Weir Rating — Compound (Auto Cd + WL Solver)

1) Datums & Geometry

g (m/s²)

Upstream apron (bottom‑intake) \(z_{a,bi}\) (masl)

Upstream apron (overflow) \(z_{a,ow}\) (masl)

Bottom‑intake crest \(z_{c,bi}\) (masl)

Overflow crest \(z_{c,ow}\) (masl)

Bottom‑intake width \(B_{bi}\) (m)

Overflow width \(B_{ow}\) (m)

Approach width \(B\) for y₁ (m)

Used for critical‑depth estimate upstream (TEL). Often equals \(B_{bi}+B_{ow}\).

Formula

Pick the convention your Excel uses so Cd matches.

Approach‑velocity factor ψ (TEL)

TEL = EL + ψ·V²/2g. Set 0 to ignore.

Submergence factor f_sub (—)

Manual multiplier on Q, if you want to reflect submergence.

2) Discharge Coefficients \(C_d\)

Mode

Cd (bottom‑intake)

Cd (overflow)

3) Target Discharges → Solve Water Level WL

Label	Q_target (m³/s)	WL (masl)	H_bi / H_ow (m)	Q_bi / Q_ow (m³/s)	EL_up (masl)	TEL (masl)	q (m²/s)	y₁ (crit) (m)	Q_calc (m³/s)	ΔQ

OK: Calculated.

Theory

For a given water level \(WL\) (masl): \(H_{bi}=\max(WL-z_{c,bi},0)\), \(H_{ow}=\max(WL-z_{c,ow},0)\). Using chosen formula convention, \[ Q_{bi}=C_{d,bi}\,B_{bi}\,F\,H_{bi}^{3/2},\qquad Q_{ow}=C_{d,ow}\,B_{ow}\,F\,H_{ow}^{3/2} \] with \(F=1\) (no √(2g)) or \(F=\sqrt{2g}\). Total \(Q=Q_{bi}+Q_{ow}\).

Auto‑calibration: Given \(n\) calibration points \((WL_i,Q_i)\), solve \(Q_i \approx C_{d,bi}(B_{bi}F H_{bi,i}^{3/2}) + C_{d,ow}(B_{ow}F H_{ow,i}^{3/2})\) by least squares. Then, for targets \(Q_t\), we **solve \(WL\)** by bisection so that \(Q(WL)=Q_t\).

TEL & y₁: \(q_B = Q/B_{app},\ y_c=(q_B^2/g)^{1/3},\ V_1=q_B/y_c,\ \mathrm{TEL}=WL+ \psi\frac{V_1^2}{2g}\). Set \(\psi=0\) to ignore velocity head.