Thermodynamics formulas — 137 solvers

Air Latent Heat (0.68 Rule)

\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)

\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)

\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.

Antoine Equation (Vapour Pressure)

\log_{10} P = A - \frac{B}{C + T}

ChemistryThermodynamicsThe three-constant fit that every handbook uses for pure-component vapour pressure. The published constants A, B and C are for pressure in mmHg and temperature in degrees Celsius; enter pressure and temperature in whatever units you like and the conversion is handled inside.

Biot Number

\mathrm{Bi} = \frac{h L_c}{k}

Heat TransferThermodynamicsRatio of internal conduction resistance to surface convection resistance; below 0.1 a body may be treated as having one uniform temperature.

Boiler Blowdown Rate from Steam Rate

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

Water TreatmentThermodynamicsContinuous blowdown a steam boiler must carry, in pounds per hour, from its steam production rate and target cycles of concentration.

Boiler Horsepower to Heat Output

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

Water TreatmentThermodynamicsConverts boiler horsepower to heat output using the ASME definition of 33,475 BTU per hour per boiler horsepower.

Boiler Horsepower to Steam Rate

S = 34.5 \; \text{BHP}

Water TreatmentThermodynamicsSteam output of a boiler from its horsepower rating, at the ASME definition of 34.5 lb/h of steam from and at 212 °F per BHP.

Boiler Makeup from Condensate Return

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

Water TreatmentThermodynamicsFresh makeup water a steam plant must treat, from the steam production rate and the fraction of condensate that comes back.

Boiler or Furnace Output from Input

\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.

Boyle's Law

P_1 V_1 = P_2 V_2

ThermodynamicsChemistryPhysicsAt constant temperature, pressure times volume stays constant for a fixed amount of gas.

Brayton Cycle Efficiency (Pressure Ratio)

\eta = 1 - \frac{1}{r_p^{\left(\gamma - 1\right)/\gamma}}

ThermodynamicsPhysicsAir-standard efficiency of a gas turbine from compressor pressure ratio alone. Turbine inlet temperature does not appear, which is why the equation says nothing about the specific work — the thing that actually limits real machines.

Calorimeter Heat (q = C_cal ΔT)

q = C_{\text{cal}} \, \Delta T

ChemistryThermodynamicsGives the heat absorbed by a calorimeter from its calibrated heat capacity and the temperature rise it records.

Capacity Rate Ratio (Cr)

C_r = \frac{\dot{m}_{min} c_{min}}{\dot{m}_{max} c_{max}}

Heat TransferThermodynamicsRatio of the smaller to the larger stream heat capacity rate ṁcₚ, the second dimensionless group the effectiveness-NTU method needs.

Carnot Efficiency

\eta = 1 - \frac{T_c}{T_h}

ThermodynamicsPhysicsThe maximum possible efficiency of any heat engine operating between two absolute temperatures.

Celsius to Fahrenheit Conversion

\theta_F = \tfrac{9}{5}\,\theta_C + 32

Everyday & HealthThermodynamicsThe Celsius and Fahrenheit scales, converted both ways. Multiply by nine fifths and add thirty-two one way, subtract thirty-two and multiply by five ninths the other.

Celsius to Kelvin Conversion

T = \theta_C + 273.15

Everyday & HealthThermodynamicsThe Celsius and kelvin scales, converted both ways. The degree is the same size on both, so only the zero moves: 273.15 of them, to absolute zero.

Charles's Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

ThermodynamicsChemistryPhysicsAt constant pressure, gas volume is directly proportional to absolute temperature.

Chiller Efficiency (kW per Ton)

\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

Q_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.

Clausius–Clapeyron Equation (Two-Point Form)

\ln\!\left(\frac{P_2}{P_1}\right) = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)

ChemistryThermodynamicsRelates two points on a liquid's vapour-pressure curve to its molar enthalpy of vaporisation, assuming ΔH is constant over the interval and the vapour behaves ideally.

Coefficient of Performance (COP)

\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.

Combined Convection and Radiation Coefficient

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.

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

ThermodynamicsChemistryPhysicsFor a fixed amount of gas, pressure times volume over absolute temperature stays constant between any two states.

Combustion (Stack) Efficiency — Siegert Formula

\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.

Compressibility Factor (Z = PV/nRT)

Z = \frac{P V}{n R T}

ChemistryThermodynamicsHow far a real gas departs from ideal behaviour, as a single multiplier on the ideal gas law. Z = 1 is ideal; below 1 attraction dominates, above 1 the molecules' own volume does.

Compressor Isentropic Efficiency

\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.

Condensate Return Percentage

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

Water TreatmentThermodynamicsPercentage of generated steam that comes back to the boiler house as condensate — the headline efficiency number for any steam plant.

Conduction Through a Pipe Wall

\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 = \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

\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.

Cooling Tower Approach

A = T_c - T_{wb}

Water TreatmentThermodynamicsCooling tower approach: how many degrees the cold basin water sits above the ambient wet-bulb temperature, the true measure of tower performance.

Cooling Tower Evaporation Rate

E = 0.001 \, R \, \Delta T

Water TreatmentFluid MechanicsThermodynamicsEvaporation loss from a cooling tower using the industry rule of 0.1% of recirculation per degree Fahrenheit of range.

Cooling Tower Heat Rejection

Q = 500 \, R \, \Delta T

Water TreatmentThermodynamicsFluid MechanicsHeat a cooling tower rejects from flow and range using the trade constant 500 = 8.34 lb/gal × 60 min/h × 1 BTU/(lb·°F).

Cooling Tower Range

\Delta T = T_h - T_c

Water TreatmentThermodynamicsCooling tower range: the temperature drop the tower achieves between the hot water returning from the plant and the cold basin water.

Critical Radius of Insulation

r_{cr} = \frac{k}{h}

Heat TransferThermodynamicsThe outer radius below which adding insulation to a small cylinder increases heat loss, because added surface beats added resistance.

Dalton's Law of Partial Pressures

P_{\text{total}} = P_1 + P_2 + P_3

ChemistryThermodynamicsPhysicsStates that each gas in a mixture exerts its own pressure independently, so the total pressure is the sum of the partial pressures.

Degree of Saturation (Moist Air)

\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

\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

\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 (Magnus Approximation)

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

Everyday & HealthThermodynamicsDew point from air temperature and relative humidity by the Magnus formula with the WMO's Sonntag coefficients (b = 17.62, c = 243.12 °C).

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}

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

\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)

\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.

Effectiveness from NTU (Counterflow)

\varepsilon = \frac{1 - e^{-\mathrm{NTU}(1 - C_r)}}{1 - C_r \, e^{-\mathrm{NTU}(1 - C_r)}}

Heat TransferThermodynamicsCounterflow effectiveness from the two dimensionless groups NTU and Cr, valid for any Cr from 0 to 1 with the balanced case handled as a limit.

Energy Cost from a Utility Rate

C_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)

\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.

Fin Efficiency (Straight Fin)

\eta_f = \frac{\tanh(mL)}{mL}

Heat TransferThermodynamicsEfficiency of a straight fin with an adiabatic tip, comparing its real duty with the duty it would give if it were all at base temperature.

Fin Heat Transfer Rate

\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.

Fin Parameter mL (Straight Fin)

mL = L \sqrt{\frac{2h}{k t}}

Heat TransferThermodynamicsThe dimensionless group governing straight-fin performance, combining fin length, thickness, material conductivity and the surface film coefficient.

Flash Steam Mass Rate

\dot{m}_f = \frac{\%F}{100} \, \dot{m}_c

ThermodynamicsWater TreatmentFlash steam a condensate stream releases when it is let down to a lower pressure, from the flash percentage and the condensate rate through the trap.

Flash Steam Percentage

\%F = \frac{h_{f1} - h_{f2}}{h_{fg2}} \times 100

Water TreatmentThermodynamicsPercentage of hot condensate that flashes to steam when let down to a lower pressure, from the saturated liquid and latent enthalpies.

Fouled Overall Coefficient

\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

\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.

Fourier Number

\mathrm{Fo} = \frac{k \, t}{\rho \, c \, L^{2}}

Heat TransferThermodynamicsDimensionless time for transient conduction, written from conductivity, density and specific heat so no diffusivity input is needed.

Gas Density from Molar Mass

\rho = \frac{PM}{RT}

ChemistryThermodynamicsPhysicsGives an ideal gas's density from its molar mass, pressure, and absolute temperature using R = 8.314462618 J/(mol·K).

Gauge and Absolute Pressure

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

Fluid MechanicsThermodynamicsPhysicsAbsolute pressure is the gauge reading plus the surrounding atmospheric pressure.

Gay-Lussac's Law

\frac{P_1}{T_1} = \frac{P_2}{T_2}

ThermodynamicsChemistryPhysicsAt constant volume, gas pressure is directly proportional to absolute temperature.

Gibbs Free Energy and the Equilibrium Constant

\Delta G^{\circ} = -RT\ln K

ChemistryThermodynamicsConverts between a reaction's standard free energy change and its equilibrium constant, the bridge joining thermodynamics to equilibrium tables.

Gibbs Free Energy Change (ΔG = ΔH − TΔS)

\Delta G = \Delta H - T\,\Delta S

ChemistryThermodynamicsCombines a reaction's enthalpy and entropy changes at a given temperature to decide whether it can happen spontaneously.

Gibbs Free Energy from Cell Potential (ΔG° = −nFE°)

\Delta G^{\circ} = -n F E^{\circ}

ChemistryThermodynamicsConverts a cell's standard potential into the standard free energy change of its reaction, using the Faraday constant F = 96485.33212 C/mol.

Glycol Loop Heat Transfer (Capacity Derate)

\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.

Heat Conduction Rate

P = \tfrac{k A \Delta T}{d}

ThermodynamicsPhysicsSteady-state heat flow through a slab by Fourier's law of conduction.

Heat Exchanger Duty (Q = U·A·F·LMTD)

\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)

\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

\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'' = \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 Index (Rothfusz Regression)

\mathrm{HI} = c_1 + c_2 T + c_3 R + c_4 T R + c_5 T^2 + c_6 R^2 + c_7 T^2 R + c_8 T R^2 + c_9 T^2 R^2

Everyday & HealthThermodynamicsThe US National Weather Service heat index, Rothfusz's nine-term regression through Steadman's apparent-temperature table (T in °F, R in percent).

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

\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.

Heat of Reaction

q = n \Delta H

ChemistryThermodynamicsScales a reaction's molar enthalpy change by the amount reacted to give the total heat released or absorbed.

Hess's Law (Three-Step Sum)

\Delta H_{\text{rxn}} = \Delta H_1 + \Delta H_2 + \Delta H_3

ChemistryThermodynamicsHess's law: the enthalpy change of a target reaction is the sum of the enthalpy changes of the steps you route it through.

Humidex (Canadian Humidity Index)

H = T + 0.5555\,(e - 10), \quad e = 6.11\,e^{5417.753\left(\frac{1}{273.16} - \frac{1}{T_d}\right)}

Everyday & HealthThermodynamicsEnvironment Canada's humidex: air temperature raised by the excess vapour pressure computed from the dew point.

Humidity Ratio from Vapour Pressure

W = 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)

\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.

Kp from Kc (Kp = Kc(RT)^Δn)

K_p = K_c (RT)^{\Delta n}

ChemistryThermodynamicsConverts a gas-phase equilibrium constant between pressure and concentration bases using the change in moles of gas, with R = 0.08206 L·atm/(mol·K).

Latent Heat

Q = m L

ThermodynamicsPhysicsHeat absorbed or released when a mass changes phase at constant temperature.

Log Mean Temperature Difference (Counterflow)

\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)

\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

\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.

Lumped Capacitance Cooling Curve

T = T_\infty + (T_0 - T_\infty) e^{-t/\tau}

Heat TransferThermodynamicsExponential temperature history of a body at uniform temperature, and the time it needs to reach any temperature between start and ambient.

Lumped Capacitance Time Constant

\tau = \frac{\rho V c}{h A}

Heat TransferThermodynamicsThermal time constant of a body cooling at uniform temperature, its stored heat per kelvin divided by the surface conductance hA.

Maximum Possible Heat Transfer (Qmax)

\dot{Q}_{max} = \dot{m}_{min} c_{min} (T_{h,in} - T_{c,in})

Heat TransferThermodynamicsThe thermodynamic ceiling on exchanger duty: the minimum capacity rate multiplied by the full inlet-to-inlet temperature difference.

Mixed Air Temperature

T_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)

\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)

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 = \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.

Napier's Steam Leak Rate

\dot{m} = \frac{A \, P}{70}

ThermodynamicsWater TreatmentFluid MechanicsSteam lost through a hole, a blowing trap or a lifted relief valve by Napier's rule — pounds per second from square inches times psia, over 70.

Net Radiation Exchange Between Surfaces

\dot{Q} = \varepsilon \sigma A (T_1^4 - T_2^4)

Heat TransferThermodynamicsNet radiant heat from a grey surface to large surroundings, using the Stefan-Boltzmann constant and the difference of fourth-power temperatures.

Net Radiative Cooling to the Sky

q_{net} = \varepsilon \sigma \left(T_s^{4} - T_{sky}^{4}\right)

Heat TransferThermodynamicsNet longwave heat lost per square metre from a surface facing the open sky, as the difference of fourth powers between the surface and the effective sky temperature. On a clear night this runs to 60-100 W/m², which is what pulls a surface below air temperature.

Newton's Law of Cooling (Q = hAΔT)

\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.

Number of Transfer Units (NTU)

\mathrm{NTU} = \frac{U A}{\dot{m} \, c_p}

Heat TransferThermodynamicsDimensionless size of an exchanger: its conductance UA divided by the heat capacity rate of the minimum stream, ṁ times its specific heat.

Nusselt Number

\mathrm{Nu} = \frac{h L}{k}

Heat TransferFluid MechanicsThermodynamicsDimensionless convection coefficient: the ratio of convective transfer at a surface to pure conduction through the same fluid layer.

Otto Cycle Efficiency (Compression Ratio)

\eta = 1 - \frac{1}{r^{\gamma - 1}}

ThermodynamicsPhysicsAir-standard efficiency of a spark-ignition engine from its compression ratio alone. Nothing about fuel, speed or displacement enters — only how hard the charge is squeezed and the heat capacity ratio of the working gas.

Overall Heat Transfer Coefficient (U)

\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 = \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.

Partial Pressure from Mole Fraction

P_i = x_i \, P_{\text{total}}

ChemistryThermodynamicsGives a gas component's partial pressure as its mole fraction times the total pressure, the practical form of Dalton's law of partial pressures.

Prandtl Number

\mathrm{Pr} = \frac{\mu c_p}{k}

Heat TransferFluid MechanicsThermodynamicsFluid property group comparing how fast momentum diffuses with how fast heat does, setting the relative thickness of the two boundary layers.

R-Value of a Snow Wall

R = \frac{L}{0.138 - 1.01\,\rho^{*} + 3.233\,\rho^{*2}}

Snow & IceHeat TransferThermodynamicsArea-specific thermal resistance of a snow wall, straight from its thickness and its density, with Sturm's conductivity fit folded in so you never have to look k up. It answers the question a shelter builder actually asks — how much wall is worth building — and nothing whatsoever about whether the shelter is safe.

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

R = \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

\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.

Rankine Cycle Thermal Efficiency

\eta = \frac{\left(h_1 - h_2\right) - \left(h_4 - h_3\right)}{h_1 - h_4}

ThermodynamicsPhysicsNet work over heat added for a steam power cycle, taken straight off the four state-point enthalpies: turbine inlet h₁, turbine exhaust h₂, condensate h₃ and feedwater leaving the pump h₄. Enter all four in the same unit; the answer is a ratio either way.

Refrigerant Mass Flow Rate

\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

\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

\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 = \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

\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

\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.

Saturation Temperature and Pressure of Steam

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)

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 = \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 (Q = mcΔT)

Q = m c \Delta T

ThermodynamicsPhysicsChemistryHeat needed to change a mass's temperature: mass times specific heat times the temperature change.

Sensible Heat Ratio (SHR)

\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 Water Equivalent (SWE)

\mathrm{SWE} = d \, \frac{\rho_s}{\rho_w}

Snow & IceWater & WastewaterThermodynamicsThe depth of water you would be left with if a snowpack melted where it lies — snow depth scaled by the ratio of snow density to water density. It is the number hydrology, irrigation and flood forecasting actually use, because depth alone says nothing about how much water is standing on the ground.

Standard Enthalpy of Reaction from Formation Enthalpies

\Delta H^{\circ}_{\text{rxn}} = \sum \Delta H^{\circ}_{f,\text{prod}} - \sum \Delta H^{\circ}_{f,\text{react}}

ChemistryThermodynamicsThe tabulated form of Hess's law: standard enthalpy of reaction equals the summed formation enthalpies of the products minus those of the reactants.

Steam Coil Condensate Load

\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.

Steam Quality from Enthalpy

x = \frac{h - h_f}{h_{fg}}

ThermodynamicsWater TreatmentDryness fraction of wet steam from its measured enthalpy, the saturated liquid enthalpy and the latent heat at the same pressure.

Steam Turbine Power Output

P = \dot{m} \, w

ThermodynamicsShaft power a steam turbine develops from its steam rate and the specific work taken out of each kilogram, before generator and gearbox losses.

Steam Turbine Specific Work

w = h_1 - h_2

ThermodynamicsWork a steam turbine takes out of every kilogram of steam: the enthalpy at the stop valve less the enthalpy at the exhaust, from the steady-flow energy equation.

Stefan-Boltzmann Law

P = \varepsilon \sigma A T^4

ThermodynamicsPhysicsPower radiated by a hot surface, using the Stefan–Boltzmann constant σ = 5.670374419×10⁻⁸ W/(m²·K⁴).

Stream Duty from Mass Flow (Q = ṁcΔT)

\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\,\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.

Thermal Conductivity of Snow (Sturm 1997)

k_{\mathrm{eff}} = 0.138 - 1.01\,\rho^{*} + 3.233\,\rho^{*2}

Snow & IceHeat TransferThermodynamicsEffective thermal conductivity of seasonal snow from its density alone, using the quadratic fit Sturm and colleagues published in 1997 from 488 measurements. This is the equation behind the fact that a snow shelter works: at 300 kg/m³ snow conducts about the same heat as softwood, and the air trapped between the grains is doing almost all of it.

Thermal Efficiency

\eta = \frac{W}{Q_h}

ThermodynamicsPhysicsFraction of heat input that a heat engine converts into useful work.

Thermal Linear Expansion

\Delta L = \alpha L_0 \Delta T

ThermodynamicsPhysicsLength change of a solid caused by a temperature change, via the linear expansion coefficient.

Thermal Resistance of a Plane Wall

R = \frac{L}{k A}

Heat TransferThermodynamicsConduction resistance of a flat slab in kelvin per watt, from its thickness, thermal conductivity and the area heat crosses.

Thermal Resistances in Series

R_{tot} = R_1 + R_2 + R_3

Heat TransferThermodynamicsTotal resistance of a composite wall, where the same heat crosses each layer in turn so the layer resistances simply add.

Tons of Refrigeration from BTU/hr

T = \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 R-Value of an Assembly

R_{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.

Turbine Isentropic Efficiency

\eta_{isen} = \frac{h_1 - h_2}{h_1 - h_{2s}}

ThermodynamicsIsentropic efficiency of a steam turbine: the actual enthalpy drop divided by the ideal drop a frictionless expansion to the same exhaust pressure would give.

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

U = \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.

van 't Hoff Equation (K at Two Temperatures)

\ln\frac{K_2}{K_1} = -\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)

ChemistryThermodynamicsMoves an equilibrium constant from one temperature to another using the standard enthalpy of reaction, with R = 8.314462618 J/(mol·K).

Van der Waals Equation of State

\left(P + \frac{a n^{2}}{V^{2}}\right)\left(V - n b\right) = n R T

ChemistryThermodynamicsThe first equation of state to describe a real gas, adding a term for molecular attraction (a) and one for the volume the molecules themselves occupy (b). Constant a is entered in Pa·m⁶/mol² and b in volume per mole.

View Factor Reciprocity

A_1 F_{1 \to 2} = A_2 F_{2 \to 1}

Heat TransferThermodynamicsReciprocity relation for radiation view factors, which lets you recover the unknown factor between two surfaces from the known one and their areas.

Wet Steam Enthalpy from h_f and h_g

h = (1 - x) \, h_f + x \, h_g

ThermodynamicsWater TreatmentEnthalpy of a wet steam mixture as the mass-weighted blend of saturated water and dry saturated steam — the lever rule on the h_f and h_g columns.

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

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.

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Modern PhysicsThermodynamicsPhysicsThe peak wavelength of thermal radiation, with b = 2.8978 × 10⁻³ m·K.

Wind Chill (2001 North American Formula)

T_{wc} = 13.12 + 0.6215\,T - 11.37\,V^{0.16} + 0.3965\,T\,V^{0.16}

Everyday & HealthThermodynamicsThe wind chill index adopted by Canada and the United States in 2001, from air temperature and wind speed at 10 m.

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