Fluid Mechanics formulas — 107 solvers

Air Changes per Hour (ACH)

\mathrm{ACH} = \frac{3600 \, \dot{V}}{V_{room}}

HVAC & HydronicsFluid MechanicsHow many times per hour a ventilation rate replaces the air in a room, the ventilation yardstick behind BTU/hr = 60 × CFM ÷ room volume.

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.

Barlow's Formula (Pipe Pressure Rating)

P = \frac{2 S t}{D}

HVAC & HydronicsFluid MechanicsMechanicsInternal pressure a pipe can hold from wall stress, wall thickness and outside diameter — the thin-wall hoop-stress relation used by pipeline codes.

Bernoulli's Equation (Two Points)

P_1 + \tfrac{1}{2}\rho v_1^{2} + \rho g z_1 = P_2 + \tfrac{1}{2}\rho v_2^{2} + \rho g z_2

Fluid MechanicsPhysicsConservation of mechanical energy along a streamline: pressure, velocity and elevation trade against one another for an incompressible fluid with no friction and no pump between the two points.

Blowdown Rate from Cycles

B = \frac{E}{\text{COC} - 1}

Water TreatmentFluid MechanicsBlowdown a cooling tower must bleed to hold a target cycles of concentration, given its evaporation rate.

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.

Buoyant Force (Archimedes' Principle)

F_b = \rho V g

Fluid MechanicsPhysicsThe upward force on a submerged body equals the weight of the fluid it displaces, with g = 9.80665 m/s².

Chemical-Consuming Loss Rate

L = B + D

Water TreatmentChemistryFluid MechanicsFlow that actually carries treatment out of a cooling tower — blowdown plus drift, because evaporation leaves every molecule of inhibitor behind.

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.

Colebrook–White Friction Factor

\frac{1}{\sqrt{f}} = -2 \log_{10}\!\left(\frac{\varepsilon}{3.7 D} + \frac{2.51}{Re \sqrt{f}}\right)

Fluid MechanicsHVAC & HydronicsPhysicsThe reference equation for turbulent friction in a rough pipe, and the curve every Moody diagram is drawn from. It has f on both sides, so this page solves it by iteration rather than by any closed formula.

Condenser Water Flow Rate

\dot{V} = \frac{\dot{Q} \cdot \mathrm{HRF}}{\rho_w c_w \, \Delta T}

HVAC & HydronicsFluid MechanicsWater TreatmentTower water flow needed to reject a chiller's load plus compressor heat, the physics behind the 3 gpm per ton at 10 °F rule of thumb.

Continuity Equation (A₁v₁ = A₂v₂)

A_1 v_1 = A_2 v_2

Fluid MechanicsWater TreatmentPhysicsFor incompressible flow, the same volume per second passes every cross-section of the pipe.

Cooling Tower Drift Loss

D = \frac{d}{100} \, R

Water TreatmentFluid MechanicsDrift (windage) loss from a cooling tower as a percentage of the recirculation rate, the fraction of basin water blown out as droplets.

Cooling Tower Evaporation Rate

E = 0.001 \, R \, \Delta T

Water TreatmentFluid MechanicsThermodynamicsEvaporation loss from a cooling tower using the industry rule of 0.1% of recirculation per degree Fahrenheit of range.

Cooling Tower Heat Rejection

Q = 500 \, R \, \Delta T

Water TreatmentThermodynamicsFluid MechanicsHeat a cooling tower rejects from flow and range using the trade constant 500 = 8.34 lb/gal × 60 min/h × 1 BTU/(lb·°F).

Cooling Tower Makeup Water Rate

M = E + B + D

Water TreatmentFluid MechanicsTotal makeup water a cooling tower needs: the sum of evaporation, blowdown to drain, and drift carried out in the air stream.

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.

Cycles of Concentration (COC = M/B)

\text{COC} = \frac{M}{B}

Water TreatmentFluid MechanicsCycles of concentration for a cooling tower from the makeup and blowdown flows — the master number every treatment program is built around.

Darcy–Weisbach Head Loss

h_f = f \, \frac{L}{D} \, \frac{v^{2}}{2g}

HVAC & HydronicsFluid MechanicsPhysicsThe rigorous pipe friction equation: head loss from friction factor, length-to-diameter ratio and velocity head, with g = 9.80665 m/s².

Darcy's Law for Groundwater Flow

Q = k\,i\,A

Soil MechanicsFluid MechanicsDarcy's law for laminar flow through soil: discharge equals hydraulic conductivity times hydraulic gradient times gross cross-sectional area.

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.

Dittus-Boelter Correlation

\mathrm{Nu} = 0.023 \, \mathrm{Re}^{0.8} \, \mathrm{Pr}^{n}

Heat TransferFluid MechanicsTurbulent tube-flow Nusselt number, valid for Re above 10,000, Pr from 0.6 to 160 and L/D over 10, with n = 0.4 heating and 0.3 cooling.

Dynamic Pressure (q = ½ρv²)

q = \tfrac{1}{2} \rho v^{2}

Fluid MechanicsPhysicsThe kinetic energy per unit volume of a moving fluid — the pressure of motion itself.

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.

Equivalent Horizontal Permeability of Layered Soil

k_{eq} = \frac{k_1 H_1 + k_2 H_2}{H_1 + H_2}

Soil MechanicsFluid MechanicsThickness-weighted equivalent permeability for flow parallel to the bedding of two soil layers, the parallel-resistance case of stratified seepage.

Equivalent Length of a Fitting

L_{eq} = \frac{K D}{f}

HVAC & HydronicsFluid MechanicsWater TreatmentConverts a fitting's K factor into the length of straight pipe that would cause the same friction loss at the same friction factor.

Equivalent Round Duct Diameter

D_e = 1.30 \frac{(ab)^{0.625}}{(a+b)^{0.25}}

HVAC & HydronicsFluid MechanicsHuebscher's equation for the round duct that has the same friction loss and airflow as a given rectangular duct of sides a and b.

Expansion Loop Leg Length (Guided Cantilever)

L = \sqrt{\frac{3 E D \, \Delta}{S_a}}

HVAC & HydronicsFluid MechanicsMechanicsLeg length an expansion loop or offset needs to absorb a given thermal movement without exceeding the pipe's allowable stress.

Expansion Tank Acceptance Volume

V_t = \frac{V_s \, e}{1 - \dfrac{P_1}{P_2}}

HVAC & HydronicsFluid MechanicsDiaphragm expansion tank size for a closed hydronic loop from system volume, water expansion and the absolute fill and relief pressures.

Fan Affinity Law — Airflow vs Speed

\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsPhysicsFan airflow in CFM changes in direct proportion to wheel speed, the first law used when re-sheaving a belt-driven air handler.

Fan Affinity Law — Power vs Speed

\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsPhysicsFan brake power varies with the cube of wheel speed — the law behind variable-air-volume energy savings and behind burnt-out re-sheaved motors.

Fan Affinity Law — Static Pressure vs Speed

\frac{SP_2}{SP_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsPhysicsFan static pressure rises with the square of wheel speed, the reason a modest re-sheave can overpressurise ductwork and blow out flex connections.

Fan Brake Horsepower

BHP = \frac{Q \cdot SP}{6356 \, \eta}

HVAC & HydronicsFluid MechanicsPhysicsShaft power a fan absorbs; the 6356 divisor assumes cubic feet per minute, inches of water gauge and horsepower at the given efficiency.

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.

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).

Gauge and Absolute Pressure

P_{abs} = P_{gauge} + P_{atm}

Fluid MechanicsThermodynamicsPhysicsAbsolute pressure is the gauge reading plus the surrounding atmospheric pressure.

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.

Hazen–Williams Head Loss

h_f = \frac{10.67 \, L \, Q^{1.852}}{C^{1.852} D^{4.8704}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe waterworks head-loss equation in SI form, with Q in m³/s and D in m; the 10.67 constant is 4.727 when working in feet and cubic feet per second.

Hazen–Williams Velocity

v = 0.849 \, C \, R^{0.63} S^{0.54}

HVAC & HydronicsFluid MechanicsWater TreatmentMean water velocity from hydraulic radius and hydraulic gradient; the 0.849 SI constant becomes 1.318 when R is in feet and v in feet per second.

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 Gradient

i = \frac{\Delta h}{L}

Soil MechanicsFluid MechanicsHydraulic gradient as the loss of total head divided by the length of the flow path, the dimensionless driving force behind all seepage.

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 Power (P = ρgQh)

P = \rho g Q h

Fluid MechanicsWater TreatmentPhysicsPower needed to lift a flow Q through a head h, with g = 9.80665 m/s².

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.

Hydronic Heat Transfer (Water)

\dot{Q} = \rho_w c_w \dot{V} \, \Delta T

HVAC & HydronicsThermodynamicsFluid MechanicsHeat carried by a water loop from flow rate and supply-to-return ΔT — the SI form of the trade rule BTU/hr = 500 × GPM × ΔT.

Hydronic Static Fill Pressure

P = \rho_w g H + P_{margin}

HVAC & HydronicsFluid MechanicsCold fill pressure a closed loop needs to lift water to its highest point plus a safety margin, the SI form of the 2.31 ft per psi rule.

Laminar Friction Factor (f = 64/Re)

f = \frac{64}{Re}

HVAC & HydronicsFluid MechanicsPhysicsIn laminar pipe flow the Darcy friction factor depends only on Reynolds number — roughness plays no part below about Re = 2300.

Loop Water Expansion Volume

\Delta V = V_0 \, \beta \, \Delta T

HVAC & HydronicsFluid MechanicsThermodynamicsVolume a hydronic loop's water gains when heated, from the starting volume, the volumetric expansion coefficient and the temperature rise.

Mach Number

Ma = \frac{v}{\sqrt{\gamma R T / M}}

Fluid MechanicsPhysicsSpeed divided by the local speed of sound in the same gas. Below about Ma = 0.3 a gas can be treated as incompressible, which is the assumption behind most duct and pipe sizing.

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.

Minor Loss from K Factor

h_L = K \, \frac{v^{2}}{2g}

HVAC & HydronicsFluid MechanicsWater TreatmentHead lost through a valve or fitting as a multiple of velocity head, with g = 9.80665 m/s² and K taken from a fitting table.

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 Positive Suction Head Available (NPSHa)

NPSH_a = h_{atm} + h_s - h_f - h_{vp}

HVAC & HydronicsFluid MechanicsWater TreatmentAbsolute head available at the pump suction above the liquid's vapour pressure — the margin that keeps a pump from cavitating.

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.

Nusselt Number

\mathrm{Nu} = \frac{h L}{k}

Heat TransferFluid MechanicsThermodynamicsDimensionless convection coefficient: the ratio of convective transfer at a surface to pure conduction through the same fluid layer.

Orifice Plate Flow

Q = \frac{C_d}{\sqrt{1 - \beta^{4}}}\cdot\frac{\pi d^{2}}{4}\sqrt{\frac{2\,\Delta P}{\rho}}

Fluid MechanicsPhysicsVolumetric flow through a square-edged orifice plate from the differential pressure across it. The 1/√(1−β⁴) term is the velocity-of-approach factor, with β the bore-to-pipe diameter ratio.

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.

Partially Filled Horizontal Cylindrical Tank

V = L \left[ r^{2} \cos^{-1}\!\left(\frac{r-h}{r}\right) - (r-h)\sqrt{2rh - h^{2}} \right]

HVAC & HydronicsFluid MechanicsGeometryLiquid volume in a horizontal cylinder from the wetted depth, using the circular segment area times the tank length.

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 Blowdown

\%B = \frac{B}{M} \times 100

Water TreatmentFluid MechanicsBlowdown expressed as a percentage of makeup water — the share of purchased water that goes straight to the sewer.

Pipe Internal Volume

V = \frac{\pi D^{2}}{4} L

HVAC & HydronicsFluid MechanicsWater TreatmentThe liquid a run of pipe holds, from inside diameter and developed length — the starting point for every flush, fill or chemical dose.

Pipe Velocity from Flow and Diameter

v = \frac{4Q}{\pi D^{2}}

HVAC & HydronicsFluid MechanicsWater TreatmentAverage velocity in a full round pipe from volumetric flow and inside diameter — the first check on any piping design.

Poiseuille's Law

Q = \frac{\pi \, \Delta P \, r^{4}}{8 \mu L}

Fluid MechanicsPhysicsLaminar flow rate through a round pipe — proportional to the fourth power of the radius.

Pore Water Pressure (u = γw zw)

u = \gamma_w z_w

Soil MechanicsFluid MechanicsHydrostatic pore water pressure at a point below a static water table, from the depth of water standing above it.

Prandtl Number

\mathrm{Pr} = \frac{\mu c_p}{k}

Heat TransferFluid MechanicsThermodynamicsFluid property group comparing how fast momentum diffuses with how fast heat does, setting the relative thickness of the two boundary layers.

Pressure Head (h = P/ρg)

h = \frac{P}{\rho g}

Fluid MechanicsWater TreatmentPhysicsConverts a pressure into the equivalent height of a fluid column, with g = 9.80665 m/s².

Pump Affinity Law — Flow vs Impeller Diameter

\frac{Q_2}{Q_1} = \frac{D_2}{D_1}

HVAC & HydronicsFluid MechanicsWater TreatmentCapacity scales directly with trimmed impeller diameter at constant speed, the classic way to de-rate an oversized centrifugal pump permanently.

Pump Affinity Law — Flow vs Speed

\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsWater TreatmentFirst affinity law: a centrifugal pump's capacity changes in direct proportion to shaft speed when the impeller diameter is unchanged.

Pump Affinity Law — Head vs Impeller Diameter

\frac{H_2}{H_1} = \left(\frac{D_2}{D_1}\right)^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentDeveloped head falls with the square of the trimmed impeller diameter, so a 10% trim sheds about 19% of the head at constant speed.

Pump Affinity Law — Head vs Speed

\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentSecond affinity law: pump head varies with the square of shaft speed, so a 20% speed cut costs 36% of the developed head.

Pump Affinity Law — Power vs Speed

\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsWater TreatmentThird affinity law: absorbed power varies with the cube of shaft speed — the single relation that pays for every variable-frequency drive.

Pump Brake Horsepower

BHP = \frac{Q \, H \, SG}{3960 \, \eta}

HVAC & HydronicsFluid MechanicsWater TreatmentShaft power the motor must actually supply; the 3960 constant assumes gpm, feet of head and horsepower, with efficiency as a fraction.

Pump Efficiency from Hydraulic and Shaft Power

\eta = \frac{P_{hyd}}{P_{shaft}}

HVAC & HydronicsFluid MechanicsWater TreatmentPump efficiency is the ratio of hydraulic power delivered to the liquid over the mechanical power absorbed at the shaft.

Pump Specific Speed (Ns)

N_s = \frac{N \sqrt{Q}}{H^{0.75}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe dimensional index that classifies impeller type, evaluated in US units with N in rpm, Q in gpm and H in feet at the best efficiency point.

Pump Water Horsepower

WHP = \frac{Q \, H \, SG}{3960}

HVAC & HydronicsFluid MechanicsWater TreatmentUseful power delivered to the liquid; the 3960 divisor assumes US gallons per minute, feet of head and horsepower output.

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.

Reynolds Number

Re = \frac{\rho v D}{\mu}

Fluid MechanicsPhysicsThe dimensionless ratio of inertial to viscous forces that decides laminar versus turbulent flow.

Round Duct Air Velocity

v = \frac{4 \dot{V}}{\pi d^{2}}

HVAC & HydronicsFluid MechanicsAir velocity in a round duct from the volume flow and the duct diameter, the check that keeps branches quiet and mains efficient.

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.

Seepage Velocity from Discharge Velocity

v_s = \frac{v}{n}

Soil MechanicsFluid MechanicsActual seepage velocity through the pores, obtained by dividing Darcy's fictitious discharge velocity by the porosity of the soil.

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.

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

SG = \frac{\rho}{\rho_{water}}

Fluid MechanicsWater TreatmentChemistryDensity expressed as a multiple of water's 1000 kg/m³.

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.

Stokes' Drag (F = 6πμrv)

F = 6\pi \mu r v

Fluid MechanicsPhysicsViscous drag on a small sphere creeping through a fluid at low Reynolds number.

Submerged (Buoyant) Unit Weight

\gamma' = \gamma_{sat} - \gamma_w

Soil MechanicsFluid MechanicsEffective or buoyant unit weight of soil below the water table, the saturated unit weight less the uplift of the water it displaces.

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.

Swamee–Jain Friction Factor

f = \frac{0.25}{\left[\log_{10}\!\left(\frac{\varepsilon}{3.7D} + \frac{5.74}{Re^{0.9}}\right)\right]^{2}}

HVAC & HydronicsFluid MechanicsPhysicsAn explicit turbulent friction factor within about 1% of the implicit Colebrook–White equation, valid for Re from 5000 to 10⁸.

System Volume from Turnover Time

V = R \, t

Water TreatmentFluid MechanicsSystem water volume estimated from the recirculation rate and the measured turnover time — the field method when no drawings exist.

Time Factor for Consolidation

T_v = \frac{c_v\,t}{H_{dr}^{2}}

Soil MechanicsFluid MechanicsDimensionless time factor of Terzaghi consolidation theory, with the coefficient of consolidation entered in m²/s and the longest drainage path.

Time Factor from Degree of Consolidation (U ≤ 60%)

T_v = \frac{\pi}{4}\left(\frac{U}{100}\right)^{2}

Soil MechanicsFluid MechanicsTerzaghi's parabolic approximation relating the time factor to the average degree of consolidation, valid for U of 60 percent or less.

Torricelli's Law (v = √(2gh))

v = \sqrt{2 g h}

Fluid MechanicsWater TreatmentPhysicsSpeed of fluid jetting from an opening a depth h below the free surface, with g = 9.80665 m/s².

Total Dynamic Head

TDH = h_s + h_f + h_v

HVAC & HydronicsFluid MechanicsWater TreatmentThe head a pump must develop: static lift plus friction losses plus velocity head, all expressed in feet or metres of the pumped liquid.

Total Water Treatment Program Cost

C = C_w + C_p + C_e

Water TreatmentChemistryFluid MechanicsTotal operating cost of a treated cooling system: the water and sewer bill, the chemical invoice and the energy bill added together.

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.

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.

Valve Flow Coefficient (Cv)

Q = C_v \sqrt{\frac{\Delta P}{SG}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe US valve-sizing relation: Cv is the gpm of 60 °F water a valve passes at 1 psi drop, so Q is in gpm and ΔP in psi.

Valve Flow Coefficient (Kv, metric)

Q = K_v \sqrt{\frac{\Delta p}{SG}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe metric valve-sizing relation: Kv is the m³/h of water a valve passes at 1 bar drop, related to Cv by Cv ≈ 1.156 Kv.

Velocity Head (h = v²/2g)

h_v = \frac{v^{2}}{2g}

Fluid MechanicsWater TreatmentPhysicsThe kinetic energy of a flow expressed as an equivalent column height, with g = 9.80665 m/s².

Venturi Meter Flow

Q = \frac{C}{\sqrt{1 - \beta^{4}}}\cdot\frac{\pi D_2^{2}}{4}\sqrt{\frac{2\,\Delta P}{\rho}}

Fluid MechanicsPhysicsFlow through a classical Venturi tube from the inlet-to-throat differential pressure. The gently converging cone recovers most of the pressure it borrows, so the discharge coefficient sits near 0.98 rather than the orifice plate's 0.61.

Volume of Water Over a Period

V = Q \, t

Water TreatmentFluid MechanicsWater a flow delivers over a period — the step that turns a makeup or blowdown rate into the daily or annual volume a customer is billed for.

Volumetric Flow Rate (Q = Av)

Q = A v

Fluid MechanicsWater TreatmentPhysicsFlow through a duct or pipe: cross-sectional area times average flow velocity.

Water Hammer Surge (Joukowsky Equation)

\Delta P = \rho \, a \, \Delta v

HVAC & HydronicsFluid MechanicsPhysicsPeak pressure surge from a sudden change in flow velocity: fluid density times pressure-wave celerity times the velocity change.