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Density ($\rho$)

Mass per unit volume of a substance, expressed as $\rho = \frac{m}{V}$, with standard SI units of $kg\,m^{-3}$.

Unit Weight ($\gamma$)

Weight per unit volume of a substance, expressed as $\gamma = \frac{W}{V} = \rho \cdot g$, with standard SI units of $N\,m^{-3}$ or $kN\,m^{-3}$.

Specific Volume ($v$)

Volume occupied by a unit mass of a fluid, defined as the reciprocal of density: $v = \frac{V}{m} = \frac{1}{\rho}$, with standard SI units of $m^3\,kg^{-1}$.

Specific Gravity ($SG$)

Dimensionless ratio of the density or unit weight of a fluid to that of a standard reference fluid (water at $4\,^\circ C$ for liquids, air for gases): $SG = \frac{\rho_{fluid}}{\rho_{water}} = \frac{\gamma_{fluid}}{\gamma_{water}}$.

Absolute Viscosity ($\mu$)

Measure of a fluid's resistance to shear deformation, defined by Newton's law of viscosity: $\tau = \mu \cdot \frac{du}{dy}$, with SI units of $Pa \cdot s$ or $N \cdot s\,m^{-2}$.

Kinematic Viscosity ($\nu$)

Ratio of absolute viscosity to mass density: $\nu = \frac{\mu}{\rho}$, representing a fluid's molecular diffusivity of momentum, with SI units of $m^2\,s^{-1}$.

Surface Tension ($\sigma$)

Tensile force acting on the surface film of a liquid per unit length, caused by unbalanced cohesive forces, with SI units of $N\,m^{-1}$.

Capillary Rise in Tubes

Vertical height $h$ a liquid rises or falls in a small circular tube of radius $r$, calculated as $h = \frac{2 \cdot \sigma \cdot \cos(\theta)}{\gamma \cdot r}$.

Capillary Rise Between Parallel Plates

Vertical height $h$ a liquid rises between two vertical parallel plates separated by distance $t$, calculated as $h = \frac{2 \cdot \sigma \cdot \cos(\theta)}{\gamma \cdot t}$.

Bulk Modulus of Compression ($E_b$)

Measure of a fluid's resistance to change in volume under pressure, defined as $E_b = -\frac{dP}{\left(\frac{dV}{V}\right)} = \rho \cdot \frac{dP}{d\rho}$, with SI units of $Pa$.

Compressibility ($\beta$)

Fractional change in volume per unit pressure increase, defined as the reciprocal of bulk modulus: $\beta = \frac{1}{E_b}$, with SI units of $Pa^{-1}$.

Unit Pressure ($P$)

Normal compressive force exerted by a fluid per unit surface area, given by $P = \frac{F}{A}$, with SI units of $Pa$ ($N\,m^{-2}$).

Absolute Pressure ($P_{abs}$)

Total pressure relative to absolute zero pressure (a perfect vacuum), calculated as $P_{abs} = P_{gage} + P_{atm}$.

Standard Air Pressure Values

Standard atmospheric pressure at sea level, equal to $101.325\,kPa$, $1.01325\,bar$, $14.7\,psi$, $760\,mmHg$, or $10.33\,m\,H_2O$.

Pascal's Law (Equality of Pressure)

Principle stating that pressure applied to a confined static fluid is transmitted undiminished equally in all directions throughout the fluid.

Hydrostatic Pressure Formula

Equation giving the fluid pressure at a specific depth $h$ in a static fluid: $P = \gamma \cdot h = \rho \cdot g \cdot h$.

Density Formula ($\rho$)

$\rho = \frac{m}{V}$ where $m$ is mass and $V$ is volume. Standard SI unit: $kg\,m^{-3}$.

Unit Weight Formula ($\gamma$)

$\gamma = \frac{W}{V} = \rho \cdot g$ where $W$ is weight, $V$ is volume, $\rho$ is density, and $g$ is acceleration due to gravity. Standard SI unit: $N\,m^{-3}$.

Specific Volume Formula ($v$)

$v = \frac{V}{m} = \frac{1}{\rho}$ where $V$ is volume, $m$ is mass, and $\rho$ is density. Standard SI unit: $m^3\,kg^{-1}$.

Specific Gravity Formula ($SG$)

$SG = \frac{\rho_{fluid}}{\rho_{water}} = \frac{\gamma_{fluid}}{\gamma_{water}}$ where $\rho_{water} = 1000\,kg\,m^{-3}$ at $4\,^\circ C$. It is a dimensionless ratio.

Absolute Viscosity Formula ($\mu$)

$\tau = \mu \cdot \frac{du}{dy}$ where $\tau$ is shear stress, $\mu$ is absolute dynamic viscosity, and $\frac{du}{dy}$ is velocity gradient. Standard SI unit: $Pa \cdot s$ or $N \cdot s\,m^{-2}$.

Kinematic Viscosity Formula ($\nu$)

$\nu = \frac{\mu}{\rho}$ where $\mu$ is absolute viscosity and $\rho$ is mass density. Standard SI unit: $m^2\,s^{-1}$.

Surface Tension Formula ($\sigma$)

$\sigma = \frac{F}{L}$ where $F$ is surface tensile force and $L$ is length along which the force acts. Standard SI unit: $N\,m^{-1}$.

Capillary Rise in Tubes Formula

$h = \frac{2 \cdot \sigma \cdot \cos(\theta)}{\gamma \cdot r}$ where $\sigma$ is surface tension, $\theta$ is contact angle, $\gamma$ is unit weight, and $r$ is tube radius.

Capillary Rise Between Parallel Plates Formula

$h = \frac{2 \cdot \sigma \cdot \cos(\theta)}{\gamma \cdot t}$ where $\sigma$ is surface tension, $\theta$ is contact angle, $\gamma$ is unit weight, and $t$ is plate separation distance.

Bulk Modulus of Compression Formula ($E_b$)

$E_b = -\frac{dP}{\left(\frac{dV}{V}\right)} = \rho \cdot \frac{dP}{d\rho}$ where $dP$ is change in pressure, $dV$ is change in volume, $V$ is initial volume, and $\rho$ is density. Standard SI unit: $Pa$.

Compressibility Formula ($\beta$)

$\beta = \frac{1}{E_b} = -\frac{1}{V} \cdot \frac{dV}{dP}$ where $E_b$ is bulk modulus of compression. Standard SI unit: $Pa^{-1}$.

Unit Pressure Formula ($P$)

$P = \frac{F}{A}$ where $F$ is normal compressive force and $A$ is surface area. Standard SI unit: $Pa$ ($N\,m^{-2}$).

Absolute Pressure Formula ($P_{abs}$)

$P_{abs} = P_{gage} + P_{atm}$ where $P_{gage}$ is gage pressure and $P_{atm}$ is atmospheric pressure.

Standard Air Pressure Equivalencies

$1\,atm = 101.325\,kPa = 1.01325\,bar = 14.7\,psi = 760\,mmHg = 10.33\,m\,H_2O$.

Pascal's Law Formula

$P_1 = P_2 \implies \frac{F_1}{A_1} = \frac{F_2}{A_2}$ where $F_1$ and $F_2$ are forces applied over surface areas $A_1$ and $A_2$.

Hydrostatic Pressure Formula

$P = \gamma \cdot h = \rho \cdot g \cdot h$ where $\gamma$ is unit weight, $\rho$ is fluid density, $g$ is gravitational acceleration, and $h$ is fluid depth.

Ideal Gas Law Formulas

$P \cdot V = n \cdot R \cdot T$ and $P = \rho \cdot R \cdot T$ where $P$ is absolute pressure, $V$ is volume, $n$ is number of moles, $R$ is gas constant, $T$ is absolute temperature, and $\rho$ is density.

Boyle's Law Formula

$P_1 \cdot V_1 = P_2 \cdot V_2$ (at constant temperature $T$ and constant gas mass).

Charles's Law Formula

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$ (at constant pressure $P$ and constant gas mass).

Gay-Lussac's Law Formula

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$ (at constant volume $V$ and constant gas mass).

Number of Moles Formula (using Mass)

$n = \frac{m}{M}$ where $m$ is total mass of substance and $M$ is molar mass. Standard SI unit: $mol$.

Number of Moles Formula (using Avogadro's Constant)

$n = \frac{N}{N_A}$ where $N$ is particle count and $N_A \approx 6.022 \times 10^{23}\,mol^{-1}$ is Avogadro's constant.

Speed of Sound in Ideal Gas Formula

$c = \sqrt{k \cdot R \cdot T}$ where $k$ is ratio of specific heats ($C_p / C_v$), $R$ is specific gas constant, and $T$ is absolute temperature in $K$.

Air Temperature Conversion Formulas

$T(K) = T(^\circ C) + 273.15$ and $T(^\circ R) = T(^\circ F) + 459.67$ where $K$ is Kelvin and $^\circ R$ is Rankine.

Hydrostatic Force on Plane Surface Formula

$F = \gamma \cdot \bar{h} \cdot A = \rho \cdot g \cdot \bar{h} \cdot A$ where $\gamma$ is fluid unit weight, $\bar{h}$ is vertical depth from free surface to centroid, and $A$ is surface area.

Eccentricity Formula for Center of Pressure

$e = \frac{I_g}{\bar{h} \cdot A}$ where $I_g$ is moment of inertia about centroidal axis, $\bar{h}$ is centroidal depth, and $A$ is submerged area.

Vertical Depth to Center of Pressure Formula

$h_p = \bar{h} + \frac{I_g \cdot \sin^2(\theta)}{\bar{h} \cdot A}$ where $\bar{h}$ is vertical depth to centroid, $I_g$ is centroidal moment of inertia, $A$ is surface area, and $\theta$ is angle of surface inclination with horizontal.

Horizontal Hydrostatic Force on Curved Surface Formula

$F_H = \gamma \cdot \bar{h}_{proj} \cdot A_{proj}$ where $A_{proj}$ is area of vertical projection of curved surface and $\bar{h}_{proj}$ is vertical depth to centroid of projected area.

Vertical Hydrostatic Force Formula (Water Above Curved Surface)

$F_V = \gamma \cdot V$ acting downward, where $V$ is volume of fluid directly above curved surface extending up to free surface.

Vertical Hydrostatic Force Formula (Water Below Curved Surface)

$F_V = \gamma \cdot V_{imag}$ acting upward, where $V_{imag}$ is imaginary volume of fluid extending vertically above curved surface to free surface level.

Resultant Hydrostatic Force on Curved Surface Formula

$F_R = \sqrt{F_H^2 + F_V^2}$ acting at angle $\theta = \arctan\left(\frac{F_V}{F_H}\right)$ relative to horizontal.

Buoyant Force Formula (Archimedes' Principle)

$F_B = \gamma_{fluid} \cdot V_{sub} = \rho_{fluid} \cdot g \cdot V_{sub}$ where $\gamma_{fluid}$ is fluid unit weight, $\rho_{fluid}$ is fluid density, $g$ is gravitational acceleration, and $V_{sub}$ is displaced fluid volume.

Apparent Weight of Submerged Body Formula

$W_{apparent} = W_{real} - F_B = V \cdot (\gamma_{body} - \gamma_{fluid})$ where $W_{real}$ is body weight in air, $F_B$ is buoyant force, and $V$ is total submerged volume.

Metacentric Height Formula ($GM$)

$GM = MB \pm GB = \frac{I}{V_{sub}} \pm GB$ where $I$ is the moment of inertia of the waterline area, $V_{sub}$ is submerged displacement volume, and $GB$ is distance between center of gravity and center of buoyancy.

Metacentric Height for Rectangular Section Formula

$MB = \frac{B^2}{12 \cdot d}$ where $B$ is the beam width and $d$ is the submerged draft of the rectangular section.

Righting Moment Formula

$M_R = W \cdot GM \cdot \sin(\theta)$ where $W$ is total weight of the floating body, $GM$ is metacentric height, and $\theta$ is angle of heel.

Overturning Moment Formula

$M_O = F_H \cdot y$ where $F_H$ is overturning horizontal force and $y$ is vertical moment arm relative to pivot axis.

Eccentricity at Base of Dam Formula

$e = \left| \frac{B}{2} - x_d \right|$ where $B$ is base width and $x_d = \frac{\sum M_R - \sum M_O}{R_y}$ is location of vertical resultant force from toe.

Base Pressure Formula ($e < B/6$)

$q = \frac{R_y}{B} \cdot \left(1 \pm \frac{6 \cdot e}{B}\right)$ resulting in trapezoidal compressive pressure distribution across entire base width.

Base Pressure Formula ($e = B/6$)

$q_{max} = \frac{2 \cdot R_y}{B}$ and $q_{min} = 0$, resulting in triangular compressive pressure distribution with zero stress at heel.

Base Pressure Formula ($e > B/6$)

$q_{max} = \frac{2 \cdot R_y}{3 \cdot a}$ where $a = \frac{B}{2} - e$, resulting in tension crack detachment at heel and triangular stress over effective length $3 \cdot a$.

Factor of Safety Against Sliding Formula ($FOS_{sliding}$)

$FOS_{sliding} = \frac{\mu \cdot R_y}{\sum F_H}$ where $\mu$ is coefficient of friction, $R_y$ is total vertical force, and $\sum F_H$ is total horizontal force.

Factor of Safety Against Overturning Formula ($FOS_{overturning}$)

$FOS_{overturning} = \frac{\sum M_R}{\sum M_O}$ where $\sum M_R$ is sum of resisting righting moments about toe and $\sum M_O$ is sum of overturning moments about toe.

Shearing Stress at Base Formula ($\tau$)

$\tau = \frac{\sum F_H}{B \cdot L}$ where $\sum F_H$ is total horizontal sliding force, $B$ is base width, and $L$ is unit length of dam.

Horizontal Acceleration of Moving Vessels Formula

$\tan(\theta) = \frac{a}{g}$ where $\theta$ is the angle of inclination of the liquid surface relative to horizontal, $a$ is horizontal acceleration, and $g$ is acceleration due to gravity.

Inclined Acceleration of Moving Vessels Formula

$\tan(\theta) = \frac{a_x}{g \pm a_y}$ where $a_x = a \cdot \cos(\alpha)$ and $a_y = a \cdot \sin(\alpha)$ for acceleration $a$ inclined at angle $\alpha$ to the horizontal.

Vertical Motion Fluid Pressure Formula

$P = \rho \cdot g \cdot h \cdot \left(1 \pm \frac{a}{g}\right)$ where positive sign indicates upward vessel acceleration, negative sign indicates downward acceleration, and $h$ is fluid depth.

Paraboloid Height in Rotating Vessels Formula

$y = \frac{\omega^2 \cdot r^2}{2 \cdot g}$ where $y$ is vertical height of paraboloid surface at radius $r$, $\omega$ is angular speed in $rad\,s^{-1}$, and $g$ is gravitational acceleration.

Square Property of Parabola Formula

$\frac{x_1^2}{y_1} = \frac{x_2^2}{y_2} = \frac{r^2}{y}$ relating radial distance $x$ to vertical coordinate $y$ measured from the vertex of the rotating fluid surface parabola.

Open Tank Rotation Without Liquid Spilled Formula

$y = 2 \cdot d$ where $y = \frac{\omega^2 \cdot r^2}{2 \cdot g}$ is total paraboloid height and $d$ is liquid rise at tank wall above original static liquid level.

Open Tank Rotation With Liquid Spilled Formula

$V_{spilled} = \frac{1}{2} \cdot \pi \cdot r^2 \cdot y - \pi \cdot r^2 \cdot (H - h_0)$ where $y$ is paraboloid height, $H$ is total tank height, and $h_0$ is initial static fluid height.

Angular Velocity when Parabola Touches Tank Bottom Formula

$\omega = \sqrt{\frac{2 \cdot g \cdot H}{r^2}}$ where $H$ is tank height, $r$ is tank radius, and $g$ is acceleration due to gravity.

Closed Tank Rotation Pressure at Top Lid Formula

$P = \rho \cdot g \cdot h_{eq} = \rho \cdot g \cdot \left(\frac{\omega^2 \cdot r^2}{2 \cdot g} - H\right)$ where $h_{eq}$ is equivalent head of extended imaginary paraboloid above top lid.

Closed Tank Rotation Paraboloid Condition ($y > H$) Formula

$V_{air} = \frac{1}{2} \cdot \pi \cdot r_0^2 \cdot y_0 = V_{initial\,air}$ where $r_0$ is radius of uncovered top lid region and $y_0$ is height of imaginary paraboloid

Volumetric Flow Rate Formula ($Q$)

$Q = A \cdot v = \frac{V}{t}$ where $A$ is cross-sectional flow area, $v$ is mean flow velocity, $V$ is volume, and $t$ is time. Standard SI unit: $m^3\,s^{-1}$.

Mass Flow Rate Formula ($\dot{m}$)

$\dot{m} = \rho \cdot Q = \rho \cdot A \cdot v$ where $\rho$ is fluid density and $Q$ is volumetric flow rate. Standard SI unit: $kg\,s^{-1}$.

Weight Flow Rate Formula ($\dot{W}$)

$\dot{W} = \gamma \cdot Q = \rho \cdot g \cdot Q$ where $\gamma$ is fluid unit weight and $Q$ is volumetric flow rate. Standard SI unit: $N\,s^{-1}$.

Theoretical vs. Actual Discharge Formula

$Q_{actual} = C_d \cdot Q_{theoretical}$ where $C_d = C_v \cdot C_c$ is the coefficient of discharge, $C_v$ is velocity coefficient, and $C_c$ is contraction coefficient.

Bernoulli's Energy Conservation Equation Formula

$\frac{P_1}{\gamma} + \frac{v_1^2}{2 \cdot g} + z_1 = \frac{P_2}{\gamma} + \frac{v_2^2}{2 \cdot g} + z_2 + h_L$ where $\frac{P}{\gamma}$ is pressure head, $\frac{v^2}{2 \cdot g}$ is velocity head, $z$ is elevation head, and $h_L$ is total head loss.

Energy Gradient Line Slope Formula ($S$)

$S = \frac{h_f}{L}$ representing the rate of friction head loss per unit length of pipe, where $h_f$ is friction head loss and $L$ is pipe length.

Power Output of Pump Formula ($P_{out}$)

$P_{out} = \gamma \cdot Q \cdot H_p$ where $\gamma$ is fluid unit weight, $Q$ is volumetric flow rate, and $H_p$ is total dynamic head added by the pump. Standard SI unit: $W$ or $kW$.

Pump Efficiency Formula ($\eta_{pump}$)

$\eta_{pump} = \frac{P_{out}}{P_{in}} \times 100\%$ where $P_{out}$ is fluid power output ($\gamma \cdot Q \cdot H_p$) and $P_{in}$ is mechanical power input to the pump.

Darcy-Weisbach Friction Head Loss Formula (Circular Pipe)

$h_f = f \cdot \frac{L}{D} \cdot \frac{v^2}{2 \cdot g} = \frac{8 \cdot f \cdot L \cdot Q^2}{\pi^2 \cdot g \cdot D^5}$ where $f$ is Darcy friction factor, $L$ is pipe length, $D$ is diameter, $v$ is velocity, and $Q$ is flow rate.

Darcy-Weisbach Friction Head Loss Formula (Non-Circular Pipe)

$h_f = f \cdot \frac{L}{4 \cdot R_h} \cdot \frac{v^2}{2 \cdot g}$ where $R_h = \frac{A}{P_w}$ is hydraulic radius, $A$ is cross-sectional area, and $P_w$ is wetted perimeter.

Manning's Friction Head Loss Formula (Circular Pipe)

$h_f = \frac{10.29 \cdot n^2 \cdot L \cdot Q^2}{D^{16/3}}$ (SI units) where $n$ is Manning's roughness coefficient, $L$ is pipe length, $Q$ is flow rate, and $D$ is diameter.

Manning's Velocity Formula (Non-Circular Pipe / General)

$v = \frac{1}{n} \cdot R_h^{2/3} \cdot S^{1/2}$ (SI units) where $n$ is Manning's roughness coefficient, $R_h$ is hydraulic radius, and $S$ is energy slope ($\frac{h_f}{L}$).

Hazen-Williams Friction Head Loss Formula (Circular Pipe)

$h_f = \frac{10.67 \cdot L \cdot Q^{1.852}}{C^{1.852} \cdot D^{4.87}}$ (SI units) where $C$ is Hazen-Williams roughness coefficient, $L$ is pipe length, $Q$ is flow rate, and $D$ is diameter.

Hazen-Williams Velocity Formula (Non-Circular Pipe / General)

$v = 0.8492 \cdot C \cdot R_h^{0.63} \cdot S^{0.54}$ (SI units) where $C$ is Hazen-Williams roughness coefficient, $R_h$ is hydraulic radius, and $S$ is energy

Series Pipe Flow Formulas

$Q_{total} = Q_1 = Q_2 = Q_3$ and total head loss $h_{L,total} = h_{f1} + h_{f2} + h_{f3} + \dots$ where $Q$ is volumetric flow rate and $h_f$ is friction head loss.

Parallel Pipe Flow Formulas

$Q_{total} = Q_1 + Q_2 + Q_3 + \dots$ and branch head loss $h_{L1} = h_{L2} = h_{L3} = h_{L,total}$ where $Q$ is total discharge and $h_L$ is head loss across parallel pipes.

Laminar Flow Reynolds Number Criterion

$Re = \frac{\rho \cdot v \cdot D}{\mu} = \frac{v \cdot D}{\nu} < 2000$ for pipe flow, where viscous forces dominate and fluid flows in smooth, parallel layers without lateral mixing.

Turbulent Flow Reynolds Number Criterion

$Re = \frac{\rho \cdot v \cdot D}{\mu} = \frac{v \cdot D}{\nu} > 4000$ for pipe flow, where inertial forces dominate, causing chaotic fluid motion, eddies, and rapid mixing.

Critical Flow Condition

Open-channel flow state occurring when Froude number $Fr = 1$, specific energy $E$ is at its absolute minimum for a given discharge $Q$, and flow velocity equals wave celerity ($v = \sqrt{g \cdot D_h}$).

Critical Depth Formula for Rectangular Channel ($y_c$)

$y_c = \sqrt[3]{\frac{q^2}{g}}$ where $q = \frac{Q}{b}$ is discharge per unit channel width, $Q$ is flow rate, $b$ is channel width, and $g$ is acceleration due to gravity.

Froude Number Formula ($Fr$)

$Fr = \frac{v}{\sqrt{g \cdot D_h}}$ where $v$ is mean flow velocity, $g$ is gravitational acceleration, and $D_h = \frac{A}{T}$ is hydraulic depth ($A$ is cross-sectional area, $T$ is top width).

Three-Reservoir System (Case 1: Flow into Middle Reservoir)

Occurs when piezometric head at junction $J$ exceeds middle reservoir elevation ($h_J > z_2$), yielding flow directions from highest reservoir 1 into both middle reservoir 2 and lowest reservoir 3 ($Q_1 = Q_2 + Q_3$).

Three-Reservoir System (Case 2: Flow out of Middle Reservoir)

Occurs when piezometric head at junction $J$ is below middle reservoir elevation ($h_J < z_2$), yielding flow directions from reservoirs 1 and 2 into lowest reservoir 3 ($Q_1 + Q_2 = Q_3$).

Three-Reservoir System Junction Continuity Condition

$\sum Q_{in} = \sum Q_{out}$ at junction $J$, with pipe head loss $h_{fi} = \left| z_i - h_J \right|$ where $z_i$ is water surface elevation of reservoir $i$ and $h_J = \frac{P_J}{\gamma} + z_J$ is piezometric head at junction $J$.

Orifice Theoretical Velocity Formula ($v_t$)

$v_t = \sqrt{2 \cdot g \cdot h}$ where $g$ is acceleration due to gravity and $h$ is fluid head above the center of the orifice.

Orifice Actual Velocity Formula ($v_a$)

$v_a = C_v \cdot \sqrt{2 \cdot g \cdot h}$ where $C_v$ is coefficient of velocity, $g$ is acceleration due to gravity, and $h$ is fluid head.

Orifice Theoretical Discharge Formula ($Q_t$)

$Q_t = A_o \cdot \sqrt{2 \cdot g \cdot h}$ where $A_o$ is cross-sectional area of the orifice, $g$ is gravitational acceleration, and $h$ is fluid head.

Orifice Actual Discharge Formula ($Q_a$)

$Q_a = C_d \cdot A_o \cdot \sqrt{2 \cdot g \cdot h}$ where $C_d$ is coefficient of discharge, $A_o$ is orifice cross-sectional area, and $h$ is fluid head.

Time to Empty / Lower Tank Level Formula (Constant Area Tank)

$t = \frac{2 \cdot A_T \cdot \left(\sqrt{h_1} - \sqrt{h_2}\right)}{C_d \cdot A_o \cdot \sqrt{2 \cdot g}}$ where $A_T$ is constant cross-sectional area of tank, $A_o$ is orifice area, $h_1$ is initial head, and $h_2$ is final head.

Time to Empty / Lower Tank Level Formula (Varying Area Tank)

$t = \int_{h_2}^{h_1} \frac{A_T(h)}{C_d \cdot A_o \cdot \sqrt{2 \cdot g \cdot h}}\,dh$ where $A_T(h)$ is cross-sectional area of the tank expressed as a function of liquid head $h$.