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From this, lateral loss fluxes that are proportionally shared can also be shared in proportion to the ownership ratios in live storage, as this is the same as sharing in proportion to each owner’s index flow rate.  Hence, substituting from equation (19) into equation (13) and rearranging yields:

Equation 20

Owner's outflow formula (excludes high flow losses)

As ownership is conserved, each owner’s outflow can be determined using the proportional routing, mass balance and division flow equations (note, this formula does not cater for ‘high flow losses’).

To simplify later steps, r is defined as:

Equation 21	Image Added

Combining mass balance equation (15) with equations (19) and (21) gives:

Equation 22	Image Added

Rewriting (22) by substituting equation (17) for , and rearranging in terms of owner outflow volume , gives:

Equation 23	Image Added

If high flow losses are not specified, the result of this outflow volume equation is used in the mass balance equation to determine division storage at the end of a time-step.

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Note: If the routing division is using an attenuation factor x=0 (i.e. Image Added), equation (23) can be rearranged to the continuous stirred reactor equation (equation (46)). This means that for fully forward weighted routing schemes, proportional routing is the same as fully-mixed.

High flow loss formula (Murray-style)

In the Murray, losses caused by flows in excess of the regulated flow range are shared to owners in proportion to how far each of them is above their fixed share of this range.  Hence, if q‾(o) is owner o’s current flow rate, Ratio_HFT(o) is their share of the high flow threshold, q‾_HFT is the high flow threshold,  the total flow rate, and Loss_HF the total high flow loss, then owner o’s share of the high flow loss is:

Substituting equation (19) into equation (24) enables it to be re-written in terms of the division’s storages, as follows:

High flow losses are worn only by owners where Storage(o,t)-Storage_dsMAX(o) is greater than their share of the high flow threshold (Storage_HFTRatio_HFT(o)).  Owners that will not be required to contribute the high flow loss are identified by calculating Storage(o,t) for each owner assuming that Loss_HF=0 and finding those that fall short of their share of the high flow threshold.

From equation (25), for those owners that exceed their share of the threshold: