### FDA5 - Background

Early attempts to calculate an optimum gasket position were based on the paper "Cut Vessel Flange Costs by Computer" by T.H. Korelitz. This procedure was designed to produce a minimum total moment and hence minimise the flange thickness. Whilst reducing flange thickness is one of the design considerations, the main aim is to reduce the bolting requirements. This could obviously be achieved by placing the gasket at the bore of the flange, thus producing the minimum possible bolt loads. This in turn, however, maximised one of the lever arms which could be critical in the thickness calculations.

A practical investigation into the design of flanges considering all possible gasket positions across the face of the flange showed that, for varying flange sizes and pressures, the gasket position giving the minimum volume of flange material was not constant.

The aim in all of these deliberations was to produce the smallest possible flange and, whilst the gasket plays an important role in this, other parameters have considerable effects, not least of these being the hub slope.

Various heat exchanger manufacturers have attempted to find the best hub slope to give minimum volume and various figures are quoted such as slopes of 1/3, 1/4. However, there is no documentation to justify these statements and in order to ascertain the effect of the hub slope the principle of dimensional similarity was applied to the problem using the Buckingham method. The results of this analysis were graphed into a family of parabolic curves of 'hub slope' against 'volume squared' for various values of the independent variables (pressure, gasket properties, flange material, bolt material, diameter, etc.) combined in a dimensionless form. These curves were extremely flat and for high values of the dimensionless parameter tended towards a straight line with a very small slope. There did not appear to be any relationship between the values of the dimensionless parameter and the position of each graph and hence all that could be stated was that, for each set of independent variables, there was one hub slope which produced a minimum volume, although in some cases this was only marginally less than volumes obtained using other hub slopes.

From the investigations it was decided that the hub slope should be chosen by the program in such a manner that a minimum volume flange is always obtained. Thus, since there was no explicit means of achieving this, six different hub slopes are computed and the minimum volume flange selected. It was also decided to allow the gasket to be positioned so that it met the exact requirements for an adequate number of the smallest acceptable diameter bolt.

The flange rotation calculations were taken from the paper entitled 'Design of bolted flanged joints of pressure vessels' by G.F. Lake and G. Boyd (10th May 1957) - reference should be made to this paper for the derivation of the relevant formulae.