Water & Wastewater formula solvers

Categories — 73 of them, 1787 solvers in all

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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.

Cooper-Jacob Drawdown

s = \frac{2.303 \, Q}{4 \pi T} \log_{10}\!\left(\frac{2.25 \, T \, t}{r^{2} S}\right)

Water & WastewaterTransient drawdown in a confined aquifer by the Cooper-Jacob (1946) straight-line approximation: the Theis solution with its infinite series truncated after two terms, valid once the dimensionless time u has fallen below 0.01.

Cooper-Jacob Validity Parameter u

u = \frac{r^{2} S}{4 T t}

Water & WastewaterThe dimensionless argument of the Theis well function. Cooper-Jacob's straight-line approximation is only true once u has fallen below 0.01, and this is the calculation that tells you whether you have waited long enough.

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.

Day–Evening–Night Sound Level (Lden)

L_{den} = 10\log_{10}\!\left(\frac{12 \cdot 10^{L_d/10} + 4 \cdot 10^{(L_e + 5)/10} + 8 \cdot 10^{(L_n + 10)/10}}{24}\right)

Acoustics & NoiseWater & WastewaterThe European Environmental Noise Directive's indicator: a 24-hour energy average over three periods, with 5 dB added to the evening and 10 dB to the night. It is what every EU noise map and action plan is drawn in.

Day–Night Average Sound Level (Ldn)

L_{dn} = 10\log_{10}\!\left(\frac{15 \cdot 10^{L_d/10} + 9 \cdot 10^{(L_n + 10)/10}}{24}\right)

Acoustics & NoiseWater & WastewaterThe twenty-four-hour energy average of a community's noise with a 10 dB penalty added to every night-time hour — the metric American land-use, airport and highway noise rules are written in. A source that runs only at night is charged ten times its energy.

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.

Exceedance Risk over a Design Life

R = 1 - \left(1 - \frac{1}{T}\right)^{n}

Water & WastewaterCivil & SurveyingProbabilityThe probability that a storm of return period T is equalled or exceeded at least once during n years of exposure. This is the calculation that turns a reassuring-sounding return period into the number that actually matters: a 100-year storm has a 26 percent chance of turning up during a 30-year mortgage.

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.

Flow-Weighted Blending Concentration

C = \frac{Q_1 C_1 + Q_2 C_2}{Q_1 + Q_2}

Water & WastewaterWater TreatmentChemistryThe concentration you land on when two streams of known flow and known strength join — each stream's mass rate added, then divided by the combined flow. It is the arithmetic behind blending a hard well against a soft surface source, and behind every question about what a side stream does to a plant's influent.

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).

Ghyben-Herzberg Freshwater Lens

z = \frac{\rho_f}{\rho_s - \rho_f} \, h \;\approx\; 40 \, h

Water & WastewaterDepth to the freshwater-saltwater interface below sea level in a coastal or island aquifer, from the hydrostatic balance Ghyben (1888) and Herzberg (1901) found independently. With fresh water at 1000 kg/m³ and seawater at 1025, the ratio is exactly 40, which is where the familiar 40:1 rule comes from.

Green-Ampt Infiltration Rate

f = K\left(1 + \frac{\psi \, \Delta\theta}{F}\right)

Water & WastewaterCivil & SurveyingSoil MechanicsThe Green-Ampt model: infiltration rate under ponded conditions, derived by applying Darcy's law across a sharp wetting front that advances into the soil like a piston. Unlike Horton's curve it is not a fit — every parameter is a measurable soil property.

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.

Horton Infiltration Rate

f = f_c + (f_0 - f_c) \, e^{-kt}

Water & WastewaterCivil & SurveyingSoil MechanicsHorton's empirical infiltration curve: the rate starts at a dry-soil capacity, decays exponentially as the surface layer saturates, and settles to a steady final rate. It is a fitted description of what soils are observed to do, not a derivation from soil physics.

Hvorslev Slug Test Conductivity

K = \frac{r_c^{2} \ln(L / R)}{2 \, L \, T_0}

Water & WastewaterHydraulic conductivity from a slug test by Hvorslev's method (USACE Waterways Experiment Station Bulletin 36, 1951). Drop a slug into the well, plot the head ratio on a log scale against time, and read the basic time lag T₀ where the ratio reaches 0.37 — that is 1/e, one time constant of an exponential recovery.

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.

IDF Rainfall Intensity (Three-Parameter)

i = \frac{a}{(t + b)^{c}}

Water & WastewaterCivil & SurveyingFluid MechanicsThe general intensity-duration-frequency form: design rainfall intensity from the storm duration, using the three constants a, b and c fitted to a local rain gauge record for one return period. The constants are regional and unit-bound — a is stated here for intensity in mm/h against duration in minutes — and the duration you type is converted to minutes before the fit is applied.

Jacob Distance-Drawdown Transmissivity

T = \frac{2.303 \, Q}{2 \pi (s_1 - s_2)} \log_{10}\!\frac{r_2}{r_1}

Water & WastewaterTransmissivity from two observation wells read at the same instant during a constant-rate pumping test. Subtracting one Cooper-Jacob drawdown from the other cancels the storage term entirely, which is why this is the direction that can actually be solved for T.

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.

Lime Dose for Softening (as CaCO₃)

L = \mathrm{CO_2} + \mathrm{Alk} + \mathrm{Mg} + \mathrm{Ex}

Water TreatmentChemistryWater & WastewaterThe stoichiometric lime requirement for excess-lime softening, in mg/L as CaCO₃: every acid the lime has to neutralise before it can precipitate anything, plus the magnesium it must convert, plus the excess that makes the reaction finish in a real basin.

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.

Rainfall Depth from Intensity and Duration

P = i \, t

Water & WastewaterCivil & SurveyingFluid MechanicsThe bridge between the two ways rainfall is quoted: a depth is an intensity held for a duration. An IDF curve gives an intensity, the rational method consumes an intensity, and the curve-number and soil-loss methods consume a depth — this is the one line that connects them.

Rational Method Peak Runoff

Q = C \, i \, A

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.

Return Period from Rank (Weibull)

T = \frac{n + 1}{m}

Water & WastewaterCivil & SurveyingProbabilityThe Weibull plotting position: rank the annual maxima from largest to smallest and the event at rank m in a record of n years is assigned a return period of (n+1)/m years. This is where return periods come from before any distribution is fitted — they are counted, not calculated.

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.

Sherman IDF Rainfall Intensity

i = \frac{a}{t^{c}}

Water & WastewaterCivil & SurveyingFluid MechanicsSherman's power-law IDF form: intensity falls as the duration raised to a fitted negative power, with no offset term. It is the straight line an IDF curve becomes on log-log paper, which is why a and c can be read off a fitted plot directly. Both are regional constants for a single return period, and a assumes intensity in mm/h against duration in minutes.

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.

Snow Water Equivalent (SWE)

\mathrm{SWE} = d \, \frac{\rho_s}{\rho_w}

Snow & IceWater & WastewaterThermodynamicsThe depth of water you would be left with if a snowpack melted where it lies — snow depth scaled by the ratio of snow density to water density. It is the number hydrology, irrigation and flood forecasting actually use, because depth alone says nothing about how much water is standing on the ground.

Snowpack Settlement (Viscous Compaction)

\rho_s = \rho_0 \, \exp\!\left( \frac{\sigma \, t}{\eta} \right)

Snow & IceSoil MechanicsWater & WastewaterHow much a layer of snow densifies under a constant overburden, treating snow as a Newtonian fluid in compaction — Kojima's model, in the form land-surface schemes have used since Anderson wrote SNTHERM. It is the reason a snowpack is shallower in the morning than the depth board said last night, with the same water still in it.

Soda Ash Dose for Softening (as CaCO₃)

S = \mathrm{TH} - \mathrm{Alk}

Water TreatmentChemistryWater & WastewaterSoda ash makes up the difference between the hardness a water carries and the alkalinity it carries — the noncarbonate hardness, which lime cannot remove because no bicarbonate is paired with it. Expressed, as everything in softening is, in mg/L as CaCO₃.

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.

Step-Drawdown Test: Aquifer Loss and Well Loss

s = BQ + CQ^{2}

Water & WastewaterJacob's (1947) separation of the drawdown inside a pumping well into two parts: BQ, the laminar loss the aquifer imposes, and CQ², the turbulent loss the well's own screen, gravel pack and borehole impose. Q is taken in cubic metres per day, so B is in d/m² and C in d²/m⁵.

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.

Storativity from Specific Storage

S = S_s \, b

Water & WastewaterStorativity of a confined aquifer as the specific storage multiplied by the saturated thickness. Specific storage is the volume of water a unit volume of aquifer releases per unit drop in head, and it comes from the compressibility of the skeleton and of the water itself.

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.

Talbot IDF Rainfall Intensity

i = \frac{a}{t + b}

Water & WastewaterCivil & SurveyingFluid MechanicsTalbot's two-parameter IDF form, the three-parameter curve with the exponent fixed at one. Two regional constants instead of three, fitted to a local record for a single return period; a is stated here for intensity in mm/h against duration in minutes, and the duration you type is converted before the fit is applied.

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.

Weir Loading Rate (Weir Overflow Rate)

W_L = \frac{Q}{L_w}

Water & WastewaterWater TreatmentFluid MechanicsFlow divided by the total length of weir crest it leaves over — the loading on a clarifier's collection weirs, reported here in cubic metres per day per metre of weir, which is numerically a square metre per day.

Well Efficiency

E = \frac{BQ}{BQ + CQ^{2}} = \frac{B}{B + CQ}

Water & WastewaterThe fraction of a well's drawdown that the aquifer is responsible for, rather than the well's own construction. Everything below 100 per cent is head thrown away in turbulence through the screen and the gravel pack, and unlike the aquifer, that part can be fixed.

Well Screen Entrance Velocity

V_e = \frac{Q}{\pi \, D \, L \, P}

Water & WastewaterVelocity of water entering a well screen: the pumping rate divided by the open area, which is the screen's cylindrical surface multiplied by the fraction of it that is actually slot. The long-standing design limit is 0.03 m/s, or 0.1 ft/s.