components [41,42], thus enhancing the mechanical and chemical properties of the system. Figure 7 illustrates the proposed mechanism by which LP contributes to the micro-filling effect of the model.
For P4 and P5: To demonstrate the broad applicability of Kadri's [26] isothermal heat dissipation test for SF concrete and its capacity for accurately predicting SF concrete behavior, a comparison was made between the numerical simulation results from the original multi-component hydration model and the results of the isothermal heat dissipation test. A comparison between the experimental and numerical results, as depicted in Figure 6 and corroborated by the model, is essential for understanding and optimizing SF concrete's performance, as it captures the intricate interplay between chemical reactions and physical processes during the initial hydration stages.

Effect of Limestone on Hydration
In the multi-component hydration model presented in this study, LP was included as a powder material. LP, although not actively engaged in the hydration process, has a critical role in optimizing the model's performance. The packing density and porosity are both increased and reduced by its micro-filling effect on other active components [41,42], thus enhancing the mechanical and chemical properties of the system. Figure 7 illustrates the proposed mechanism by which LP contributes to the micro-filling effect of the model.

Effect of Limestone on Hydration
In the multi-component hydration model presented in this study, LP was included as a powder material. LP, although not actively engaged in the hydration process, has a critical role in optimizing the model's performance. The packing density and porosity are both increased and reduced by its micro-filling effect on other active components [41,42], thus enhancing the mechanical and chemical properties of the system. Figure 7 illustrates the proposed mechanism by which LP contributes to the micro-filling effect of the model.  For P2 and P3: HS ij = HS ij · (1 + k H1 · r s ) For P4 and P5: For P6: where j is the point number in the reference heat rate function; Q ij and HS ij are the heat rate and accumulated heat of component i, respectively; Q imax is the maximum heat generation; p LP and p C are the weight fractions; and B LP and B C are the unit weight surface areas of LP and Portland cement, respectively. The acceleration effect is expressed by the ratio of the surface areas of LP and cement, r s , which is taken as an indicator. The coefficients k H1 , k H2 , k H3 , and k Q are multiplied by r s to represent the degree of contribution of LP. The results of the isothermal exothermic test for LP concrete, as reported by Moon [43], were compared with the numerical simulation results obtained using the original multicomponent hydration model (Figure 8). This comparison was made to demonstrate the broad applicability of the multi-component hydration model and its ability to accurately predict the behavior of various types of concrete, including those containing LP as an inert material. The comparison affirms that the model is able to precisely capture the exothermic heat discharged during the initial hydration period, which is essential for comprehending and optimizing the performance of LP concrete.
where j is the point number in the reference heat rate function; Qij and HSij are rate and accumulated heat of component i, respectively; Qimax is the maximum hea ation; pLP and pC are the weight fractions; and BLP and BC are the unit weight surfa of LP and Portland cement, respectively. The acceleration effect is expressed by of the surface areas of LP and cement, rs, which is taken as an indicator. The coe kH1, kH2, kH3, and kQ are multiplied by rs to represent the degree of contribution of L The results of the isothermal exothermic test for LP concrete, as reported b [43], were compared with the numerical simulation results obtained using the multi-component hydration model (Figure 8). This comparison was made to dem the broad applicability of the multi-component hydration model and its ability rately predict the behavior of various types of concrete, including those containin an inert material. The comparison affirms that the model is able to precisely cap exothermic heat discharged during the initial hydration period, which is essential prehending and optimizing the performance of LP concrete.

Simulation of Multi-Component Powders on Hydration
In the preceding sections, the hydration of binary powder systems (exclu ment) was analyzed in detail to examine the heat release and chemical reactions th during the early stages of hydration. The analysis yielded significant revelations

Simulation of Multi-Component Powders on Hydration
In the preceding sections, the hydration of binary powder systems (excluding cement) was analyzed in detail to examine the heat release and chemical reactions that occur during the early stages of hydration. The analysis yielded significant revelations regarding the conduct of these systems, aiding us in constructing a more thorough comprehension of their efficacy. In this section, the analysis was extended to ternary powder systems by using the developed multi-component hydration model to verify the heat release during hydration. Thongsanitgarn [44] delved into the data that were generated from an investigation of ternary powder systems to gain further understanding of the behavior of these systems as well as any distinctions or resemblances between the binary powder systems under study. Comparing and contrasting the outcomes of these analyses allowed for a more profound comprehension of the elements that affect the hydration of powder systems and how they can be improved for better performance.
Based on the literature data, the isothermal exothermic test results were compared with the numerical simulation results of the multi-component hydration model in this study ( Figure 9). The overall exothermic fitting is more accurate, which reflects the wide applicability of the multi-component hydration model. Based on the literature data, the isothermal exothermic test results were compare with the numerical simulation results of the multi-component hydration model in thi study ( Figure 9). The overall exothermic fitting is more accurate, which reflects the wid applicability of the multi-component hydration model.

Microscopic Influence of Hydration Products
Considering that it is the solid phase rather than the porous phase that sustain strength development, the volume ratio of hydration products to the initial capillary spac Dhyd.out must be taken into account, and the volume of hydrates formed outside the prim tive cement particles was used as an index describing strength development [36].
where Dhyd.out is the ratio of the initial capillary space to the space occupied by a larg amount of external hydrates; Vhyd.out is the volume of hydrates formed outside the primitiv cement particles; Vhyd.total is the total volume of hydration products formed inside the orig inal cement particles; Vhyd.in is equivalent to the reaction volume fraction of mineral com pounds; and Vcap.ini is the volume of the initial capillary space. The volume changes of var ious hydration inner and outer products are shown in Figure 10.

Microscopic Influence of Hydration Products
Considering that it is the solid phase rather than the porous phase that sustains strength development, the volume ratio of hydration products to the initial capillary space D hyd.out must be taken into account, and the volume of hydrates formed outside the primitive cement particles was used as an index describing strength development [36].
where D hyd.out is the ratio of the initial capillary space to the space occupied by a large amount of external hydrates; V hyd.out is the volume of hydrates formed outside the primitive cement particles; V hyd.total is the total volume of hydration products formed inside the original cement particles; V hyd.in is equivalent to the reaction volume fraction of mineral compounds; and V cap.ini is the volume of the initial capillary space. The volume changes of various hydration inner and outer products are shown in Figure 10.
where Dhyd.out is the ratio of the initial capillary space to the space occupied by a large amount of external hydrates; Vhyd.out is the volume of hydrates formed outside the primitive cement particles; Vhyd.total is the total volume of hydration products formed inside the original cement particles; Vhyd.in is equivalent to the reaction volume fraction of mineral compounds; and Vcap.ini is the volume of the initial capillary space. The volume changes of various hydration inner and outer products are shown in Figure 10. The Schiller model [36] shows that the volume of the initial capillary space V cap.ini is equivalent to the porosity without strength. Although Schiller determined the value by fitting the estimated value with the experimental results, in the model proposed in this study, the volume of the initial capillary space V cap.ini was calculated based on the water to binder ratio of the mixture as follows: where W/C is the water to binder ratio and ρ c is the cement density. The compressive strength is impacted by the magnitude of the hydration products created by cement particles. These particles are shaped like spheres, and the mineral elements of cement clinker react with free water during the hydration process to create products. These spheres are divided into the non-hydration inner core, original boundary, hydrated inner product, and hydrated outer product. As shown in Figure 11, a lower water to binder ratio results in a higher compressive strength, regardless of whether D hyd.out is the same, due to the smaller spacing between particles in the mixture with a reduced water to binder ratio. The Schiller model [36] shows that the volume of the initial capillary space Vcap.ini is equivalent to the porosity without strength. Although Schiller determined the value by fitting the estimated value with the experimental results, in the model proposed in this study, the volume of the initial capillary space Vcap.ini was calculated based on the water to binder ratio of the mixture as follows: where W/C is the water to binder ratio and ρc is the cement density. The compressive strength is impacted by the magnitude of the hydration products created by cement particles. These particles are shaped like spheres, and the mineral elements of cement clinker react with free water during the hydration process to create products. These spheres are divided into the non-hydration inner core, original boundary, hydrated inner product, and hydrated outer product. As shown in Figure 11, a lower water to binder ratio results in a higher compressive strength, regardless of whether Dhyd.out is the same, due to the smaller spacing between particles in the mixture with a reduced water to binder ratio.

Effect of Hydration Products on Packing Density
The change factor ki of the hydration products' accumulation state was taken into account by considering two parameters: the alteration in the volume and the gap between cement particles. The ki is calculated as follows:

Effect of Hydration Products on Packing Density
The change factor k i of the hydration products' accumulation state was taken into account by considering two parameters: the alteration in the volume and the gap between cement particles. The k i is calculated as follows: where D hyd.out is the ratio of the space occupied by a large amount of external hydrates to the initial capillary space; θ is the gap effect between cement particles; k i is the hydration factor of different powders i, including SF, FA, and LP; ϕ i is the mass ratio of powders in the mixture; and m c is the mass of different powders i (kg/m 3 ). The volume of hydration products dynamically changes according to the hydration reaction; thus, the stacking state of the entire structure changes accordingly. An accurate prediction of concrete strength should be quantified by considering the influence of the change in hydration product volume on stacking.

Empirical Strength Prediction Model
A model of strength prediction based on alterations in the porosity and hydration product was proposed in a previous study [36]. The model equations are as follows: where f ∞ is the ultimate strength (N/mm 2 ); α and β are the material constants; and P c , P SF , P FA , and P LP represent the weight fractions of Portland cement, SF, FA, and LP, respectively. A, B, C, D, and E, are the material constants, indicating the contribution factor of each component on the strength development. This model has A at 120, B at 70, C at 210, D at 240, and E at 180, which represent the critical measure of concrete's strength and the water to binder ratio while taking into consideration the alterations in hydration products. In this model, the strength prediction of UHPC is limited to 0.14-0.22.
The accuracy of the model was verified using 15 sets of strength analysis data, which were obtained through experiments using different mix proportion design combinations ( Table 2).
By comparing the measured compressive strength values of different mix designs with the compressive strength values simulated by the empirical model, the experiment proves that the empirical model can successfully quantify the compressive strength values, as shown in Figure 12. The model can accurately predict the compressive strength of various mix proportions and has wide applicability and practicability in improving the performance of concrete.

Simplified Mix Design Process
To enable an accurate prediction of concrete strength with minimal experimentation, a strength prediction equation was developed based on the changes in the volume and heat rate of water. By establishing a parameter system using only a few experiments, it is possible to use the concrete strength to accurately predict and optimize the mix design process for improved performance. The resulting simplified mix design process, shown in Figure 13, provides a practical and efficient method for designing and constructing concrete structures with predictable and reliable strength characteristics. By combining our strength prediction equation with the simplified mix design process, a comprehensive framework is offered for optimizing the performance of concrete, ensuring its durability and stability over time. These results provide a dependable and effective method for forecasting and refining the concrete strength and efficiency, thereby significantly influencing the design and construction of concrete structures.
By comparing the measured compressive strength values of different mix designs with the compressive strength values simulated by the empirical model, the experiment proves that the empirical model can successfully quantify the compressive strength values, as shown in Figure 12. The model can accurately predict the compressive strength of various mix proportions and has wide applicability and practicability in improving the performance of concrete.

Simplified Mix Design Process
To enable an accurate prediction of concrete strength with minimal experimentation, a strength prediction equation was developed based on the changes in the volume and heat rate of water. By establishing a parameter system using only a few experiments, it is possible to use the concrete strength to accurately predict and optimize the mix design process for improved performance. The resulting simplified mix design process, shown in Figure 13, provides a practical and efficient method for designing and constructing concrete structures with predictable and reliable strength characteristics. By combining our strength prediction equation with the simplified mix design process, a comprehensive framework is offered for optimizing the performance of concrete, ensuring its durability and stability over time. These results provide a dependable and effective method for forecasting and refining the concrete strength and efficiency, thereby significantly influencing the design and construction of concrete structures.

Validation
The effectiveness of the simplified mix design process was verified by means of 23 sets of experimental data from various papers [45][46][47][48]. The purpose of this verification was to demonstrate the precision and dependability of the simplified mix design process in predicting the concrete strength and optimizing the mix designs for enhanced performance. The verification results, as depicted in Figure 14, affirm the efficiency of the sim-

Validation
The effectiveness of the simplified mix design process was verified by means of 23 sets of experimental data from various papers [45][46][47][48]. The purpose of this verification was to demonstrate the precision and dependability of the simplified mix design process in predicting the concrete strength and optimizing the mix designs for enhanced performance. The verification results, as depicted in Figure 14, affirm the efficiency of the simplified mix design process in precisely forecasting the concrete strength while simultaneously optimizing the mix designs. A simulation was conducted to evaluate the efficacy of the simplified m cess with the aim of ascertaining the ideal mix proportion and forecastin value of concrete. The simulation was designed to demonstrate the dependa cision of the simplified mix design process in forecasting concrete strength a mix designs for enhanced performance. The simulation results, as shown confirm the effectiveness of the simplified mix design process in accurat concrete strength values while simultaneously optimizing mix proportions

Conclusions
In this study, SF, FA, and LP were used as raw materials to conduc experiments to determine the different original PSD properties of UHPC multi-component hydration, which was based on the pozzolanic reaction LP, was formulated to quantify both the rate and total heat release of the h cess. Additionally, a microscopic pore development model based on the ac hydration products was created to measure the effect of these products o packing characteristics. A UHPC strength prediction model was designe strength, which was based on the abovementioned model and only used d experiments. The precision of the proposed model was confirmed by contr pirical outcomes with existing experimental outcomes. The conclusions of prospects are as follows: A simulation was conducted to evaluate the efficacy of the simplified mix design process with the aim of ascertaining the ideal mix proportion and forecasting the strength value of concrete. The simulation was designed to demonstrate the dependability and precision of the simplified mix design process in forecasting concrete strength and optimizing mix designs for enhanced performance. The simulation results, as shown in Figure 14, confirm the effectiveness of the simplified mix design process in accurately predicting concrete strength values while simultaneously optimizing mix proportions.

Conclusions
In this study, SF, FA, and LP were used as raw materials to conduct 15 groups of experiments to determine the different original PSD properties of UHPC. A model of multi-component hydration, which was based on the pozzolanic reaction of SF, FA, and LP, was formulated to quantify both the rate and total heat release of the hydration process. Additionally, a microscopic pore development model based on the accumulation of hydration products was created to measure the effect of these products on the particle packing characteristics. A UHPC strength prediction model was designed to measure strength, which was based on the abovementioned model and only used data from a few experiments. The precision of the proposed model was confirmed by contrasting the empirical outcomes with existing experimental outcomes. The conclusions of this study and prospects are as follows: (1) The compressive strength tests using different powders were carried out. The results show that SF can significantly improve the compressive strength of concrete, and FA is better than LP in terms of improving the strength. The MAA model is used to simulate the experimental results of compressive strength with different mix designs, which can predict the trend of compressive strength under different mix designs well.
(2) A model of hydration heat release, which was composed of multiple components, was formulated to quantitatively gauge the influence of SF, FA, and LP. Verification through experimentation affirmed that the simulation of the reaction rate and heat release during the hydration process was very accurate.
(3) A micro-pore development model was developed based on the accumulation of hydration products. Estimates were made regarding the fluctuation in the volume of distinct hydration products and the dynamic alteration of the cement particle gap. Based on these results, a complete set of formulas was established to create the parameter system.
(4) The verification of the proposed strength prediction method was accomplished through the utilization of experimental data obtained from the literature. The results suggest that the proposed method can achieve accurate strength prediction by utilizing data from only a few experiments.
(5) In this study, the proposed method could reduce the amount of practical engineering experiments and guide the mix design; however, the UHPC strength model in this paper is only suitable for compressive strength, and the test results of flexural strength have not been further analyzed. It is suggested that in a follow-up study, the prediction model of the bending strength of UHPC with different powders could be considered to better guide the mix design's mechanical properties in UHPC.