Oppenheim method

The radiative conductances can be calculated using the Oppenheim method.

Oppenheim method, or otherwise known as the radiosity method, is the recommended approach. Oppenheim method consists of creating an artificial Oppenheim element I for each radiating element i, which may be visualized as an imaginary element floating above element i, and intercepting all radiation to it.

A radiative coupling is created between I and i, and each view factor VF_ij is transformed into a radiative coupling A_iVF_ij between I and the Oppenheim element J of j equal to:

Oppenheim method is an alternative to Gebhardt method for calculating radiative couplings. It has several advantages:

• Temperature-dependent emissivities are handled more accurately and efficiently. This is because during temperature calculations only the radiative conductance A_iϵ_i/(1−ϵ_i) between the Oppenheim element and the element itself needs to be modified.
• CPU performance is generally much improved for two reasons: the matrix inversion process of Gebhardt method is bypassed, and the number of radiative couplings is much generally much smaller. This is because in a given enclosure Gebhardt method creates a radiative coupling between every element, even if they cannot view each other.
• Accuracy is improved because the RK elimination of insignificant radiative couplings is bypassed.
• File sizes are generally smaller.

With Oppenheim method significant performance degradation can be expected during transient analysis with the explicit forward differencing method. This is because the Oppenheim elements are assigned zero capacitance, and the forward differencing method requires a full steady-state solution for the surface elements at each integration time step.

However, no accuracy degradation is observed with the implicit time integration methods.

For each radiating element, I, a corresponding Oppenheim element, i+T1, is created. A radiative coupling between I and its Oppenheim element is created with a magnitude equal to Area(I) * emissivity(I) / (1 - emissivity(I)). Each Oppenheim element is assigned an element number equal to i+T1. In case of numbering conflict, an unused element number is assigned.

Radiosity method

The radiosity method is used to model radiative heat exchange between surfaces. It defines the radiosity as the total energy flux leaving a surface, which includes both emitted and reflected radiation:

where:

• q_o,k is the outgoing (radiosity) flux from surface element k.
• ε_k is the emissivity of surface k.
• σ is the Stefan-Boltzmann constant.
• T is the absolute temperature of the surface.
• ρ_k is the reflectivity of surface k.
• q_i,k is the incoming radiative flux to surface k.

The incoming flux is computed as a sum of contributions from all other surfaces using view factors:

Where F_kj is the view factor from surface k to surface j.

The net heat transfer to surface element k is then given by the difference between outgoing and incoming fluxes, multiplied by the surface area:

The radiosity equation can be expressed in a form that relates the outgoing flux from a surface element to the difference in radiosity between that element and all others in the system. The net radiative heat exchange for surface k:

Oppenheim elements are created to carry the unknown radiosities. The thermal solver uses a “temperature,” rather than a radiosity as the unknown for the Oppenheim element.

The radiosity equations are linearized and integrated into the thermal conductance matrix. This allows the radiative exchange to be solved simultaneously with the temperature field.

To ensure energy balance across all surfaces, the following equation is used:

These equations are assembled into a global matrix system of the form where first matrix is the conductance matrix: