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Perhaps I misunderstand the question or issue. But if the need is to calculate the chord opening in the ellipsoidal head at the nozzle opening this can be found from geometry.
We need to find the "(x,y)" coordinates of the two points at which the ID of the nozzle intersects the head inside surface, then calculate the chord length between these two points.
Point "A" will have an "x" value equal to: DR + Rn ie: (distance from vessel axis to center of nozzle, as measured perpendicular to the axis) + (radius of nozzle)
Point "B" will have an "x" value equal to: DR - Rn
Now determine the "y" values from the mathematical curve of the head.
This latter step can get tricky. The ellipsoidal head has elliptical curve on inside surface in new condition but "parallel" curves are no longer ellipses! By this I mean that the curve on the outside surface of the head (assuming that the head thickness is constant, which it may not be) is not an ellipse, nor is the inside surface when a constant corrosion allowance is applied. Depending upon how precise one wants to be you can take this change in mathematical curve into account.
Once the (x,y) values for both points are found, the length of the chord connecting them is easily found.
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