Quantitative Correlation between Interfacial Heat Transfer Coefficient and Pressure for 19Cr-14Mn-0.9N High Nitrogen Steel Cylindrical Ingot

Abstract

This research is aimed to correlate heat transfer coefficient to pressure at the interface between 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot and cast iron mould, during the pressurized solidification process of cylindrical ingot. The correlations were obtained by mathematical inverse model of heat conduction problem. Validation results indicate that this model is applicable to investigate the change in interfacial heat transfer coefficient during the pressurized solidification process of 19Cr-14Mn-0.9N high nitrogen steel, and guarantee the correlation accuracy. Combing with theoretical derivation for cylindrical steel ingot, the change in interfacial heat transfer coefficient with time can be described by h_f,0.5 = 679.68t^−0.12 W/(m³·K) for 0.5 MPa, h_f,0.85 = 753.53t^−0.12 W/(m³·K) for 0.85 MPa and h_f,1.2 = 790.39t^−0.12 W/(m³·K) for 1.2 MPa, quantitatively. Meanwhile, an empirical formula was presented to correlate interfacial heat transfer coefficient to pressure, which can be taken as heat transfer boundary to simulate the change in solidification state of 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot with pressure.

1. Introduction

High pressure metallurgy is one of the most effective methods to manufacture high nitrogen steel, including pressurized induction melting (PIM) and pressurized electroslag remelting (PESR) etc.^{1,2,3,4,5,6,7)} And, it has been applied to manufacture industrial production. In Germany, the industrial pressurized electric slag remelting (4.2 MPa, 20 t) has been used to manufacture large generator retaining ring (P900 and P2000 high nitrogen steel).^1,2,3,8) During the manufacture process of high nitrogen steel, the high pressure plays many roles,⁹⁾ such as increasing the solubility of nitrogen,^{10,11,12,13,14)} accelerating cooling rate of ingot,^15,16) and suppressing the appearance of pores defects,^{10,15,17,18,19,20,21,22)} and so on. Recently, during the pressurized solidification process of high nitrogen steel, the most research has focused on the effect of pressure on solidification structure and defects.^{10,15,16,17,18,19,20,21,22,23)} The investigations are relatively less about the interfacial heat transfer between ingot and mould, especially about quantitative relationship between interfacial heat transfer coefficient and pressure. It caused the lack of accurate thermal boundary conditions to investigate pressurized solidification process of high nitrogen steel.¹⁸⁾

The solidification state of ingot is controlled by temperature distribution. And, the temperature distribution of ingot is closely related to the interfacial heat transfer between ingot and mould.²⁴⁾ Thus, the investigation of interfacial heat transfer is seriously important to research the solidification state, such as the change in dendrite spacing and segregation.^25,26,27) The more accurate is interfacial heat transfer coefficient , the more true is the research results of solidification state by simulation method.²⁶⁾ Up to now, the change in interfacial heat transfer coefficient with pressure is commonly obtained by mathematical inverse model of heat conduction problem with measured temperature.^{28,29,30,31,32,33)} Those researches mostly focus on the pressurized solidification of nonferrous metals.³⁴⁾ And it was indicated that the inverse model is accurate for the pressurized solidification of nonferrous metals,^{31,32,33,35,36,37,38)} such as aluminum alloy.^{16,31,32,33,37,38,39,40,41)} However, further verification of inverse model is required to approve its prediction accuracy for steel ingot, because it treats higher pour temperature, larger temperature difference between those of ingot and mould, and faster cooling rate of ingot and so on, comparing with nonferrous metals.^42,43,44,45)

In addition to the above mentioned facts, it needs to be quantitatively investigated by the numerical model, for the change in the temperature of 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot with pressure obtained in the former experiments of 1st anthor.¹⁵⁾ This research is aimed to obtain the quantitative relation between interfacial heat transfer coefficient and pressure during the pressurized solidification process of high nitrogen steel. Taking 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot as a research object, the mathematical inverse model of heat conduction problem has been verified, and then used to obtain interfacial heat transfer coefficients. Combining with theoretical derivation, an empirical formula was presented for correlation between interfacial heat transfer coefficient and pressure during the pressurized solidification process of 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot.

2. Mathematical Model

The mathematical model used to obtain interfacial heat transfer coefficients is divided into two parts. They are forward heat conduction model (FHCM) and inverse heat transfer conduction model (IHTCM), respectively. Forward heat conduction model is used to obtain temperature field with a given heat transfer coefficient, and the purpose of inverse heat transfer conduction model is to determine heat transfer coefficient by a given temperature distribution.

2.1. Forward Heat Conduction Model (FHCM)

Based on the law of conservation of energy and symmetry of cylindrical ingot, one dimensional transient heat conduction can be used to obtain the temperature distribution of ingot,⁴⁶⁾ which is described by:   

$$\rho C_p\frac{\partial T}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left(kr\frac{\partial T}{\partial r}\right)+q$$(1)

Where, ρ is the mass density of the material, kg/m³; r is radius, m; C_p is specific heat capacity, J/(kg·K); k is thermal conductivity, W/m·K; and T is temperature, K; q is volumetric heat source during solidification and given by:⁴⁶⁾   

$$q=\rho L_m\frac{\partial f_s}{\partial t}=\rho L_m\frac{\partial f_s}{\partial T}\frac{\partial T}{\partial t}$$(2)

Where, L_m is the latent heat of solidification, J/kg; f_s is solid fraction; and t is time, s. Assuming that k is independent of r, substituting Eq. (2) into Eq. (1) results in   

$$\rho C_{p,eq}\frac{\partial T}{\partial t}=k\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)$$(3)

Where, $C_{p,eq}=C_p-L_m\frac{\partial f_s}{\partial T}$ for ingot. The corresponding boundary conditions to Eq. (3) are as follows:

$k\frac{\partial T}{\partial r}|{}_{r=R}=h_i(T_{w,m}-T_{w,i})$ at the ingot/mould interface.

Where, h_i is macroscopic average ingot/mould interfacial heat transfer coefficient, W/(m³·K); T_w,i and T_w,m is the surface temperature of ingot and the inner wall temperature of mould, respectively. Similarly, Eq. (3) can be used to calculate the temperature field of mould, when C_p,eq is substituted by the specific heat capacity of mould C_p.m.

2.2. Inverse Heat Transfer Conduction Model (IHTCM)

It is well known,^{28,29,30,31,32,33)} the relationship between the temperature field of ingot/mould and macroscopic average ingot/mould interfacial heat transfer coefficient can be characterized by:   

$$T=T(h_i)$$(4)

Equation (4) can be approximated by using the first order approximation of Taylor expansion, and recast into:   

$$T_n(h_i)=T_n(h_{i,0})+(h_i-h_{i,0})\frac{\partial T_n(h_i)}{\partial h_i}$$(5)

Where, h_i,0 is the reference value of h_i, n is the number of temperature node. In order to investigate deviation between the true value and numerical one of macroscopic average ingot/mould interfacial heat transfer coefficient, Eq. (6) is proposed on the basis of least square method.^32,46)   

$$f(h_i)=\sum _{n=1}^N{(T_{m,n}-T_n(h_i))}^2+\beta {(h_i-h_{i,0})}^2$$(6)

Where, β is the scaling factor, T_m,n is the true value of temperature, and T_n(h_i) is the numerical value of temperature obtained with h_i. When h_i is infinitely close to the true value of macroscopic average ingot/mould interfacial heat transfer coefficient, $\frac{\partial f(h_i)}{\partial h_i}=0$, and Eq. (6) can be recast into:   

$$\sum _{n=1}^N{(T_{m,n}-T_n(h_i))}^2\frac{\partial T_n(h_i)}{\partial h_i}=\beta {(h_i-h_{i,0})}^2$$(7)

Substituting Eq. (7) into Eq. (6) results in:   

$$h_i=h_{i,0}+{\displaystyle \frac{\sum _{n=1}^N(T_{m,n}-T_n(h_{i,0})){\displaystyle \frac{\partial T_n(h_i)}{\partial h_i}}}{{\sum _{n=1}^N\left({\displaystyle \frac{\partial T_n(h_i)}{\partial h_i}}\right)}^2+\beta }}$$(8)

Equation (8) can be used to calculate the interfacial heat transfer coefficient between ingot and mould with a given temperature distribution.

2.3. Numerical Solution

The conservation equations (Eqs. (3) and (8)) correspond to two unknown T and h_i. In order to obtain T and h_i by numerical solution, Eq. (3) is discretized by the implicit finite difference method, and can be rewritten as:   

$$\begin{array}{l}\left(1+{\displaystyle \frac{2k\mathrm{\Delta }t}{{\rho }_n{C}_{p,n}\mathrm{\Delta }{r}^2}}\right)T_n^t=\left({\displaystyle \frac{k\mathrm{\Delta }t}{{\rho }_n{C}_{p,n}\mathrm{\Delta }{r}^2}}-{\displaystyle \frac{k\mathrm{\Delta }t}{2{\rho }_n{C}_{p,n}{r}_n\mathrm{\Delta }r}}\right)T_{n-1}^t\\ +\left({\displaystyle \frac{k\mathrm{\Delta }t}{{\rho }_n{C}_{p,n}\mathrm{\Delta }{r}^2}}+{\displaystyle \frac{k\mathrm{\Delta }t}{2{\rho }_n{C}_{p,n}{r}_n\mathrm{\Delta }r}}\right)T_{n+1}^t+T_n^{t-\mathrm{\Delta }t}\end{array}$$(9)

Meanwhile, based on nonlinear estimation technique used by Beck,^28,30) Eq. (8) can be written as   

$$h_{i,t}=h_{i,0,t}+{\displaystyle \frac{\sum _{n=1}^N\sum _{\tau =0}^{\delta -1}\left(T_{m,n,t+\tau \times \mathrm{\Delta }t}-T_{n,t+\tau \times \mathrm{\Delta }t}(h_{i,0,t})\right){\mathrm{\Phi }}_{i,n,t+\tau \times \mathrm{\Delta }t}}{\sum _{n=1}^N\sum _{\tau =0}^{\delta -1}{\mathrm{\Phi }}_{i,n,t+\tau \times \mathrm{\Delta }t}{}^2+\beta }}$$(10)

  

$${\mathrm{\Phi }}_{i,n,t+\tau \times \mathrm{\Delta }t}={\displaystyle \frac{T_{n,t+\tau \times \mathrm{\Delta }t}(h_{i,t}+\alpha h_{i,t})-T_{n,t+\tau \times \mathrm{\Delta }t}(h_{i,t})}{\alpha h_{i,t}}}$$(11)

Where, Φ_{i,n,t+τ×Δt} is sensitivity coefficient, α is a constant, Δt and Δr are the interval of both time and space, respectively. During inverse calculation process of heat conduction problem, Eqs. (9), (10) and (11) are solved by the program compiled with Intel Visual Fortran. Firstly, the temperature is obtained with boundary condition and the reference value of interfacial heat transfer coefficient h_i,0 by sloving Eq. (9) (IHTCM), and then taken as input data to calculate sensitivity coefficient Φ_{i,n,t+τ×Δt} combing with the experimental temperature according to Eq. (11). Based on the reference value h_i,0 and sensitivity coefficient Φ_{i,n,t+τ×Δt}, the macroscopic average ingot/mould interfacial heat transfer coefficient h_i,n is obtained by Eq. (10) (IHTCM). This procedure, taking h_i,n as the new reference value, will be repeated for a new macroscopic average ingot/mould interfacial heat transfer coefficient, and is continued until the convergence criteria $\left(\frac{h_{i,n}-h_{i,0}}{h_{i,n}}<{10}^{-3}\right)$ is met. The overview of the solution procedure is given by the flow chart, as shown in Fig. 1.

Fig. 1.

Flow chart for the determination of metal/mold heat transfer coefficient.

3. Results and Discussion

3.1. Model Validation

The model validation includes two parts. They are the validations of forward heat conduction model (FHCM) and inverse heat transfer conduction model (IHTCM), respectively. Comparing with aluminum or other low melting point alloys,^{31,33,37,38,39,41)} it is more difficult to measure cooling curves of steel ingot, because of the higher liquidus temperature and faster cooling process of ingot.^42,43,44,45) Meanwhile, the sharp rises and oscillations in measured values of cooling curves result in the lower accuracy and larger errors of validation results.^15,44) Thus, in order to improve the accuracy of FHCM and IHTCM, the temperature obtained by ProCast software and the change in interfacial heat transfer coefficient with time h_i=2000t^−0.5 have been used, instead of measured values.

The validation of forward heat conduction model (FHCM) is achieved with the comparison of temperature values (T_n,P and T_n,C). And these temperature values are obtained with ProCast software and forward heat conduction model, according to a given interfacial heat transfer coefficient (h_i =2000t^−0.5), respectively. Figure 2 shows the overview of the validation procedure.

Fig. 2.

Flow chart for the validation procedure of forward heat conduction model.

In order to investigate the accuracy of forward heat conduction model, the relative errors of temperature R_T,n has been calculated by:   

$$R_{T,n}=\frac{\left|T_{n,P}-T_{n,C}\right|}{T_{n,P}}$$(12)

Where, T_n,P and T_n,C are ingot temperature obtained by ProCast software and forward heat conduction model (FHCM), respectively. In the model validation, 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot and cast iron are used as ingot and mould, respectively.¹⁸⁾ The changes in ingot temperature T_n,P with time are obtained by ProCAST software, as shown in Fig. 3(a). FHCM treats the solidification process of cylindrical steel ingot as one dimensional geometric model to calculate T_n,C. Two temperature nodes in ingot (P1 and P2) are chosen to investigate the relative errors of temperature (R_T,1 and R_T,2), as shown in Fig. 3(a). And the changes in the relative errors (R_T,1 and R_T,2) with time are exhibited in Fig. 3(b).

Fig. 3.

Changes in relative errors and interfacial heat transfer coefficients with time. (Online version in color.)

At the initial stage of solidification, there exists a sharp drop in R_T,2, which is primarily caused by the sharp drop in the given interfacial heat transfer coefficient (h_i =2000t^−0.5) and the difference in interval of time Δt between ProCast software and FHCM. The sharp drop in the given interfacial heat transfer coefficient leads to a sharp decrement of ingot temperature. In this case, the calculation error of ingot temperature is greater for larger interval of time Δt. The interval of time Δt in ProCast software is about 10⁻⁴s, and that in FHCM is generally above 0.1 s due to the limit of nonlinear estimation technique.^{30,32,43,47,48)} So the huge difference in interval of time Δt results in the great relative errors of ingot temperature obtained by FHCM. At the sharp drop stage of given interfacial heat transfer coefficient, the accuracy of FHCM is low, and the relative errors of temperature reach maximum value 0.07. Subsequently, the relative errors decrease gradually with the disappearance of sharp drop in the given interfacial heat transfer coefficient. Meanwhile, the effects of given interfacial heat transfer coefficient and interval of time are weaker in a position further away from the ingot surface. Thus, the drop rate of R_T,1 is almost immeasurably smaller than that in R_T,2. Except for that at the initial stage of solidification, R_T,1 and R_T,2 are always smaller than 0.01, and both become smaller and smaller over solidification time. It is indicated that the FHCM is a reliable model with good accuracy except for the rapid change process in interfacial heat transfer coefficient, and can be used to calculate the temperature distribution of ingot during the inverse calculation process of heat transfer coefficient.

Inverse heat transfer conduction model takes T_n,P and T_n,C as input data to calculate interfacial heat transfer coefficient (h_i,C). And it is verified by the comparison of h_i,C and the given interfacial heat transfer coefficient (h_i=2000t^−0.5). The overview of model validation procedures is shown in Fig. 4.

Fig. 4.

Flow chart for the validation procedure of inverse heat transfer conduction model.

Similarly, the relative errors of interfacial heat transfer coefficient R_h have been calculated to evaluate the accuracy of inverse heat transfer conduction model (IHTCM).   

$$R_h=\frac{\left|h_{i,}-h_{i,C}\right|}{h_i}$$(13)

Where, h_i,C is interfacial heat transfer coefficient obtained by inverse heat transfer conduction model (IHTCM). And changes in the relative errors R_h and interfacial heat transfer coefficients (h_i,C and h_i) with time are exhibited in Fig. 3(b). R_h is always kept decreasing over the range of solidification times. IHTCM takes the temperature of ingot T_n,C obtained by FHCM as the input data to calculate interfacial heat transfer coefficient h_i,C. It results in that the accuracy of IHTCM is largely influenced by the errors of ingot temperature obtained by FHCM. The change tendency of R_h is identical with that in R_T,1, and the sharp drop in R_h is primarily caused by the great the relative errors of ingot temperature obtained by FHCM. Except for that at the initial stage of solidification, R_h is always smaller than 0.03, which suggests that the accuracy of IHTCM will become higher and higher as the solidification proceeds. For normal heat transfer at the interface between ingot and mould, there barely exists the sharp drop in interfacial heat transfer coefficient,^49,50,51) like the given interfacial heat transfer coefficient (h_i=2000t^−0.5). Thus, the errors of ingot temperature obtained by FHCM is tiny, and the interfacial heat transfer coefficients obtained by IHTCM is seriously accurate.

Based on the relative errors of temperature R_T and interfacial heat transfer coefficient R_h, the validity of FHCM and IHTCM has been illustrated. It also suggests that the FHCM and IHTCM can be used to accurately calculate interfacial heat transfer coefficient, during the solidification process of 19Cr-14Mn-0.9N nitrogen steel cylindrical ingot.

3.2. Quantitative Relations between Interfacial Heat Transfer Coefficient and Pressure

The applicability of FHCM and IHTCM to a cylindrical ingot has been demonstrated with the relative errors of temperature R_T and interfacial heat transfer coefficient R_h. In order to obtain interfacial heat transfer coefficient during the pressurized solidification process of 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot, the cooling curves (T_M) of ingot and mould, as input data of FHCM and IHTCM, have been measured with platinum rhodium thermocouples of “B” type under different pressure (0.5, 0.85 and 1.2 MPa), which has been reported in previous research of 1st author.¹⁵⁾

Figures 5(a) 5(b) and 5(c) show the calculated values (T_C) obtained by FHCM, the measured values (T_M) and relative errors (R_T,C) of ingot temperature under 0.5, 0.85 and 1.2 MPa. T_1,M and T_2,M are the measured value of cooling curves at 10 and 5 mm away from ingot surface, respectively. And T_1,C and T_2,C are the corresponding calculated values obtained by FHCM. Under 0.5, 0.85 and 1.2 MPa, the maximum differences between measured value (T_M) and calculated values (T_C) are 53, 48 and 60 K, respectively. And the corresponding maximum relative errors are 0.041, 0.041 and 0.048. Due to the thermocouple heating, turbulent liquid flow and the increment of cooling rate, the measurement errors of temperature become bigger and bigger with the distance closer to ingot surface, especially in the early stage of solidification.¹⁵⁾ Thus, the maximum values of differences and relative errors obtained with T_2,M and T_2,C both are at 5 mm away from ingot surface. And, comparing with that at 5 mm, the relative errors of temperature (R_T1,C), at 10 mm away from ingot surface, is smaller and doesn’t exceed 0.03 over the range of solidification times. Meanwhile, Fig. 5(d) shows the calculated values (T_mould,C) obtained by FHCM, the measured values (T_mould,M) and relative errors (R_mould) of mould temperature under 0.5, 0.85 and 1.2 MPa. T_mould,M is the measured value of mould temperature at 8 mm away from mould inner wall. Under 0.5, 0.85 and 1.2 MPa, the maximum differences between measured (T_mould,M) and calculated values (T_mould,C) are 22, 11 and 14 K, respectively. And the corresponding maximum values of relative errors (R_mould) are 0.054, 0.026 and 0.023. As the solidification proceeds for 19Cr-14Mn-0.9N ingot, the relative errors decrease gradually, and then are always smaller than 0.025. It is indicated that the calculated temperature obtained by FHCM shows a good agreement with measured temperature, FHCM is effective to calculate temperature of a cylindrical ingot and provides the accurate input data to IHTCM.

Fig. 5.

Calculated values, measured values and relative errors of temperature under (a) 0.5 MPa, (b) 0.85 MPa, (c) 1.2 MPa for ingot and (d) mould. (Online version in color.)

The calculation results of interfacial heat transfer coefficient between cylindrical ingot and mould is obtained by IHTCM. And, the change in interfacial heat transfer coefficient with time is shown in Fig. 6. The interfacial heat transfer coefficient decreases gradually as the solidification proceeds.¹⁵⁾ And the interfacial heat transfer coefficient increases with the increment of pressure, which is consistent with the conclusion reported by Jiang^15,52) and Ilkhchy.³⁴⁾

Fig. 6.

Calculation results and profile curves of interfacial heat transfer coefficient. (Online version in color.)

According to the research reported by Cheung,³³⁾ the temperature T_i near the surface of ingot can be obtained by the following correlation:   

$$T_i=T_{w,m}+\left(T_{w,i}-T_{w,m}\right)e^{-\frac{h_i}{k_i}(r_i-R)}$$(14)

Where, R is the radius of ingot, m; T_w,i and T_w,m are the temperature of ingot surface and mould inner wall, respectively. Substituting Eq. (14) into Eq. (3) yields:   

$$\begin{array}{l}{\rho }_i{C}_{p,eq}{\displaystyle \frac{d{T}_{w,i}}{dt}}={\displaystyle \frac{d}{dr}}\left(-h_i\left(T_{w,i}-T_{w,m}\right)e^{-{\displaystyle \frac{h_i}{k_i}}(r_i-R)}\right)\\ \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}-{\displaystyle \frac{h_i}{r_i}}\left(T_{w,i}-T_{w,m}\right)e^{-{\displaystyle \frac{h_i}{k_i}}(r_i-R)}\end{array}$$(15)

Equation (15) can be simplified as:   

$$\frac{\text{d}{T}_{w,i}}{(T_{w,i}-T_{w,m})}=\left(\frac{h_i^2}{{\rho }_i{C}_{p,eq}{k}_i}{e}^{-\frac{h_i}{k_i}(r_i-R)}-\frac{h_i}{{\rho }_i{C}_{p,eq}{r}_i}{e}^{-\frac{h_i}{k_i}(r_i-R)}\right)\text{d}t$$(16)

At the surface of ingot r_i=R, Eq. (16) integrated in time can be expressed as:   

$$\frac{t}{{\rho }_i{C}_{p,eq}{k}_i}{h}_i^2-\frac{t}{{\rho }_i{C}_{p,eq}R}{h}_i-ln\left(\frac{T_{w,i}-T_{w,m}}{T_{w,i,0}-T_{w,m}}\right)=0$$(17)

Equation (17) can be solved for interfacial heat transfer coefficient to obtain the following:   

$$h_i=\frac{-B}{2A}+\frac{\sqrt{B^2-4AC{t}^{-1}}}{2A}$$(18)

Where, $A={\displaystyle \frac{1}{{\rho }_i{C}_{p,eq}{k}_i}}$, $B=-{\displaystyle \frac{1}{{\rho }_i{C}_{p,eq}R}}$, $C=-ln\left({\displaystyle \frac{T_{w,i}-T_{w,m}}{T_{w,i,0}-T_{w,m}}}\right)$. Equation (18) is the correlation between interfacial heat transfer coefficient and time. When the the radius R is sufficiently large, B is too tiny to be ignored, and Eq. (18) can be simplified as:   

$$h_i=a{t}^m$$(19)

Where, a and m are constants for a given solidification condition. Equation (19) suggests the profile equation of heat transfer coefficient, which is the consistent with that reported by Cheung³³⁾ and Ilkhchy.³⁴⁾ The corresponding profile curves have been fitted as shown in Fig. 6. And the fitting formulas have same expression for different pressure. They are h_f,0.5 = 679.68t^−0.12 (for 0.5 MPa), h_f,0.85 = 753.53t^−0.12 (for 0.85 MPa) and h_f,1.2 = 790.39t^−0.12 (for 1.2 MPa), respectively. The corresponding R-squares are 0.9558, 0.9716 and 0.9692. The fitting results indicate that IHTCM and profile equation are accurate and credible to investigate the change in interfacial heat transfer coefficient as the solidification proceeds of 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot.

It is known that the relationship between heat transfer coefficient and pressure can be described by polynomial at a given time.³⁴⁾ In order to obtain the quantitative correlation between heat transfer coefficient and pressure, the fitting formulas for different pressure were used to propose an empirical formula by polynomial fitting. And, the empirical formula correlation between heat transfer coefficient and pressure is given by:   

$$h_i=\left(-1.51P^2+41.48P+510.01\right)t^{-0.12}$$(20)

Where, P is pressure, MPa; h_i is the heat transfer coefficient. At a given time, the empirical formula is consistent with that reported by Ilkhchy.³⁴⁾ Meanwhile, both the empirical formula Eq. (20) and that reported by Ilkhchy can be expressed as the following polynomial function consisting of pressure:   

$$h_i=a_3{P}^3-a_2{P}^2+a_{1}P+a_0$$(21)

Where, a₃, a₂, a₁ and a₀ are constants at a given time. According to correlation between the heat transfer coefficient and external pressure proposed by Ilkhchy,³⁴⁾ the effect of a₃P³ on h_i can be ignored at low pressure due to a₃ is too small, compared with that of a₂P², a₁P and a₀. And, a₂P², a₁P and a₀ play dominant roles to correlate interfacial heat transfer coefficient to pressure. Thus, Eq. (20) is an accurate empirical formula, which is applicable to be taken as heat transfer boundary conditions to investigate the change in solidification state of 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot with pressure. Based on the investigation on the correlation between interfacial heat transfer coefficient and pressure for 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot, Eq. (20) can be applicable under the conditions that: 1) the maximum value of pressure is around 1.2MPa under the pressurized solidification, 2) the minimum radius value of cylindrical ingot is great than or equal to 63 mm, and 3) the alloy is high nitrogen steel.

4. Conclusion

The applicability of the mathematical inverse model of heat conduction problem has been investigated, for the pressurized solidification process of 19Cr-14Mn-0.9N high nitrogen steel. And then, the model was used to propose an empirical formula correlating interfacial heat transfer coefficient to pressure, combing with theoretical derivation. The main conclusions can be obtained as follows:

(1) The mathematical inverse model is a reliable model with good accuracy except for the rapid change process in interfacial heat transfer coefficient. And it is accurate to investigate the change in interfacial heat transfer coefficient, during the pressurized solidification process of 19Cr-14Mn-0.9N high nitrogen steel.

(2) The heat transfer coefficient decreases as solidification proceeds for 19Cr-14Mn-0.9N high nitrogen steel. And the relationship between heat transfer coefficient and time is written below:

h_f,0.5 = 679.68t^−0.12 (for 0.5 MPa), h_f,0.85 = 753.53t^−0.12 (for 0.85 MPa), h_f,1.2 = 790.39t^−0.12 (for 1.2 MPa).

(3) An empirical formula correlating interfacial heat transfer coefficient to pressure has been proposed, which can be taken as accurate heat transfer boundary conditions to investigate the effect of pressure on solidification state of 19Cr-14Mn-0.9N high nitrogen steel cylindrical ingot.   

$$h_i=\left(-1.51P^2+41.48P+510.01\right)t^{-0.12}$$

Acknowledgements

The present research was financially supported by National Natural Science Foundation of China (No. U1960203, 51904065 and 51774074), Project funded by China Postdoctoral Science Foundation (No. 2019M651133), Fundamental Research Funds for the Central Universities (No. N182503028, N172512033 and N2024005-4), and Talent Project of Revitalizing Liaoning (XLYC1902046).

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