Heat Mass Transfer DOI 10.1007/s00231-015-1632-1

ORIGINAL

Determination of the heat transfer coefficient at the metal–sand mold interface of lost foam casting process Liqiang Zhang1,2 · Wenfang Tan1 · Hao Hu1 

Received: 30 October 2014 / Accepted: 9 July 2015 © Springer-Verlag Berlin Heidelberg 2015

Abstract  For modeling solidification process of casting accurately, a reliable heat transfer boundary condition data is required. In this paper, an inverse conduction model was established to determine the heat flux and heat transfer coefficient at the metal–sand mold interface for cylindrical casting in the lost foam process. The numerically calculated temperature was compared with analytic solution and simulation solution obtained by commercial software ProCAST to investigate the accuracy of heat conduction model. The instantaneous cast and sand mold temperatures were measured experimentally and these values were used to determine the interfacial heat transfer coefficient (IHTC). The IHTC values during lost foam casting were shown to vary from 20 to 800 W m−2 K−1. Additionally, the characteristics of the time-varying IHTC have also been discussed in this study.

1 Introduction As one of the boundary conditions in the simulation of casting, the IHTC is necessary for accurate simulation of casting processes [1–3]. However, the values of IHTC are difficultly obtained by using the experimental or analytic methods as it is influenced by various factors, such as alloy type, latent heat, roughness of surfaces, thermo-physical * Liqiang Zhang [email protected] 1

College of Mechanical and Electrical Engineering, Central South University of Forestry and Technology, Changsha 410004, Hunan, People’s Republic of China

2

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, Hunan, People’s Republic of China





properties of mold, coating material, coating thickness and gap formation due to the deformation of casting and mold [4–6], etc. In recent years, the IHTC was usually determined by using an inverse approach, which aims at minimizing the difference between the measured and calculated temperatures [7, 8]. For example, Ilkhchy et al. [9] determined the IHTC at the interface between the aluminum alloy and the metallic mold during the solidification of casting under different pressures using an inverse heat conduction problem (IHCP) method. Their research revealed the pressure had remarkable effect on the heat transfer condition in the metal/ mold interface, and the main effect of the pressure relied in the variation of metal/mold contact. Kovacˇević et al. [10] estimated the IHTC between the thin wall aluminum alloy plate casting and the sand mold by an iterative algorithm based on the function specification method. The IHTC values were described as a function of casting surface temperature at the interface. Silva et al. [11] proposed a theoretical method to predict the heat transfer coefficients at the metal– mold interface during horizontal unsteady-state directional solidification of casting. The research revealed the transient metal–mold heat transfer behavior can be expressed as a power function of time. Arunkumar et al. [12] employed a serial-inverse heat conduction algorithm to estimate the multiple heat flux transients along the metal–mold interface. The IHTC was found to vary for the different mold segments analyzed and in particular the peak IHTC was found to occur at different times, indicating that the initiation of the air gap varied with position along the mold wall. Long et al. [13] studied the heat transfer behavior in the high pressure die casting process based upon the generated Magmasoft simulation model. The research indicated that the interfacial heat transfer behavior was significant affected by temperature, pressure and metal speed during the filling of high pressure die casting.

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Heat Mass Transfer

Though the IHTC has been studied extensively in literature, little attention has been paid to the IHTC during the lost foam casting process. Moreover, little research has been published about the establishment of inverse heat conduction model for the casting with a cylindrical geometry. In the present paper, a cylindrical geometry was selected for the determination of the IHTC between the cast iron and the surrounding sand mold in the lost foam casting process based on the inverse heat conduction method. After acquiring IHTC time dependent curve, the heat transfer behavior between the cast iron and the sand mold was discussed. In addition, the effect of casting diameter on the IHTC was investigated.

2 Experimental procedures For determining the IHTC using the IHCP method, two lost foam casing experiments with the cylindrical casting were selected to obtain the temperature data in this paper. All process conditions of the two experiments except the casting diameter were the same. The casting diameters were 50 and 70 mm, respectively. The thicknesses of sand mold during each experiment were both 80 mm. The schematic diagram of experimental setup with the casting arrangement and the position of the thermocouples are shown in Fig. 1. For each experiment, three K-type armored thermocouples with the diameter of 5 mm were inserted into the sand mold cavity (see Fig. 1). The thermocouple locations in the casting and the sand mold are shown in Fig. 1. T1 and T2 are the positions of the placed thermocouples for the experiment. For the inverse heat conduction method to be applied successfully, it is also necessary to optimize the location of the thermocouples. It should not be too close to the interface because the temperature may not be representative and the inverse method might amplify the noise present in the sampled data (stochastic response). It should also not be too far from the interface because the inverse heat conduction method could become unstable [12]. In this study, these thermocouples were placed located at 5 mm from the casting–sand mold interface. The reason is that the degradation foam pattern from the moving metalfoam interface produces a chilling effect on the temperature field during the filling process of loss foam casting (especially the effect on the temperature change close to the

Fig. 1  Schematic diagram of the experimental setup (all dimensions in mm)

mold surface). Therefore, these thermocouples in the present work were not too close to the sand mold surface for reducing the effect on the measured temperature data. The casting and sand mold temperatures were measured with a thermocouple at 0.125 s of time intervals. All thermocouples were connected by cables to Measurement Computing USB-2416 data acquisition system and a computer, which consisted of a MCT-2 data acquisition/switch unit and a computer. The experiments were performed with a grey cast iron. The grey cast iron was melted in an electric furnace and then was poured into the cylindrical mold. The pouring temperature was about 1350 °C. The initial sand mold temperature of each experiment was the same, about 25 °C. The mold is furan resin-bonded sand. Its dimensions are shown in Fig. 1. The thermo-physical parameters of grey cast iron and furan resin-bonded sand used in the experiment are given in Table 1. Before the pouring begins, the sand mold was placed on the adiabatic wool with 10 mm thickness to prevent/minimize the heat flow along the axial direction. The heat flow was permitted along the radial direction of the casting. Such heat isolation environment was designed so that the heat flux in

Table 1  Thermo-physical parameters of materials Material

Density (kg m−3) Specific heat [kJ (kg k)−1]

Casting (HT300) (6.3–7.4) × 103 Sand mold

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1.52 × 103

Latent heat (kJ kg−1) Thermal conductivity [W (m K)−1]

Solidus_T (K) Liquidus_T (K)

0.532

60–1458

23.6–36.4

1406

1522

1.22

–

0.53

–

–

Heat Mass Transfer

the casting and sand mold mainly transferred vertically against the interface.

3 Mathematical modeling for cylindrical casting

For the heat flux load vector:  e At bottom and top faces : fq m   2 + J2 = 2πrwi {N}qheat J11 12

(4)

i=1

In the inverse heat conduction method adopted for this work, a forward heat conduction model of the cylindrical casting needs to be developed and then coupled to an inverse algorithm. Forward heat conduction problems are defined as those in which the temperature in a body or domain is calculated by solving the heat transport equation subject to the appropriate boundary conditions and initial conditions (if transient). The solution domain of physical model in this paper is cylindrical geometry, as shown in Fig.  1. The applicable form of the direct heat conduction equation assuming heat transport by diffusion for the casting in r, z coordinates are usually written as [14]:

  ∂T ∂ 2T ∂T k ∂ r + k 2 + Q = ρCp r ∂r ∂r ∂z ∂t

(1)

where T is the temperature (K), t is the time (s), k is the thermal conductivity (W m−1 K−1), Cp is the specific heat (J Kg−1 K−1), r and z (m) are the radial and axial distance, respectively, Q is a volumetric source term associated with the latent heat of solidification (W m−3). Among the thermo-physical parameters required to solve Eq. (1), the volumetric heat associated with solidification, Q, is needed. The approach adopted in the present work is to modify the specific heat to account for the latent heat of solidification. The detailed description is given in Ref. [15]. In this case, the finite element method (FEM) is used to solve the problem described above subject to a set of boundary and initial conditions. The element size used for the mesh is 1.0 mm. In r, z coordinates, since each side of an element represents a “ring” with a different radius, the radius r is multiplied in the matrices and load vector. For the stiffness and heat capacity matrix:

[Kc ] = 2π

1 1

=

[B]T [K][B]|J|rdudv

2πrwi wj [B]T [K][B]|J|

(5)

i=1

For the distributed heat source term load vector:  e fQ = 2π

1 1

{N}|J|Qrdudv =

−1 −1

m  m 

2π rwi wj {N}|J|Q

i=1 i=1

(6)

In the above equations, {N} is the shape function, [B] is the derivative matrix, [J] is the Jacobian matrix. These matrices used in the finite element method are very important to determine the element properties. J11, J12, J21, J22 is the element of the matrix [J], respectively. The detailed description of these equations is given in Ref. [14]. After the above forward heat conduction model is constructed, the interfacial heat flux between the casting and the sand mold can be determined by using an inverse algorithm based on the temperature histories at interior points of the casting. The IHCP is a heat conduction problem where boundary conditions or thermo-physical properties is unknown. The goal in solving an IHCP is to find the unknown quantity using known temperatures from the interior of the body. The mathematical description of IHCP is given in Ref. [15]. After the heat flux transients is estimated by solving the IHCP, The heat flow across the casting-sand mold interface can be characterized by Eq. (7) and h can be determined provided that all the other terms of the equation, namely q, TC and TM, are known [16]: (7)

h = q/(TC − TM ) −2

−1 −1 m  m 

 e At left and right faces : fq m   2 + J2 = 2πrwi {N}qheat J21 22

(2)

−1

where h is the casting-sand mold IHTC (W m  K ); q is the heat flux across the interface (W m−2); TC and TM are casting and sand mold surface temperature (K), respectively.

i=1 j=1

[C]e = 2π

=

1 1

−1 −1 m m 

4 Results and discussion {N}T ρCp {N}|J|rdudv

2πrwi wj {N}T ρCp {N}|J|

i=1 j=1

(3)

4.1 Verification of forward heat conduction algorithm for cylindrical casting For solving each heat flux q, a number of forward heat conduction problems need to be calculated. So, the

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Heat Mass Transfer

Fig. 2  Transient heat transfer problem of cylindrical casting a 3D cylinder and b cylinder coordinates

accuracy of forward heat conduction model would affect directly the calculated result in the inverse calculation procedure. In this work, the forward heat conduction algorithm for cylindrical casting before solving inverse problem was firstly applied into a typically physical model to test its accuracy, reliability and stability. The solution domains of physical model are shown in Fig. 2. The dimension is Ø4 mm × 4 mm (R = L = 2 mm, r  = z = 1 mm). The element size used for the mesh is 1.0 mm. The algorithm accuracy was verified by comparing the calculated temperature with analytic solution and simulation result obtained by commercial software ProCAST. The compared result between the analytic and numerical solution is shown in Fig. 3a. It can be obviously seen that a good agreement is obtained between the analytical and calculated temperature. There is only a little difference between 1 and 3 s. The reason of error is that some approximate algorithm is adopted in the process of numerical calculation based on FEM. However, the result of comparison is enough to confirm the accuracy and reliability of established heat conduction algorithm for the calculating the temperature distributions of the assumed physical model in this paper. Figure 3b is the comparison between the numerical solution and the one obtained from commercial software ProCAST. It can be clearly seen that a better agreement is obtained. This is because both the adopted algorithm for the forward heat conduction model in this paper and the one in ProCAST belong to numerical calculation technology and the solving procedure both are similar. So, the agreement between the numerical calculated temperature and the one obtained from commercial software is better. The result confirms further that the developed

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heat conduction model in this paper for the cylindrical casting is accurate and reliable. 4.2 Experimental cooling curve The experimental setup and procedure are briefly described in Sect. 2. Figure 4 shows the cooling curves of two experiments measured at the location T1 in the sand mold and T2 in the casting. All process conditions of the two experiments except the casting diameter are the same. The details of locations are described in Fig. 1. At the location T1 and T2, the distance to the interface between the casting and the sand mold is both 5 mm. The variation trends of temperatures in two experiments are the similar. The temperature curve at the location T2 in Fig. 4 may be taken as a typical cooling curve during sand gravity die casting. In the initial period, the decrease of casting temperatures is due to the cooling of sand mold. When the casting temperatures decrease down to near solidus curve about 1133 °C, the cooling curves begin to change slightly. This can be attributed to the latent heat released during eutectic reaction of molten metal. Then, the gradients decrease by a second drop in temperature. When the casting temperatures decrease down to about 725 °C, the cooling curves change slightly again. The eutectoid reaction is the main contribution to the slight change of casting temperature. Afterwards, the casting temperature continues to decrease until the heat equilibrium reaches between the casting and the sand mold. In Fig. 4, it can still be obviously observed that the temperature variation in the sand mold has an inverse trend compared with the corresponding casting in each experiment. This is reasonable because the heat transfer mainly exist the interface between the casting and the sand mold with adequate insulation of the sand mold and casting chamber, as shown in Fig. 1. Such surrounding heat

Heat Mass Transfer

insulation environment is designed so that the sand mold cooling may be considered as the main contribution to the decreasing of casting temperature. In addition, the cooling rate of casting with the diameter of 70 mm in Fig. 4 is lower than that of 50 mm in diameter. Inversely, the increase of sand mold temperature is faster when the casting is 70 mm in diameter. This is reasonable because the heat diffusion in the metal with bigger volume need more time compared with the small volume. Thus, more heat flux is transferred to the sand mold because of slighter decrease of casting temperature with the diameter of 70 mm so that the temperature in sand mold increases more quickly. 4.3 Interfacial heat flux and heat transfer coefficient

Fig. 3  The comparison of temperature obtained from different methods a analytic and numerical solution and b ProCAST and numerical solution

Fig. 4  Measured temperatures versus time curves at deferent locations

To estimate the interfacial heat flux transients, the temperatures measured at the location T2 inside the casting are used as an input to an inverse heat conduction model. The mathematical description of IHCP is given in Ref. [15]. The heat flux transients estimated by solving the IHCP are shown in Fig. 5. The variation trends of interfacial heat flux for different casting diameter are the similar. In the initial step, the inverse model predicts a heat flux of approximately 800 kW m−2 that develops very rapidly within approximately 100 s. Then, the heat flux quickly drops throughout the solidification temperature range and by the time the solidus temperature is reached, at approximately 200 s, is down to a value of around 20 kW m−2. Thereafter, the interfacial heat fluxes keep constant. In addition, the interfacial heat flux for the casting with the diameter of 70 mm lags behind that of 50 mm. The result is consistent with the variation of measured temperature. This is also attributed to the slower heat diffusion in the metal with bigger volume compared with the small volume.

Fig. 5  Calculated interfacial heat flux versus time

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Using the temperature at the interface and the surface heat flux output from the inverse heat conduction code the IHTC for different casting diameter has been evaluated for the experimental casting configuration. The results together with the variation in casting temperature with time are presented in Fig. 6. As can be seen the behavior is similar to that observed for the heat flux—i.e. a rapid drop from a initial peak value of approximately 800–30 W m−2 K−1 over the solidification temperature range, followed by a slow change to a constant value of 35 W m−2 K−1 over the balance of the time plotted. Moreover, the IHTC for the casting with the diameter of 70 mm also lags behind that of 50 mm. The IHTC results clearly show two regimes in Fig. 6a: (1) during solidification and (2) shortly after the end of solidification. This result points to the need to clearly understand the underlying mechanism influencing interfacial resistance during development of the solidification structure and again shortly after solidification is complete. The previous body of work suggests that at various times heat transfer can occur across the interface by one or more of the following mechanisms: contact conduction, radiation and convection within a gas phase if present. Factors influencing these processes include the surface roughness of the mold, thermal conductivity of mold and casting thickness [17, 18]. Results of this work would appear to indicate that the transition from contact conduction to some other form of heat transfer during solidification is particularly important. However, it is different from the heat flux variation that there are two troughs in which the IHTC is varying with time, as shown in Fig. 6a. First trough for casting of 70 mm in diameter in Fig. 6b occurs at approximately 530 s. Figure 6b shows that the casting temperatures decrease down to near solidus curve at this time. So, this trough can be attributed to the latent heat released during eutectic reaction of molten metal discussed above. Second trough occurs at approximately 1940 s by the time the casting temperatures decrease down to about 725 °C, as shown in Fig. 6b. Therefore, this is most likely associated with the latent heat released of the eutectoid reaction, which occurs in this alloy at ±727 °C a few degrees depending on the cooling rate. IHTC for casting of 50 mm in diameter in Fig. 6c has the same trend with casting of 70 mm in diameter. 4.4 Verification of the identified IHTC In order to confirm the validity of the proposed model in the present study and test the accuracy of the calculated IHTC, the temperature variation with time at the location T1 in the sand mold was predicted with the forward conduction model using the identified IHTC, shown in Fig. 6a.

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Fig. 6  The variation of identified IHTC and casting temperature with time a comparison of IHTCs, b IHTC with casting_70 mm and c IHTC with casting_50 mm

The results were then compared with the measured temperature data, which are shown in Fig. 7. As can be seen the agreement between the calculated and experimental temperatures is good. The relative error between the numerical calculation and measurements is lower than 4 °C. Therefore, the feasibility of using the proposed inverse method to calculate the IHTC can be verified.

Heat Mass Transfer

appear that understanding the degradation in contact conduction that occurs during solidification is critical to developing relationships for the IHTC that are more universally applicable. 3. The identified IHTC varies with time during casting solidification and the values have varied in the range of about 20–800 W m−2 K−1. Moreover, the casting diameter has a great influence on the change velocity of IHTC. The variation of IHTC for the casting with the diameter of 70 mm lags behind that of 50 mm in diameter. Acknowledgments  The research supports from Scientific Research Fund of Hunan Provincial Education Department (No. 13B145), Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province and Open Research Fund Program of the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body (No. 31415003) are gratefully acknowledged.

References

Fig. 7  Comparison of calculated and measured temperatures a T with casting_50 mm and b T with casting_70 mm

5 Conclusions In the present work, an inverse heat conduction model has been used to estimate the interfacial heat flux and IHTC between the casting and the sand mold during solidification of cylindrical casting. The major conclusions can be summarized as follows: 1. The accuracy of forward heat conduction model for cylindrical casting has been investigated by the comparison of numerical calculated temperature field and the ones obtained from commercial software. The results confirm that the proposed model can be applied to identify the heat flux at the interface for heat conduction problems of the cylindrical casting accurately and effectively. 2. The results clearly show two regimes in which the IHTC is varying with time: (1) during solidification; and (2) shortly after the end of solidification. It would

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