Nanofocus diamond X-ray windows: Thermal modeling of nano-sized heat source systems

•A steady-state analytical model of diamond thermal windows for nanofocus X-ray tubes has been developed.•This model replaces the boundary conditions by a series of virtual sources.•An analysis is performed to truncate the series of the solution and tabulate the solution with a controlled error.•It considers the thermal conductivity thermal dependence of the materials.•The model has been validated with experimental data of the literature.

An analytical thermal model is proposed to study heat transfers occurring at high power density in X-ray tubes with micron to submicron sized source. The use of a simple analytical approach instead of a complex numerical simulation allows readily modeling of more and more challenging systems such as multi-source X-ray tubes. By significantly reducing the computing time, it enables a wider parameter range evaluation for engineering phase. We focused our work on tubes integrating a transmission window that is mainly edge-cooled. Our approach can be generally used in cylindrical lateral heat spreaders with distributed small sized source systems. The model enables an efficient estimation of temperature distributions for a large range of parameters and source designs and is, for instance, well-suited to described nanosized heat source systems. It is developed from an electrostatic analogy with the point charge particle model and uses of a series of virtual sources. A multiscale resolution of the heat equation is proposed, hence providing the temperature distribution at any point within the whole system. Moreover, the non-linearity of equations caused by temperature-dependent thermal conductivities is solved by using Kirchhoff's transformation, giving a more realistic approach of heat conduction in diamond X-ray windows, where temperature in excess of 1000 °C can be encountered. The influence of convective and radiative transfers has been discussed and the physical accuracy of the predicted temperature is controlled by adjusting the number of virtual sources of the model.

Graphical abstractA virtual source model was developed to study heat transfers occurring at high power density in X-ray tubes with micrometric to nanometric sized source. A simple analytical approach allows modeling of more and more challenging systems. It reduces the computing time and allows a wide parameter range evaluation for engineering phase.Figure optionsDownload full-size imageDownload as PowerPoint slide