Prediction of Temperature Distribution in High-Strength Concrete Using Hydration Model

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

High-strength concrete is being used increasingly in the construction of high-rise buildings, bridge foundations, and marine structures. If high-strength concrete is used for columns or other large section members of massive concrete structures, the center temperature of the members will rise rapidly at early ages due to hydration heat, and the high temperatures will remain in the member for a relatively long period of time due to the low thermal conductivity of concrete. Meanwhile, the surface will release its heat directly into the surrounding environment. This causes a significant difference between the center and the surface in the member even though the concrete members are small.1 This temperature variation can result in tensile stresses within the member restrained by external conditions that may cause cracking, loss of strength, excessive seepage, and reduced durability of the structure.2,3

The prediction of temperature history in hardening concrete is essential to estimate the thermal stress as well as to prevent thermal cracking. The ability to predict the expected temperature history would thus be useful to structural engineers and designers interested in producing a durable concrete structure. In the past, several researchers4-6 have attempted to predict the temperature rise occurring in hardening concrete, but the focus of these models has been on following the relationships between hydration heat rate and concrete maturity function under adiabatic conditions. The models did not take into account the effect of the water-cement ratio (w/c) on hydration heat release. The heat generation and varying mechanical properties of concrete at early ages are strongly related to the degree of hydration of each mineral compound consisting of cement. Thus, it is desired to consistently predict them with a single correlating parameter, that is, degree of hydration.

This paper presents a computer model to predict the temperature history in hardening concrete at any time. The numerical simulation procedure starts with a hydration model that describes the evolution of cement paste microstructure as a function of the changing composition of the hydration products. The coefficients for the formulation of the model are determined by pertaining data about materials, mixture proportions, and environmental conditions into an artificial neural network. A three-dimensional (3D) finite element thermal analysis is applied to model the transient heat transfer between the concrete and the surroundings as affected by the concrete mixture, thermal boundary, and environmental conditions.

RESEARCH SIGNIFICANCE

In recent years, high-strength concrete has proven to be an extremely sensitive material regarding thermal cracking at an early age. To estimate the thermal stress as well as to prevent thermal cracking, the prediction of temperature profile in hardening concrete is essential. To solve this problem, a universal model is necessary to comprehensively describe the early-age behavior. The authors believe that this detailed study dealing with the short-term temperature rise will be very useful to concrete technology.

UNIFIED SOLIDIFICATION MODEL OF CEMENT-BASED MATERIALS

Unit cell model

The unit cell-based 3D cement hydration and microstructural model has been described in a number of recent publications.7,8 In this paper, the authors precisely modified some equations in a previous paper.8 The cement paste is modeled as a unit cell that consists of three portions: the unhydrated cement grain, gel, and capillary pores. To simplify this aspect of the model, the cement particles are assumed to be spherical, uniformly sized, and uniformly distributed throughout the cement paste. As the hydration proceeds, the hydration products grow uniformly at the surface of a cement particle and so the overall shape remains spherical over time. As shown in Fig. 1, the model is formulated geometrically in terms of a unit cell. The basic elements of the model include the following.

The length of the cube that corresponds to the volume of cement paste, 1 cm^sup 3^, is

... (1)

where χ is the w/c, ρ^sub c^ is the density of cement (g/cm^sup 3^), and r0 is the average radius of a cement particle.

... (2)

where S is the specific surface area of cement (cm^sup 2^/g).

When the cement particle is not in contact with an adjacent particle, the degree of hydration α is

... (3)

where ri is the radius of the unhydrated cement particle (= r0 - t) and t is the layer depth of inner hydration products.

The rate of volume growth of the hydration products ? is

... (4)

where R is the radius of the cement particle including the outer hydration products.

From Eq. (3) and (4), R is a function of the degree of hydration and the rate of volume growth

R = [1 + (ξ - 1)α]^sup 1/3^r^sub 0^ (5)

In this paper, the rate of volume growth9 is assumed have a constant value of 2.0. Therefore, R can be calculated from the hydration model. As shown in Fig. 1, R changes gradually as the hydration reaction progresses. At any time, the volume of the cement particle including the outer hydration product Vs is

... (6)

... (7)

... (8)

... (9)

Vs changes gradually with R as shown in Fig. 2.

Hydration model

In the unit cell, the volume fractions of the constituents quantified with a single function of the degree of hydration. The degree of hydration of paste was calculated with the hydration model that describes the rate of hydration. The basic concepts of the model are based on a classic, shrinking, or unreacted core model,10 as shown in Fig. 3. The reaction rate is expressed as a single equation composed of four rate coefficients: B, C, D, and kr. These coefficients11 determine the rate of formation and destruction of the initial impermeable layer, the activated chemical reaction process, and the diffusion controlled process. The process is considered to be mass transfer controlled so that the rate of water diffusion determines the hydration rate and is expressed as follows

... (10)

where α is the degree of hydration, ? is the stoichiometric ratio by mass of water to cement, ρ is the density of unhydrated cement, r0 is the radius of unhydrated cement particles, De is the effective diffusion coefficient of water in the cement gel, kr is the coefficient of reaction rate per unit area of reaction surface, Cw is the imaginary concentration of water at the outer region of gel, and kd is the mass transfer coefficient.

The value of kd is assumed to be a function of the degree of hydration to account for the change in permeability of the hydration products as the reaction progresses and is expressed as

... (11)

The effect of temperature on the hydration rate is introduced via the mass transfer coefficient. Accurate relationships are considered to be obtainable by experimentally determining the changes in the reaction rate. Herein it is assumed that B^sub 20^, C^sub 20^, D^sub 20^, and k^sub r20^ follow an Arrhenius-type law. The coefficients at T are expressed as follows

B = B^sub 20^ × exp[-β^sub 1^(1/T - 1/293)] (12a)

C = C^sub 20^ × exp[-β^sub 2^(1/T - 1/293)] (12b)

D = D^sub 20^ × exp[-β^sub 3^(1/T - 1/293)](12c)

k^sub r^ = K^sub r20^ × exp[-E/R(1/T - 1/293)] (12d)

where B^sub 20^, C^sub 20^, D^sub 20^, and kr20 are the frequency factors evaluated of B, C, D, and kr at 297 K; T is the absolute temperature (K); and β^sub 1^, β^sub 2^, β^sub 3^, and E/R are the reduced activation energies (K).

The model is expressed the entire hydration process of cement paste using a single kinetic expression (Eq. (10)). This unreacted core model, however, applies only to the case where a cement particle is immersed in an infinite pool of water. If the w/c is low, this model cannot properly explain the entire process of cement hydration. To be successful, a hydration model must explain the decrease in the hydration rate when there is insufficient free water.

Water reduction model

The proposed model represents the reduction of water considering two mechanisms; the decreased permeability of water through the hydration products due to the decrease in contact area between the unreacted cement and the free water (refer to Fig. 4). When the hydration products surrounding individual cement particles are not in contact with each other, the interfacial area between the free water and the cement particle is

... (13)

If the hydration products around the particles are in contact with each other

... (14)

... (15)

The interfacial area decreases gradually as shown in Fig. 5. It is assumed that the hydration rate decreases as the interfacial area is reduced. The reduction of the interfacial area is expressed in terms of the concentration of water in the bulk via the hydration model.

C^sub w^ = ηC^sub wo^ (16)

where η is the efficiency of reduction in the contact area, C^sub wo^ is the concentration of water in the bulk, and Cw is the imaginary concentration of water that participates in the hydration reaction.

The relationship between η and R is given by

... (17)

... (18)

Under these assumptions, the reduction of water was described as a reduction of the imaginary concentration of water at the outer region of gel (Cw in Eq. (10)).

Determination of coefficients of hydration model by neural networks

The proposed hydration model includes four unknown coefficients: B, C, D, and kr. These coefficients for the formulation of the hydration model are determined by inputting the pertinent data on the materials, the mixture proportions, and the environmental conditions into an artificial neural network. Back propagation network, a multi-input-singleoutput, was used.12 This network has four intermediate layers, as shown in Fig. 6. A sigmoid function (Fig. 7) that spans a range between [0,1] was used as the input-output function of the individual neural units.13 The number of units in the input layer is 7, and the number of units in the intermediate layers is 15 (4, 4, 4, and 3). An error back propagation method was adopted as the learning algorithm. The sign of the training data for the neural network is 35. The input data values include the mineral composition of the cement (C3S, C2S, C3A, C4AF), the average radius of the cement particles, the cement density, and the w/c, as listed in Table 1. The kinetic factors B, C, D, and kr of the hydration model are given by the output of the net. The effect of temperature on the hydration rate is then introduced using Eq. (12a) through (12d) to determine the reduced activation energies β^sub 1^, β^sub 2^, β^sub 3^, and E/R. According to this approach, the coefficients of the hydration model are given in Table 2.

MODELING OF CONCRETE TEMPERATURE PREDICTION

Description of temperature increasing model

The temperature distribution is determined by resolving the following heat equation

... (19)

where ρ is the density (kg/m3), C is the specific heat capacity (kcal/kg °C), T is the concrete temperature (°C), t is the time (hours), and H is the heat generation rate for cement paste (kcal/(m3?h)).

In this paper, C is assumed to be the function of the degree of hydration and concrete mixture composition14 and the parameter C is expressed as

... (20)

where W, Ce, S, and G are the unit weight of water, cement, sand, and gravel, respectively (kg/m3), and α is the hydration rate.

The heat generation rate for cement paste, H, is assumed to be proportional to hydration rate, as

... (21)

Boundary condition and initial condition

The boundary condition can be described as follows

k?T ? Ts T8 - ( ) = (22)

where ? is the heat convection coefficient between concrete and the surrounding environment (10.0 kcal/m^sup 2^h°C), Ts is the temperature of concrete surface, and T8 is the temperature of the surrounding environment.

The initial condition can be described as follows

T t 0 = T0 (23)

The numerical method to solve temperature increasing equation

In this paper, a finite element method (FEM) is adopted both in time and in space to solve Eq. (19) numerically. The eight-node isoparametric element was built to discrete mass volume concrete in 3D spaces. After the space discretization with a Galerkin's procedure,15 Eq. (19) can be rewritten as follows

... (24)

where [ ] and { } indicate matrix and vector, respectively.

In Eq. (24), the global matrix [B], [C], and {P} are obtained from the integration of the element matrix as follows

... (25a)

... (25b)

... (25c)

Furthermore, based on the time space discretization, Eq. (25) can be written as

([B]+[C]θΔt){T}^sub n+1^=([B]-[C](1-θ)ΔT){T}^sub n^+{P}^sub n^Δt (26)

Generally, the value of parameter ? should be greater than 0.5 to confirm the stability of the numerical calculation. In this paper, based on Galerkin's procedure in time space, the value of parameter ? is adopted as 2/3.

VERIFICATION OF PROPOSED MODEL BY FIELD TEST

Outline of experiment

Experimental results from Tomosawa et al.16 are used for the verification of temperature prediction. The chemical compositions of the cements and the mixture proportions of concrete used in the construction are given in Table 3 and 4, respectively. Two types of concrete were investigated: concrete containing ordinary portland cement (OPC) and concrete containing belite-rich portland cement (BPC), which has been increasingly used for massive concrete and high-strength concrete members due to its low hydration heat generation and low water-reducer demand. highstrength concrete was experimentally placed in a three-story reinforced concrete building having a basement floor with a building area and total floor area of 615 m^sup 2^ (735.5 yd^sup 2^) and 1353 m^sup 2^ (1618 yd^sup 2^), respectively. The concrete temperature was measured at columns (cross section of 550 x 600 mm [1.8 x 1.97 ft]) on the second and third floors (floor height of 3.5 and 3.75 m [11.5 and 12.3 ft]) and beams (cross section of 600 x 600 mm [1.97 x 1.97 ft] and span of 6 m [19.68 ft) on the third floor and roof. Second floor columns and a third floor beams were placed with OPC and third floor columns and a roof beam were placed with BPC. The concrete temperature of actual columns and beams were monitored using thermocouples. The measuring points are as shown in the cross-sectional view in Fig. 8(a). Measurements were made at three height positions (refer to Fig. 8(b)), but this paper only deals with the central and surface position.

Comparison of prediction and measured data of temperature

As shown in Fig. 9, the FEM, with 3D elements was used for the temperature analysis. Because of the symmetry, the FEM calculation and mesh is only performed on a quarter of the column member and on half of the beam members. Three-thousand five-hundred eight node-rectangular elements were used for the analytical discretization of the column and beam. By assuming perfect bond between reinforcement and concrete, the elements of reinforcement are assumed to be truss elements. The heat transfer coefficient of a section through plywood formwork is assumed to be 8.0 kcal/m^sup 2^h°C (1.64 Btu/ft2h°F). The Poisson's ratio for the high-strength concrete is assumed to be 0.17. The temperature history of concrete members is estimated with the flow process diagram, as shown in Fig. 10. The member is first divided into discrete elements, and the reaction rate of each element during a minimal period of time is calculated using the hydration model based on the temperature at the time. Figure 11 shows the comparison of calculated and measured temperature history at the center (C1) of the column and the surface (C3) of the concrete members. The calculated results for both OPC and BPC satisfactorily reproduce the actual temperature history. Figure 11 also shows the temperature distribution of the concrete members at 21 hours after casting. It shows that the maximum temperature in the members generated in the connection part between the column and beam.

CONCLUSIONS

A mathematical model, based on a single kinetic equation to predict the extent of hydration and the microstructure evolution of a cement particle, was modified. Based on the mathematical model, a 3D computer model was developed to predict the temperature rise in hardening concrete. The 3D computer model developed herein clearly demonstrates that it is possible to predict with reasonable accuracy the temperature distribution in hardening concrete in the field. Currently, the 3D computer model cannot predict the temperature history of concrete, which includes mineral admixtures such as fly ash, slag, and silica fume, because a mathematical model does not solve the hydration and microstructure development of their effects. Considering the mineral admixture, the 3D computer model can be applied to the prediction of the temperature rise of various concretes.

ACKNOWLEDGMENTS

The authors acknowledge the financial support of the Sustainable Building Research Center, Hanyang University, which was selected by the Engineering Research Center Program of Ministry of Science and Technology (No. R11-2005-056-04003-0) in South Korea.