Temperatures of continuously operated mobile x-ray focal spots

Abstract

We use the adjective "mobile" to describe a focal spot in an x‐ray tube which moves cyclically in a closed path relative to the target surface on which it is generated quite without reference to whether the spot be fixed or in motion with respect to the tube. Alex Müller and also A. Bouwers have made calculations of the thermal effects in such spots energized for very short fractions of one cycle of motion. This paper treats the case where the focal spot is energized continuously so that a "steady" state of thermal oscillation is set up. The solutions obtained refer only to conditions after this steady state of oscillation is reached. Only the ideal case of a spot of uniform intensity with sharp boundaries is treated. For simplicity the flow of heat from front target surface to cooling water is assumed one dimensional and normal to the front surface over an everywhere uniform thickness d. Curves are plotted showing the ratio of temperature rise in moving spots to that in fixed spots of the same size as a function of r the size of the spot relative to the length of path it describes and of θ a variable depending on the speed of rotation, the thickness d and the thermal constants of the target material. Curves of the ratio of permissible energy input for moving and fixed spots, respectively, as a function of the aforementioned variables are also shown. The paper gives a solution reduced to figures and curves of the heat flow equation for certain boundary conditions which to the author's knowledge has never before been obtained and therefore has some interest beyond its immediate application to x‐ray tubes. Certain mathematical difficulties of practical interest are also overcome in a way which may be helpful in other problems of similar nature. For the reader uninterested in mathematics the results are independently discussed in a separate section. Approximate methods are given for applying the results of the paper to targets consisting of two materials such as tungsten and copper.

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