Abstract
The paper presents a mathematical theory of stability of small oscillations in surge tanks of the type where two water surfaces are present. The name ‘parallel-branch’ surge tank is coined to describe such a system, and it is shown that results for orifice and differential surge tanks can be obtained from it. On the assumption of ideal governing and small oscillations of flow and water levels, equations giving the limits of stability are deduced for all cases. It is an easy matter from these to test the stability of any particular design by inserting appropriate numerical values. In general, the results confirm the well-known Thoma formula for the case of a simple surge tank and give results identical with those obtained by Professor Escande for the case of an orifice surge tank. For differential surge tanks, the limits of stability are given in an equation covering the area factors of the riser and the outer chamber as well as the resistance factors of the main conduit and orifices respectively. Graphs showing this relation between the above quantities indicate that: (1) if the combined area of riser and outer chamber is equal to the Thoma area calculated for a simple surge tank, then the system is stable provided the resistance factor of the orifice is below a certain limiting value; (2) when the above combined area is less than the Thoma area, then, in general, small but finite oscillations of the water level will persist. By suitable choice of orifices the amplitudes of these oscillations can be made negligible; (3) if the resistance factor of the orifices exceeds the previously mentioned limiting value, then, in general, it is not possible for stability to be achieved with riser areas less than a specified value. Methods of calculating the frequency and amplitude of permanent oscillations are presented in the paper. The general case of a true parallel-branch system is best solved numerically, but the results indicate that in all practical cases the phenomenon of finite amplitude oscillations will exist. Attention is drawn, however, to the necessity of considering the period at which instability is possible as, on occasion, this is too short for the assumption of ideal governing to be applicable.
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