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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.

Van der Waals Equation of State

\left(P + \frac{a n^{2}}{V^{2}}\right)\left(V - n b\right) = n R T

ChemistryThermodynamicsThe first equation of state to describe a real gas, adding a term for molecular attraction (a) and one for the volume the molecules themselves occupy (b). Constant a is entered in Pa·m⁶/mol² and b in volume per mole.

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.

Vapour Pressure Deficit (Air)

VPD = e_s(T)\left(1 - \frac{RH}{100}\right)

Indoor & GreenhouseCrop ProductionHow much more water vapour the air could hold, from air temperature and relative humidity. The number that governs transpiration, and the one growers steer a room by instead of humidity alone.

Variance and Standard Deviation

\sigma^2 = \sigma \cdot \sigma

StatisticsAlgebraConverts between variance and standard deviation: the variance is the square of the standard deviation, and the deviation is its square root.

Variance of a Sum of Dice

\sigma^{2} = \frac{n \, (d^{2} - 1)}{12}

Games & RatingsStatisticsThe spread of the total from n identical dice of d faces. Because the dice are independent their variances simply add, which is why rolling several small dice gives a much tighter total than rolling one large die of the same average — the classic reason a designer chooses three six-sided dice over one twenty-sided one.

Vector Subtraction (Component Form)

c_x = a_x - b_x

Vectors & MatricesAlgebraPhysicsSubtracts one vector from another one component at a time, the operation behind every relative-velocity and displacement-change problem.

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.

Velocity-Displacement Relation (v² = v₀² + 2ad)

v^2 = v_0^2 + 2 a d

MechanicsPhysicsLinks initial and final speeds to acceleration and displacement without involving time.

Ventilation-Limited Burning Rate (Kawagoe)

\dot{m} = 0.5 \, A_0 \sqrt{H_0}

Fire Protection & Fire DynamicsKunio Kawagoe's 1958 result, and one of the oldest quantitative facts in fire science: in a fully developed compartment fire the fuel burns at a rate set by the opening, not by the fuel. Half a kilogram of wood per second for every square metre of opening times the square root of its height.

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.

Vertex x-Coordinate of a Parabola

h = -\frac{b}{2a}

AlgebraGeometryThe x-coordinate of a parabola's vertex, sitting midway between the two roots and marking the axis of symmetry.

Vertex y-Coordinate of a Parabola

k = c - \frac{b^2}{4a}

AlgebraGeometryThe y-coordinate of a parabola's vertex — its minimum value when a is positive, and its maximum when a is negative.

Vertical Curve Length from K Value

L = K\,A

Civil & SurveyingLength of a parabolic vertical curve as the design K value times the algebraic grade change, K being length per percent of grade.

Vertical Precision from Base-to-Height Ratio

\sigma_Z = \frac{H}{B} \cdot \frac{H}{f} \cdot \sigma_p

Photogrammetry & Remote SensingThe height precision a stereo geometry can deliver: the parallax measuring precision, magnified by the image scale and again by the inverse of the base-to-height ratio. This is the number that should be quoted as a survey's vertical accuracy, and it is several times larger than the ground sample distance everyone quotes instead.

Vertical Velocity Component

v_y = v \sin\theta

MechanicsPhysicsVertical component of a projectile's launch velocity — the part of the speed that fights gravity and sets the time aloft.

Vibration Isolation Efficiency

I = 1 - TR

Machine DesignTransmissibility restated the way a specification writes it: the percentage of the shaking force the mounts keep off the floor. A negative answer is the honest report that the mounts are making it worse.

Vickers Hardness

HV = \dfrac{2F \sin(136^\circ/2)}{d^{2}} = \dfrac{1.8544\,F}{d^{2}}

Metallurgy & Heat TreatmentPress a square-based diamond pyramid into the metal, measure the two diagonals of the mark it leaves, and divide the load by the SURFACE AREA of that mark. The 1.8544 is pure geometry from the indenter's 136° face angle, and because the pyramid is self-similar, the number is independent of the load — which no other common hardness scale can claim.

View Factor Reciprocity

A_1 F_{1 \to 2} = A_2 F_{2 \to 1}

Heat TransferThermodynamicsReciprocity relation for radiation view factors, which lets you recover the unknown factor between two surfaces from the known one and their areas.

Vogel-Fulcher-Tammann Viscosity

\log_{10}\eta = A + \frac{B}{T - T_0}

Ceramics & GlassGlass viscosity spans roughly fifteen orders of magnitude between the melt and the annealed solid, and no Arrhenius line goes near it. The Vogel-Fulcher-Tammann form — published independently by Vogel in 1921, Fulcher in 1925 and Tammann and Hesse in 1926 — bends the line by putting a divergence temperature in the denominator, and it fits.

Void Ratio and Porosity (e = n/(1 − n))

e = \frac{n}{1 - n}

Soil MechanicsStrength of MaterialsConverts between void ratio, the void volume per unit of solid, and porosity, the void volume per unit of total soil volume.

Voltage Divider

V_{out} = V_{in} \frac{R_{2}}{R_{1} + R_{2}}

Electrical TradeElectricity & MagnetismOutput of two resistors in series across a source: the input voltage split in proportion to the lower resistor's share.

Voltage Drop, Single Phase

V_{d} = \frac{2 \rho L I}{A}

Electrical TradeElectricity & MagnetismVoltage lost in a single-phase run from conductor resistivity, one-way length, current and area — the 2 counts both conductors.

Voltage Drop, Three Phase

V_{d} = \frac{\sqrt{3} \, \rho L I}{A}

Electrical TradeElectricity & MagnetismLine-to-line voltage drop on a balanced three-phase run, using √3 rather than 2 because there is no return conductor.

Voltage Regulation

\%VR = \frac{100 \, (V_{nl} - V_{fl})}{V_{fl}}

Electrical TradeElectricity & MagnetismHow far a source's terminal voltage sags between no load and full load, stated as a percentage of the full-load voltage.

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 Energy Density (Laser Powder Bed Fusion)

E_v = \frac{P}{v \, h \, t}

Additive ManufacturingLaser power divided by scan speed, hatch spacing and layer thickness: the joules the beam spends on every cubic millimetre of powder bed. Every machine log records it and every parameter paper quotes it — and it is a bookkeeping number, not a process window. Four independent variables go in and one number comes out, so the information that decides whether the track is sound, balled or keyholing is destroyed on the way through.

Volumetric Flow Rate (Q = Av)

Q = A v

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

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.