# Transient thermal analysis takes one-tenth the time

-February 07, 2002

For transient-thermal-analysis problems, running a full CFD (computational-fluid-dynamics) analysis to solve for mass, momentum, and energy equations using finite volume requires a lengthy computation time and can use a tremendous amount of computer-disk storage space. When the analysis includes a number of customer-required power on/off scenarios (duty cycles), performing a full CFD analysis for each duty cycle can be a tedious process. Also, running the full CFD model for each duty cycle and ensuring that the model reaches periodic steady state is time-consuming.

Fortunately, a simplified heat-transfer-coefficient CFD model, or h-model, dramatically reduces computation time. For steady-state problems, this h-model reduces the computation time from 4.5 hours to a few minutes. For transient thermal analysis, the h-model reduces the computation time by a factor of 10 from the full CFD analysis. The temperature-rise correlation between the two models is within 5%. As an added bonus, the smaller computational domain for the simplified model requires less disk storage space.

This simplified model solves only for the energy equation and incorporates heat-transfer coefficients (h) that you determine from full CFD steady-state runs at various power-dissipation levels. The following example shows how you can derive linear approximations of h as a function of wall temperature exposed to the air fluid, then apply the approximations as boundary conditions directly coincident with the outer surface of the solid model. You can then collapse the computational domain to the boundaries of the model, which reduces the mesh region to one-sixth the full CFD model.

You can apply this simplified h-model to any similar transient thermal analysis. Using a simplified h-model rather than running a full CFD analysis significantly reduces effort with no loss in accuracy. Because the two models provide the same temperature results, it becomes unnecessary to run a full CFD analysis for various duty-cycle simulations. This methodology reduces computational time by a factor of 10, which can reduce both development costs and time to market.

## Study of an electronic module

A common problem in transient thermal analysis is determining the maximum temperatures of a die during cyclic electrical loading. This study analyzes the behavior of an electronic module designed by Motorola Automotive Communications and Electronic Systems Group (www.motorola.com). The module is a controller measuring 95×217×62.75 mm. The ambient-temperature environment during standard operation is 38°C.

The module's power-dissipation values range from 28W in a steady-state mode to 200W transient peak power. The power-stack region consists of six silicon bare-die transistors and one temperature-sense resistor soldered to a DBC (direct-bond-copper) substrate. The DBC is soldered to a copper base plate, which then attaches to an aluminum heat sink. Aluminum-wire bonding carries the power and signal to and from the bare-die transistors (Figure 1).

## Determine heat-transfer coefficient

Convection heat transfer occurs between a fluid in motion, in this case air, and a boundary surface, when the two are at different temperatures. For natural or free convection, you typically use two values to calculate the heat-transfer coefficient: the Rayleigh number and the Nusselt number. You use the Rayleigh number (Ra) to choose the flow regime—either turbulent or laminar. The flow is laminar if Ra&1×109. Ra is a function of the temperature difference between the fluid and the solid-model surface. The Nusselt number (Nu) is a function of the Rayleigh number. Reference 1 presents calculations for Nu for vertical and horizontal plates, as Table 1 shows.

You can calculate the heat-transfer coefficient, h, from the Nusselt number as follows, where k is the thermal conductivity of air, and L is the length of the plate:

Note that h is a function of the surface temperature (TS) because you calculate Nu from Ra, which is a function of TS.

To simplify the transient analysis, it is tempting to use a manual calculation of h as in Equation 1. Note, however, that the Nusselt-number equations assume that the walls are isothermal. In reality, the walls do not exhibit a uniform temperature. Using a simple calculation of h from the Nusselt number would result in erroneous calculations during the transient analysis. A more accurate method, therefore, is to run a steady-state full CFD model and obtain h values of surfaces in contact with the airflow.

## Run the simulation

This example uses Icepak, a CFD-based thermal-design software program from Fluent Inc (www.fluent.com), to model the electronic module. The CFD model incorporates only the power stack (FET transistors), which is the primary region of power dissipation, the heat sink, and the module cover.

For this CFD analysis, the modes of heat transfer are conduction and natural convection. This example uses the following procedure to determine an approximation for the h values:

1. Perform a steady-state full CFD analysis, solving for mass, momentum, and energy balance.

2. From the heat-sink base, sides, and covers that are exposed to the airflow, obtain the mean heat-transfer coefficient (h) as well as mean temperature values. You can accomplish these tasks using the summary-report feature in Icepak.

3. Using three additional power-dissipation levels, repeat steps 1 and 2.

4. Plot h values versus the surface temperatures of the walls exposed to the fluid for the various power-dissipation levels, generating a graph for each wall surface in contact with the fluid. Figure 2 shows typical graphs for the heat-sink base and top cover at dissipation levels of 12.3 to 75W. Note that temperature units are kelvin.

5. For each wall surface, obtain a linear approximation equation of h as a function of temperature by generating a best-fit line through the data points, as in Figures 2a and 2b.

For this example, the following procedure yields the equations for h to create the simplified h-model:

1. First, simplify the full CFD model by deactivating the heat-sink fins.

2. To reduce the computational domain, move walls so they become coincident with solid-model surfaces in contact with the fluid. For the full CFD model, the computational domain size is 220×320×300 mm; for the h-model, the domain size is 62.75×217×95 mm.

3. To each coincident wall, apply its linear approximation equation of h as a function of temperature. Note that the model size remains the same; however, the computational domain is reduced to become coincident with the solid model's outer surfaces.

4. To further reduce the number of elements in the mesh, apply hollow blocks where no meshing occurs and where no solution is required. These hollow blocks are regions outside the solid model and within the new smaller computational domain.

5. Solve only for the energy equation, which means solving only for temperatures and performing no velocity or pressure calculations.

6. Turn off gravity; the linear equations (his a function ofwall temperature) imposed on these walls include the effect of gravity (buoyancy forces).

7. As a sanity check, run the full CFD transient model and simplified h-model to determine whether they agree. Once you determine that the two models provide the same temperature results, you no longer need to run the full CFD model for various duty-cycle simulations.

Note that the h-model treats h as a linear function of the surface temperature of objects exposed to the fluid. In reality, however, h is a function of the Nusselt number, which is dependent on the Rayleigh number and is typically a power function. At the time of this study, Icepak did not allow a nonlinear function for h. Comparing the temperature results for the h-model with the full CFD model determines the validity of the linear approximation. In this study, the two methods were in agreement. For even better temperature accuracy, the current version of Icepak allows the user to enter h values as a piecewise linear function of temperature, better representing the nonlinear relationship between h and surface temperature. You can use the previously described methods to determine the piecewise linear functions for h. You can also successfully apply the h-model to surfaces for which h varies considerably as a function of hotter and cooler locations by partitioning these surfaces into segmented walls.

## Analysis results versus CFD simulations

Performing both transient and steady-state analyses of the electronic module using the simplified h-model and comparing the results with full CFD simulations reveals the advantages of the h-model. In all analyses, the total power dissipation is 75W, with each FET bare-die transistor dissipating 12W, and the temperature sensor dissipating 3W. The ambient temperature is 38°C. For the transient analyses, the duty cycle is 10 seconds on and 50 seconds off, with data taken and stored every 5 seconds.

The full CFD models have 272,360 elements and 284,831 nodes. The simplified h-model has 42,379 elements and 46,294 nodes, less than one-sixth the original mesh size. Both simulations use an identical solid-model mesh for the electronic module; the smaller computation domain for the h-model results in a reduced mesh size.

For a steady-state analysis, the computation time for the full CFD model is 4.5 hours. The computation time for the steady-state h-model is 5 minutes.

During the transient analysis, the full CFD simulation analyzes 2650 seconds of duty cycle before halting because the computer hard drive becomes full. The simulation does not reach periodic steady state. If the full simulation had finished, the computation time would have been about 25 hours. The simplified h-model runs for the entire 3600 seconds of duty cycle; computation time is only 2.5 hours, which is a tenfold reduction.

Each simulation requires disk space for result files. For the full CFD model, each result file is 7 Mbytes. If the full CFD model runs the entire 3600 seconds, saving every 5 seconds and yielding 720 result files, more than 5 Gbytes of computer-hard-drive storage space is necessary. Using the same number of result files for the h-model requires less than 1 Gbyte of hard-drive storage space.

Fast results are useful only if they are accurate, and the h-model transient results are similar to the results obtained from the full CFD analysis. Figure 3 shows the temperature-versus-time results for one of the transistors (FET A_Hi) for the full CFD and h-model. The difference in total temperature rise between the two models is within 5%.

The full CFD model and the h-model also produce similar temperature-distribution results. Figures 4a and 4b show transient temperature results for the outside of the electronic module at 1805 seconds for the full CFD and h-model simulations, respectively. Both figures show the maximum x face (HS_Base.1) and maximum y face (cover_top). The figures also illustrate the computational domain size for the full CFD model and smaller domain size for the h-model. In Figure 4b, the simulation applies the equation h=0.0852T–20.592 to the wall at HS_Base.1 and the equation h=0.018T+0.5566 to the wall at cover_top. The full CFD model results in a temperature range within the model of 51.7 to 74.9°C, and the h-model results in a nearly identical range of 50.0 to 75.4°C.

Figures 5a and 5b show the bare-die transistor temperatures at 1805 seconds for the full CFD model and the h-model, respectively. Arrows indicate the locations of the FET A_Hi transistor and the temperature-sense resistor. The maximum bare-die transistor temperature is 74.9°C in the full CFD model and, again, a nearly identical temperature of 75.4°C in the h-model.

The accurate temperature results for the h-model, when compared with those of the full CFD model, demonstrate the validity of using a linear approximation for h as a function of surface temperature.