Heat Mass Transfer DOI 10.1007/s00231-014-1304-6

ORIGINAL

Development of an inverse heat conduction model and its application to determination of heat transfer coefficient during casting solidification Liqiang Zhang • Carl Reilly • Luoxing Li Steve Cockcroft • Lu Yao

•

Received: 29 January 2013 / Accepted: 28 January 2014 Ó Springer-Verlag Berlin Heidelberg 2014

Abstract The interfacial heat transfer coefficient (IHTC) is required for the accurate simulation of heat transfer in castings especially for near net-shape processes. The large number of factors influencing heat transfer renders quantification by theoretical means a challenge. Likewise experimental methods applied directly to temperature data collected from castings are also a challenge to interpret because of the transient nature of many casting processes. Inverse methods offer a solution and have been applied successfully to predict the IHTC in many cases. However, most inverse approaches thus far focus on use of in-mold temperature data, which may be a challenge to obtain in cases where the molds are water-cooled. Methods based on temperature data from the casting have the potential to be used however; the latent heat released during the solidification of the molten metal complicates the associated IHTC calculations. Furthermore, there are limits on the maximum distance the thermocouples can be placed from the interface under analysis. An inverse conduction based method have been developed, verified and applied successfully to L. Zhang  L. Li (&) State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, Hunan, People’s Republic of China e-mail: [email protected] L. Zhang  L. Li College of Materials Science Engineering, Hunan University, Changsha 410082, Hunan, People’s Republic of China L. Zhang  C. Reilly  S. Cockcroft (&)  L. Yao Department of Materials Engineering, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada e-mail: [email protected] L. Yao e-mail: [email protected]

temperature data collected from within an aluminum casting in proximity to the mold. A modified specific heat method was used to account for latent heat evolution in which the rate of change of fraction solid with temperature was held constant. An analysis conducted with the inverse model suggests that the thermocouples must be placed no more than 2 mm from the interface. The IHTC values calculated for an aluminum alloy casting were shown to vary from 1,200 to 6,200 Wm-2 K-1. Additionally, the characteristics of the time-varying IHTC have also been discussed.

1 Introduction During the last two decades the use of solidification simulation software for both the design of casting processes and their optimization from a quality standpoint has greatly increased with improvements in the computational technology [1, 2]. One of the boundary conditions in the simulation of shape castings, the interfacial heat transfer coefficient (IHTC), is particularly critical from the standpoint of accurate modeling of the solidification process. Unfortunately, the boundary condition is complex as its behavior is influenced by a number of factors including, the relative amounts of thermal contraction of the casting and thermal expansion of the mold. Consequently, the IHTC cannot be easily quantified experimentally nor does it lend itself to a theoretical approach [3–6]. In recent years, inverse heat conduction (IHC) algorithms, which utilize time–temperature data measured during casting, have been applied successfully to determine the IHTC [7–9]. This methodology was initially proposed by Beck for aerospace applications [10], but has subsequently been refined and applied in variety of ways in materials engineering including for thermo-physical property characterization

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[11], heat flux characterization during impingement water cooling [12] and for characterization of convective heat transfer coefficients [13]. Focusing on those studies directly related to IHTC characterization, the published literature indicates there are a number of factors that can influence the IHTC and that the IHC method is the dominant method to determine the heat transfer coefficient at a metal–mold interface in shape castings. For example, Gafur et al. [14] estimated the heat flux and IHTC during solidification of commercially pure aluminum and analyzed the effect of superheat and chill thickness by using a non-linear inverse analysis method. Their analysis revealed a peak heat flux at the end of mold filling. The effect of superheat on the IHTC was shown to be minimal after filling. It was found that the chill thickness had a significant effect on the heat flux after the occurrence of the peak heat flux. Arunkumar et al. [15] employed a serial-IHC 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. Hamasaiid et al. [16] studied the effect of mold coating materials and thickness on casting/mold interfacial heat transfer using an inverse method. The experimental data revealed that while the alloy was liquid the coating materials had only a weak influence on heat flow and that the IHTC decreased as the coating thickness increased. After solidification took place and the IHTC decreased, the effect of the coating material became negligible. Coates and Argyropoulos [17] investigated the effects of surface roughness and metal temperature on the heat transfer coefficient at the metal/mold interface. The experimental results revealed that as the mold surface became rougher the heat transfer decreased. Also, for a given surface roughness, the IHTC was seen to increase as the liquid metal temperature increased. Broucaret et al. [18] analyzed the influence of the initial temperature of the mold and the mold surface coating on the IHTC. The analysis procedure was based upon an inverse methodology, which utilized the Laplace Transform of heat conduction equation. The findings showed that the IHTC increased with the thermal conductivity of the coating and with lower initial mold temperatures. Prabhu and Ravishankar [19] studied the effect of sodium modification treatment of the melt on casting/mold interfacial heat transfer during directional solidification of an aluminum alloy casting with metallic chills using thermal analysis and inverse modeling techniques. The result from this research revealed that the melt modification treatment has a significant effect on the casting/chill IHTC.

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The modified melt had a higher IHTC than the unmodified alloy. Rajaraman and Velraj [20] calculated the IHTC using two inverse calculation methods based on measured temperatures in a mold. The two inverse methods: the control volume [21] and Beck’s [10] IHC method were then compared. The calculated IHTC using the two methods were found to be close. The majority of previous research has calculated the heat flux and IHTC based on the measured temperature in the mold. Little research has been published which is based upon the measured temperature within the casting [14–18]. One study appears to be have been completed by Prabhu and Ravishankar [19], however details of the inverse conduction methodology employed are unclear. In another by Martorano and Capocchi [22] the whole domain was used for the inverse solution (casting and mold). Alternatively, there have been studies that do not use and inverse methodology. For example, Sun and Chao [23] employed a lump capacitance method to calculate the IHTC for application in the green sand mold casting process. In this work, the interface heat flux and subsequently the IHTC have been estimated using the IHC method based on temperatures measured in an aluminum alloy casting directionally solidified against a water-cooled copper chill. This method has the potential to be useful in situations where it is difficult to measure temperatures within a mold, such as may arise when the mold is water-cooled. The methodology is first developed, verified and then applied to experimental results.

2 Experimental procedure Tapered cylinder castings, directionally solidified against a copper chill, were used to obtain the temperature data required for input to the inverse model. A schematic representation of the experimental setup including the position of the thermocouples is shown in Fig. 1. The mold sides were insulated by ceramic wool and the copper chill was watercooled. The flow rate of water was 0.4 L s-1. Three K-type thermocouples with the diameter of 0.1 mm protected by a 2.3 mm diameter ceramic sheath were positioned within the mold at 2, 5 and 10 mm from the chill, as shown in Fig. 1, to measure the evolution of temperature with time. Due to physical constraints the position of the 2 and 5 mm thermocouples were place off center by 0.5 and 1.0 mm, respectively. The data from the thermocouples was recorded at a rate of 50 Hz. The thermocouples tips were twisted and spot-welded. The thermocouple data was recorded using a Measurement Computing USB-2416 data acquisition system and PC. The combined error in the temperature measurement (thermocouple and data acquisition system) is estimated to be

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

Table 1 Thermo-physical parameters of A356 alloy [25]

Material

Density kg m-3

Specific heat J (kg K)-1

Latent heat J kg-1

Thermal conductivity W (m K)-1

Solidus temperature K

Liquidus temperature K

Casting (A356)

2,685

967

249,000

166.12

829

889

±1.5 °C [24]. The experiments were performed with an A356 alloy (Al–7Si–0.4Mg). The alloy was melted in a resistance furnace using a graphite crucible and was hand poured at 750 °C into the mold. It took approximately 9 s to pour. The thermo-physical parameters for the alloy are summarized in Table 1. A number of castings suffered from leakage at the casing/chill interface. The results from these castings were discarded. The ‘‘good’’ castings were found to have good reproducibility in terms of the temperature behavior. One of these castings was used for the analysis.

can be reasonably approximated as a one-dimensional heat transfer problem with the predominant heat-flow direction toward the chill.

3 Inverse model

where q is the density of the melt (kg m-3), Cp is the specific heat (J kg-1 K-1), T is the temperature (K), t is the time (s), k is the thermal conductivity (W m-1 K-1) and Q is a volumetric source term associated with the latent heat of solidification (W m-3). This expression assumes 1-D heat conduction, which is a reasonable approximation at the location of the thermocouples used in the experiment as they located in close proximity to the chill (maximum

In the IHC method adopted for this work, a forward heat conduction model of the casting needs to be developed and then coupled to an inverse algorithm. Referring to Fig. 1, due to adequate insulation of the mold wall around the circumference of the casting and limited heat transfer from the top, it is assumed that the heat flow through the casting

3.1 The forward heat conduction model The applicable form of the direct heat conduction equation assuming heat transport by diffusion for the casting can be written as:  2  oT o T qCp ¼k þQ ð1Þ ot ox2

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Heat Mass Transfer Fig. 2 a Schematic illustration of calculated variation in heat flux q(t) with time [26]; b constant heat flux assumption used over time period RDh, adopted in the inverse algorithm

distance 10 mm). The radial heat losses to the mold area estimated to be very low owing to comparatively low conductivity of the fireclay and ceramic wool. At the top of the casting there is heat loss via natural convection to the environment, which is estimated to be two-orders of magnitude less than the heat flux to the chill. The finite element method (FEM) was used to solve for the transient temperature field within the casting subject to specified heat-flux condition at the boundary adjacent to the mold at the bottom of the domain. The boundary at the top of the chill was assumed to be adiabatic. A linear temperature element was used to mesh the geometrical model and the element size was 1 mm. 3.2 The inverse heat conduction (IHC) methodology

augmented to m ? 1 and the time in the inverse algorithm is then updated to tm ? Dh. The process is repeated until the desired analysis time tmax is achieved. The careful selection of R can help stabilize the estimation procedure. In this study, the value of R of 20 was found to give good results. An algorithm flow chart for the IHC method is shown in Fig. 3. In this work a large value of R was used to compensate the delayed heat propagation due to the released latent heat during phase transformation. Normally a much small R is used when no phase transfer occurs [10]. The test for goodness-of-fit described above uses an objective function, F(qim(t)), defined in the following expression to assess the error in qim(t): 3  2   X i TmþRDh;k  YmþRDh;k ð2Þ F qim ðtÞ ¼ k¼1

The IHC method seeks to estimate the variation in heat flux at the bottom of the casting with time, qm(t), as shown schematically in Fig. 2a, using as input the discrete temperature data obtained from the thermocouples solidified into the casting. In qm(t), m represents the time-step counter associated with integration in time (the maximum m is therefore linked to the temporal resolution of the solution). Because the temperature response of a thermocouple located within the casting will lag behind a change in heat flux imposed at the casting boundary (due to the finite rate of heat diffusion) determination of an accurate estimate of the heat flux is a challenge. Starting at time t1, the approach adopted is to solve the heat conduction equation for R time steps into the future while holding the estimated heat flux constant, as shown in Fig. 2b. At the end of RDh seconds, where Dh is the time-step size, an assessment of the error in ith iteration qim(t) is made. The error is determined by comparing the predicted temperature, at the location of the thermocouple(s), with an objective function that utilizes the measured temperature (described below). Depending on the level of agreement between the two, the estimated heat flux at the ith iteration is either updated (refined) and the process rerun for the ith ? 1 iteration or the estimated heat flux, qim(t), becomes the predicted heat flux a time t1 or q1(t). In the event that the second condition holds, the time-step counter is

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Tim?RDh,k

is the calculated temperature at time where m ? RDh, at the location of thermocouple k, Ym?RDh,k is the measured temperature at time m ? RDh at the location of thermocouple k. The error tolerance in the inverse model is 0.5 °C. Once the difference between the measured and predicted temperatures is less than or equal to the tolerance criterion, an estimate of qi is obtained. In the present work there are three thermocouples. In a multi-dimensional analysis involving a spatial distribution of surface heat fluxes, the objective function would be summed, a second time, over the number of individual discrete surface heat fluxes. A solution is sought where the function F is below a threshold value through a simple iterative process—see Fig. 3. In the iteration scheme, the heat flux is increased or decreased by an increment Dqi, which is calculated as follows: Dqi ¼ S

1

i

 DT mþRDh

ð3Þ

where S is a the sensitivity coefficient vector and is defined as the temperature change or response at the measurement point with respect to a change in surface heat flux (in cases where there is more than one surface heat flux this would be a matrix) and DT is the vector of

Heat Mass Transfer Fig. 3 Flow chart of inverse heat conduction algorithm

i temperature errors or Tm?RDh,k - Ym?RDh,k, k = 1, 3. The sensitivity coefficients can be calculated by perturbing the surface heat flux in the model and quantify the effect on temperature. The calculation for each sensitivity coefficient in the matrix is performed using the following expression:

S¼

oTk Tk ðq þ eqÞ  Tk ðqÞ ¼ eq oq

ð4Þ

where k = 1, 3 and e is set equal to 0.001. The updated i ? 1 estimate of the heat flux at time step m, is calculated by adding Dqi, to the ith estimate as follows: i qiþ1 m ðtÞ ¼ qm ðtÞ þ lDqi

ð5Þ

A dampening factor, l, has been used to avoid overcorrection and is set to a value close to but less than 1. 3.3 Equivalent specific heat method Among the thermophysical properties 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. With this modification, the expression for the effective specific heat within the phase change interval becomes:

CP0 ¼ Cp  L

ofs oT

ð6Þ

where L is the latent heat of solidification (J kg-1), fs is the solid fraction and Cp is the nominal specific heat of A356 (J kg-1 K-1). Based on empirical data describing the variation in fs with temperature [27, 28], the resulting relationship for the effective specific heat, C0P , is shown in Fig. 4. As can be seen, there are two peaks in the effective specific heat term, one associated with the initial stages of growth of the primary phase, between approximately 615 and 605 °C and a second larger one between approximately 575 and 565 °C, associated with growth of the eutectic phase. In the context of normal forward conduction problems, these peaks would not create numerical instabilities, as they would translate to decreases in the thermal diffusivity of the material, which would tend to stabilize the solution—see Eq. (7), where a is the thermal conductivity (m2 s-1). However, in the inverse problem the large decreases in thermal diffusivity would both increases the thermal lag time and reduce the magnitude of the sensitivity coefficients utilized in Eq. (5). As a result there are potentially problems in achieving convergence [9, 23, 29]. k ð7Þ qCp To help overcome this difficulty, an equivalent specific heat, C00P , also shown in Fig. 4, was defined that yields the

a¼

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Heat Mass Transfer Fig. 4 Effective and equivalent specific heat variation with temperature

Fig. 5 Calculated temperatures comparison with different Cp at three locations a Thermocouple T1, b Thermocouple T2, c Thermocouple T3

123

Heat Mass Transfer 1800 1600

q(kW/m 2)

1400 1200 1000 800 600 400 200 0 0

20

40

60

80

100

120

140

Time (s) Fig. 6 Idealized interface heat flux variation with time

same integrated area (the same amount of latent heat is released), but assumes that the derivative of fs w.r.t T is constant over the phase change interval.

4 Results and discussions 4.1 Calculated cooling curve with different Cp To begin, the effect of holding the rate of change of fraction solid with temperature constant over the phase change interval is assessed. Figure 5a–c show the calculated cooling curves at the locations of the thermocouples, 3, 5 and 10 mm from the chill interface, respectively, using both the effective specific heat (varying rate of change of fraction solid with temperature) and the equivalent specific heat (constant rate of change of fraction solid with temperature). The same initial and boundary conditions were used in both cases. As can be seen from Fig. 5 the results for the two cases are similar. There is however a small difference within the solidification temperature range of the order or 3 °C or 0.5 %. The error is largest at the location of T3, which is 10 mm from the chill—see Fig. 5c. The results indicate that adopting the equivalent specific heat should have only a small affect on the forward conduction calculation within the overall inverse code. 4.2 Validation of the inverse method using equivalent Cp Prior to application to the practical problem, the inverse code was first validated using as input hypothetical (fictitious) thermocouple data generated from the forward conduction code using a known variation in surface heat flux with time. Figure 6 shows the known heat flux variation

used. The curve represents an idealized version of what might be expected for the variation in heat flux with time at the interface with the mold—i.e. a rapid initial increase in heat flux associated with mold filling, a rapid decrease associated with the formation of a gap at the interface due to shrinkage of the casting and finally a period of little change with time. Therefore, the assumed interfacial heat flux has a shape of quadratic function followed by a line with constant heat flux. Figure 7a, b show comparisons between the applied heat flux and the heat flux calculated using the inverse code, using as input the hypothetical thermocouple data at 2 and 5 mm from the interface, respectively. The results shown were obtained using the equivalent specific heat term applied over the solidification temperature range. As can be seen, the inverse result based on the 2 mm data is in good agreement with the applied heat flux whereas the result based on the 5 mm data clearly shows some instability in the solution, particularly within the first 5 s. Analysis with the IHC code revealed the problem to be related to the thermal diffusivity. Use of the nominal specific heat in place of the equivalent specific heat, in both the forward code used to generate the fictitious thermocouple data and the inverse code (i.e. ignoring the latent heat of solidification), eliminated the problem and use of the effective specific heat in place of the equivalent specific heat exacerbated the problem. A sensitivity analysis conducted with the model placed an upper bound of 2 mm on the distance of the thermocouples from the chill to avoid problems with stability when the latent heat of solidification is included via an equivalent specific heat term. This issue is not encountered to the same extent (greater thermocouple/interface distances can be tolerated) when using data collected from in-mold thermocouples for an inverse analysis as the mold material does not in general go through a phase change. 4.3 Experimental cooling curve Figure 8 shows the cooling curves measured at the three locations 2, 5, and 10 mm from the interface with the chill. It can be seen that the general trend in the data is similar for all three positions in that there is an initial rapid drop in temperature in the first 10 s, which is followed by a much slower rate of decline. Moreover, the cooling rate at location T1 is clearly the highest and T3 the lowest consistent with their relative positions in relation to the chill. Based on the results from Sect. 4.2 above, only the data from thermocouple T1 was input to the IHC—i.e. the sensitivity coefficients for the other two thermocouple positions are set to zero and they are not employed in the objective function defined in Eq. (2). The measured temperature curves at T2 and T3 were used instead to validate the calculated IHTC, which appears in Sect. 4.5.

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(b)

1800

1600

1400

1400

1200

1200

1000 800

400

Calculated q

1000 800 600

Assumed q

600

Assumed q

400

Calculated q

200

200 0

0 0

20

40

60

80

Time (s)

Fig. 8 Measured temperatures vs time curves at three locations in a water-cooled casting following smoothing

Before performing the IHC analysis, the experimental data was smoothed using an adjacent-averaging method [30]. This method uses a specified number n to control the degree of smoothing. In the present work a value of n = 5 was found to be effective in both smoothing the data to avoid convergence problems in the inverse code while still maintaining the proper trends in the original data. The temperature curves shown in Fig. 8 were generated from the smoothed data. 4.4 Interfacial heat flux and heat transfer coefficient Figure 9a shows the variations in the calculated interfacial heat flux and interfacial temperature with time. The interfacial temperature represents the predicted temperature of the cast metal surface at the interface and was output from the inverse model. Turning first to the interfacial heat flux behavior, the inverse model predicts a flux of approximately 3,000 kW m2 that develops very rapidly within approximately 1 s. Shortly thereafter, the heat flux quickly drops throughout the solidification

123

1800

1600

q(kW/m 2 )

(a)

q(kW/m 2 )

Fig. 7 Comparison of calculated and input idealized heat flux—a based on the temperature data at 2 mm from the interface and b based on the temperature data at 5 mm from the interface

100

120

140

0

20

40

60

80

100

120

140

Time (s)

temperature range and by the time the solidus temperature is reached, at approximately 5 s, is down to a value of around 1,400 kW m2. The heat flux then more gradually decreases to approximately 750 kW m2 at around 10 s and then very slowly declines to a value of approximately 500 kW m2 at 80 s, which is maximum time shown in the plot. There is evidence of a small secondary peak at approximately 4 s. This is most likely associated with the formation of the primary eutectic, which occurs in this alloy at 575 °C ±a few degrees depending on the cooling rate [28]. The adoption of the equivalent specific heat in the inverse code, will result in some small error in the calculation previously described, particularly in situations where there is a large variation in amount of heat released with temperature such as occurs during the transition from primary phase growth to primary eutectic growth in A356. Thus, the small secondary peak is likely an artifact of the inverse methodology and not real. Overall however, the variation in the heat flux predicted with time is very similar to those previously reported in the literature based on inmold temperature acquisition [15, 19, 31]. For example, Sahin et al. [31] reported the heat flux rising steeply to the maximum value after pouring and then decreasing to a very low value after the solidification of casting is completed. While the estimation of the interface heat flux between the mold and the casting is of value its use for applications other than those for which it has been evaluated is limited. A more widely applicable way of evaluating heat transfer at the interface is via a Cauchy-type boundary condition, shown in Eq. (8). The advantage of this approach is that it decouples the interface resistance (expressed as the reciprocal of h) and the driving force for heat transfer (Tcasting - Tchill). In this manner, changes in the driving force, which will in general occur during cooling of the casting and will be different from casting-to-casting can be accommodated. The Cauchy-type still suffers from the need to specify the heat transfer coefficient, which is

Heat Mass Transfer

  h ¼ q= Tcasting  Tchill

Fig. 9 Variation of calculated quantities in time: a interface heat flux, b temperature at the chill surface and c IHTC and temperature at the casting surface

strongly dependent on the behavior of gap at the interface and therefore will vary from one casting configuration to the next. Nonetheless it represents an improvement in range of applicability over the Neumann or specified heat flux type. An additional advantage of the Cauchy-type is that because the IHTC can be directly related to the resistance of the interface to heat transport, it has the potential to be evaluated as a function of physical phenomena such as interface pressure, interface displacement (gap), surface morphology and coatings.

ð8Þ

Using the temperature at the interface and the surface heat flux output from the IHC code the IHTC has been evaluated for the experimental casting configuration. This calculation required augmentation of the conduction model described in Sect. 3.1 to include the chill— refer to Fig. 1. The chill was modeled from the interface to the location of the water-cooling channels as a 1-D domain. The heat flux variation predicted with the inverse model was applied at the top boundary (the interface with the casting) and a constant temperature or Dirichlet boundary was applied at the bottom of 10 °C. This approach allowed estimation of the variation in Tchill in Eq. (8) with time, which in turn allowed estimation of the variation in the heat transfer coefficient with time. The calculated Tchill together with the casting interface temperature Tcasting are shown in Fig. 9b. The variation in the heat transfer coefficient with time together with Tcasting, are presented in Fig. 9c. As can be seen the behavior for the variation in the heat transfer coefficient is similar to that observed for the heat flux—i.e. an initial rapid increase to a peak, a large drop from approximately 6,250–2,850 W m-2 K-1 over the solidification temperature range, followed by a more gradual drop to 1,500 W m-2 K-1 over 5 s and then a slow decline to 1,250 W m-2 K-1 over the balance of the time plotted. The maximum values of IHTC are higher than those reported for an aluminum alloy solidified against a steel chill reported in the literature, but the overall trend is the same [15, 30]. It is interesting to note that the initial peak in heat transfer is better delineated in the IHTC plot than in the surface heat flux plot as a result of de-convolution with the driving force term. The results clearly show two regimes in which the IHTC is varying strongly with time: (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 wettability of the liquid metal on the mold surface, surface roughness of the mold, thermal conductivity of mold and hydrostatic pressure [14–17, 22, 32]. 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.

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Heat Mass Transfer

Fig. 10 Comparison of simulated and measured temperatures a Thermocouple T2, b Thermocouple T3

4.5 Verification of the identified heat transfer coefficients As discussed above, the measured temperature data from the thermocouple locations T2 and T3 were used to test the accuracy of the calculated IHTC. For verification of the calculated IHTC, the temperature variation with time at the locations of T2 and T3 were predicted with the forward conduction model using a Cauchy-type boundary condition with the IHC-calculated IHTC, shown in Fig. 9c. The results were then compared with the measure variation in temperature with time measured with TC2 and TC3, which are shown in Fig. 10a, c, respectively. As can be seen the agreement at both locations is good. The maximum temperature differences between the numerical calculation and measurements are 6 °C (*0.9 %) at T2 and 4 °C (*0.7 %) at TC3. The error is well within the range of that previously deemed to be accepted [30]. During the casting experiment, the thermocouples were not exactly in the center due to limitations in positioning thermocouples within the mold. Moreover, the error is only slightly higher in magnitude to that observed by adoption of the equivalent heat capacitance, shown in Fig. 5.

sensors. This necessitated a large value of R (the number of future time steps) in the inverse algorithm than would normally be required, or is recommended in the literature. The major conclusions from this work can be summarized as follows: 1.

2.

3.

4.

An inverse conduction methodology has been developed, verified and successfully applied to quantify the casting–metal mold IHTC in an aluminum alloy casting solidified against a water-cooled copper chill based on the measured temperatures within the casting. It is critical for convergence that the temperature data obtained from within the casting is acquired within approximately 2 mm from the interface, owing to a strong thermal lag associated with the latent heat of solidification. In the current model an equivalent specific heat method was needed to account for latent heat and obtain solution convergence. The results clearly show two regimes in which the IHTC is varying strongly with time: (1) during solidification; and (2) shortly after the end of solidification. It would 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.

5 Conclusions By using temperature data measured in the casting, the IHC method presented in this work can be used in situations where it is difficult to measure temperatures within a mold. This represents a particularly challenging problem as the thermocouples are placed within a material undergoing a phase change (with the associated release of heat), which increases the thermal inertia of the material surrounding the

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Acknowledgments The research supports from the Hunan Science Fund for Distinguished Young Scholars No. 09JJ1007, The China Scholarship Council (CSC), International Cooperation and Exchanges Program No. 2008DFA50990 and the Science Fund of the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body No. 60870005 are gratefully acknowledged. Funding from the Natural Sciences and Engineering Research Council of Canada is also gratefully acknowledged. The thoughtful discussions on the subject with Dr. Jindong Zhu at the Department of Materials Engineering, University of British Columbia are also greatly appreciated.

Heat Mass Transfer

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