A gas accumulator (shown in Figure 1) is an enclosed tank filled with gas and has a free surface. It is used as a surge suppressor (Wylie, et al., 1993, pp. 125-128Wylie, E.B., V.L. Streeter & L. Suo, Fluid Transients in Systems, Prentice Hall, Englewood Hills, New Jersey, 1993.).The flow balance on the accumulator junction is similar to a branch junction, with the deficit equaling the flow out of the accumulator, ṁ_A.

Similar to a branch, it can be shown that for all connecting pipes to the junction,

where SC and SB are given by Equation 2 and Equation 3 from Branch Waterhammer Theory. The potential connector pipe and orifice effect is incorporated by the lumped inertia relationship as follows

where C1 and C2  are defined in conjunction with lumped inertia. Refer to Equation 1 and Equation 2 from the Lumped Inertia Pipe topic. The Δh_liquid is the height of liquid in the accumulator from the bottom of the accumulator, and is assumed to be zero if the tank geometry is not defined.

The gas volume and pressure in the accumulator can be related by the thermodynamic law

where n is the polytropic constant (1.4 for isentropic air, 1.0 for isothermal) and CA is a constant.

Therefore,

and

By substitution

or

Combining equations and simplifying,

where

This is a non-linear equation that can be solved using Newton-Raphson iteration. Newton-Raphson has the following form:

where xi is the current value of x, xi+1 is the new value, F is the function of x to drive to zero, and F´ is the derivative.

In this case

When the value of ṁ_A,new has been found that causes F to equal zero, the equation will be satisfied. To solve by Newton-Raphson the derivative of F is given by,

where

By iteration, ṁ_A,new is calculated. This value can then be back-substituted to obtain the new junction pressure and new gas volume.

Gas Accumulator Vapor Cavitation Theory

Any short-lived vapor cavities that develop at a Gas Accumulator junction are assumed to have a negligible effect compared to the accumulator itself. Conceptually, vapor bubbles would rise by buoyancy to the accumulator surface and evaporate into the accumulator gas. Their effect on the system would thus be negligible compared to that of the accumulator gas. Therefore, vapor cavitation is ignored at gas accumulators.