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HVAC & Hydronics formula solvers

Categories — 73 of them, 1787 solvers in all
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Air Changes per Hour (ACH)

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

HVAC & HydronicsFluid MechanicsHow many times per hour a ventilation rate replaces the air in a room, the ventilation yardstick behind BTU/hr = 60 × CFM ÷ room volume.

Air Latent Heat (0.68 Rule)

Q˙l=ρaV˙hfg ΔW\dot{Q}_l = \rho_a \dot{V} h_{fg} \, \Delta W

HVAC & HydronicsThermodynamicsLatent heat from dehumidifying an airstream, set by airflow and the change in humidity ratio — the trade rule BTU/hr = 0.68 × CFM × Δgrains.

Air Sensible Heat (1.08 Rule)

Q˙s=ρacaV˙ ΔT\dot{Q}_s = \rho_a c_a \dot{V} \, \Delta T

HVAC & HydronicsThermodynamicsSensible heat carried by an airstream from CFM and dry-bulb ΔT, assuming standard air — the trade rule BTU/hr = 1.08 × CFM × ΔT.

Air Total Heat (4.5 Rule)

Q˙t=ρaV˙ Δh\dot{Q}_t = \rho_a \dot{V} \, \Delta h

HVAC & HydronicsThermodynamicsTotal (sensible plus latent) heat carried by an airstream from airflow and enthalpy change — the trade rule BTU/hr = 4.5 × CFM × Δh.

Air Velocity Pressure (the 4005 Rule)

VP=(V4005)2\mathrm{VP} = \left(\frac{V}{4005}\right)^{2}

HVAC & HydronicsFluid MechanicsThe pressure a pitot tube reads as air is brought to rest, and the velocity that reading implies — the balancer's 4005 rule, worked honestly in whichever units you type.

Barlow's Formula (Pipe Pressure Rating)

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

HVAC & HydronicsFluid MechanicsMechanicsInternal pressure a pipe can hold from wall stress, wall thickness and outside diameter — the thin-wall hoop-stress relation used by pipeline codes.

Boiler or Furnace Output from Input

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

HVAC & HydronicsThermodynamicsUsable heat delivered by a boiler or furnace from its fuel input rate and its efficiency rating, the nameplate arithmetic behind AFUE.

Chiller Efficiency (kW per Ton)

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

HVAC & HydronicsThermodynamicsThe chiller-plant efficiency metric: kilowatts drawn per ton of cooling produced, where lower is better and 0.5 kW/ton is excellent.

Chiller Heat Rejection

Qr=Qe HRFQ_r = Q_e \, \mathrm{HRF}

Water TreatmentThermodynamicsHVAC & HydronicsHeat a chiller's tower has to reject: the evaporator load times the heat rejection factor that adds the compressor's own work to the load.

Coefficient of Performance (COP)

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

HVAC & HydronicsThermodynamicsEfficiency of a heat pump or chiller: useful heating or cooling delivered divided by the electrical power drawn to deliver it.

Colebrook–White Friction Factor

1f=−2log⁡10 ⁣(ε3.7D+2.51Ref)\frac{1}{\sqrt{f}} = -2 \log_{10}\!\left(\frac{\varepsilon}{3.7 D} + \frac{2.51}{Re \sqrt{f}}\right)

Fluid MechanicsHVAC & HydronicsPhysicsThe reference equation for turbulent friction in a rough pipe, and the curve every Moody diagram is drawn from. It has f on both sides, so this page solves it by iteration rather than by any closed formula.

Combined Convection and Radiation Coefficient

ht=hc+εσ(Ts+Tsur)(Ts2+Tsur2)h_t = h_c + \varepsilon \sigma (T_s + T_{sur})(T_s^2 + T_{sur}^2)

Heat TransferThermodynamicsHVAC & HydronicsTotal surface coefficient adding a linearised radiation term to the convective film, so one h covers both mechanisms over a modest ΔT.

Combustion (Stack) Efficiency — Siegert Formula

η=100−A ΔTCO2\eta = 100 - A \, \frac{\Delta T}{\mathrm{CO_2}}

HVAC & HydronicsThermodynamicsClassic flue-gas efficiency estimate from net stack temperature and flue CO₂ percentage, with a fuel constant A of about 0.66 for gas.

Compressor Isentropic Efficiency

ηs=h2s−h1h2−h1\eta_{s} = \frac{h_{2s} - h_{1}}{h_{2} - h_{1}}

HVAC & HydronicsThermodynamicsHow much of the work a compressor actually consumes went into raising the refrigerant's enthalpy, measured against the frictionless compression to the same discharge pressure.

Condenser Water Flow Rate

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

HVAC & HydronicsFluid MechanicsWater TreatmentTower water flow needed to reject a chiller's load plus compressor heat, the physics behind the 3 gpm per ton at 10 °F rule of thumb.

Conduction Through a Pipe Wall

Q˙=2πkL ΔTln⁡(r2/r1)\dot{Q} = \frac{2 \pi k L \, \Delta T}{\ln(r_2 / r_1)}

Heat TransferThermodynamicsHVAC & HydronicsRadial conduction through a cylindrical pipe or insulation layer, where the area grows outward so the resistance follows a logarithm.

Convection Film Resistance

R=1hAR = \frac{1}{h A}

Heat TransferThermodynamicsHVAC & HydronicsThermal resistance of a boundary-layer film in kelvin per watt, the reciprocal of the film coefficient times the wetted surface area.

Cooling Coil Bypass Factor and Apparatus Dew Point

BF=tla−tadptea−tadp\mathrm{BF} = \frac{t_{la} - t_{adp}}{t_{ea} - t_{adp}}

HVAC & HydronicsThermodynamicsThe fraction of the air that leaves a cooling coil as though it never touched it, and the apparatus dew point the rest of it was brought to — the model behind every coil selection.

Darcy–Weisbach Head Loss

hf=f LD v22gh_f = f \, \frac{L}{D} \, \frac{v^{2}}{2g}

HVAC & HydronicsFluid MechanicsPhysicsThe rigorous pipe friction equation: head loss from friction factor, length-to-diameter ratio and velocity head, with g = 9.80665 m/s².

Degree of Saturation (Moist Air)

μ=WWs\mu = \frac{W}{W_s}

HVAC & HydronicsThermodynamicsThe humidity ratio present divided by the humidity ratio at saturation for the same temperature and pressure — close to relative humidity but not equal to it, and the quantity older charts plot as percentage saturation.

Degrees of Superheat

ΔTsh=T−Ts(p)\Delta T_{sh} = T - T_s(p)

ThermodynamicsWater TreatmentHVAC & HydronicsHow many degrees a steam temperature sits above saturation at its own pressure, with the saturation temperature taken from IAPWS-IF97 Region 4.

Desuperheater Water Injection Rate

m˙w=m˙1 h1−h2h2−hw\dot{m}_w = \dot{m}_1 \, \frac{h_1 - h_2}{h_2 - h_w}

ThermodynamicsWater TreatmentHVAC & HydronicsSpray water a desuperheater needs to bring superheated steam down to a target enthalpy, straight from the mass and energy balance across the station.

Dew Point from Humidity Ratio

Td=243.04 γ17.625−γ,γ=ln⁡ ⁣pv610.94,pv=W p0.62198+WT_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}

HVAC & HydronicsThermodynamicsAir Quality & DispersionThe temperature at which air of a given humidity ratio and barometric pressure starts to condense, by running the vapour-pressure chain backwards through the Magnus fit.

EER to COP Conversion

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

HVAC & HydronicsThermodynamicsConverts between the two efficiency scales, since one watt of input equals 3.412 BTU/hr and both ratios describe the same machine.

Effective R-Value with Framing (Parallel Path)

1Reff=ffrRfr+1−ffrRcav\frac{1}{R_{eff}} = \frac{f_{fr}}{R_{fr}} + \frac{1 - f_{fr}}{R_{cav}}

Heat TransferHVAC & HydronicsThermodynamicsWhole-wall R-value once the studs are counted, area-weighting the framing and cavity paths as parallel conductances rather than averaging their R-values.

Energy Cost from a Utility Rate

Ce=E peC_e = E \, p_e

Water TreatmentThermodynamicsHVAC & HydronicsCost of the energy a system consumes: kilowatt-hours or fuel BTUs times the utility rate, for tower fans, pumps and boiler gas.

Energy Efficiency Ratio (EER)

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

HVAC & HydronicsThermodynamicsCooling efficiency as BTU/hr of capacity per watt of electrical input, a deliberately mixed-unit ratio equal to 3.412 times the COP.

Equivalent Length of a Fitting

Leq=KDfL_{eq} = \frac{K D}{f}

HVAC & HydronicsFluid MechanicsWater TreatmentConverts a fitting's K factor into the length of straight pipe that would cause the same friction loss at the same friction factor.

Equivalent Round Duct Diameter

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

HVAC & HydronicsFluid MechanicsHuebscher's equation for the round duct that has the same friction loss and airflow as a given rectangular duct of sides a and b.

Excess Air from Flue Gas Oxygen

EA=O220.9−O2EA = \frac{O_2}{20.9 - O_2}

Air Quality & DispersionHVAC & HydronicsHow much air beyond stoichiometric is passing through a burner, read straight off the oxygen in the flue gas. The single most useful number a combustion analyser gives you, because everything about efficiency follows from it.

Expansion Loop Leg Length (Guided Cantilever)

L=3ED ΔSaL = \sqrt{\frac{3 E D \, \Delta}{S_a}}

HVAC & HydronicsFluid MechanicsMechanicsLeg length an expansion loop or offset needs to absorb a given thermal movement without exceeding the pipe's allowable stress.

Expansion Tank Acceptance Volume

Vt=Vs e1−P1P2V_t = \frac{V_s \, e}{1 - \dfrac{P_1}{P_2}}

HVAC & HydronicsFluid MechanicsDiaphragm expansion tank size for a closed hydronic loop from system volume, water expansion and the absolute fill and relief pressures.

Fan Affinity Law — Airflow vs Speed

Q2Q1=N2N1\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsPhysicsFan airflow in CFM changes in direct proportion to wheel speed, the first law used when re-sheaving a belt-driven air handler.

Fan Affinity Law — Power vs Speed

P2P1=(N2N1)3\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsPhysicsFan brake power varies with the cube of wheel speed — the law behind variable-air-volume energy savings and behind burnt-out re-sheaved motors.

Fan Affinity Law — Static Pressure vs Speed

SP2SP1=(N2N1)2\frac{SP_2}{SP_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsPhysicsFan static pressure rises with the square of wheel speed, the reason a modest re-sheave can overpressurise ductwork and blow out flex connections.

Fan Brake Horsepower

BHP=Q⋅SP6356 ηBHP = \frac{Q \cdot SP}{6356 \, \eta}

HVAC & HydronicsFluid MechanicsPhysicsShaft power a fan absorbs; the 6356 divisor assumes cubic feet per minute, inches of water gauge and horsepower at the given efficiency.

Fin Heat Transfer Rate

Q˙f=ηf hAf ΔTb\dot{Q}_f = \eta_f \, h A_f \, \Delta T_b

Heat TransferThermodynamicsHVAC & HydronicsDuty of a fin or finned surface: the ideal convective rate over the whole fin area, derated by the fin efficiency.

Fouled Overall Coefficient

1Uf=1Uc+1hf\frac{1}{U_f} = \frac{1}{U_c} + \frac{1}{h_f}

Heat TransferThermodynamicsHVAC & HydronicsAdds a fouling deposit as one more resistance in series, reducing the clean overall coefficient to the fouled value used for design margin.

Fouling Factor on an Overall Coefficient

1Uf=1Uc+Rf\frac{1}{U_f} = \frac{1}{U_c} + R_f

Heat TransferThermodynamicsHVAC & HydronicsService-condition U of a heat exchanger, adding the TEMA fouling factor as an extra area-specific resistance on top of the clean coefficient.

Glycol Loop Heat Transfer (Capacity Derate)

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

HVAC & HydronicsThermodynamicsWater TreatmentHeat carried by a glycol loop using the actual mix density and specific heat, which is how the 500 constant derates for antifreeze.

Hazen–Williams Head Loss

hf=10.67 L Q1.852C1.852D4.8704h_f = \frac{10.67 \, L \, Q^{1.852}}{C^{1.852} D^{4.8704}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe waterworks head-loss equation in SI form, with Q in m³/s and D in m; the 10.67 constant is 4.727 when working in feet and cubic feet per second.

Hazen–Williams Velocity

v=0.849 C R0.63S0.54v = 0.849 \, C \, R^{0.63} S^{0.54}

HVAC & HydronicsFluid MechanicsWater TreatmentMean water velocity from hydraulic radius and hydraulic gradient; the 0.849 SI constant becomes 1.318 when R is in feet and v in feet per second.

Heat Exchanger Duty (Q = U·A·F·LMTD)

Q˙=UAF ΔTlm\dot{Q} = U A F \, \Delta T_{lm}

Heat TransferThermodynamicsHVAC & HydronicsThe LMTD design equation with the correction factor F, which derates the counterflow driving force for shell-and-tube or crossflow arrangements.

Heat Exchanger Effectiveness (ε = Q/Qmax)

ε=Q˙Q˙max\varepsilon = \frac{\dot{Q}}{\dot{Q}_{max}}

Heat TransferThermodynamicsHVAC & HydronicsEffectiveness as the ratio of actual duty to the thermodynamic maximum, the performance figure that needs no outlet temperatures to interpret.

Heat Flow from Thermal Resistance

Q˙=ΔTR\dot{Q} = \frac{\Delta T}{R}

Heat TransferThermodynamicsHVAC & HydronicsOhm's law for heat: the flow through an assembly equals the temperature difference across it divided by its total thermal resistance.

Heat Flux Through Insulation (q = ΔT/R)

q′′=ΔTRq'' = \frac{\Delta T}{R}

Heat TransferHVAC & HydronicsThermodynamicsHeat flow per unit area through an insulated assembly, straight from the temperature difference and the R-value, with no area needed.

Heat Loss Through an Assembly (Q = A·ΔT/R)

Q˙=A ΔTRtot\dot{Q} = \frac{A \, \Delta T}{R_{tot}}

Heat TransferHVAC & HydronicsThermodynamicsSteady heat loss through a wall, roof or floor from its area, the inside-to-outside temperature difference and the assembly's total R-value.

Humidity Ratio from Vapour Pressure

W=0.62198 pvp−pvW = 0.62198\,\frac{p_v}{p - p_v}

HVAC & HydronicsThermodynamicsAir Quality & DispersionKilograms of water vapour per kilogram of dry air, from the vapour pressure and the barometric pressure — the one humidity variable that does not move when you heat the air.

Hydronic Heat Transfer (Water)

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

HVAC & HydronicsThermodynamicsFluid MechanicsHeat carried by a water loop from flow rate and supply-to-return ΔT — the SI form of the trade rule BTU/hr = 500 × GPM × ΔT.

Hydronic Static Fill Pressure

P=ρwgH+PmarginP = \rho_w g H + P_{margin}

HVAC & HydronicsFluid MechanicsCold fill pressure a closed loop needs to lift water to its highest point plus a safety margin, the SI form of the 2.31 ft per psi rule.

Laminar Friction Factor (f = 64/Re)

f=64Ref = \frac{64}{Re}

HVAC & HydronicsFluid MechanicsPhysicsIn laminar pipe flow the Darcy friction factor depends only on Reynolds number — roughness plays no part below about Re = 2300.

Log Mean Temperature Difference (Counterflow)

ΔTlm=ΔT1−ΔT2ln⁡(ΔT1/ΔT2)\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}

Heat TransferThermodynamicsHVAC & HydronicsEffective driving temperature difference in a counterflow exchanger, from the terminal differences at the hot and cold ends of the shell.

Log Mean Temperature Difference (Parallel Flow)

ΔTlm=ΔT1−ΔT2ln⁡(ΔT1/ΔT2)\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}

Heat TransferThermodynamicsHVAC & HydronicsEffective driving temperature difference when both streams enter at the same end, pairing the two inlets and the two outlets.

Loop Water Expansion Volume

ΔV=V0 β ΔT\Delta V = V_0 \, \beta \, \Delta T

HVAC & HydronicsFluid MechanicsThermodynamicsVolume a hydronic loop's water gains when heated, from the starting volume, the volumetric expansion coefficient and the temperature rise.

Minor Loss from K Factor

hL=K v22gh_L = K \, \frac{v^{2}}{2g}

HVAC & HydronicsFluid MechanicsWater TreatmentHead lost through a valve or fitting as a multiple of velocity head, with g = 9.80665 m/s² and K taken from a fitting table.

Mixed Air Temperature

Tm=f Toa+(1−f) TraT_m = f \, T_{oa} + (1-f) \, T_{ra}

HVAC & HydronicsThermodynamicsTemperature of the blend leaving a mixing box, weighted by the outdoor air fraction — the reading that verifies an economizer's damper position.

Moist Air Density at Altitude (and the 1.08 Correction)

ρ=pz (1+W)Rda T (1+1.6078 W),pz=101 325 (1−2.25577×10−5z)5.25588\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}

HVAC & HydronicsThermodynamicsAir Quality & DispersionDensity of moist air from altitude, temperature and humidity ratio, and the correction factor it forces on the 1.08, 0.68 and 4.5 rules — which are sea-level, standard-air numbers and nothing else.

Moist Air Enthalpy (per kg DRY air)

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

HVAC & HydronicsThermodynamicsTotal heat content of moist air per kilogram of DRY air — the sensible term for the air plus the latent and sensible terms for the water it carries. The basis is per kg of dry air, not per kg of mixture.

Moist Air Specific Volume (per kg DRY air)

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

HVAC & HydronicsThermodynamicsCubic metres of moist air per kilogram of DRY air, from temperature, humidity ratio and barometric pressure — the number that turns a fan's volume flow into a mass flow.

Net Positive Suction Head Available (NPSHa)

NPSHa=hatm+hs−hf−hvpNPSH_a = h_{atm} + h_s - h_f - h_{vp}

HVAC & HydronicsFluid MechanicsWater TreatmentAbsolute head available at the pump suction above the liquid's vapour pressure — the margin that keeps a pump from cavitating.

Newton's Law of Cooling (Q = hAΔT)

Q˙=hA ΔT\dot{Q} = h A \, \Delta T

Heat TransferThermodynamicsHVAC & HydronicsConvective heat rate from a surface, set by the film coefficient, the wetted area and the surface-to-fluid temperature difference.

Overall Heat Transfer Coefficient (U)

1U=1hi+Lk+1ho\frac{1}{U} = \frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}

Heat TransferThermodynamicsHVAC & HydronicsOverall coefficient U for a plane wall with fluid on both sides, adding the inside film, the wall and the outside film as resistances in series.

Overall U from Total Resistance

U=1RtotAU = \frac{1}{R_{tot} A}

Heat TransferThermodynamicsHVAC & HydronicsConverts an assembly's total resistance in kelvin per watt into the overall coefficient U quoted on exchanger and envelope datasheets.

Partially Filled Horizontal Cylindrical Tank

V=L[r2cos⁡−1 ⁣(r−hr)−(r−h)2rh−h2]V = L \left[ r^{2} \cos^{-1}\!\left(\frac{r-h}{r}\right) - (r-h)\sqrt{2rh - h^{2}} \right]

HVAC & HydronicsFluid MechanicsGeometryLiquid volume in a horizontal cylinder from the wetted depth, using the circular segment area times the tank length.

Pipe Internal Volume

V=πD24LV = \frac{\pi D^{2}}{4} L

HVAC & HydronicsFluid MechanicsWater TreatmentThe liquid a run of pipe holds, from inside diameter and developed length — the starting point for every flush, fill or chemical dose.

Pipe Velocity from Flow and Diameter

v=4QπD2v = \frac{4Q}{\pi D^{2}}

HVAC & HydronicsFluid MechanicsWater TreatmentAverage velocity in a full round pipe from volumetric flow and inside diameter — the first check on any piping design.

Pump Affinity Law — Flow vs Impeller Diameter

Q2Q1=D2D1\frac{Q_2}{Q_1} = \frac{D_2}{D_1}

HVAC & HydronicsFluid MechanicsWater TreatmentCapacity scales directly with trimmed impeller diameter at constant speed, the classic way to de-rate an oversized centrifugal pump permanently.

Pump Affinity Law — Flow vs Speed

Q2Q1=N2N1\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsWater TreatmentFirst affinity law: a centrifugal pump's capacity changes in direct proportion to shaft speed when the impeller diameter is unchanged.

Pump Affinity Law — Head vs Impeller Diameter

H2H1=(D2D1)2\frac{H_2}{H_1} = \left(\frac{D_2}{D_1}\right)^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentDeveloped head falls with the square of the trimmed impeller diameter, so a 10% trim sheds about 19% of the head at constant speed.

Pump Affinity Law — Head vs Speed

H2H1=(N2N1)2\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentSecond affinity law: pump head varies with the square of shaft speed, so a 20% speed cut costs 36% of the developed head.

Pump Affinity Law — Power vs Speed

P2P1=(N2N1)3\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsWater TreatmentThird affinity law: absorbed power varies with the cube of shaft speed — the single relation that pays for every variable-frequency drive.

Pump Brake Horsepower

BHP=Q H SG3960 ηBHP = \frac{Q \, H \, SG}{3960 \, \eta}

HVAC & HydronicsFluid MechanicsWater TreatmentShaft power the motor must actually supply; the 3960 constant assumes gpm, feet of head and horsepower, with efficiency as a fraction.

Pump Efficiency from Hydraulic and Shaft Power

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

HVAC & HydronicsFluid MechanicsWater TreatmentPump efficiency is the ratio of hydraulic power delivered to the liquid over the mechanical power absorbed at the shaft.

Pump Operating Point

Qop=H0−Hstk+cQ_{op} = \sqrt{\frac{H_{0} - H_{st}}{k + c}}

HVAC & HydronicsFluid MechanicsWater TreatmentFlow at which a pump actually runs: the crossing of its own head curve, H = H₀ − cQ², with the system curve H = H_st + kQ².

Pump Specific Speed (Ns)

Ns=NQH0.75N_s = \frac{N \sqrt{Q}}{H^{0.75}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe dimensional index that classifies impeller type, evaluated in US units with N in rpm, Q in gpm and H in feet at the best efficiency point.

Pump System Curve

H=Hst+kQ2H = H_{st} + k Q^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentHead a piping system demands at any flow: the static lift, which never changes, plus a friction term that grows as the square of the flow.

Pump Water Horsepower

WHP=Q H SG3960WHP = \frac{Q \, H \, SG}{3960}

HVAC & HydronicsFluid MechanicsWater TreatmentUseful power delivered to the liquid; the 3960 divisor assumes US gallons per minute, feet of head and horsepower output.

R-Value of an Insulation Layer (R = L/k)

R=LkR = \frac{L}{k}

Heat TransferHVAC & HydronicsThermodynamicsArea-specific thermal resistance of one layer, from its thickness and thermal conductivity — the RSI or R-value quoted on every insulation label.

Radiator Output at Non-Rated Temperature

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

HVAC & HydronicsThermodynamicsCorrects a radiator or baseboard's catalogue output to the actual water-to-air temperature difference using the emitter exponent n.

Refrigerant Mass Flow Rate

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

HVAC & HydronicsThermodynamicsRefrigerant circulated per unit time from the cooling capacity and the enthalpy change across the evaporator, the basis of compressor sizing.

Refrigerant Subcooling

SC=Tsat−Tliquid\mathrm{SC} = T_{sat} - T_{liquid}

HVAC & HydronicsThermodynamicsDegrees of subcooling at the condenser outlet: how much colder the liquid is than its saturation temperature at the same pressure.

Refrigerant Superheat

SH=Tsuction−Tsat\mathrm{SH} = T_{suction} - T_{sat}

HVAC & HydronicsThermodynamicsDegrees of superheat at the compressor suction: how much warmer the vapour is than its saturation temperature at the same pressure.

Refrigeration COP from Enthalpies

COP=h1−h4h2−h1COP = \frac{h_1 - h_4}{h_2 - h_1}

ThermodynamicsHVAC & HydronicsCoefficient of performance of a vapour-compression cycle read straight off a pressure-enthalpy chart: refrigerating effect over compressor work. h₁ is the compressor suction, h₂ the discharge, h₄ the enthalpy leaving the expansion device.

Relative Humidity from a Sling Psychrometer

φ=pws(twb)−A p (tdb−twb)pws(tdb)\varphi = \frac{p_{ws}(t_{wb}) - A\,p\,(t_{db} - t_{wb})}{p_{ws}(t_{db})}

HVAC & HydronicsThermodynamicsAir Quality & DispersionRelative humidity from a dry-bulb and a wet-bulb reading, using the psychrometric equation with A = 6.66×10⁻⁴ K⁻¹ — the calculation behind every paper psychrometric slide rule.

Relative Humidity from Vapour Pressure

φ=pvpws\varphi = \frac{p_v}{p_{ws}}

HVAC & HydronicsThermodynamicsAir Quality & DispersionRelative humidity as what it actually is: the vapour pressure present divided by the saturation vapour pressure at the same temperature — a ratio to a target that moves whenever the air is heated.

Round Duct Air Velocity

v=4V˙πd2v = \frac{4 \dot{V}}{\pi d^{2}}

HVAC & HydronicsFluid MechanicsAir velocity in a round duct from the volume flow and the duct diameter, the check that keeps branches quiet and mains efficient.

Saturation Temperature and Pressure of Steam

Tsat=Ts(psat)psat=ps(Tsat)T_{sat} = T_s(p_{sat}) \qquad p_{sat} = p_s(T_{sat})

ThermodynamicsWater TreatmentHVAC & HydronicsSaturation temperature from pressure and saturation pressure from temperature, on the IAPWS-IF97 Region 4 line the printed steam tables come from.

Saturation Vapour Pressure (Magnus / Alduchov–Eskridge)

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

HVAC & HydronicsThermodynamicsAir Quality & DispersionThe vapour pressure of water at saturation, from temperature alone, by the Alduchov–Eskridge Magnus fit — the ceiling every other psychrometric quantity is measured against.

Seasonal Heating Energy (Degree-Day Method)

E=Q˙d ΔTm tΔTd ηE = \frac{\dot{Q}_d \, \Delta T_m \, t}{\Delta T_d \, \eta}

HVAC & HydronicsThermodynamicsEstimates seasonal fuel energy by scaling the design heat loss with the average temperature deficit, season length and equipment efficiency.

Sensible Heat Ratio (SHR)

SHR=Q˙sQ˙s+Q˙l\mathrm{SHR} = \frac{\dot{Q}_s}{\dot{Q}_s + \dot{Q}_l}

HVAC & HydronicsThermodynamicsThe fraction of a cooling coil's total load that is sensible, the number that decides whether a room ends up cool or merely cold and clammy.

Snow Gun Output Rate

V˙s=V˙w ρwρs\dot{V}_s = \dot{V}_w \, \frac{\rho_w}{\rho_s}

Snow & IceHVAC & HydronicsTrades & ConstructionSnow production rate from the water flow a gun is fed and the density of the snow it makes. Guns are specified in gallons or litres per minute of water and hills are planned in cubic metres or acre-feet of snow, and this is the conversion between the two.

Snow Made in a Wet-Bulb Window

Vs=V˙w t (1−f) ρwρsV_s = \dot{V}_w \, t \, (1 - f) \, \frac{\rho_w}{\rho_s}

Snow & IceHVAC & HydronicsTrades & ConstructionHow much snow a given water flow actually puts on the ground over a night, once the water lost to evaporation and drift is taken off. THE FACT THIS PAGE EXISTS TO TEACH: snowmaking is gated on WET-BULB temperature, not on the thermometer. You can make snow at +2 °C if the air is dry enough, and you cannot make it at −1 °C if the air is damp.

Snow Volume from Water Volume

Vs=Vw ρwρsV_s = V_w \, \frac{\rho_w}{\rho_s}

Snow & IceHVAC & HydronicsTrades & ConstructionHow much snow a given volume of water becomes, at whatever density the snow is made to. It is a mass balance and nothing more — the water does not change amount, only how much space it occupies — which is why the snow density on the bottom is the only interesting number in it.

Stack Draft Pressure (Chimney Effect)

Δp=h g (ρa−ρs)\Delta p = h \, g \, (\rho_a - \rho_s)

Air Quality & DispersionHVAC & HydronicsThe pressure a chimney generates on its own, from the height of the column and the density difference between cold outside air and hot flue gas. The same equation explains why a tall building's lobby doors are hard to open in January.

Stack Exit Velocity

vs=4Qvπd2v_s = \frac{4 Q_v}{\pi d^2}

Air Quality & DispersionHVAC & HydronicsThe speed exhaust leaves a round stack, from the volumetric flow and the inside diameter. Every plume-rise calculation starts here, and so does the check against stack-tip downwash.

Steam Coil Condensate Load

m˙=Q˙hfg\dot m = \frac{\dot Q}{h_{fg}}

HVAC & HydronicsThermodynamicsWater TreatmentSteam a coil condenses per unit time from its heat duty and the latent heat at the coil pressure — the same figure that sizes the trap and the condensate return.

Stream Duty from Mass Flow (Q = ṁcΔT)

Q˙=m˙ cp ΔT\dot{Q} = \dot{m} \, c_p \, \Delta T

Heat TransferThermodynamicsHVAC & HydronicsHeat picked up or given off by one exchanger stream, from its mass flow, specific heat and the temperature change across the unit.

Superheated Steam Enthalpy h(p, T)

h=R T τ(γτ∘+γτr),τ=540 KTh = R\,T\,\tau\left(\gamma^{\circ}_{\tau} + \gamma^{r}_{\tau}\right), \quad \tau = \frac{540\ \mathrm{K}}{T}

ThermodynamicsHVAC & HydronicsSpecific enthalpy of superheated steam from its pressure and temperature, straight off IAPWS-IF97 Region 2 — the equation the printed superheated tables are generated from, without the interpolation between rows.

Swamee–Jain Friction Factor

f=0.25[log⁡10 ⁣(ε3.7D+5.74Re0.9)]2f = \frac{0.25}{\left[\log_{10}\!\left(\frac{\varepsilon}{3.7D} + \frac{5.74}{Re^{0.9}}\right)\right]^{2}}

HVAC & HydronicsFluid MechanicsPhysicsAn explicit turbulent friction factor within about 1% of the implicit Colebrook–White equation, valid for Re from 5000 to 10⁸.

Tons of Refrigeration from BTU/hr

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

HVAC & HydronicsThermodynamicsConverts a cooling load in BTU/hr (or kW) to tons of refrigeration, where one ton is 12,000 BTU/hr or 3.5169 kW.

Total Dynamic Head

TDH=hs+hf+hvTDH = h_s + h_f + h_v

HVAC & HydronicsFluid MechanicsWater TreatmentThe head a pump must develop: static lift plus friction losses plus velocity head, all expressed in feet or metres of the pumped liquid.

Total R-Value of an Assembly

Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3

Heat TransferHVAC & HydronicsThermodynamicsTotal R-value of a wall, roof or floor built up from three layers in series, where the same heat crosses each layer so the R-values simply add.

U-Factor from Total R-Value (U = 1/R)

U=1RtotU = \frac{1}{R_{tot}}

Heat TransferHVAC & HydronicsThermodynamicsConverts an assembly's total R-value into the U-factor used by energy codes and window labels, and back — the two are simple reciprocals.

Valve Flow Coefficient (Cv)

Q=CvΔPSGQ = 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=KvΔpSGQ = 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.

Water Hammer Surge (Joukowsky Equation)

ΔP=ρ a Δv\Delta P = \rho \, a \, \Delta v

HVAC & HydronicsFluid MechanicsPhysicsPeak pressure surge from a sudden change in flow velocity: fluid density times pressure-wave celerity times the velocity change.

Wet-Bulb Temperature (Stull 2011)

Tw=T arctan⁡ ⁣[0.151977RH+8.313659 ]+arctan⁡(T+RH)−arctan⁡(RH−1.676331)+0.00391838 RH3/2arctan⁡(0.023101 RH)−4.686035T_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

Snow & IceHVAC & HydronicsThermodynamicsWet-bulb temperature from dry-bulb temperature and relative humidity in a single closed-form expression, fitted by Roland Stull in 2011 to replace the iterative psychrometric solve. It is the lowest temperature evaporation alone can reach, which makes it the control variable for cooling towers, evaporative coolers, heat-stress limits and snowmaking alike.