Product: Abaqus/Standard
This example demonstrates the following Abaqus features and techniques:
computing steady-state heat transfer in an exhaust manifold,
comparing results for radiation heat transfer formulations using approximate and fully implicit methods, and
using film conditions to simulate the convective heat transfer from the exhaust gases.
Heat transfer in engine exhaust manifolds is governed by three effects: conduction through the metal, convection from the hot exhaust gases, and radiative exchange between different parts of the metal surface. This example illustrates the computation of the equilibrium thermal state of a manifold subject to these effects. The units of length in this example are millimeters; otherwise, standard metric units are used.
The procedure consists of a single heat transfer step in which the thermal loading conditions are ramped up from zero. The boundary constraints on the manifold flanges are a simplification of those experienced under operating conditions: the temperatures at the cylinder head and the outlet are fixed. Convection due to heat transfer from the hot exhaust is applied at the internal surfaces of the manifold tubes. Radiation is modeled between the internal surfaces of the tubes using two methods: the fully implicit cavity radiation method and an approximate cavity radiation method.
The exhaust manifold part being analyzed is depicted in Figure 5.1.5–1. It consists of a four tube exhaust manifold with three flanges, as in “Exhaust manifold assemblage,” Section 5.1.3.
The manifold is cast from gray iron with a thermal conductivity of 4.5× 10–2 W/mm/°C, a density of 7800 × 10–9 kg/mm3, and a specific heat of 460 J/kg/°C. The manifold begins the analysis with an initial temperature of 20°C. The part is dimensioned in millimeters, and the temperature is measured in °C, so the Stefan Boltzmann constant is taken as 5.669 × 10–14 W/mm2/K4 and absolute zero is set at 273.15°C below zero. The surface emissivity of gray iron is taken as a constant value of 0.77.
The hot exhaust gases create a heat flux applied to the interior tube surfaces. In this example this effect is modeled using a surface-based film condition, with a constant temperature of 816°C and a film condition of 500 × 10–6 W/mm2/°C. A temperature boundary condition of 355°C is applied at the flange surfaces attached to the cylinder head, and a temperature boundary condition of 122°C is applied at the flange surfaces attached to the exhaust.
The radiative transfer between the interior surfaces of the manifold tubes is modeled using two methods for comparison: the fully implicit cavity radiation method and the approximate cavity radiation method (see “Cavity radiation,” Section 37.1.1 of the Abaqus Analysis User's Manual). In the fully implicit method, geometric viewfactors are computed in Abaqus between each facet of the mesh on the exposed interior tube surface. These viewfactors quantify the effect of radiative transfer between each facet and each of the other facets in the user-defined cavity. The viewfactors, in turn, are used to compute a fully populated interaction matrix to compute the radiation flux between each pair of facets in the model. In the approximate method, the geometric viewfactors are approximated by assuming that each facet has an equal view of all other facets; this has the effect of modeling the flux at each facet as equal to that resulting from a black enclosure, held at the average temperature in the cavity, enclosing the facet. In the fully implicit method, some of the facets on the interior of the manifold have a view of the exterior, which is not modeled in this example. The exterior ambient temperature is taken to be the average of the temperatures used for the cylinder head and exhaust boundary conditions. In the approximate method, only the temperatures on the surface are considered, so an ambient temperature does not need to be defined. For simplicity, both the fully implicit method and the approximate method are defined using a single surface that includes all of the interior facets of the manifold tubes.
| Case 1 | Steady-state heat transfer with film and radiation effects; radiation modeled using the fully implicit method. |
| Case 2 | Steady-state heat transfer with film and radiation effects; radiation modeled using the approximate method. |
The following sections discuss analysis considerations that are applicable to both cases.
Due to the fourth-order dependence of the radiation flux on the surface temperatures, this example problem is intrinsically nonlinear. For both cases the steady-state heat transfer procedure is used. This is a general analysis step in Abaqus, chosen because iteration is required for convergence. An initial increment is chosen as one-tenth of the final value.
Figure 5.1.5–2 shows the nodal temperature field for the manifold. On the left, the analysis results using the fully implicit method are shown; on the right, results from the approximate method are shown. In this problem we observe good agreement between the two methods, although some differences can be discerned in the plots.
The peak temperature in the field is higher when using the fully implicit method. The effect of radiation heat transfer is to smooth out the temperature field in the equilibrium solution: high-temperature zones radiate more heat, which is absorbed by the cooler areas. In the fully implicit method, this smoothing effect is limited and affected by the geometric viewfactors: the distance and orientation of the surface facets affects the degree to which radiation exchange can occur. In the approximate method, this is not the case. Each facet absorbs or emits radiative heat flux based on its temperature and the averaged cavity temperature only; the localizing effects of viewfactors are ignored. Therefore, the average method results reflect the greater smoothing effect of the radiation model used, resulting in lower peak values.
Figure 5.1.5–3 shows the flux magnitude results. The flux field shows even greater agreement than the temperature field.
Because the fully implicit radiation algorithm uses a fully populated matrix operator to model the interactions of each facet, it is significantly more computationally expensive than the approximate radiation method. Table 5.1.5–1 illustrates the differences between the two methods. In this problem the cavity surface contained 4505 facets—it consists of the entire interior of the manifold. The savings in memory are quite significant and directly reflect the cost of the large operator used in the fully implicit method. The memory and timing results were obtained on a desktop computer using Xeon processors, but the relative comparisons between run times are more pertinent than the specific run times.
Input data for the analysis using the approximate radiation method.
Input data for the analysis using the fully implicit radiation method.
Table 5.1.5–1 Relative computational costs of the approximate and fully implicit methods.
| Approximate method | Fully implicit method | Approximate ratio | |
|---|---|---|---|
| Minimum memory required (MB) | 31 | 501 | 16 |
| Memory to minimize I/O (MB) | 52 | 1052 | 20 |
| User time (sec) | 14.6 | 621.5 | 42 |
| System time (sec) | 0.3 | 10.5 | 35 |
| Total CPU time (sec) | 14.9 | 632 | 42 |
| Wallclock time (sec) | 16 | 653 | 41 |
| Increments | 6 | 6 | 1 |
| Total iterations | 10 | 7 | 0.7 |
| Wallclock time per iteration | 1.6 | 93.3 | 58 |
