ESTIMATION AND VALIDATION OF INTERFACIAL HEAT TRANSFER COEFFICIENT DURING SOLIDIFICATION OF SPHERICAL SHAPED ALUMINUM ALLOY (AL 6061) CASTING USING INVERSE CONTROL VOLUME TECHNIQUE

Solidification of casting is a complex phenomenon which requires accurate input to simulate for real time applications. Interfacial heat transfer coefficient (IHTC) is an important input parameter for the simulation process. The IHTC is varying with respect to time during solidification and the exact value is to be given as input for the accurate simulation of the casting process. In this work an attempt is made to estimate the IHTC during solidification of spherical shaped aluminum alloy component with sand mould. The mould surface heat flux and mould surface temperatures are estimated by inverse control volume technique using the temperature measured at different locations in the mould. The IHTC is calculated using these values. The estimated value of mould surface temperature is validated with the available measured mould temperature at specified location using direct heat conduction problem.


INTRODUCTION
During the solidification of casting process, there is an air gap formation between the metal and the mould due to shrinkage of solidifying metal. This air gap acts as a barrier for the heat to flow from the cast metal to the mould. The heat transfer between the cast and the mould is one of the important parameter that influences the solidification time, cooling rate and quality of the casting. This heat transfer rate has been dependent on various factors like thickness of surface coatings, casting size, chill or mould material, applied pressure, alloy type and composition, liquid alloy surface tension, mould and chill preheat, alloy superheat and chill surface roughness etc. These factors make the solidification phenomenon more complex and difficult to analyze the heat transfer at the metalmould interface. The drop in temperature between the cast surface and the mould surface is represented by an important parameter called interfacial heat transfer coefficient (IHTC). Predicting the heat transfer at the metal -mould interface is one of the important boundary conditions during the solidification.
Most of the commercial software's available for the solidification process should provide reliable results only when the correct material properties and the initial and boundary values are given. The current commercially available software assumes constant value of IHTC (film coefficient). However the IHTC is not constant during the solidification process and it varies with time. Hence there is a need for estimation of accurate value of interfacial heat transfer coefficient for different geometries. This paper deals with the estimation IHTC using inverse control volume technique for spherical shaped Aluminum alloy cast part with surrounding sand mould.
During the solidification of casting process the surface heat flux and the mould surface temperature are varying with respect to time. These two are the primary factors acting as boundary conditions for the evaluation of interfacial heat transfer coefficient. The surface heat flux and the mould surface temperature are calculated with the help of the thermocouple conveniently placed some distance away from the mould surface. Since the boundary condition described here is the "cause" (surface heat flux) is calculated from the "effect" measured temperature histories at different locations on the mould surface during solidification of casting (Grysa, 2011;Ozisik et al., 2004). Therefore from the effect (measured temperature histories), the cause at the metal mould interface (surface heat flux or temperature) is calculated which makes the problem as an inverse heat conduction problem (IHCP). The IHCP is also known as Ill posed problem. The direct heat conduction problems or well posed problem is one in which the effect is calculated from the known boundary condition. Generally the analysis of solidification of casting process falls under inverse heat conduction problem (Ozisik et al., 2004).
Many authors tried the varieties of numerical solution for the IHCP due to the fast development in computer applications. There are 14 solution methodologies are available to solve the IHCP but the selection of a particular method is mainly based on easy programming. Some of the numerical methods for IHCP are Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM) and Control Volume Technique. The other methods available are genetic algorithm, Becks function specification method and lumped capacitance method etc. (Grysa, 2011).
The solution for an inverse heat conduction problem (IHCP) of the solidification of casting at the metal -mould interface is based on the minimization of the objective function containing both estimated and measured temperatures (Beck et al., 1977). Sahin (Sahin et al., 2006) has reported, the estimation of IHTC for cylindrical geometry aluminum alloy casting with two different mould materials copper and HS13 steel using FDM. The upward directional solidification is mainly affected by the contact area and roughness of the material. The IHTC value was very high when pouring the material in liquid stage and during solidification it dropped down. The value of IHTC ranged from 19 -9.5 kW/m 2 K for copper and for HS13, it ranged from 6.5 -5 kW/m 2 K.
The IHTC is very much affected by the release of latent heat and evaporation of moisture content of green sand during solidification. These characteristics are analyzed by Chao (Chao et al., 2007) for the Aluminum and Tin-lead alloys using lump capacitance method and they reported that the peak value of IHTC was due to the latent heat release of molten material.
The increase in initial temperature of a coated material increases the IHTC and increasing the die coating thickness resulted in reduction of IHTC by Hallam (Hallam, 2004). A decreasing trend of IHTC was observed when the temperature of the contacting metal with the mould decreased was reported by Bezhenov (Bezhenov et al., 2017). Jose (Jose et al., 2006) evaluated transient heat transfer coefficient using control volume technique for Al-Si alloys by comparing the experimental data with the theoretical temperature profile of a numerical model.
Ranjbar (Ranjbar et al., 2009) estimated the IHTC by optimizing the solidification experiment and using inverse heat conduction problem. They performed experiment on Sn-10%Pb alloy in a metallic mould and used a Conjugate Gradient Method (CGM). Zhang et al., 2013, estimated the interfacial heat flux for cylindrical shaped casting and verified the accuracy of the solution by comparing the temperature data obtained from commercial ProCast software. Zhang (Zhang et al., 2013) developed a model to determine IHTC between casting and metal chill by measuring temperatures in the chill, cast and using the inverse heat conduction method. They also verified the experimental temperatures with the numerically calculated temperatures using FEA and found good agreements.
Zhang (Zhang et al., 2017) conducted a detailed study for a heat transfer phenomena on the solidification of 5 step squeeze casting of wrought aluminum alloy with a different hydraulic pressure. The authors observed that the increase in hydraulic pressure increases the IHTC values due to the reduction in the gap between the solidifying metal and die surface. Vaisleiou (Vaisleiou et al., 2017) developed a genetic algorithm (GA) approach to determine the value of heat transfer coefficient (HTC) between Al-Si cast and steel mould, considering two different geometries. The GA simulated the temperature by getting the different values of HTC as input and the simulation continued until there was a small difference between simulated and actual measured temperature being obtained.
Few research papers are available for comparing the results of two different inverse approaches. Rajaraman (Rajaraman and Velraj, 2008) estimated and compared the IHTC results for cylindrical geometry casting with sand mould using Beck's function specification approach and control volume approach. They reported that the control volume approach gives accurate results as it does not involve iterative processes. Rajaraman (Rajaraman et al., 2018) also estimated IHTC for rectangular geometry aluminum alloy casting using two different inverse methods.
The results obtained by both the inverse methods have identical trend in many places with minimum deviation in some regions. From the literature it is found that many researchers considered various geometries, materials and solution techniques for the estimation of IHTC. The value of IHTC is also differs based on the geometry, thickness, materials and types of coatings etc. Only very few researchers validated the IHTC obtained. In this paper an attempt is made to estimate the IHTC between the spherical shaped Aluminum alloy casting and spherical shaped sand mould during the solidification. The estimated mould surface temperature is given as input for the direct heat conduction problem to validate the results of IHCP.

PROBLEM DEFINITION
The IHTC has a major influence on the heat transfer between the metal and the mould. The schematic representation of IHTC during solidification of casting is shown in Fig. 1.

Fig. 1 Schematic representation of IHTC during solidification of casting
The heat transfer at the metal mould interface is calculated based on mould surface and cast surface temperatures as given by equation (1): Where q (t) is heat flux at the mould surface (W/m 2 ), h(t) is interfacial heat transfer coefficient (W/m 2°C ) , Tc(t) cast surface temperature (°C) and Tm(t) mould surface temperature (°C)

Fig. 2
Experimental set up for acquiring temperature data from cast and mould The experimental setup for the estimation of IHTC for spherical geometry cast with the surrounding sand mould is shown in Fig.2. A spherical cavity of 30 mm diameter was prepared in the green sand using split pattern, the runner and risers were located at appropriate locations. The entire outer surface of the mould was exposed to atmosphere to ensure one dimensional heat transfer. There were four K type thermocouples used to acquire the temperature history from the cast and the mould. One thermocouple was directly inserted at convenient location and nearby the cast surface to measure the cast temperature. This temperature was taken as cast surface temperature of the solidifying metal due to high thermal conductivity of Aluminum alloy. The remaining three thermocouples were located in the mould at distances of 20, 30 and 40 mm from the mould surface to measure the temperature of the mould with respect to time. All these thermocouples were connected to a 16 channel NI 9213 data acquisition system interfaced with a computer and LABVIEW software to measure the temperatures on the cast and the mould as shown in Fig.2.

MELTING AND POURING
The Al 6061 aluminum alloy is melted in a 15 kW, 10 kg electric furnace with high degree of super heat. The high degree of superheat was maintained to avoid solidification of cast metal during pouring and to maintain good fluidity. The melted alloy was poured into the spherical cavity mould, the temperatures were recorded by data acquisition system (DAQ) for every 3 sec and stored in computer till solidification process was complete. This data acquired process continued till the temperature difference between the mould and atmosphere was significantly less. The properties of cast metal and sand mould are listed in table1. The estimation of interfacial heat transfer coefficient during solidification of casting process is highly transient in nature. Solving such problem is highly complex in nature if the properties are considered as dependent on temperature. More over the variation of these properties with respect to temperature is not much significant and hence they are treated as constant for the present analysis.

Mathematical Model for Inverse Heat Conduction Problem
The governing equations for the 1-D spherical coordinate system is given by Where, = is the thermal diffusivity (m 2 /s), k is the thermal conductivity (W/m°C), ρ is the density (kg/m 3 ) and c is the specific heat (J/kg°C) of the mould material. The above equation can be solved with the help of two boundary conditions and one initial condition. The boundary and initial conditions for the above governing equation are as follows:

Initial condition and Boundary Conditions
The mould is considered as isotherm at time t = 0 i.e T(r,t) = Tinitial (known condition) The boundary conditions are The temperature at any time t is known at location E (r = r3+∆r/2) = T3 (Measured temperature) Temperature at any time t is known at location G (r = r5+ ∆r/2) = T5 (Measured temperature -used for the validation of results by FDM).
The unknown surface heat flux can be calculated using inverse control volume technique by applying energy balance at different control volume as described below.
The heat flux may be calculated with the help of the measured temperature at the mould in T3 and T4 locations.
Since the value of ∆ is very small the heat conduction equation for spherical geometry can be approximated using Fourier law of heat conduction. The heat flow in the mould is calculated using the following expression: Where km is the thermal conductivity of the mould material in W/m°C. the heat flux is calculated by dividing the heat flow with the area perpendicular to the heat flow direction. The calculated heat flux at various time intervals are used for the calculation of IHTC as given in equation (4) The interfacial heat transfer coefficient is calculated by rearranging equation (1) is given below: Where, q is the surface heat flux in W/m 2 , Tc and Tm are the temperatures of the cast and the mould surfaces in °C respectively.

Fig. 3 Control volume grid for Inverse heat conduction problem
The control volume method is based on the energy balance between the control volumes drawn over the individual nodal points. Here the mould is divided into the direct region and inverse region. In Fig. (3) the region right side of thermocouple location T3 (point E) is called direct region since temperatures are known at discrete time intervals. The region left side of portion E is called inverse region, since the mould surface temperature and heat flux are not known in these regions. The mould surface temperature and heat flux are expressed in terms of known measured temperature T3 by applying energy balance over different control volume with the help of inverse method.
In this approach an energy balance is first applied to control volume over nodal point E, as the temperature of nodal point E is measured experimentally at discrete time interval. The heat flow in the mould at point E, QE is calculated using the following expression: (5) On rearranging equation (5), we obtain equation (6).
The heat flow at point E was determined using the measured temperatures in the direct region at location T3 and T4. Therefore at point E, the temperature (T3) and heat flux were known for various time steps but at location S the boundary conditions were unknown. These unknown boundary conditions at S were determined from the known values at E, making use of the Fourier law of heat conduction at mould surface After rearranging the above equation (7), we obtained equation (8): Applying energy balance over control volume 3 control volume On substitution of T4 in equation (9), we obtain equation (10) Equation (11) is obtained by applying energy balance in control volume 2 On substitution of T2 and 2 in eqn (11) and on rearranging, we obtain equation (12) Where, 1 = 3 3 − 2 3 3 2 2 , 2 = 4 3 − 3 3 3 2 , 3 = 2 2 + 3 2 2 2 By applying energy balance over control volume 1 equation (13) is obtained On substitution of T0, T1, T2 and 1 in equation (13) Where, = � 2 The higher order derivatives in the above equations are evaluated with the help of Stirling's interpolation formulae (Veerarajan et al., 2004). Equation (13) gives the surface heat flow (Qs) and the surface heat flux (qs) is calculated by dividing the heat flow (Qs) by area (A) of the mould surface at S. The IHTC is calculated from mould surface heat flux (qs), mould surface temperature (T1) and cast surface temperature (Tc) using equation (4).

Finite Difference formulation for validation
The calculated mould surface temperature (T1) at particular time interval was given as input for the direct heat conduction problem. The temperature distribution at different nodes from point S to G is calculated by FDM technique for the given surface temperature (T1) as shown in figure (4). These calculated temperature at location G in the direct region were used to validate the results of control volume technique.
The finite difference approximation may be derived using the Taylor series expansion for the 1 D transient heat conduction problem for the derivatives 2 2 , in the equation (2). The dependent variable values in a transient heat conduction problem for the future time steps were evaluated from the known set of initial conditions using explicit method. The values of temperature available at the current time step't' was denoted by the superscript 'i' as . The next calculated temperature values for the next time step't+∆t' was represented by the superscript 'i+1' as +1 is given in Fig.4.   Fig. 4 Finite difference grids to find the temperature distribution in the mould for DHCP The finite difference grid points for the estimation of the next time step temperature distribution in the mould is given in equation (15): The calculated temperature using the equation (15) was compared with the temperature measured at the same node (G) as described in the flow chart for validation of results in Fig.5. Fig.6 shows the variation of temperature with respect to time recorded by a thermocouples located in the cast (Tc) and in the mould at 20 mm, 30 mm, and 40 mm respectively. The temperature readings were recorded until the difference between the cast temperature and the mould temperature is considerably less. The cast temperature measured by thermocouple Tc suddenly increases after pouring and reaches a maximum value of 655ºC after 22 sec and then decreases gradually as shown. From the graph it is observed that the maximum temperature recorded by a thermocouple located at 10 mm from mould surface is 544.5ºC after 65 sec from pouring and reduced as time increases. The maximum temperature attained by the mould at this location was mainly due to the high latent heat release, high thermal conductivity of aluminum cast and also the position of thermocouple.

Calculated mould surface temperature, heat flux and IHTC
The mould surface temperature calculated using inverse control volume technique with respect to time is shown in Fig. 7. From the graph it is observed that the mould surface temperature increased to a maximum value of 508ºC in a short duration of 34 sec and then suddenly drops to 200ºC at 106 sec. This sudden drop in temperature is due to large difference in temperature between the solidifying cast and sand mould. Thereafter the decrease in mould surface temperature is gradual and reaches approximately 100ºC at 180 sec. This is due to high thermal inertia of the moulding sand. Beyond this, the mould surface temperature is almost constant. The calculated heat flux at the mould surface is shown in Fig.8. The trend of calculated heat flux is same as that of mould surface temperature. The maximum value is found to be 55,000 W/m 2 at 35 sec and decreases gradually and reaches a value of 3000 W/m 2 in 400 sec.  The variations of IHTC with respect to time and cast surface temperature are shown in Fig.9 and Fig.10. From the figures it is clear that the IHTC reaches peak value of 8870 W/m 2 K immediately after pouring. This is due to more contact of the metal with the mould surface at the initial stage. The value of IHTC decreases gradually reaches to value of 1335 W/m 2 K at 250 sec. The gradual decrease in IHTC is due to the increase in gap between the solidifying metal and mould due to shrinkage as well the presence of gases in the gap. Thereafter the IHTC gradually decreases and almost maintain a constant trend after 375 sec. This shows that the solidification process is almost completed and there is no change in the properties interfacial of the region.

Validation
The main objective of the present work is to validate the estimated value of mould temperature by comparing with the experimentally measured temperature data. The estimated mould surface temperature was given as input for the direct heat conduction problem and the temperature history was calculated at various location of the mould. The nodal points were chosen in such a way that any one measured temperatures coincide with the nodal point G (Fig.3). Finite difference method is used to estimate the temperature history.
The estimated temperature at known location (T5) was compared with the measured temperature data available. Fig.11 shows the comparison of the measured and calculated temperature at location G (T5). It was confirmed from the figure that the calculated temperature were in good agreement with the measured temperature data. This showed that the accuracy of control volume technique is good as compared to other techniques for the estimation of IHTC by Rajaraman (Rajaraman et al., 2008(Rajaraman et al., , 2018.

CONCLUSION
The following conclusion were arrived in the present work • The IHTC is calculated between aluminum alloy and sand mould using inverse control volume technique • The IHTC calculated has a peak value 8870 W/m 2 K at 80 sec and then gradually decreases to 1335 W/m 2 K at 250 sec • The calculated mould temperature is validated by using direct heat conduction problem and obtained temperature history.

•
The estimated and the measured temperature are in good agreement and this testifies to the better accuracy of the selected control volume approach.