Equation Sheet Builder

Pressure at depth h in a static fluid

Upward force on a fully or partially submerged object

Pressure difference from a manometer height reading

Height of liquid rise in a capillary tube due to surface tension

Force exerted along a contact line of length L

Resultant hydrostatic force on a flat submerged surface; acts at the centre of pressure (below centroid)

Depth of resultant force below the free surface; always deeper than the centroid for inclined/vertical plates

Initial stability indicator for a floating body; GM > 0 means stable, GM < 0 means unstable

Conservation of mass for steady, incompressible flow

Mass of fluid passing a cross-section per unit time

Energy conservation along a streamline (inviscid, incompressible)

Kinetic energy per unit volume; also called velocity pressure

Net force equals rate of change of momentum (control volume)

Bernoulli with head loss between two cross-sections

Total mechanical energy per unit weight at any cross-section; conserved along a streamline in ideal flow

Rate of momentum transport through a cross-section; appears in control volume force balances

Equation of state for a perfect gas; links pressure, density, and absolute temperature

Ratio of inertial to viscous forces; classifies laminar vs turbulent

Head loss due to pipe wall friction

Volumetric flow rate for laminar flow in a circular pipe

Darcy friction factor for laminar pipe flow (Re < 2300)

Implicit equation for turbulent friction factor (Moody chart basis)

Explicit approximation of Colebrook-White (error < 3%)

Head loss from fittings, valves, bends, entries and exits

Equivalent diameter for non-circular cross-sections

Pressure surge from instantaneous valve closure

Empirical velocity formula for water supply pipes

Volumetric flow through a sharp-edged orifice plate; C_d ≈ 0.61 for standard orifices

Volumetric flow through a Venturi meter; C_v ≈ 0.98, β = throat/pipe diameter ratio

Free-stream velocity from the difference between stagnation and static pressure

Exit velocity from a tank orifice at depth h below the free surface (Bernoulli applied to large tank)

Sum of static and dynamic pressure; constant along a streamline for incompressible inviscid flow

Fraction of dynamic pressure recovered as static pressure in a diffuser

Dimensionless local pressure margin above vapour pressure; cavitation begins when σ falls below the critical value

Head loss per unit pipe length; equals the slope of the hydraulic grade line for uniform pipe flow

Pipes in series share the same flow rate; total head loss is the sum of individual losses

Pipes in parallel share the same head loss; total flow is the sum of branch flows

Aerodynamic drag on a body immersed in flow

Aerodynamic lift on an airfoil or wing

Drag on a sphere at very low Reynolds number (Re < 1)

Laminar boundary layer thickness on a flat plate at distance x

Terminal velocity of a small sphere settling under gravity at very low Reynolds number (Re < 1)

Average skin-friction drag coefficient over an entire laminar flat plate (Blasius solution)

Effective reduction in flow passage width due to the slower-moving fluid inside the boundary layer

Ratio of inertial to gravitational forces; governs open channel flow

Ratio of flow speed to local speed of sound

Ratio of inertial to surface tension forces

Ratio of pressure forces to inertial forces

Ratio of unsteady to steady inertia; governs vortex shedding

Ratio of convective to conductive heat transfer

Ratio of momentum to thermal diffusivity

Ratio of buoyancy to viscous forces; governs natural convection

Product of Grashof and Prandtl; determines natural convection regime

Ratio of momentum to mass diffusivity; analogous to Prandtl for mass transfer

Ratio of advective to diffusive heat transport; Pe >> 1 means convection dominates over conduction

Dimensionless mass transfer coefficient; mass-transfer analogue of the Nusselt number

Ratio of thermal to mass diffusivity; Le ≈ 1 for air, important in simultaneous heat and mass transfer

Dimensionless heat transfer rate at the wall relative to the thermal capacity of the flow

Acoustic velocity in an ideal gas

Total temperature when flow is isentropically decelerated to rest

Static-to-stagnation temperature ratio for isentropic flow

Static-to-stagnation pressure ratio for isentropic flow

Mach number downstream of a normal shock (M₁ must be supersonic)

Cross-sectional area ratio relative to the sonic throat area A*

Maximum (choked) mass flow rate through a sonic throat

Maximum pipe length from Mach M to choking (Fanno line)

Static pressure ratio across a normal shock; always greater than 1 (pressure always jumps)

Turning angle for isentropic expansion around a convex corner; Δν = ν₂ − ν₁ gives the flow deflection

Stagnation temperature ratio for frictionless heat addition in a constant-area duct (Rayleigh line)

Theoretical head imparted by an impeller (no hydraulic losses)

Shaft power required by a pump, or delivered by a turbine

Scale pump flow, head, and power with rotational speed

Classifies pump type: radial (low Ωs) → mixed → axial (high Ωs)

Available suction head above vapour pressure — must exceed NPSH required

Peripheral velocity at the impeller outer diameter

Ratio of actual pump head to theoretical Euler head; accounts for internal hydraulic losses

Index relating pump speed and flow to NPSH; keep N_ss < 8500 (US units) to avoid cavitation

Ratio of available NPSH to pump head; cavitation occurs when σ drops below the critical value σ_c

Scale fan flow, pressure rise, and power with rotational speed at constant geometry and density

Mean velocity for uniform flow in open channels

Older form of Manning's for uniform open channel flow

Flow depth at minimum specific energy in a rectangular channel

Sequent depth ratio across a hydraulic jump (rectangular channel)

Total energy per unit weight measured from the channel bed

Discharge over a sharp-crested rectangular weir

Dimensionless bed shear stress; τ* > 0.047 initiates sediment motion

Implicit equation for normal depth in uniform open channel flow; solved iteratively or by software

Area and wetted perimeter for a trapezoidal cross-section; z = horizontal run per unit vertical rise

Discharge over a sharp-crested V-notch weir; H^{2.5} relationship makes it accurate at low flows

Convective heat transfer rate from a surface to a fluid

Conductive heat flux through a material

Log Mean Temperature Difference — effective driving force in a heat exchanger

Combined thermal resistance for a flat wall with fouling on both sides

Nusselt number for turbulent pipe flow (n = 0.4 heating, 0.3 cooling)

Efficiency of a straight fin with uniform cross-section

Actual heat transfer as a fraction of thermodynamic maximum

Resistance of conductive and convective layers (add in series)

Number of Transfer Units — dimensionless heat exchanger size used in the ε-NTU method

Average Nusselt number for natural convection on an isothermal vertical plate (all Rayleigh numbers)

Links heat transfer (Stanton number) to skin friction; allows h to be estimated from friction measurements