In casting process simulation, the heat transfer coefficients between materials describe how and to what extent heat will transfer across the boundaries between the two materials, the casting and the mold. Many variables can affect the heat transfer between two materials that are in contact. In the permanent mold process, the application of mold coating plays a large role in how heat moves between the casting and mold surface, and the coating itself can be influenced by factors such as the application method, coating thickness and coating type.
The process of obtaining appropriate heat transfer coefficient values has been challenging in the past due to the uncertainty of heat transfer between materials. However, advances in casting simulation modeling have made an inverse methodology possible. Foundries can match thermocouple curves taken during actual testing with virtual thermocouples placed in a virtual simulation model, thus determining precisely the heat transfer coefficient of different types of coatings and their influence on the casting process.
This article focuses on the steps involved in the inverse method, and it will detail the outcome of an inverse optimization study that was performed for the gravity permanent mold process.
To verify this inverse optimization method, a case study was performed showing how a wheel foundry used it to measure and evaluate heat transfer coefficients for the mold coating application.
Experimental Trials
Four experiments were performed using a heating unit (Fig. 1) to measure the temperatures in the mold and surrounding environment. In all four experiments, the same test cup was used. The test cup contains 11 thermocouple bores at different depths (Fig. 2).
For each experiment, a different condition was applied (Table 1). In the first experiment, the test cup was placed bare in the heating unit. For the second experiment, insulation was placed around the outer surface of the mold. The third experiment included a proprietary mold coating that was applied on the inner surface of the test mold cup while retaining the insulation. In the fourth experiment, the insulation was removed from the outer surface of the test cup while mold paint was applied to the inner surface of the test mold cup.
The experimental cycle contains three stages: delay prior to pouring, pouring, and cooling. The “delay” time represents a period after preheating and prior to pouring. During the delay stage the degassed metal is tapped from the bull ladle with a ceramic ladle. This process takes between 20-50 seconds. The pouring time is the time it takes to pour the tapped aluminum in the ceramic ladle into the cup mold inside the heating chamber. Finally, the observation time is the period when the system slowly cools down.
With the recorded thermocouple and process data from the original experiments, an inverse optimization was set up using a proprietary software. The exact CAD model of the test cup mold with the insulation (when applicable) was considered in the simulation with virtual thermocouples placed in the exact location as the real ones in the test (Table 2). In the simulation, the experimental process with appropriate datasets, A356 aluminum and 4140 steel, were simulated with the same initial temperature as recorded by the thermocouples in the test (Table 3). The mold preparation time, pouring time of five seconds and solidification and cooling times of 2,000 seconds were all entered into the software accordingly.
To represent the heat transfer coefficient behavior, a hyperbolic tangent function was selected (Figure 3). With this selection, only five variables needed to be addressed, resulting in a smaller number of total possible designs to consider. The shape also makes intuitive sense as it was anticipated a lower heat transfer coefficient would be needed at lower temperatures when the metal contracts and consequently loses contact with the mold wall. This loss of contact between the two materials will result in an air gap between the two materials that will transfer heat at a slower rate than when the two materials have better contact at higher temperatures.
Two of the variables for the hyperbolic tangent function that can be modified are the minimum and maximum ranges for the heat transfer coefficient values (Y0 and Y1). The three remaining variables control the temperature for each interval of the curve (X0, W0 and W1). W1 is the solidification interval in which the metal is transitioning from liquid to solid and consequently contracting and generating a gap between the melt and the mold wall. Finally, W0 is the cooling interval after the end of solidification. Table 4 shows the ranges and steps used for each variable considered in the initial inverse optimization tests.
Because the total design space, or total number of possible designs from the ranges and steps shown in Table 4 is 11,979, it was decided the genetic algorithms in the program would be used to schedule and assess a smaller subset of designs. To efficiently search this design space, the software user must define objectives that work well. This involves selecting a reasonable number of objectives as well as defining objectives that effectively complete the task at hand. In this case, the goal of the optimization is to match the measured and simulated cooling curves as closely as possible.
The more objectives pursued in an optimization, the more difficult to find a compromise between the objectives. So, only one of the cooling curves from the 11 thermocouples placed in the mold, was used for the initial inverse optimization run. This reduced the required number of objectives and resulted in:
Less required simulation time.
An evaluation of the quality of the mold data set.
An analysis of the heat flow in various parts of the mold.
Thermocouple 1 was selected for the initial analysis due to its proximity to the melt-mold interface and because it was positioned in the center of the mold. The measured curve was transferred to the software containing the time in seconds and temperature in degrees Celsius. Since the measurements were recorded up to 5,300 seconds, the curve was adjusted to consider only the first 2,000 seconds (Figure 4). In addition, the comparison interval was set to consider only the solidification and cooling stage.
To numerically describe the difference between the measured and simulated cooling curves a number of predefined objectives can be chosen. In order to minimize the difference between the two curves, two main objectives based on error terms are defined.
The first objective selected is the Delta Integral, which is the area between the measured and the simulated curve. This error term takes into account the areas in between the curves. The goal is to have the sum of the areas tend to zero (Figure 5).
The second objective, Delta Gradient, focuses on the slope of the simulated curve and is the sum of the difference in the first derivatives (Figure 3). Such a gradient error term accumulates all absolute values of the gradients between the intersections of measured and calculated curves.
The last step in defining the inverse optimization is to determine the appropriate method for selecting the first sequence of designs (or generation) to be evaluated, as well as defining the size of each generation and how many generations to calculate. In this case, a generation size of 24 was selected and five generations were considered. In the end 120 designs were run out of a possible total of 11,979 designs.
Result Analysis
The first step in analyzing the results is to make sure that the simulation experiment is representative to a point where the results can be used. When using genetic algorithms it is necessary to evaluate if designs were being selected in an effective manner by the software. The lack of convergence to a solution can be traced back to an incorrect process definition, an incorrect design interval definition, an incorrect curve selection and/or inaccurately measured experimental data. If there is no convergence to the ultimate goal of the simulation, the results would not be adequate for obtaining a definitive HTC curve.
The history chart can be used to evaluate the convergence of the algorithm because it indicates whether the algorithm is approaching the objective described in the definition phase. As the number of designs increase, the values for the objective, in this case Delta Integral, should decrease. Figure 6 illustrates the convergence of the algorithm in one of the experiments. The designs shown in Figure 6 are progressively approaching lower Delta Integral values as time increases, and it can be inferred the inverse optimization experiment was set up correctly and the results match reality.
Once it has been determined the inverse optimization is converging, the second phase of evaluation is to visually check the simulated cooling curve vs. the measured cooling curve. In Figure 7, the progression of finding an appropriate heat transfer coefficient curve for the simulation can be seen. Design 21 is far from matching the original measured curve depicted in blue. As the optimization progresses, the simulated curve of Design 57 tends to the measured curve. This behavior is encouraging, even if an exact match is not yet obtained, because it illustrates that the initial assumptions and input data are correct and, most likely, the variable intervals are not yet within the actual value of the heat transfer coefficient. Figure 8a shows the same test but with a corrected HTC interval applied.
Figure 8 shows a comparison of data from the thermocouple 1 measured in two different situations, T2 and T3. The first is the measured curve and simulated curve when no coating is applied. The second is the same test but with coating applied to the surface of the mold. As expected, the thermocouple recorded higher temperatures earlier for the uncoated cup mold than when coating was considered (Fig. 7). In both cases, a close correlation was obtained through the inverse optimization test (red curves). For test T3 (continuous line where coating is applied), at 1,250 seconds, the simulated curve and measured curve start to diverge, reaching a separation value of 57F (14C) at 2,100 seconds.
Once correlation with the chosen thermocouple was achieved, a second simulation with the same setup as the inverse optimization was run to compare the measured and simulated cooling curves for other thermocouples. This second simulation investigated if it is necessary to generate a separate heat transfer coefficient for the wall thermocouples.
The measured curves in Figures 9a and 9b present a distinct shape, with a clear peak and a drop of temperature to a plateau. After a few seconds, the thermocouple starts to record a drop in temperature. This behavior is seen throughout all experiments. When this is overlaid with the respective virtual thermocouple, the same behavior is observed without the need to adjust the thermocouple curve (Fig. 9a). In addition, all remaining thermocouples placed radially and in different levels than thermocouple 1 show the same behavior as its measured counterpart (Fig. 9b). This supports the early assumption that the material properties are adequate.
The simulation procedure was repeated for all of the experiments listed in Table 5, which shows the proportional values for the heat transfer coefficient achieved when comparing the use of coating and insulation around the mold wall. In this case, the values were a quarter lower when coating was applied to the cup mold surface as compared to when coating was not applied on the mold. As expected, the use of insulation around the mold does not influence the heat transfer coefficient of the interface.
The heat transfer coefficient curve created using inverse optimization indicated that when coating is applied, the minimum and maximum values for the heat transfer coefficient are four times lower than if the mold had no coating. Furthermore, adding insulation to the outer wall of the mold did not influence the value of the heat transfer coefficient.
When using the inverse optimization methodology for developing a temperature dependent heat transfer coefficient curve, in a controlled environment, multiple thermocouples are not needed. In addition, for permanent mold processes with gravity pour, the heat transfer coefficient can be considered uniform for the whole mold-casting interface, as long as the coating is applied with a uniform thickness and the same coating material.
By using autonomous optimization, the uncertainty of the heat transfer coefficient can be calculated and resolved with a total simulation time for all simulations run of under five hours. With this methodology the foundry is able to test different types of coatings and apply the computed heat transfer coefficients to simulations without having to assume the heat transfer coefficient.