# Optimal Heat Transfer Coefficient Distributions during the Controlled Cooling Process of an H-Shape Steel Beam

Authors: Yu-Feng Gan; Jiin-Yuh Jang
Journal: Advances in Materials Science and Engineering
 (2017)

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## Abstract

Three-dimensional thermal-mechanical models for the prediction of heat transfer coefficient distributions with different size beams are investigated. H300 × 300, H250 × 250, and H200 × 200 H-shape steel beams are investigated in a controlled cooling process to obtain the design requirements for maximum uniform temperature distributions and minimal residual stress after controlled cooling. An algorithm developed with the conjugated-gradient method is used to optimize the heat transfer coefficient distribution. In a comparison with the three group results, the numerical results indicate that, with the same model and under the same initial temperature (T=850°C) and final temperature (T=550±10°C), the heat transfer coefficients obtained with the conjugated-gradient method can produce more uniform temperature distribution and smaller residual web stress, with objective functions of the final average temperature Tave±ΔT and maximum temperature difference to minimum min⁡ΔTmax(x,y). The maximum temperature difference is decreased by 57°C, 74°C, and 75°C for Case  1, Case  2, and Case  3, respectively, the surface maximum temperature difference is decreased by 60~80°C for three cases, and the residual stress at the web can be reduced by 20~40 MPa for three cases.


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## Body

## 1. Introduction

H-shape steel beams are widely utilized in industry and on construction sites due to their safety characteristics and the fact that they are economic cross-section steel with a wide flange, thin web, and superior cross-section properties [1, 2]. H-shape steel beams have uneven temperature distribution in the cooling process due to their complex geometry, which probably produces warping and breaking, so controlled cooling plays a very important role after rolling to obtain high-performance, high-strength H-beams.As early as the late 1950s and 1960s, studies on the improvement of the mechanical properties of steel were performed mostly in Europe, where in particular, such studies were being performed actively in the BISRA (British Iron and Steel Research Association). Then, at the end of the 1970s, accelerated cooling equipment was first installed in a Japanese steel mill to provide high-strength, high-toughened steel [3]. JFE Steel developed TMCP (thermomechanical control process) technology in 1980, which is a microstructural control technique combining controlled rolling and cooling [4]. In 1998, JFE Steel developed Super-OLAC, an advanced accelerated cooling system capable of cooling plates homogeneously at high cooling rates close to the theoretical limits [5].Recently, there have been many researchers committed to controlling the temperature distribution and the temperature differences occurring in the cooling process through simulations and experiments intended to improve steel mechanics. In 2003, Liu et al. [6] studied the heat transfer boundary conditions and analyzed the distribution and changes in the temperature field of H-beams during the cooling process, and their results were compared with the available experimental data. In 2008, Liu et al. [7] analyzed the temperature field of H-beams during cooling using a numerical simulation and discussed the temperature distribution and character in different parts of the beams during both air cooling and spray cooling. It was found that the appropriate cooling mode can obtain a fairly uniform temperature field and thus improve the steel mechanical properties. Zhao et al. [8] focused on how to improve H-beam cooling distortion and discussed the uneven temperature distribution occurring in the cooling process causing uneven distortion phenomenon in an H-beam cross-section.In 2013, Guo et al. [9] analyzed temperature fields during both controlled cooling and air cooling and the microstructural mechanical properties for different parts of an H-beam after it was control cooled for 4.5 s, and the lowest and the highest temperature were measured at the edge of the flange and at the R corner, respectively. The microstructure for the different parts of the H-beam consisted of ferrite and pearlite, and the grain size at the R corner was coarser than that at the flange and web. The changes in the microstructure were in good agreement with those of the temperature field, and the results indicated that the uniformity of the microstructure and mechanical properties can be improved by increasing the water flow rate at the R corner. In 2014, Ma et al. [10] designed four kinds of cooling schemes with spray cooling to probe the transient temperature field and the stress field, which have different water flow rate at different parts of an H-beam, and it was found that the main reason for residual thermal stress is section temperature differences in the H-shape steel beam during cooling process after rolling.There have been many studies discussing simulations of a thermal stress field. In 2008, Zhu et al. [11] simulated the thermal residual stress field of H-beams with different lengths using air cooling, where the results indicated that the residual stress at the web exhibited compressive stress, while that at the web center and the fillet exhibited tensile stress. The residual stress condition in the direction of the width influenced the distribution of the residual stress in the direction of the length. Liu et al. (2008) [12] considered heat transfer and the thermoelastic plastic theory together with the finite element method to simulate the temperature and the thermal stress of an H-beam during the cooling process and found that the appropriate cooling mode and parameters can effectively enable the acquisition of a fairly uniform temperature field and lower residual stress. In 2011, Zhao et al. [13] analyzed the distribution of the stress and strain field under different thermal boundary and cooling time conditions, and their results showed that the temperature of an H-beam will decrease with increases in the density of the water flow, and the maximum stress will increase. In the same year, Zhu et al. [14] simulated the residual stress distribution of hot rolled H-beams after cooling and the transformations in residual stress during flange cutting. Jang and Gan [15] recently proposed an algorism to calculate the optimum heat transfer coefficient combination for an H-beam.Based on previous studies, it can be inferred that the temperature field significantly affect the stress of H-beams. Therefore, the heat transfer coefficients of an H-beam as the thermal boundary condition are very important operating parameters to control the temperature field. In order to optimize the temperature field and temperature differences, optimization of the heat transfer coefficients is investigated and solved numerically using a commercial CFD code ANSYS FLUENT solver along with the conjugate-gradient method (CGM) that has been successfully applied to other applications [16–19]. The design variables in the present study are the heat transfer coefficients, which strongly affect the temperature field and the residual stress field of an H-beam. The optimal temperature field and stress field with the heat transfer coefficients are thus determined using the developed optimization algorithm.


## 2. Mathematical Analysis

In this work, the controlled cooling process of H-beams from the China Steel Corporation (CSC), Taiwan, is simulated via transient heat transfer using a three-dimensional model to control cooling after the rolling process. The H-beams are composed of three parts, the web, flange, and fillet, for which the physical model and the grid meshing are shown in Figures1 and 2. Three sizes of H-beam models have been established, and the sizes of the H-beams (H300 × 300, H250 × 250, and H200 × 200) are listed in Figure 1, which correspond to Case  1, Case  2, and Case  3 in this paper, respectively. The H-beams used in this study were a high strength, low alloy steel ASTM A572 beam. Based on the cooling and the geometrical characteristics of the H-beams, symmetry physical models were built which symmetry with x-axis, y-axis, and z-axis, respectively, and the controlled cooling of the H-beams was divided into five parts corresponding to the heat transfer coefficients h1, h2, h3, h4, and h5, as shown in Figure 3. In this research, two approaches were used to obtain the objective temperature fields of the H-beams, where one adopted the constant heat transfer coefficients and the other one adopted the CGM search for the optimal heat transfer coefficients as the boundary condition by which to compare the temperature fields and stress fields with the same average cooling temperature.Figure 1
The physical model.Figure 2
The meshing grid of the physical model.Figure 3
The heat transfer coefficients distribution.


### 2.1. Governing Equations

The following transient heat conduction equation governs the temperature field of an H-beam [20]:(1)∂∂xjk∂T∂xj=ρ·Cp∂T∂t,where ρ, Cp, and k are the density, heat capacity, and thermal conductivity of the H-beam, respectively. The phase transformation occurs at T=550°C, which brings latent heat and internal stress. These effects are considered in the variations of thermal-mechanical physical properties (specific heat, thermal conductivity, thermal expansion coefficient, Young’s modulus, and yield stress) with temperature as shown in Figure 4 [21]. The diagrams of the variation in microstructures are also shown in Figure 4(d). The density ρ is a constant equal to 7750 kg/m3 and the initial temperature of the H-beam is assumed to be uniform at 850°C.Figure 4
Variations in the thermal-mechanical properties with temperature.
(a)
Specific heat and thermal conductivity
(b)
Thermal expansion
(c)
Young’s modulus and yield stress
(d)
Phases volumeThe incremental theory of plasticity governs the residual stress field of an H-beam. The plastic deformation of the materials is assumed to obey the von Mises yield criterion and the associated flow rule. The relationship of the rate components between thermal stresses,σij, and strains, εij, is described by [20](2)ε˙ij=1+υEσ˙ij-υEσ˙kkδij+λsij+α+∂α∂TTs-T∞T˙,sij=σij-13σkkδij,σe=32sijsij1/2,dλ=3dεi2σi,where E is Young’s modulus; υ is the Poisson’s ratio; α is the thermal expansion coefficient; sij are the component of deviatoric stresses; and λ is the plastic flow factor. λ=0 for elastic deformation or σe<σs, and λ>0 for plastic deformation or σe≥σs, where σs is the yield stress and σe is the von Mises effective stress. Poisson’s ratio υ is a constant equal to 0.3, where Young’s modulus E, the thermal expansion coefficient α, and the yield stress σs vary with the temperature and have similar significantly change in about 550°C with the heat capacity and the thermal conductivity, as shown in Figures 4(b) and 4(c) [20]. The initial residual stress of the H-beam is assumed to be uniform and equal to zero.


### 2.2. Initial and Boundary Conditions

The convective heat transfer between the H-beams surface and the surroundings is evaluated by using the equation(3)q′′=-k∂T∂nsurface=hTs-T∞,where h is the convective heat transfer coefficient for the surface of the H-beam and varies with the different parts of the H-beam. In addition, Ts is the surface temperature of the H-beam, and T∞ is the environmental temperature of 25°C in the water cooling and air cooling process. The cooling conditions require water cooling for 15 s and then air cooling to the ambient temperature (about 100 minutes), where the water cooling rate is about 20°C/s, the air cooling heat transfer coefficient is 30 W/m2K, and the water cooling heat transfer coefficients are determined by the constant and optimization values of the CGM.


## 2.1. Governing Equations

The following transient heat conduction equation governs the temperature field of an H-beam [20]:(1)∂∂xjk∂T∂xj=ρ·Cp∂T∂t,where ρ, Cp, and k are the density, heat capacity, and thermal conductivity of the H-beam, respectively. The phase transformation occurs at T=550°C, which brings latent heat and internal stress. These effects are considered in the variations of thermal-mechanical physical properties (specific heat, thermal conductivity, thermal expansion coefficient, Young’s modulus, and yield stress) with temperature as shown in Figure 4 [21]. The diagrams of the variation in microstructures are also shown in Figure 4(d). The density ρ is a constant equal to 7750 kg/m3 and the initial temperature of the H-beam is assumed to be uniform at 850°C.Figure 4
Variations in the thermal-mechanical properties with temperature.
(a)
Specific heat and thermal conductivity
(b)
Thermal expansion
(c)
Young’s modulus and yield stress
(d)
Phases volumeThe incremental theory of plasticity governs the residual stress field of an H-beam. The plastic deformation of the materials is assumed to obey the von Mises yield criterion and the associated flow rule. The relationship of the rate components between thermal stresses,σij, and strains, εij, is described by [20](2)ε˙ij=1+υEσ˙ij-υEσ˙kkδij+λsij+α+∂α∂TTs-T∞T˙,sij=σij-13σkkδij,σe=32sijsij1/2,dλ=3dεi2σi,where E is Young’s modulus; υ is the Poisson’s ratio; α is the thermal expansion coefficient; sij are the component of deviatoric stresses; and λ is the plastic flow factor. λ=0 for elastic deformation or σe<σs, and λ>0 for plastic deformation or σe≥σs, where σs is the yield stress and σe is the von Mises effective stress. Poisson’s ratio υ is a constant equal to 0.3, where Young’s modulus E, the thermal expansion coefficient α, and the yield stress σs vary with the temperature and have similar significantly change in about 550°C with the heat capacity and the thermal conductivity, as shown in Figures 4(b) and 4(c) [20]. The initial residual stress of the H-beam is assumed to be uniform and equal to zero.


## 2.2. Initial and Boundary Conditions

The convective heat transfer between the H-beams surface and the surroundings is evaluated by using the equation(3)q′′=-k∂T∂nsurface=hTs-T∞,where h is the convective heat transfer coefficient for the surface of the H-beam and varies with the different parts of the H-beam. In addition, Ts is the surface temperature of the H-beam, and T∞ is the environmental temperature of 25°C in the water cooling and air cooling process. The cooling conditions require water cooling for 15 s and then air cooling to the ambient temperature (about 100 minutes), where the water cooling rate is about 20°C/s, the air cooling heat transfer coefficient is 30 W/m2K, and the water cooling heat transfer coefficients are determined by the constant and optimization values of the CGM.


## 3. Numerical Method

### 3.1. Numerical Algorithm

In this study, the commercial software ANSYS is adopted to solve the governing equations and the finite element method is used to simulate the temperature field and the thermomechanical analysis. The computation process is transient, so the first-order implicit method was adopted to maintain the numerical stability of every time step. The time step was 0.1 s for the CGM and the program controlled for thermomechanical analysis to satisfy the requirements of the computational accuracy. The computational grids for the three-dimensional models, which are composed of 3,544,272, 107,848, and 94,248 cells for Case  1, Case  2, and Case  3, respectively, were typically adopted in the computational domain, as shown in Figure2 and Table 2. However, a careful check of the grid independence of the numerical solutions was made to ensure the accuracy and validity of the numerical results. For this purpose, the symmetry model grid systems were tested, which comprised 541,323, 443,034, and 295,556 cells for Case  1, 394,611, 359,214, and 262,200 cells for Case  2, and 385,120, 245,916, and 129,276 cells for Case  3, respectively. It was found that the relative errors under the same conditions among the temperature solutions obtained with three types of grids were less than 0.5% for the three cases. The discretized system was solved iteratively until it satisfied the following residual convergence criterion.(4)max⁡Ti,jn-Ti,jn-1Ti,jn≤10-6,where Ti,jn-1 is the previous temperature value of Ti,jn at the same time level.The simulations were performed as a parallel calculation using sixteen core central processing units for the three-dimensional models. The computer computation times were approximately 10, 8, and 6 minutes for each search step for Case  1, Case  2, and Case  3, respectively.


### 3.2. Optimization

In this study, the conjugate-gradient method (CGM) was combined with a finite volume method (FVM) code as an optimizer to search for the optimum heat transfer coefficientsh1,h2,h3,h4,h5. The objective functions J1obj(h1,h2,h3,h4,h5) and J2obj(h1,h2,h3,h4,h5) were defined as the range of the final average temperature (Tave±δ=550±10°C) calculated by vertex average temperature of the H-beam and the maximum temperature difference as a minimum (min⁡ΔTmaxx,y), calculated by the vertex maximum temperature subtracting the vertex minimum temperature of the H-beam.Above all, the CGM method evaluates the gradient of the objective functions, and then it sets up a new conjugate direction for the updated design variables with the help of a direct numerical sensitivity analysis. The initial guess for the value of each search variable was made, and in the successive steps, the conjugate-gradient coefficients and the search directions were evaluated to estimate the new search variables. The solutions obtained from the finite difference method were then used to calculate the value of the objective functions, which were further transmitted back to the optimizer for the purpose of calculating the consecutive directions. The procedure for applying this method is described in the following:(1)
Generate an initial guess for five design variablesx1,x2,x3,x4,x5: the heat transfer coefficients h1,h2,h3,h4,h5.(2)
Adopt the finite difference method to predict the temperature fieldsTx,y associated with the last h1,h2,h3,h4,h5, and then calculate the objective functions J1obj(h1,h2,h3,h4,h5) and J2obj(h1,h2,h3,h4,h5).(3)
When the value ofJ1obj(h1,h2,h3,h4,h5) reaches the object range and J2obj(h1,h2,h3,h4,h5) reaches the minimum, the optimization process is terminated. Otherwise, proceed to step (4).(4)
Determine the gradient functions,∂J1obj/∂xim and ∂J2obj/∂xim (i=1,2,3,4,5), by applying a small perturbation Δx1,Δx2,Δx3,Δx4,Δx5 to each value of x1,x2,x3,x4,x5, and calculate the corresponding change in objective functions J1obj(h1,h2,h3,h4,h5) and J2obj(h1,h2,h3,h4,h5). Then the gradient functions with respect to each value of the design variables x1,x2,x3,x4,x5 can be calculated by the direct numerical differentiation as(5)∂J1obj∂x1=ΔJ1objΔx1,∂J1obj∂x2=ΔJ1objΔx2,∂J1obj∂x3=ΔJ1objΔx3,∂J1obj∂x4=ΔJ1objΔx4,∂J1obj∂x5=ΔJ1objΔx5∂J2obj∂x1=ΔJ2objΔx1,∂J2obj∂x2=ΔJ2objΔx2,∂J2obj∂x3=ΔJ2objΔx3,∂J2obj∂x4=ΔJ2objΔx4,∂J2obj∂x5=ΔJ2objΔx5.(5)
Calculate the conjugate-gradient coefficients,γm, and the search directions, ξ1m+1, ξ2m+1, ξ3m+1, ξ4m+1, ξ5m+1, for the search variables. For the first step with k=1, γ1=0.(6)γm=∑n2∂Jobjm/∂xn∑n2∂Jobjm-1/∂xn2,n=1,2,3,4,5,ξ1m+1=∂Jobjm∂x1+γmξ1m,ξ2m+1=∂Jobjm∂x2+γmξ2m,ξ3m+1=∂Jobjm∂x3+γmξ3m,ξ4m+1=∂Jobjm∂x4+γmξ4m,ξ5m+1=∂Jobjm∂x5+γmξ5m.(6)
Assign values to the coefficients in a descending directionβ1,β2,β3,β4,β5 for all values of the design variables x1,x2,x3,x4,x5. Specifically, these values are chosen with a trial and error process. In general, the coefficients in the descending direction β1,β2,β3,β4,β5 range from 1 to 50.(7)
Update the design variables with(7)x1m+1=x1m+β1ξ1m,x2m+1=x2m+β2ξ2m,x3m+1=x3m+β3ξ3m,x4m+1=x4m+β4ξ4m,x5m+1=x5m+β5ξ5m.A flowchart of the CGM optimization process is plotted as Figure 5.Figure 5
A flowchart of the CGM optimization process.


## 3.1. Numerical Algorithm

In this study, the commercial software ANSYS is adopted to solve the governing equations and the finite element method is used to simulate the temperature field and the thermomechanical analysis. The computation process is transient, so the first-order implicit method was adopted to maintain the numerical stability of every time step. The time step was 0.1 s for the CGM and the program controlled for thermomechanical analysis to satisfy the requirements of the computational accuracy. The computational grids for the three-dimensional models, which are composed of 3,544,272, 107,848, and 94,248 cells for Case  1, Case  2, and Case  3, respectively, were typically adopted in the computational domain, as shown in Figure2 and Table 2. However, a careful check of the grid independence of the numerical solutions was made to ensure the accuracy and validity of the numerical results. For this purpose, the symmetry model grid systems were tested, which comprised 541,323, 443,034, and 295,556 cells for Case  1, 394,611, 359,214, and 262,200 cells for Case  2, and 385,120, 245,916, and 129,276 cells for Case  3, respectively. It was found that the relative errors under the same conditions among the temperature solutions obtained with three types of grids were less than 0.5% for the three cases. The discretized system was solved iteratively until it satisfied the following residual convergence criterion.(4)max⁡Ti,jn-Ti,jn-1Ti,jn≤10-6,where Ti,jn-1 is the previous temperature value of Ti,jn at the same time level.The simulations were performed as a parallel calculation using sixteen core central processing units for the three-dimensional models. The computer computation times were approximately 10, 8, and 6 minutes for each search step for Case  1, Case  2, and Case  3, respectively.


## 3.2. Optimization

In this study, the conjugate-gradient method (CGM) was combined with a finite volume method (FVM) code as an optimizer to search for the optimum heat transfer coefficientsh1,h2,h3,h4,h5. The objective functions J1obj(h1,h2,h3,h4,h5) and J2obj(h1,h2,h3,h4,h5) were defined as the range of the final average temperature (Tave±δ=550±10°C) calculated by vertex average temperature of the H-beam and the maximum temperature difference as a minimum (min⁡ΔTmaxx,y), calculated by the vertex maximum temperature subtracting the vertex minimum temperature of the H-beam.Above all, the CGM method evaluates the gradient of the objective functions, and then it sets up a new conjugate direction for the updated design variables with the help of a direct numerical sensitivity analysis. The initial guess for the value of each search variable was made, and in the successive steps, the conjugate-gradient coefficients and the search directions were evaluated to estimate the new search variables. The solutions obtained from the finite difference method were then used to calculate the value of the objective functions, which were further transmitted back to the optimizer for the purpose of calculating the consecutive directions. The procedure for applying this method is described in the following:(1)
Generate an initial guess for five design variablesx1,x2,x3,x4,x5: the heat transfer coefficients h1,h2,h3,h4,h5.(2)
Adopt the finite difference method to predict the temperature fieldsTx,y associated with the last h1,h2,h3,h4,h5, and then calculate the objective functions J1obj(h1,h2,h3,h4,h5) and J2obj(h1,h2,h3,h4,h5).(3)
When the value ofJ1obj(h1,h2,h3,h4,h5) reaches the object range and J2obj(h1,h2,h3,h4,h5) reaches the minimum, the optimization process is terminated. Otherwise, proceed to step (4).(4)
Determine the gradient functions,∂J1obj/∂xim and ∂J2obj/∂xim (i=1,2,3,4,5), by applying a small perturbation Δx1,Δx2,Δx3,Δx4,Δx5 to each value of x1,x2,x3,x4,x5, and calculate the corresponding change in objective functions J1obj(h1,h2,h3,h4,h5) and J2obj(h1,h2,h3,h4,h5). Then the gradient functions with respect to each value of the design variables x1,x2,x3,x4,x5 can be calculated by the direct numerical differentiation as(5)∂J1obj∂x1=ΔJ1objΔx1,∂J1obj∂x2=ΔJ1objΔx2,∂J1obj∂x3=ΔJ1objΔx3,∂J1obj∂x4=ΔJ1objΔx4,∂J1obj∂x5=ΔJ1objΔx5∂J2obj∂x1=ΔJ2objΔx1,∂J2obj∂x2=ΔJ2objΔx2,∂J2obj∂x3=ΔJ2objΔx3,∂J2obj∂x4=ΔJ2objΔx4,∂J2obj∂x5=ΔJ2objΔx5.(5)
Calculate the conjugate-gradient coefficients,γm, and the search directions, ξ1m+1, ξ2m+1, ξ3m+1, ξ4m+1, ξ5m+1, for the search variables. For the first step with k=1, γ1=0.(6)γm=∑n2∂Jobjm/∂xn∑n2∂Jobjm-1/∂xn2,n=1,2,3,4,5,ξ1m+1=∂Jobjm∂x1+γmξ1m,ξ2m+1=∂Jobjm∂x2+γmξ2m,ξ3m+1=∂Jobjm∂x3+γmξ3m,ξ4m+1=∂Jobjm∂x4+γmξ4m,ξ5m+1=∂Jobjm∂x5+γmξ5m.(6)
Assign values to the coefficients in a descending directionβ1,β2,β3,β4,β5 for all values of the design variables x1,x2,x3,x4,x5. Specifically, these values are chosen with a trial and error process. In general, the coefficients in the descending direction β1,β2,β3,β4,β5 range from 1 to 50.(7)
Update the design variables with(7)x1m+1=x1m+β1ξ1m,x2m+1=x2m+β2ξ2m,x3m+1=x3m+β3ξ3m,x4m+1=x4m+β4ξ4m,x5m+1=x5m+β5ξ5m.A flowchart of the CGM optimization process is plotted as Figure 5.Figure 5
A flowchart of the CGM optimization process.


## 4. Result and Discussion

In order to examine the validity of the results of the CGM method, two initial guess values were used to search for the optimal heat transfer coefficient distributions of the H-beams for three cases, and they all obtained approximately the same result. Based on the optimal heat transfer coefficient distributions using CGM method, the temperature field and the residual stress of the H-beam were calculated to compare with the constant heat transfer coefficient results. Taking into account the operability in the cooling process, the area of the up surface of the flange is small and not in the water cooling range; therefore, the heat transfer coefficienth5 is a significantly smaller value than h1 to h4.Figure6 shows the final average temperature and the maximum temperature difference search path, where the initial guess value 1 was h1=h2=h3=h4=1600 (W/m2K), h5=160 (W/m2K) and the initial guess value 2 was h1=h2=h3=h4=1200 (W/m2K), h5=120 (W/m2K) for Case  1. The initial guess values for heat transfer coefficients h1=h2=h3=h4=1600 W/m2K in water are estimated from the water jet heat transfer coefficient correlations by Wendelstorf et al. [22], while the h5=160 W/m2K in air is estimated from the combined natural and radiation heat transfer theory [23]. The search trend of the final average temperature for these two initial guess values was close to the objective value step by step and were within the error range. The search trend for the temperature difference for the two initial guess values was obtained by searching for the lowest value step by step with the final average temperature in the range of the objective value until the temperature difference variable was stable. The different search steps were obtained for different initial guess values, but they all obtained almost the same result, where the resultant deviation for the two initial values was below 1°C. It is apparent that Case  2 and Case  3 both had the same search path as that of Case  1. The initial guess value 1 was h1=h2=h3=h4=1500 (W/m2K), h5=150 (W/m2K) and the initial guess value 2 was h1=h2=h3=h4=1000 (W/m2K), h5=100 (W/m2K) for Case  2; the initial guess value 1 was h1=h2=h3=h4=1200 (W/m2K), h5=120 (W/m2K), and the initial guess value 2 was h1=h2=h3=h4=800 (W/m2K), h5=80 (W/m2K) for Case  3, respectively. For the three cases, there were different search steps for the difference cases and initial guess values, where the search steps were simply related to the initial guess value and the search step size.Figure 6
The search path with the different initial values for the three cases.
(a)
Case  1
(b)
Case  2
(c)
Case  3Tables1–3 list the different distributions of the heat transfer coefficients for an example with ten points in the optimization search process, where the final average temperatures were all in a range of 540°C to 560°C, but the temperature difference decreased from 201°C to 144°C, from 204°C to 130°C, and from 179°C to 104°C for Case  1 to Case  3, respectively. The optimization distribution of the heat transfer coefficients is the tenth point in Tables 1–3. For Case  1, the optimization distribution of heat transfer coefficients h1 to h5 is 1227, 1745, 1732, 1479, and 101 (W/m2K), where the average temperature is 560°C, and the temperature difference is 144°C. For Case  2, the optimization distribution of the heat transfer coefficients h1 to h5 is 1091, 1705, 1666, 1445, and 37 (W/m2K), where the average temperature is 560°C, and the temperature difference is 130°C. For Case  3, the optimization distribution of heat transfer coefficients h1 to h5 is 913, 1579, 1556, 1166, and 225 (W/m2K), where the final average temperature is 560°C, and the temperature difference is 104°C.Table 1
The optimization search ofh1–h5 distribution for Case  1 (e.g., with ten points).
Number
h
1
h
2
h
3
h
4
h
5
T
a
v
e
Δ
T
W/m2K
W/m2K
W/m2K
W/m2K
W/m2K
°C
°C
(1) (constant)
1500
1500
1500
1500
150
550
201
(2)
1546
1595
1596
1557
157
540
205
(3)
1485
1547
1548
1507
107
546
195
(4)
1419
1556
1555
1508
107
550
180
(5)
1335
1576
1427
1415
167
555
176
(6)
1309
1565
1497
1482
163
556
170
(7)
1316
1572
1568
1507
106
557
160
(8)
1264
1642
1635
1504
111
558
150
(9)
1240
1702
1695
1490
105
559
145
(10) (optimization)
1227
1745
1732
1479
101
560
144Table 2
The optimization search ofh1–h5 distribution for Case  2 (e.g., with ten points).
Number
h
1
h
2
h
3
h
4
h
5
T
a
v
e
Δ
T
W/m2K
W/m2K
W/m2K
W/m2K
W/m2K
°C
°C
(1) (constant)
1400
1400
1400
1400
140
548
204
(2)
1340
1449
1450
1419
100
548
198
(3)
1365
1453
1452
1420
99
549
192
(4)
1315
1460
1458
1421
98
551
178
(5)
1289
1464
1461
1421
96
553
172
(6)
1213
1482
1475
1465
92
555
151
(7)
1181
1545
1485
1486
67
556
142
(8)
1133
1610
1550
1477
64
558
136
(9)
1109
1610
1555
1456
44
559
132
(10) (optimization)
1091
1705
1666
1445
37
560
130Table 3
The optimization search ofh1–h5 distribution for Case  3 (e.g., with ten points).
Number
h
1
h
2
h
3
h
4
h
5
T
a
v
e
Δ
T
W/m2K
W/m2K
W/m2K
W/m2K
W/m2K
°C
°C
(1) (constant)
1200
1200
1200
1200
120
543
179
(2)
1213
945
934
1087
101
552
210
(3)
1168
1013
1000
1140
151
553
189
(4)
1121
1081
1065
1192
201
553
168
(5)
1074
1148
1130
1243
201
555
147
(6)
1038
1213
1191
1267
182
556
133
(7)
1017
1278
1254
1280
204
556
131
(8)
991
1353
1323
1265
203
557
123
(9)
959
1435
1398
1223
181
559
111
(10) (optimization)
913
1579
1556
1166
225
560
104Figure7 shows that the variation of the average temperature in the different parts of the beam and that the maximum temperature difference with time for the three cases under two heat boundary conditions satisfied the average temperature of H-beams in the range of 540~560°C. For Case  1, when the heat transfer coefficients were constant (h1=h2=h3=h4=1500 (W/m2K), h5=150 (W/m2K)), the different parts had different trends in variation. For the web, the cooling rate was very fast at first and then gradually slowed with the decrease of the web surface temperature, and when the time was over 12 seconds, the cooling rate became as fast as it was originally. In the case of the flange, the cooling rate was slower than that of the web, where the cooling rate went from fast to slow. In the case of the fillet, the cooling rate was the slowest and exhibited the same variable trends as that of the flange. The maximum temperature difference increased initially in the first 6 seconds and then decreased slightly until the maximum temperature difference was 201°C at 15 seconds. When the heat transfer coefficients were optimized (h1=1227 (W/m2K), h2=1745 (W/m2K), h3=1732 (W/m2K), h4=1479 (W/m2K), and h5=101 (W/m2K)), the variation of the average temperature with time in the different parts exhibited the same trends compared with the results of the constant heat transfer coefficients, except the fillet cooling rate was slightly larger than that of the constant heat transfer coefficients; the flange cooling rate was slightly less than that of the constant heat transfer coefficients; the web cooling rate was obviously slower than that of the constant heat transfer coefficients, and the cooling rate of the web went from fast to slow. The temperature difference for the optimal heat transfer coefficients increased initially and then decreased until the maximum temperature difference was 144°C at 15 seconds, and the peak of the temperature difference was controlled at 6 seconds. For Case  2 and Case  3, the constant heat transfer coefficients were h1=h2=h3=h4=1400 (W/m2K), h5=140 (W/m2K), and h1=h2=h3=h4=1200 (W/m2K), h5=120 (W/m2K), and the optimal heat transfer coefficients were h1=1091 (W/m2K), h2=1705 (W/m2K), h3=1666 (W/m2K), h4=1444 (W/m2K), h5=37 (W/m2K), and h1=913 (W/m2K), h2=1579 (W/m2K), h3=1556 (W/m2K), h4=1166 (W/m2K), and h5=225 (W/m2K), respectively. The variation of the average temperature with time exhibited the same trends as those found for Case  1, regardless of whether it was determined with constant heat transfer coefficients or optimal heat transfer coefficients. The cooling rates of Case  2 among the web, flange, and fillet were closer than in Case  1, and Case  3 was the closest because of the physical dimensions. The temperature difference for the constant heat transfer coefficients reached a peak at 6.5 seconds and 7 seconds for Case  2 and Case  3 and then slowly decreased until the temperature differences were 204°C and 180°C at 15 seconds, respectively. The maximum temperature difference of the optimal heat transfer coefficients reached a peak at 4 seconds and at 3 seconds for Case  2 and Case  3 and then slowly decreased to 130°C and 104°C at 15 seconds, respectively.Figure 7
The temperature in different parts and the temperature difference variable with time.
(a)
Case  1
(b)
Case  2
(c)
Case  3Figure8 displays the temperature field for three cases with constant heat transfer coefficients and optimal heat transfer coefficients at 15 seconds. For Case  1, the temperature ranged from 667°C to 466°C with the constant heat transfer coefficients, and the temperature ranged from 654°C to 510°C with the optimal heat transfer coefficients. For Case  2, the temperature ranged from 658°C to 454°C with the constant heat transfer coefficients, and the temperature ranged from 639°C to 509°C with the optimal heat transfer coefficients. For Case  3, the temperature ranged from 646°C to 466°C with the constant heat transfer coefficients, and the temperature ranged from 618°C to 514°C with the optimal heat transfer coefficients; the surface maximum temperature difference is decreased by 60~80°C for three cases. For the three cases, the temperature field for the optimal heat transfer coefficients was more uniform than for the temperature field of the constant heat transfer coefficients. Also, from Case  1 to Case  3, the sizes became gradually smaller, so the temperature differences were also smaller.Figure 8
The distribution of temperature with the different heat transfer coefficients at 15 seconds in a section of the H-beams.
(a)
Case  1
(b)
Case  2
(c)
Case  3Figure9 and Table 4 display the residual stress distribution at the web for three cases with constant heat transfer coefficients and optimal heat transfer coefficients based on the analysis of the temperature field. From Figure 9, the flange and the fillet of the residual stress are tensor in nature; the web of the residual stress is compressive in nature, with the maximum compressive stress occurring at the junction of the web and the fillet. For Case  1, the residual stress in the web is in the range of −125 MPa to −107 MPa with the constant heat transfer coefficients and is about −90 MPa with optimal heat transfer coefficients. For Case  2, the residual stress in the web is in the range of −100 MPa to −80 MPa with the constant heat transfer coefficients and is about −60 MPa with optimal heat transfer coefficients. For Case  3, the residual stress in the web is in the range of −80 MPa to −70 MPa with the constant heat transfer coefficients and is about −50 MPa with optimal heat transfer coefficients. From Table 4, it is observed that optimum heat transfer coefficients result in smaller differences in maximum temperature and thereby generated a lower residual stress distribution in the web across all sizes of the beams researched in this paper.Table 4
A comparison of the results for the three cases.
Case
Case  1
Case  2
Case  3
Constant
Optimal
Constant
Optimal
Constant
Optimal
h
1 (W/m2K)
1500
1227
1400
1091
1200
913
h
2 (W/m2K)
1500
1745
1400
1705
1200
1579
h
3 (W/m2K)
1500
1732
1400
1666
1200
1556
h
4 (W/m2K)
1500
1479
1400
1445
1200
1166
h
5 (W/m2K)
150
101
140
37
120
225
Δ
T
m
a
x (°C)
201
144
204
130
180
104
Residual stress at web (MPa)
-
114
-
91
-
89
-
62
-
73
-
51Figure 9
The residual stresses distribution at the web.
(a)
Case  1
(b)
Case  2
(c)
Case  3


## 5. Conclusions

The conjugate-gradient method (CGM) was used here to predict the optimization heat transfer coefficient distribution of H-beams with different sizes. To this end, three 3D numerical models were analyzed to estimate the temperature field and the residual stresses field developed during the cooling process. The results from the optimal heat transfer coefficients were compared with those constant heat transfer coefficients. Based on the numerical results, the conclusion can be summarized as follows:(1) The conjugate-gradient method (CGM) has accurately predicted the heat transfer coefficients, and the different initial guess values have resulted in very similar result. In the search process, we can get many heat transfer coefficients to satisfy the average temperature with different temperature differences, and there is a group of heat transfer coefficients that has the lowest temperature difference using the CGM optimization search method.(2) A comparison of the constant heat transfer coefficients and the optimal heat transfer coefficients reveals obvious improvement in the predicted uniform temperature field and temperature difference with the optimal heat transfer coefficients, where the maximum temperature difference and the surface maximum temperature difference are decreased for all the cases analyzed.(3) The cooling rate was fastest in the web, followed by the flange and the fillet.(4) After the water cooling and the air cooling to the ambient temperature, the residual stress at the web was calculated, and a comparison of the three results for the constant heat transfer coefficients and optimal heat transfer coefficients indicates that the residual stress can be reduced by 20~40 MPa for the three cases.



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*Source: 9873283-2017-09-27.xml*