Int J Adv Manuf Technol (2010) 47:1159–1166 DOI 10.1007/s00170-009-2240-3
ORIGINAL ARTICLE
Prediction of heat transfer coefficient of steel bars subjected to Tempcore process using nonlinear modeling Isukapalli B. Sankar & K. Mallikarjuna Rao & A. Gopala Krishna
Received: 23 March 2009 / Accepted: 28 July 2009 / Published online: 27 August 2009 # Springer-Verlag London Limited 2009
Abstract This paper deals with the prediction of heat transfer coefficient of steel bars subjected to Tempcore process. A nonlinear mathematical model, in terms of process variables, is developed using response surface methodology. Three significant control parameters are considered. Central composite design of experiments is structured and conducted using finite element method to formulate the predictive nonlinear model. Statistical analysis and experimental results suggest that the proposed model could be used for predicting heat transfer coefficient with adequate accuracy. The knowledge of heat transfer coefficient makes it possible to predict temperature evolution in the steel rods. As the temperature distribution affects the mechanical properties of steel rods, the proposed methodology can be effectively employed in controlling the quality of products. Keywords Finite element analysis . Tempcore process . Heat transfer coefficient . Response surface methodology . Central composite design
1 Introduction In Tempcore process, a bar emerging from the last rolling stand is rapidly cooled with water to form martensite in the surface layer [1, 2]. After the quenching stage, heat flows I. B. Sankar (*) 104/A/sector-3, Ukkunagaram, Visakhapatnam 530032, Andhra Pradesh, India e-mail:
[email protected] K. Mallikarjuna Rao : A. Gopala Krishna Department of Mechanical Engineering, University College of Engineering, Jawaharlal Nehru Technological University, Kakinada 533 003, Andhra Pradesh, India
from the center to the surface, and a self-tempering treatment of martensite is achieved. At the same time, austenite core of the bar is cooled and transforms generally into ferrite and pearlite. Water flux, water pressure, water temperature, inside diameter of the tube through which bar travels, diameter of the steel rod, quenching time, and surface temperature of the steel rod affect the heat transfer coefficient. Various researchers carried out the modeling temperature evolution of stock for a cooling system by solving the following heat transfer equation [3]: @ @T k @T @T k ð1Þ þ þ qtr ¼ rCp @r @r r @r @t where k is the thermal conductivity, T is the temperature, qtr is the latent heat released due to phase change, ρ is the density, and Cp is specific heat of material of bar. The equations governing physical processes in cylindrical geometries are described analytically in terms of cylindrical coordinates. When the geometry, loading, and the boundary conditions are dependent only on radial direction and independent of the other two coordinates, the governing equations are one-dimensional. In the present problem, the bar can be assumed to be infinitely long (10.3 m) and travels at high speed (min 18 m/s) and is cooled by quenching. Hence, the heat flow in axial direction is considered negligible compared to the rapid heat transfer in radial direction. Therefore, the present problem is considered as an axisymmetric shape. Equation 1 is subjected to the following boundary conditions: 1. At the center line
t 0;
r ¼ 0;
@T ¼0 @r
ð2Þ
1160
Int J Adv Manuf Technol (2010) 47:1159–1166
determined from water spray cooling impinging on the upper side of a steel plate are helpful in simulating the cooling process where water-filled cooling tubes are employed. Carnahan et al. [6] and Incropera et al. [7] used finite difference method to calculate temperature variation within a steel bar cross-section, formulating as a onedimensional axisymmetric heat transfer equation. Hernandez et al. [8] measured the temperature response inside the steel rod and converted it inversely into heat transfer coefficient using inverse technique to determine heat transfer coefficient in steel rods. Morales et al. [9, 10] summarized the experimental conditions and the heat transfer coefficient model for water-cooling and air-forced cooling for steel rods. Ozsoyeller [11] predicted the strength and the selftempering temperature of steel rods by varying the quenching time. The detailed literature survey indicates the following:
Fig. 1 Photographic view
2. At the rod surface
t > 0;
r ¼ ro ;
k
@T ¼ hðT Ta Þ @r
ð3Þ
3. While the initial condition is
t ¼ 0;
0 r ro ;
1. Most of the researchers studied heat transfer coefficient for steel plates and bars. Heat transfer coefficient for Tempcore process needs to be explored. 2. Most of the published works on the heat transfer coefficient are based on water pressure and flow rate and for steel rods of higher diameters (above 20 mm). Hence, the present work is aimed at:
T ¼ Tin
ð4Þ
where h is the heat transfer coefficient, and Tin is the initial temperature of the rod before it enters the water-cooling sections. To find a heat transfer coefficient for forced convection, most of the researchers carried out on hot steel plates. Mitsutsuka [4] evaluated the heat transfer coefficient as a function of water flux, water temperature, and steel temperature. Mitsutsuka et al. [5] showed that correlations
1. Conducting a comprehensive study of effects of variables on heat transfer coefficient for Tempcore process, for lower diameters; 2. Exploring the possibility of minimizing errors in prediction by using nonlinear model. Experiments are conducted on steel rods, and heat transfer coefficient is calculated using finite element method. In this research, it is proposed to employ secondorder polynomial in the response surface method. Central composite design (CCD) of experiments is used in combination with finite element method to develop the nonlinear model. The selection of second-order response surface methodology (RSM) is to find out the interactions between the variables, to understand the nonlinear behavior (if it exists) of the processing variables, and also minimize the errors in the prediction.
Table 1 Thermo-physical properties of steel employed in the model
Convective heat transfer
z
Property
Unit
Correlation
Heat capacity
J kgK
Cp ¼ 281:4 þ 0:5066 T
550 K T 950 K
Cp =1,400
950 K≤T≤1,075 K
Cp ¼ 493:71 þ 2:3 T
1; 075 K T 1; 350 K
Thermal
r
Conductivity Density
Fig. 2 Finite element model of cylindrical shape
J mK:s kg m3
Observation
K ¼ 75:42 0:047 T
830 K T 1; 075 K
K ¼ 9:84 þ 0:013 T
1,075 K≤T≤1,350 K
ρ=7,850
Constant
Int J Adv Manuf Technol (2010) 47:1159–1166 Table 2 Control factors and their levels
S. No.
1161 Control factor
1 2 3
Bar diameter Quenching time Diameter of tube
Symbol for coded value
A B C
Levels
Units
−1
0
+1
8 0.2 18
10 0.4 21
12 0.6 24
mm s mm
2 Experimental setup
3 Methodology
Figure 1 shows the photographic view of experimental setup. IS1786 steel rods with chemical composition of 0.17% C, 22% Si, 79% Mn, 036% P, and 0.041% S in different diameters (8, 10, and 12 mm) are chosen as different specimen. Pipes of different diameters (18, 21, and 27 mm) are used in the cooling zone. Online pyrometers are installed in the pipes, along the length, to measure the temperatures. Siemens R-30 process computer is used to compute the quenching time.
The present study is aimed at establishing the input– output relationships for prediction of heat transfer coefficient. Experiments are conducted using the most significant variables using CCD-based RSM. The temperatures of steel rod are measured using pyrometers installed along the length of cooling zone. The heat transfer coefficients are calculated using the finite element method. A nonlinear model is developed, and the model is tested for accuracy.
Table 3 Experimental results
3.1 Finite element method
Run order
A
B
C
h(w/m2 K)
1 2 3 4 5 6 7 8 9 10 11 12 13 14
10 12 8 12 12 12 8 10 8 8 8 12 10 10
0.4 0.6 0.2 0.2 0.6 0.4 0.6 0.6 0.4 0.2 0.4 0.2 0.6 0.2
18 21 21 21 24 24 18 18 21 24 18 18 21 18
3,940,000 3,210,000 5,830,000 9,910,000 2,720,000 3,570,000 3,910,000 4,210,000 4,340,000 5,500,000 5,240,000 10,500,000 3,230,000 7,940,000
15 16 17 18 19 20 21 22 23 24 25 26 27
10 10 8 12 12 8 12 10 10 8 12 8 10
0.6 0.4 0.6 0.4 0.2 0.2 0.4 0.2 0.2 0.6 0.6 0.4 0.4
24 24 24 18 24 18 21 24 21 21 18 24 21
2,750,000 3,030,000 2,840,000 4,190,000 9,650,000 6,100,000 3,810,000 7,490,000 7,630,000 3,210,000 3,410,000 3,790,000 3,420,000
Equation 1 with its boundary and initial conditions is solved by the finite element method. The finite element formulation for the computation of the transient temperature distribution T(r, z, t) for solids with general surface heat transfer is given as [12]
½ M þ aΔtsþ1 ½ K sþ1 fugsþ1 ¼ ½½ M ð1 aÞΔtsþ1 ½Ks fugs þ Δtsþ1 afF gsþ1 þ ð1 aÞfF gs
ð5Þ
where [M] is the mass matrix, [K] is the conduction matrix, {u} is the element temperature matrix, and {F} is the element matrix for boundary conditions. Suffix s denotes the time, Δt denotes the time increment, and α is used for numerical integration scheme. Galerkin’s method is applied for the time integration of equations [12]. A finite element bar model is developed as an axisymmetric shape as shown in Fig. 2. Totally, the model is meshed into 32 elements comprising 65 nodes. The thermo-physical properties of steel used are tabulated in Table 1 [13]. The time-dependent properties of thermal conductivity and specific heats of austenite, ferrite, pearlite, bainite, and martensite are obtained from literature [13–15] and included in the model. Steel density is assumed to be constant over the temperature range of interest. The value of enthalpy for austenite decomposition into pearlite at 1,000 K is 77 kJ/kg [16]. In the finite element method, heat transfer equations are added
1162 Table 4 Analysis of variance for heat transfer coefficient
Int J Adv Manuf Technol (2010) 47:1159–1166 Source Model A B C AB AC BC A2 B2 C2 Residual Total
Sum of squares
df
Mean square
F value
P value Prob>F
8.161E+013 3.752E+012 5.007E+013 1.563E+012 1.024E+013 7.813E+009 4.513E+009 3.412E+012 1.344E+013 1.114E+012 1.282E+013 9.444E+013
9 1 1 1 1 1 1 1 1 1 10 19
9.068E+012 3.752E+012 5.007E+013 1.563E+012 1.024E+013 7.813E+009 4.513E+009 3.412E+012 1.344E+013 1.114E+012 1.282E+012
7.07 2.93 39.05 1.22 7.98 0.0006 0.0003 2.66 10.48 0.87
0.0026 (significant) 0.1179 0.0001 0.2954 0.0180 0.9393 0.9539 0.1339 0.0089 0.3733
to the equations determining the transition from isothermal time temperature transformation to continuous cooling transformation and are solved by a Fortran 77 program. To determine the heat transfer coefficients, an inverse modeling approach is followed. This involves recasting the finite element model to solve for the sum of squares of errors between the calculated and the measured temperatures at each pyrometer sampling points given the convective heat transfer coefficient as the input. The value of h is varied until the minimum sum of the squares of the errors is obtained using a minimization technique, Brent’s method [17].
Fig. 3 Interaction plot for diameter of rod vs quenching time
3.2 RSM-CCD of experiments CCD is one of the most important experimental designs used for optimizing process parameters. CCD is nothing but 2k factorial design augmented with center points and axial points. CCD is far more efficient than running 3k factorial design with quantitative factors [18]. Hence, the CCD approach is selected for the present study. The CCD of experiments for three factors consists of the following: 1. Full factorial design for the three factors (23) 2. Six center point runs
Fig. 4 Interaction plot of diameter of rod vs inside diameter of the tube
Int J Adv Manuf Technol (2010) 47:1159–1166
1163
Equation 7 is differentiated with respect to X and equated to zero in order to obtain different coefficients of the second-order polynomial equation. In the present work, Minitab software [19] is used for calculation of coefficients and for analysis. 3.3 Control variables The control variables are classified into two groups: 1. Fixed processing factors: not intended to vary after each run. 2. Variable processing factors: intended to vary to study the effects.
Fig. 5 Interaction effect of quenching time and inside diameter of the tube
3. Eight axial points with β value of 2. The β value represents the distance between the center point and some additional axial points to fit a nonlinear curve. A second-order polynomial equation is developed to study the effects of the variables on the heat transfer coefficient. This equation is of the following general form: Y ¼ Ao þ
k X
Ai Xi þ
XX
Aij Xi Xj þ
i
i¼1
k X
Aii Xi2
ð6Þ
i¼1
The above equation contains linear terms, squared terms, and interaction terms. Equation 6 can be rewritten in matrix form as Y ¼ Ao þ X T b þ X T CX ð7Þ where X T ¼ X1; X2:::; Xk ; b ¼ A1; A2 ; ::; Ak , and C is the k×k symmetric matrix and is given as 0 1 A11 . . . 1=2A1k B .. C .. C ¼ @ ... ð8Þ . A .
Akk
Table 5 R2 analysis Standard deviation Mean CV% PRESS
4 Experimental results and analysis This section discusses the experiment, its results, the nonlinear mathematical model developed, and its adequacy. 4.1 RSM-CCD of experiments Table 2 lists the chosen control variables and their levels. A total number of 27 experiments are conducted, and a set of data is collected as per the structure of the CCD of the experiments. Table 3 shows the experimental results of the CCD of experiments. MINITAB, 14 V [19] statistical analysis software, is used to compute the regression coefficients of the secondNormal Probability Plot of the Residuals (response is actual) 99 95 90 Percent
1=2A1k
The values of quenching parameters are water pressure is 6 bar, water flow rate is 30 m3/h, and water temperature is 303 K. The initial steel rod temperature is 1,300 K, and length of steel rod is 10.3 m. The above parameters are fixed throughout the experimentation. Steel rod diameter (A), quenching time (B), and inside diameter of the tube through which bar travels (C) are varied to study the heat transfer coefficient.
80 70 60 50 40 30 20 10 5
1.132E+006 4.563E+006 24.82 9.728E+013
R-squared Adj R-squared Pred R-squared Adeq precision
0.8642 0.7420 −0.0301 8.835
1 -500000
-250000
0
250000
Residual
Fig. 6 Normal probability plot for residuals
500000
750000
1164
Int J Adv Manuf Technol (2010) 47:1159–1166
Table 6 Comparison of predicted vs actual values
Run order
A
B
3,940,000 3,210,000 5,830,000 9,910,000 2,720,000 3,570,000 3,910,000 4,210,000 4,340,000 5,500,000 5,240,000 10,500,000 3,230,000 7,940,000 2,750,000 3,030,000 2,840,000
4,046,380 2,568,000 5,223,680 9,047,830 2,409,920 3,105,900 3,393,880 3,797,420 4,010,160 5,192,000 4,721,240 9,933,000 2,713,200 6,749,000 2,409,000 2,530,050 2,379,920
2.7 20 10.4 8.7 11.4 13 13.2 9.8 7.6 5.6 9.9 5.4 16 15 12.4 16.5 16.2
18 19
12 12
0.4 0.2
18 24
4,190,000 9,650,000
3,360,380 8,260,400
19.8 14.4
Actual
8000000 6000000 4000000 2000000 0 7
9
Percentage deviation
18 21 21 21 24 24 18 18 21 24 18 18 21 18 24 24 24
10000000
5
Predicted value
0.4 0.6 0.2 0.2 0.6 0.4 0.6 0.6 0.4 0.2 0.4 0.2 0.6 0.2 0.6 0.4 0.6
Predicted
3
Actual value
10 12 8 12 12 12 8 10 8 8 8 12 10 10 10 10 8
12000000
1
Heat transfer coefficient
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
order polynomial model for the heat transfer coefficient (h). Because of the lower predictability of the first-order models for the present problem, the second-order models are postulated. The analysis of variance (ANOVA) is used to check the adequacy of the developed models. Table 4 shows the ANOVA for the heat transfer coefficient. The P value for the model is lower than 0.05 (i.e., at 95% confidence level), which indicates that the model is considered to be statistically significant [20]. Values of “Prob>F” less than 0.0500 indicate that model terms are significant. It is observed form the table that A, AB, and B2 are significant model terms.
h
C
11 13
15
17
TEST RUNS
Fig. 7 Comparison of predicted vs actual values
19
21
23
25
27
Quenching time (B) has nonlinear relationship with the heat transfer coefficient. Figures 3, 4, and 5 show contour plots for the responseheat transfer coefficient. From the figures, it is seen that quenching time in combination with other factors has positive influence on increase of heat transfer coefficient. From Fig. 3, it is observed that reduced quenching time and increased diameter of the bar increases the heat transfer coefficient. To check whether the fitted model actually describes the experimental data, the regression coefficient (R2) is computed. Table 5 shows the R2 analysis. The regression coefficient (R2) for the heat transfer coefficient is found to be 0.8642. This shows that the second-order model can explain the variation in the heat transfer coefficient up to the extent of 86.42%. The R2 of 0.8642 is in reasonably agreement with the “Adj R2” of 0.742. “Adeq precision” measures the signal to noise ratio. A ratio greater than 4 is desirable. The ratio of 8.835 indicates an adequate signal. The stastistical anaysis shows that the developed nonlinear model based on CCD is statistically adequate and can be used to navigate the design space. The normal probability plot of the residuals for the output response is shown in Fig. 6. A check on the plot reveals that the residuals are located on a straight line,
Int J Adv Manuf Technol (2010) 47:1159–1166
1165
Fig. 8 Predicted values vs actual best-line plot
11000000 10000000 9000000
Actual
8000000 7000000 6000000 5000000 4000000 3000000 2000000 2000000
3000000
4000000
5000000
6000000
7000000
8000000
9000000 10000000
Predicted
which means that the errors are distributed normally, and the regression model is fairly well fitted with the observed
values. The following equation is obtained for the heat transfer coefficient in terms of coded values:
h ¼ 3:381E þ 006 þ 5:242E þ 005 A 1:915E þ 006 B 3:383E þ 005 C 1:131E þ 006 A B þ 31250:00 23750:00 B C þ 4:866E þ 005 A2 þ 9:657E þ 005 B2 þ 2:780E þ 005 C2
4.2 Confirmation Performance of the developed model is tested on 19 randomly generated runs. Table 6 the predicted responses, the measured heat transfer coefficients, and the percentage deviation for each run. Figure 7 shows the graphical comparison of predicted values versus actual values. The best-fit plot of the chosen 19 points is drawn in Fig. 8 and is found to be close to the ideal line, Y=X. The above
%deviation 25 20 15 10 5 0 -5
0
5
10
15
20
25
Test run Fig. 9 Percentage deviation with respect to experimental runs
30
graphs indicate that the predicted responses have good agreement with the actual values. Figure 9 shows the plot of percentage deviation with respect to individual test runs. The overall average absolute percentage deviation is found to be 12%.
5 Conclusion Nonlinear model for the heat transfer coefficient based on CCD of experiments through finite element method is successful developed for prediction of heat transfer coefficient. To validate the model, some randomly generated test runs are taken. The overall average deviation between predicted and actual results is found to be 12%. Because of the lower deviation value, the proposed methodology can be adopted to estimate the heat transfer coefficient in Tempcore process. With the knowledge of heat transfer coefficient, temperature evolution in the steel rods is predicted. As the temperature distribution affects the mechanical properties of steel rods, the proposed methodology can be effectively employed in controlling the quality of products. The same methodology can be adopted for different materials taking into consideration their thermophysical properties.
1166
References 1. Economopoulos M, Respen Y, Lessel G, Steffes G (1975) Application of the Tempcore process to the fabrication high yield strength concrete- reinforcing bars. Metallurgical Reports, CRM, No.45, December 1:1–17 2. Simon P, Economopoulos, Nilles P (1984) Tempcore a new process for production of high quality reinforcing bars. Iron Steel Eng 61(3):55–57 3. Kreith F, Black WZ (1980) Basic heat transfer. Harper and Row, NY, p 49 4. Mitsutsuka M (1968) Study of water spray cooling of steel plate at high temperature. Tetsu-to-Hagane 54:157–162 5. Mitsutsuka M, Fukuda N (1983) Cooling characteristics and heat transfer coefficients during water-spray cooling of hot steel plate. Tetsu-To-Hagane 69:262–267 6. Carnahan B, Luther HA, Wilkes WO (1979) Applied numerical methods. John Wiley, New York, pp 446–471 7. Incropera FP, Dewitt DP (1996) Introduction to heat transfer, 3rd edn. Wiley, New York 8. Hernandez-Morales B, Brimacombe JK, Hawbolt EB (1992) Application of inverse technique to determine heat transfer coefficients in heat-treating operations. J Mater Eng Perform 1:763–772 9. Morales RD, Lopez AG, Olivares IM (1990) Heat transfer analysis during water spray cooling of steel rods. ISIJ Int 30:48– 57
Int J Adv Manuf Technol (2010) 47:1159–1166 10. Morales RD, Toledo M, Lopez AG, Tapia AC, Peterson C (1992) Mathematical model for on-line heat treatment of steel bars. Steel Res 62:433–440 11. Özsoyeller L (1993) The effect of Tempcore process on mechanical properties of reinforced low alloy steel bars. Ph.D thesis. Dokuz Eylül University, I zmir 12. Reddy JN (2005) An introduction to the Finite element method. McGraw-Hill, New York 13. BISRA (1978) Physical constants of some commercial steels at elevated temperatures. Butterworths Scientific Publications, London 14. Garken S, Gurry RW (1953) Physical chemistry of metals. Mcgraw-Hill, New York 397 15. Brain I, Knacke O, Kubbaschrwski O (1977) Thermo-mechanical properties of inorganic substances; supplement. Springer Verlag, New York, pp 245–246 16. Taniguchi S, Murakami T, Watanabe A, Kikuchi A (1988) Phase transformation and heat transfer during cooling of steel. Tetsu-toHagane 74(2):318–325 17. Press WB, Flannery BP, Teukolsky SA, Vetterling WT (1986) Numerical recipes, the art of scientific computing. Cambridge University Press, Cambridge 18. Montgomery DC (2005) Design and analysis of experiments, 5th edn. Wiley, Singapore 19. Minitab Inc (2003) Minitab Software Version 14, User’s Guide, Technical Manual. Minitab Inc, Pennsylvania 20. Montgomery DC, Peck EA, Vining GG (2003) Introduction to linear regression analysis, 3rd edn. Wiley, New York
