To prevent its Newton-Raphson Jacobian matrix from becoming ill-conditioned when solving network hydraulics, EPANET replaces the nonlinear Hazen-Williams (and Chezy-Manning) head loss equation with a linear approximation whenever pipe flow is below the flow q* that produces a head loss gradient of RQtol(equal to 1e-7). From 0 to q* the head loss should equal (RQtol / Hexp) * q, where Hexpis the exponent used in the H-W (or C-M) equation. However the 1/Hexp factor is missing in the current code (see line 556 in hydcoeffs.c). This same bug occurs for emitter head loss (missing a 1/Qexp factor) and for pressure dependent demand head loss (missing a Pexp factor).
The derivation of the (RQtol / Hexp) coefficient for low pipe flow is as follows. The H-W head loss equation can be expressed as:
hloss = R * q^Hexp
where hlossis head loss, Ris a resistance factor, q is flow rate and Hexpis 1.852. Its derivative with respect to flow, hgrad, is:
To prevent its Newton-Raphson Jacobian matrix from becoming ill-conditioned when solving network hydraulics, EPANET replaces the nonlinear Hazen-Williams (and Chezy-Manning) head loss equation with a linear approximation whenever pipe flow is below the flow
q*that produces a head loss gradient ofRQtol(equal to 1e-7). From 0 toq*the head loss should equal(RQtol / Hexp) * q, whereHexpis the exponent used in the H-W (or C-M) equation. However the1/Hexpfactor is missing in the current code (see line 556 in hydcoeffs.c). This same bug occurs for emitter head loss (missing a1/Qexpfactor) and for pressure dependent demand head loss (missing aPexpfactor).The derivation of the
(RQtol / Hexp)coefficient for low pipe flow is as follows. The H-W head loss equation can be expressed as:where
hlossis head loss,Ris a resistance factor,qis flow rate andHexpis 1.852. Its derivative with respect to flow,hgrad, is:At the flow
q*wherehgradequalsRQtolwe have:Using this point to define a linear head loss relation from 0 to
q*yieldsRQtol / Hexpas the coefficient (or slope) for this relation.