Fluid Mechanics, HVAC & Refrigeration formula sheet — printable

Density

\rho = \tfrac{m}{V}

Specific Gravity

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

Hydrostatic Pressure (P = ρgh)

P = \rho g h

Pressure Head (h = P/ρg)

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

Gauge and Absolute Pressure

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

Volumetric Flow Rate (Q = Av)

Q = A v

Continuity Equation (A₁v₁ = A₂v₂)

A_1 v_1 = A_2 v_2

Pipe Velocity from Flow and Diameter

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

Dynamic Pressure (q = ½ρv²)

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

Velocity Head (h = v²/2g)

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

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

Torricelli's Law (v = √(2gh))

v = \sqrt{2 g h}

Reynolds Number

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

Laminar Friction Factor (f = 64/Re)

f = \frac{64}{Re}

Buoyant Force (Archimedes' Principle)

F_b = \rho V g

Darcy–Weisbach Head Loss

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

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}}

Colebrook–White Friction Factor

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

Hazen–Williams Head Loss

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

Hazen–Williams Velocity

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

Minor Loss from K Factor

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

Equivalent Length of a Fitting

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

Valve Flow Coefficient (Cv)

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

Valve Flow Coefficient (Kv, metric)

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

Orifice Plate Flow

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

Venturi Meter Flow

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

Barlow's Formula (Pipe Pressure Rating)

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

Water Hammer Surge (Joukowsky Equation)

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

Pipe Internal Volume

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

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]

Total Dynamic Head

TDH = h_s + h_f + h_v

Hydraulic Power (P = ρgQh)

P = \rho g Q h

Pump Water Horsepower

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

Pump Brake Horsepower

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

Pump Efficiency from Hydraulic and Shaft Power

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

Net Positive Suction Head Available (NPSHa)

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

Cavitation Number (Margin above Vapour Pressure)

\sigma_c = \frac{p - p_v}{\tfrac{1}{2} \rho v^{2}}

Pump Affinity Law — Flow vs Speed

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

Pump Affinity Law — Head vs Speed

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

Pump Affinity Law — Power vs Speed

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

Pump Affinity Law — Flow vs Impeller Diameter

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

Pump Affinity Law — Head vs Impeller Diameter

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

Fan Affinity Law — Airflow vs Speed

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

Fan Affinity Law — Static Pressure vs Speed

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

Fan Affinity Law — Power vs Speed

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

Fan Brake Horsepower

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

Pump Specific Speed (Ns)

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

Saturation Vapour Pressure (Magnus / Alduchov–Eskridge)

p_{ws} = 610.94 \exp\!\left(\frac{17.625\,t}{t + 243.04}\right)

Relative Humidity from Vapour Pressure

\varphi = \frac{p_v}{p_{ws}}

Humidity Ratio from Vapour Pressure

W = 0.62198\,\frac{p_v}{p - p_v}

Dew Point (Magnus Approximation)

T_d = \frac{c\,\gamma}{b - \gamma}, \quad \gamma = \ln\!\frac{\mathrm{RH}}{100} + \frac{b\,T}{c + T}

Dew Point from Humidity Ratio

T_d = \frac{243.04\,\gamma}{17.625 - \gamma}, \quad \gamma = \ln\!\frac{p_v}{610.94}, \quad p_v = \frac{W\,p}{0.62198 + W}

Wet-Bulb Temperature (Stull 2011)

T_w = T\,\arctan\!\left[0.151977\sqrt{\mathrm{RH} + 8.313659}\,\right] + \arctan(T + \mathrm{RH}) - \arctan(\mathrm{RH} - 1.676331) + 0.00391838\,\mathrm{RH}^{3/2}\arctan(0.023101\,\mathrm{RH}) - 4.686035

Relative Humidity from a Sling Psychrometer

\varphi = \frac{p_{ws}(t_{wb}) - A\,p\,(t_{db} - t_{wb})}{p_{ws}(t_{db})}

Moist Air Enthalpy (per kg DRY air)

h = 1.006\,t + W\,(2501 + 1.86\,t)

Moist Air Specific Volume (per kg DRY air)

v = \frac{0.287042\,(t + 273.15)\,(1 + 1.6078\,W)}{p}

Moist Air Density at Altitude (and the 1.08 Correction)

\rho = \frac{p_z\,(1 + W)}{R_{da}\,T\,(1 + 1.6078\,W)}, \quad p_z = 101\,325\,(1 - 2.25577 \times 10^{-5} z)^{5.25588}

Mixed Air Temperature

T_m = f \, T_{oa} + (1-f) \, T_{ra}

Degree of Saturation (Moist Air)

\mu = \frac{W}{W_s}

Air Total Heat (4.5 Rule)

\dot{Q}_t = \rho_a \dot{V} \, \Delta h

Air Sensible Heat (1.08 Rule)

\dot{Q}_s = \rho_a c_a \dot{V} \, \Delta T

Air Latent Heat (0.68 Rule)

\dot{Q}_l = \rho_a \dot{V} h_{fg} \, \Delta W

Sensible Heat Ratio (SHR)

\mathrm{SHR} = \frac{\dot{Q}_s}{\dot{Q}_s + \dot{Q}_l}

Round Duct Air Velocity

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

Equivalent Round Duct Diameter

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

Air Changes per Hour (ACH)

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

Hydronic Heat Transfer (Water)

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

Glycol Loop Heat Transfer (Capacity Derate)

\dot{Q} = \rho c \dot{V} \, \Delta T

Loop Water Expansion Volume

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

Expansion Tank Acceptance Volume

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

Hydronic Static Fill Pressure

P = \rho_w g H + P_{margin}

Radiator Output at Non-Rated Temperature

\dot{Q} = \dot{Q}_r \left(\frac{\Delta T}{\Delta T_r}\right)^{n}

Seasonal Heating Energy (Degree-Day Method)

E = \frac{\dot{Q}_d \, \Delta T_m \, t}{\Delta T_d \, \eta}

Boiler or Furnace Output from Input

\dot{Q}_{out} = \dot{Q}_{in} \, \eta

Energy Cost from a Utility Rate

C_e = E \, p_e

Tons of Refrigeration from BTU/hr

T = \frac{\dot{Q}}{12{,}000\ \text{BTU/hr}}

Coefficient of Performance (COP)

\mathrm{COP} = \frac{\dot{Q}}{\dot{W}}

Energy Efficiency Ratio (EER)

\mathrm{EER} = \frac{\dot{Q}\ [\text{BTU/hr}]}{\dot{W}\ [\text{W}]}

EER to COP Conversion

\mathrm{EER} = 3.412 \times \mathrm{COP}

Chiller Efficiency (kW per Ton)

\mathrm{kW/ton} = \frac{\dot{W}\ [\text{kW}]}{\dot{Q}\ [\text{tons}]}

Refrigeration COP from Enthalpies

COP = \frac{h_1 - h_4}{h_2 - h_1}

Refrigerant Superheat

\mathrm{SH} = T_{suction} - T_{sat}

Refrigerant Subcooling

\mathrm{SC} = T_{sat} - T_{liquid}

Refrigerant Mass Flow Rate

\dot{m} = \frac{\dot{Q}}{\Delta h}

Chiller Heat Rejection

Q_r = Q_e \, \mathrm{HRF}

Condenser Water Flow Rate

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

Cooling Tower Range

\Delta T = T_h - T_c

Cooling Tower Approach

A = T_c - T_{wb}

Cooling Tower Heat Rejection

Q = 500 \, R \, \Delta T

Cooling Tower Evaporation Rate

E = 0.001 \, R \, \Delta T

Cycles of Concentration (COC = M/B)

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

Blowdown Rate from Cycles

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

Cooling Tower Makeup Water Rate

M = E + B + D

Boiler Horsepower to Heat Output

Q = 33{,}475 \; \text{BHP}

Boiler Horsepower to Steam Rate

S = 34.5 \; \text{BHP}

Boiler Blowdown Rate from Steam Rate

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

Condensate Return Percentage

\%CR = \frac{S_c}{S} \times 100

Boiler Makeup from Condensate Return

M = S\left(1 - \frac{\%CR}{100}\right)