Water & Wastewater formulas — 55 solvers

Alkalinity Remaining After Alum

A_f = A_0 - 0.45 \, D

Water & WastewaterWater TreatmentChemistryAlkalinity left after coagulation, since each mg/L of alum destroys 0.45 mg/L of alkalinity as CaCO₃ in forming aluminium hydroxide floc.

Backwash Water Volume

V_{bw} = v_b \, A \, t

Water & WastewaterWater TreatmentFluid MechanicsWater consumed by one filter backwash, from the backwash rise rate, the filter bed area and the duration of the wash.

BOD Mass Loading

W = Q \, C

Water & WastewaterWater TreatmentChemistryMass of BOD, COD or solids arriving per unit time from a flow and its concentration, on the sanitary basis of 1 mg/L = 1 g/m³.

BOD Removal Efficiency

E = \frac{C_i - C_e}{C_i} \times 100

Water & WastewaterWater TreatmentPercent removal across a treatment unit or a whole plant, from the influent and effluent concentrations of BOD, TSS or any pollutant.

Broad-Crested Weir Discharge

Q = C_d \left(\tfrac{2}{3}\right)^{3/2} \sqrt{g} \; b \, H^{3/2}

Fluid MechanicsCivil & SurveyingWater & WastewaterDischarge over a broad-crested weir or a level spillway crest, derived by forcing critical depth over the crest. In SI the constant group is 1.705, so Q = 1.705 C_d b H^1.5 with lengths in metres.

Chezy Equation

V = C \sqrt{R \, S}

Fluid MechanicsCivil & SurveyingWater & WastewaterAntoine Chezy's 1769 uniform-flow equation: mean velocity equals the Chezy coefficient times the square root of the hydraulic radius times the energy slope.

Clarifier Solids Loading Rate

\text{SLR} = \frac{(Q + Q_r) \, X}{A}

Water & WastewaterWater TreatmentSolids applied to a secondary clarifier per unit of surface area, in kg of MLSS per square metre per day, including the return flow.

Cost of Water Over a Period

C_w = V \, p_w

Water TreatmentFluid MechanicsWater & WastewaterCost of the water a system buys over a period: the metered volume times the utility's rate, in whatever currency that rate was in.

Critical Depth in a Rectangular Channel

y_c = \left(\frac{Q^2}{g \, b^2}\right)^{1/3}

Fluid MechanicsCivil & SurveyingWater & WastewaterCritical depth of a rectangular channel from the discharge and the bed width — the depth of minimum specific energy, where the Froude number is exactly one.

CT Value for Disinfection Credit

\text{CT} = C \, t

Water & WastewaterWater TreatmentChemistryDisinfection CT: residual concentration multiplied by contact time, returned in the regulatory unit of mg·min/L.

Detention Storage from a Routing Step

\Delta S = (Q_{in} - Q_{out}) \, \Delta t

Water & WastewaterCivil & SurveyingFluid MechanicsOne step of the reservoir continuity equation: the storage a detention basin accumulates is the difference between inflow and outflow multiplied by the length of the step.

Energy Loss in a Hydraulic Jump

\Delta E = \frac{(y_2 - y_1)^{3}}{4 \, y_1 \, y_2}

Fluid MechanicsCivil & SurveyingWater & WastewaterSpecific energy destroyed by a hydraulic jump in a rectangular channel, obtained by subtracting the downstream specific energy from the upstream one and simplifying with the momentum equation.

Filtration Rate (Filter Loading Rate)

v_f = \frac{Q}{A}

Water & WastewaterWater TreatmentFluid MechanicsFiltration rate through a granular media filter: flow divided by filter bed area, the approach velocity reported in metres per day.

Food-to-Microorganism (F/M) Ratio

\frac{F}{M} = \frac{Q \, S_0}{V \, X}

Water & WastewaterWater TreatmentFood-to-microorganism ratio: BOD applied per day divided by the mass of mixed liquor solids holding it, in kg BOD per kg MLVSS per day.

Francis Formula: Rectangular Weir

Q = 3.33 \, L \, H^{3/2}

Water & WastewaterFluid MechanicsFlow over a suppressed rectangular weir by the Francis formula, with the trade constant 3.33 for crest length and head in feet.

Froude Number (Open Channel)

Fr = \frac{V}{\sqrt{g \, D}}

Fluid MechanicsCivil & SurveyingWater & WastewaterFroude number of an open channel: the ratio of flow velocity to the speed of a shallow-water wave, with D the hydraulic depth (flow area divided by top width).

Harmon Peaking Factor

PF = 1 + \frac{14}{4 + \sqrt{P/1000}}

Water & WastewaterWater TreatmentFluid MechanicsHarmon peaking factor for sanitary sewer design: the ratio of peak hourly to average daily flow for a served population.

Henry's Law (Gas Solubility)

C = H\,P

ChemistryWater & WastewaterAt a fixed temperature the concentration a gas reaches in a liquid is proportional to its partial pressure above the liquid. H is the solubility form of the Henry constant, in moles per cubic metre per pascal.

Hydraulic Detention Time

t = \frac{V}{Q}

Water & WastewaterWater TreatmentFluid MechanicsTheoretical detention time of a tank, clarifier or contact basin: the working volume divided by the flow passing through it.

Hydraulic Jump Conjugate Depths

\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^{2}} - 1\right)

Fluid MechanicsCivil & SurveyingWater & WastewaterBelanger's momentum solution for a hydraulic jump in a rectangular channel: the sequent (conjugate) depth downstream of the jump from the depth and Froude number upstream of it.

Hydraulic Radius

R = \frac{A}{P}

Fluid MechanicsCivil & SurveyingWater & WastewaterHydraulic radius of any conduit or channel: the flow area divided by the wetted perimeter, the single geometric number that Manning, Chezy and every friction equation actually consume.

Jar Test Dose Scale-Up

D = \frac{V_{st} \, C_{st}}{V_{s}}

Water & WastewaterWater TreatmentChemistryConverts millilitres of stock solution added to a jar test beaker into the equivalent plant dose in mg/L of raw water.

Kirpich Time of Concentration

t_c = 0.0078 \, L^{0.77} \, S^{-0.385}

Water & WastewaterCivil & SurveyingTime of concentration of a small rural catchment by Kirpich's 1940 fit, published for channel length in feet and returned here in minutes whatever units you type.

Manning's Equation for Flow

Q = \frac{1}{n} A R^{2/3} S^{1/2}

Water & WastewaterFluid MechanicsOpen-channel discharge by Manning's equation in SI form, from flow area, roughness, hydraulic radius and slope.

Manning's Equation for Velocity

v = \frac{1}{n} R^{2/3} S^{1/2}

Water & WastewaterFluid MechanicsOpen-channel velocity by Manning's equation in SI form, from the roughness coefficient, hydraulic radius and channel slope.

Mean Cell Residence Time (Sludge Age)

\text{SRT} = \frac{V \, X}{Q_w \, X_w}

Water & WastewaterWater TreatmentSolids retention time of an activated sludge plant: the mass of solids under aeration divided by the mass wasted each day.

Muskingum Channel Storage

S = K\left[X I + (1 - X) O\right]

Water & WastewaterCivil & SurveyingFluid MechanicsStorage held in a river reach by the Muskingum method: a prism term set by the outflow plus a wedge term set by the inflow, weighted by X and scaled by the travel time K.

Net Water and Sewer Cost of a Cooling Tower

C = V_m \, p_w + (V_m - V_e) \, p_s

Water TreatmentFluid MechanicsWater & WastewaterFull water and sewer bill for a cooling tower: makeup charged at the water rate, plus only the volume actually discharged at the sewer rate.

Parshall Flume Free Flow

Q = 4 \, W \, H^{1.522 \, W^{0.026}}

Water & WastewaterFluid MechanicsFree-flow discharge through a Parshall flume of 1 to 8 ft throat width, with W and the head H in feet and Q in cubic feet per second.

Per-Capita Wastewater Flow

q = \frac{Q}{P}

Water & WastewaterWater TreatmentFluid MechanicsAverage wastewater contributed per person per day, from the plant flow and the population served, reported in gallons per capita per day.

Percent Backwash Water

\%BW = \frac{V_{bw}}{V_f} \times 100

Water & WastewaterWater TreatmentShare of a filter's production consumed by its own backwash, the housekeeping ratio that flags short runs and poor coagulation.

Population Equivalent

PE = \frac{W}{w}

Water & WastewaterWater TreatmentPopulation equivalent of a waste stream: its pollutant load divided by the load one person contributes, typically 60 g BOD per day.

Rational Method Peak Runoff

Q = \frac{C \, i \, A}{3.6 \times 10^{6}}

Water & WastewaterCivil & SurveyingFluid MechanicsPeak runoff from a small catchment: the runoff coefficient times the rainfall intensity times the drainage area, in the SI form that takes intensity in mm/h and area in square metres.

Return Activated Sludge Rate

Q_r = \frac{Q \, X}{X_r - X}

Water & WastewaterWater TreatmentReturn sludge flow needed to hold a target mixed liquor concentration, from a solids balance around the aeration basin.

SCS Curve Number Runoff Depth

Q = \frac{(P - 0.2S)^2}{P + 0.8S}, \quad S = \frac{25400}{CN} - 254

Water & WastewaterCivil & SurveyingDirect runoff depth from a storm by the SCS (NRCS) curve-number method, with the potential maximum retention S in millimetres taken from the curve number and the initial abstraction fixed at 0.2S.

SCS Lag Time of Concentration

t_c = \frac{L^{0.8} \, (S + 1)^{0.7}}{1140 \, Y^{0.5}}

Water & WastewaterCivil & SurveyingTime of concentration from the SCS watershed lag equation, in which the curve number carries the retardance: lag equals flow length to the 0.8 power times the retention term over 1900 times the root of the percent slope, and the time of concentration is the lag divided by 0.6.

SCS Triangular Unit Hydrograph Peak

q_p = \frac{2 \, A \, Q}{2.67 \, T_p}

Water & WastewaterCivil & SurveyingFluid MechanicsPeak discharge of the SCS synthetic triangular unit hydrograph, from the runoff volume spread over a base length of 2.67 times the time to peak.

Sewer Credit for Evaporated Water

C_c = V_e \, p_s

Water TreatmentFluid MechanicsWater & WastewaterSewer credit for water a tower evaporates: the volume that never reaches the drain, valued at the municipal sewer rate.

Sludge Volume from Dry Solids

V = \frac{m}{S \, \rho_w \, (P/100)}

Water & WastewaterWater TreatmentVolume of wet sludge holding a given mass of dry solids, from the solids percentage and the sludge specific gravity, at 1000 kg/m³ water.

Sludge Volume Index (SVI)

\text{SVI} = \frac{SV_{30}}{X}

Water & WastewaterWater TreatmentSludge volume index in mL/g: the volume a gram of mixed liquor solids occupies after settling thirty minutes in a litre cylinder.

Solids Capture in Dewatering

R = \frac{C_c (C_f - C_{fil})}{C_f (C_c - C_{fil})} \times 100

Water & WastewaterWater TreatmentPercent solids capture of a belt press, centrifuge or filter from the feed, cake and filtrate solids concentrations by mass balance.

Space Time and Space Velocity

\tau = \frac{V}{v_0}

ChemistryWater & WastewaterHow long the feed nominally spends in the reactor: vessel volume divided by volumetric feed rate. Space velocity is simply its reciprocal, 1/τ, and is the number catalyst vendors quote as LHSV or GHSV in reciprocal hours.

Specific Capacity of a Well

S_c = \frac{Q}{s}

Water & WastewaterCivil & SurveyingSpecific capacity of a water well: the pumping rate divided by the drawdown it produces, reported in cubic metres per day per metre of drawdown.

Specific Energy in an Open Channel

E = y + \frac{V^2}{2g}

Fluid MechanicsCivil & SurveyingWater & WastewaterSpecific energy: the total head measured from the channel bed, being the flow depth plus the velocity head. It is the quantity that stays constant across a smooth transition.

Specific Gravity of Sludge

\frac{1}{S_{sl}} = \frac{P/100}{S_s} + \left(1 - \frac{P}{100}\right)

Water & WastewaterWater TreatmentSpecific gravity of a wet sludge from its percent solids and the specific gravity of the dry solids, by a reciprocal mass balance.

Stokes Settling Velocity

v_s = \frac{g (\rho_s - \rho) d^2}{18 \mu}

Water & WastewaterFluid MechanicsTerminal settling velocity of a small sphere in laminar flow by Stokes' law — the grit chamber and clarifier design relation.

Submerged Culvert Inlet (Orifice) Headwater

Q = C_d \, A \sqrt{2g\left(HW - \tfrac{D}{2}\right)}

Civil & SurveyingFluid MechanicsWater & WastewaterCapacity of a culvert in inlet control with the inlet submerged, treated as an orifice discharging under the head between the headwater surface and the centroid of the barrel opening.

Surface Overflow Rate

v_o = \frac{Q}{A}

Water & WastewaterWater TreatmentFluid MechanicsSurface overflow (surface loading) rate of a settling basin — flow divided by plan area, reported here in metres per day.

Thiem Steady-State Well Drawdown

s = \frac{Q}{2 \pi T} \ln\frac{R}{r}

Water & WastewaterCivil & SurveyingSteady-state drawdown around a well fully penetrating a confined aquifer, by the Dupuit-Thiem equilibrium solution. Transmissivity is taken in square metres per day and the pumping rate is converted to a daily volume inside the brain.

Transmissivity from Conductivity and Thickness

T = K \, b

Water & WastewaterCivil & SurveyingTransmissivity of a confined aquifer as the hydraulic conductivity multiplied by the saturated thickness, returned in square metres per day for a conductivity entered in any speed unit.

Trickling Filter Hydraulic Loading

L_h = \frac{Q + Q_r}{A}

Water & WastewaterWater TreatmentFluid MechanicsHydraulic loading on a trickling filter including recirculation — total flow per unit of media surface area, in metres per day.

Trickling Filter Recirculation Factor

F = \frac{1 + R}{(1 + 0.1R)^2}

Water & WastewaterWater TreatmentNRC recirculation factor for trickling filter design, converting a recirculation ratio into the effective number of passes through the media.

V-Notch (Triangular) Weir Flow

Q = \frac{8}{15} C_d \sqrt{2g} \, \tan\!\frac{\theta}{2} \, H^{5/2}

Water & WastewaterFluid MechanicsDischarge over a sharp-crested triangular weir from the notch angle, head and discharge coefficient, in the standard theoretical form.

Van Kleeck Volatile Solids Reduction

R = \frac{V_{in} - V_{out}}{V_{in} - V_{in} V_{out}} \times 100

Water & WastewaterWater TreatmentPercent volatile solids destroyed in a digester by the Van Kleeck equation, using only the volatile fractions in and out.

Volumetric Organic Loading Rate

L_v = \frac{Q \, S_0}{V}

Water & WastewaterWater TreatmentOrganic load applied per unit of reactor volume, returned in kg BOD per cubic metre per day — the sizing number for basins and digesters.