Formula quick-reference index

γ = ρg

γ	specific weight — weight per unit volume
ρ	mass density — mass per unit volume
g	gravitational acceleration, 32.2 ft/s² (9.81 m/s²)

SG = γ_substance / γ_water

SG	specific gravity, dimensionless

ν = μ / ρ

μ	dynamic (absolute) viscosity
ν	kinematic viscosity

p = γh

p	gauge pressure at depth h
h	vertical depth below the free surface

F_R = γ h_c A

F_R	magnitude of hydrostatic force on a plane surface
h_c	vertical depth of the surface's centroid
A	area of the surface

↑ Topic 2: Buoyancy & Flotation

y_p = y_c + I_xx,cy_cA

y_p	distance from the free surface to the centre of pressure, measured along the incline
y_c	distance from the free surface to the centroid, measured along the incline
I_xx,c	second moment of area about the centroidal axis (rectangle: bh³/12, with h along the incline)

Topic 2: Buoyancy & Flotation

F_b = γ_fluid · V_displaced

F_b	buoyant force, acting vertically upward through the centroid of the displaced volume
V_displaced	volume of fluid displaced = submerged volume of the body only

↑ Topic 2: Buoyancy & Flotation

Floating equilibrium: W = F_b

W	total weight of the floating body

↑ Topic 2: Buoyancy & Flotation

GM = MB − GB stable if GM > 0

G	centre of gravity of the body
B	centre of buoyancy (centroid of submerged volume)
M	metacentre — where the tilted buoyant-force line crosses the body's centreline
MB = I₀/V_sub	distance from B to M; I₀ is the second moment of the waterplane area about its centroidal axis

Topic 3: Continuity, Energy & Momentum

Q = A₁V₁ = A₂V₂

Q	volumetric flow rate (discharge)
A	cross-sectional area normal to the flow
V	mean velocity through the section

ρ₁A₁V₁ = ρ₂A₂V₂

Used when density changes (gases, compressible flow). For water, ρ cancels and you get Q = AV.

p₁/γ + V₁²/2g + z₁ + h_A = p₂/γ + V₂²/2g + z₂ + h_T + h_L

p/γ	pressure head
V²/2g	velocity head
z	elevation head (same datum for both points!)
h_A	head added by a pump
h_T	head removed by a turbine
h_L	total head loss between 1 and 2

ΣF = ρQ(β₂V₂ − β₁V₁)

ΣF	vector sum of forces on the fluid (pressure + weight + reaction), in the chosen direction
β	momentum correction factor, ≈ 1.0 for turbulent flow — the exam usually lets you drop it

Topic 4: Pipe Flow

h_f = f LD V²2g

h_f	friction head loss
f	Darcy friction factor (dimensionless — not the Fanning factor, which is f/4)
L, D	pipe length and diameter
V	mean velocity

Re = VDν laminar: f = 64Re

Re	Reynolds number; laminar below ≈ 2,300, turbulent above ≈ 4,000
ν	kinematic viscosity of the fluid

Turbulent f: Haaland approximation 1/√f ≈ −1.8 log[(ε/3.7D)^1.11 + 6.9/Re]

ε	absolute roughness of the pipe wall (commercial steel ≈ 0.00015 ft)
Use the Moody chart or Colebrook equation equivalently; Haaland is calculator-friendly.

V = 1.318 · C · R^0.63 · S^0.54 (English units, V in ft/s)

C	Hazen-Williams roughness coefficient (≈ 150 new PVC, 130 new steel, 100 old cast iron)
R	hydraulic radius = A/P (D/4 for a full circular pipe)
S	slope of the energy grade line = h_f/L

h_m = K V²2g

K	minor loss coefficient (entrance ≈ 0.5, exit = 1.0, valves/fittings from tables)
Equivalently, fittings can be converted to an equivalent length of straight pipe.

Series pipes: Q is the same everywhere; head losses add. Parallel pipes: head loss is the same in each branch; discharges add.

Topic 5: Pumps & System Curves

H_sys = H_static + KQ²

H_sys	total head the system demands at flow Q
H_static	static head: elevation lift + (p_discharge − p_suction)/γ
KQ²	friction + minor losses, which scale with the square of flow

Operating point: the (Q, H) where the pump curve crosses the system curve.

Q₂Q₁ = N₂N₁ H₂H₁ = (N₂N₁)² P₂P₁ = (N₂N₁)³

Q	discharge
H	head
P	power
N	rotational speed (or impeller diameter D)

Water power: P = γQHη English shortcut: bhp = Q(gpm) × H(ft)3960 × η

η	pump efficiency as a decimal
The 3960 shortcut already includes unit conversions for water — do not multiply by γ again.

NPSH_A = p_atm/γ − p_v/γ ± h_s − h_L,suction > NPSH_R

NPSH_A	net positive suction head available from the system
NPSH_R	net positive suction head required by the pump (from its curve)
p_v	vapour pressure of the liquid at its temperature
h_s	static suction head: + if the supply sits above the pump, − if the pump must lift (suction lift)
h_L,suction	head loss in the suction piping

Topic 6: Open-Channel Flow

V = (k/n) · R^2/3 · S^1/2 Q = AV

V	mean velocity
Q	discharge
k	unit constant: 1.486 for English units (ft/s), 1.0 for SI
n	Manning's roughness coefficient (concrete ≈ 0.013, earth ≈ 0.022, natural channel ≈ 0.03–0.05)
R	hydraulic radius = A/P, where P is the wetted perimeter
S	longitudinal slope of the channel (energy slope for uniform flow)

Rectangular: A = by, P = b + 2y Trapezoidal: A = (b + zy)y, P = b + 2y√(1+z²) Full circular: R = D/4

b	bottom width
y	flow depth
z	side slope, horizontal:vertical
Do NOT use R = D/4 for a partly full pipe — recompute A and P for the actual depth.

Fr = V / √(g·D_h) with D_h = A/T

Fr	Froude number: < 1 subcritical, = 1 critical, > 1 supercritical
D_h	hydraulic depth = area / top width T (for a rectangle, D_h = y)

Rectangular channels: y_c = (q²/g)^1/3 and E_min = 3y_c/2

y_c	critical depth
q = Q/b	discharge per unit width
E = y + V²/2g	specific energy — minimised at critical depth

Topic 7: Hydrology & Runoff

Q_p = C · i · A

Q_p	peak runoff rate (cfs when i is in in/hr and A in acres — the units work out)
C	runoff coefficient (0–1); use an area-weighted composite for mixed land use
i	rainfall intensity for a duration equal to the time of concentration, t_c
A	drainage area

Composite C = Σ(C_jA_j) / ΣA_j

t_c

time of concentration — travel time from the hydraulically most distant point

S = 1000/CN − 10 (inches) I_a = 0.2S

CN	curve number, 0–100 (higher = more runoff); from TR-55 tables by soil group and cover
S	potential maximum retention after runoff begins
I_a	initial abstraction — interception, depression storage, early infiltration

Q = (P − I_a)²(P − I_a) + S for P > I_a; Q = 0 otherwise

Q	direct runoff depth (inches)
P	storm rainfall depth (inches)

t_p = ΔD/2 + 0.6 t_c q_p = 484 A / t_p

t_p	time to peak (hr)
q_p	peak of the unit hydrograph (cfs)
ΔD	unit storm duration (hr)
A	area in square miles

Topic 8: Groundwater Flow

Q = K · i · A with i = dh/dl

Q	discharge through the porous medium
K	hydraulic conductivity (permeability)
i	hydraulic gradient — head loss per unit length, dimensionless
A	bulk cross-sectional area normal to flow (solids + voids)

v = Q/A (Darcy flux) v_s = v/n (seepage velocity)

v	discharge per unit bulk area — not the actual pore-water speed
v_s	true average velocity through the pores
n	porosity

Confined (constant thickness b): q = T · (dh/dl), T = K·b

T	transmissivity — the aquifer's headline property for confined flow
b	saturated thickness of the confined aquifer

Unconfined (Dupuit): q = (K/2L)·(h₁² − h₂²) per unit width

h	saturated thickness (water-table height above the impermeable base)
Assumes nearly horizontal flow — the Dupuit approximation; fine for gentle gradients.

Confined: Q = 2πT(h₂ − h₁)ln(r₂/r₁)

h₁, h₂	hydraulic heads at radial distances r₁, r₂ from the well
T = Kb	transmissivity

Unconfined: Q = πK(h₂² − h₁²)ln(r₂/r₁)

Heads h are water-table heights above the aquifer base — and they enter squared.

Topic 9: Water Treatment

Coagulation/flocculation: rapid mix then gentle stirring; design by detention time t = V/Q and the Camp number Gt

Typical: rapid mix 30–60 s; flocculation 20–40 min; Gt ≈ 10⁴–10⁵.

Sedimentation: surface overflow rate = QA weir loading = QL_weir

Surface overflow rate (gpd/ft²)	the controlling design parameter — particles settle if their settling velocity exceeds it
Weir loading (gpd/ft)	checked separately so settled sludge is not scoured over the weirs

Filtration: filtration rate = Q/A (gpm/ft²); backwash reverses the flow to clean the media

Rapid sand filters run ≈ 2–4 gpm/ft²; head loss grows as the bed clogs, triggering backwash.

Disinfection: CT = C × T

C	disinfectant residual concentration (mg/L)
T	contact time (min) — use T₁₀, the time 90% of the water exceeds, for credit
CT (mg·min/L)	compared against regulatory tables for the target pathogen and disinfectant

Topic 10: Wastewater Treatment

BOD exerted at time t: y = L₀(1 − e^−kt) (base e) or y = L₀(1 − 10^−Kt) (base 10)

y	oxygen consumed by time t
L₀	ultimate BOD
k, K	deoxygenation rate constants — k = 2.303K, so check which base the question uses

BOD₅ ≈ 0.68 × L₀ (for the standard k = 0.23 day⁻¹, base e, at 20°C)

Handy when a question gives one and asks for the other.

F/M = Q · S₀V · X

Q	influent flow
S₀	influent BOD₅
V	aeration tank volume
X	MLSS — mixed liquor suspended solids
Units: day⁻¹; conventional plants run ≈ 0.2–0.5 day⁻¹.

MCRT = mass of solids in the systemmass of solids wasted per day = V·XQ_wX_w + Q_eX_e

Q_w, X_w	waste sludge flow and concentration
Q_e, X_e	effluent flow and suspended solids (often negligible)
MCRT in days; conventional ≈ 5–15 days, extended aeration 20–30+.

Surface overflow rate = QA Solids loading rate = (Q + Q_r) · XA

Q_r	return sludge flow
Design SOR ≈ 400–700 gpd/ft²; solids loading ≈ 20–30 lb/day/ft² for conventional plants.