# Free Engineering Formulas/Equations

Here are some of the basic engineering formulas/equations related to energy conversion systems which are built into the Engineering Software product line:

Continuity Equation

m = vA

Momentum Equation

F = (vm + pA)out - in

Energy Equation

Q - W = ((h + v2/2 + gh)m)out - in

Ideal Gas State Equation

pv = RT/MW

Perfect Gas

cp = constant

k = cp / cv

Isentropic Compression

T2 / T1 = (p2 / p1)(k-1)/k

T2 / T1 = (V1 / V2)(k-1)

p2 / p1 = (V1 / V2)k

Combustion -- Flame Temperature

hreactants = hproducts

Combustion -- HHV

HHV = hreactants - hproducts

Isentropic Expansion

T1 / T2 = (p1 / p2)(k-1)/k

T1 / T2 = (V2 / V1)(k-1)

p1 / p2 = (V2 / V1)k

Sonic Velocity

vs = (kRT/MW)1/2

Mach Number

M = v/vs

Isentropic Flow

Tt / T = (1 + M2(k - 1)/2)

pt / p = (1 + M2(k - 1)/2)k/(k-1)

ht = (h + v2/2)

Tt = (T + v2/(2cp))

Thrust

Thrust = vm + (p - pa)A

Cycle Efficiency

Cycle Efficiency = Net Work/Heat

Carnot Cycle Efficiency

Carnot Cycle Efficiency = 1 - Theat rejection / Theat addition

Brayton Cycle Efficiency

Brayton Cycle Efficiency = 1 - 1/(p2 / p1)(k-1)/k

Otto Cycle Efficiency

Compression Ratio = V1 / V2

Otto Cycle Efficiency = 1 - 1/Compression Ratio(k-1)

Diesel Cycle Efficiency

Compression Ratio (CR) = V1 / V2

Cut-Off Ratio (COR) = V3 / V2

Diesel Cycle Efficiency = 1 - (CORk - 1)/(k*CR(k-1) *(COR - 1))

Fuel Cell

Fuel Cell Efficiency = - (Gout - Gin)/HHV

Heat Rate

Heat Rate = (1/Cycle Efficiency)*3,412

Heat Exchanger (Heat Transfer)

mhot(Thot inlet - Thot outlet)Cp-hot = mcold(Tcold outlet - Tcold inlet)Cp-cold

Air Conditioner (AC) Operation

AC Coefficient of Performance (COP) = 1/(Thigh temperature/Tlow temperature -1)

Heat Pump Operation

HP Coefficient of Performance (COP) = 1/(1 - Tlow temperature/Thigh temperature)

Physical Properties

For each reaction species, the thermodynamic functions specific heat, enthalpy and entropy as
functions of temperature are given in the form of least squares coefficients as follows:

Cp/R = A1 + A2T + A3T2 + A4T3 + A5T4

H/(R*T) = A1 + A2T/2 + A3T2/3 + A4T3/4 + A5T4/5 + A6/T

S/R = A1lnT + A2T + A3T2/2 + A4T3/3 + A5T4/4 + A7

or

S/R = A1lnT + A2T + A3T2/2 + A4T3/3 + A5T4/4 + A7 - lnp

For each species, two sets of coefficients are included for two adjacent temperature intervals, 273 to 1,000 [K]
and 1,000 to 5,000 [K]. The data have been constrained to be equal at 1,000 [K].

 Species C H2 S O2 N2 CO2 H2O SO2 CH4 Molecular Weight [kg/kmol] 12 2 32 32 28 44 18 64 16

 Gas Gas Kappa[/] Gas Constant[J/kg*K] Specific Heat[J/kg*K] Air 1.4 286.7 1,004 Ammonia NH3 1.3 488.2 2,092 Argon Ar 1.67 208.1 519 Carbon Dioxide CO2 1.28 188.9 845 Carbon Monoxide CO 1.4 296.8 1,042 Helium He 1.66 2,078.5 5,200 Hydrogen H2 1.4 4,124 14,350 Methane CH4 1.32 518.2 2,223 Nitrogen N2 1.4 296.9 1,038 Oxygen O2 1.4 259.8 916 Water Vapor H2O 1.33 188.5 1,690

 Fuel HHV [Btu/lbm] Hydrogen 60,000 Natural Gas 20,000 through 24,000 Oil 18,000 through 21,000 Carbon 14,000 Coal 8,000 through 12,000 Wood 6,000

Also,

U = H - p*v*MW or U = H - R*T

G = H - S*T

and

u = h - p*v or u = h - R*T/MW

g = h - s*T

Legend:

m -- Mass Flow Rate [kg/s]

-- Density [kg/m3]

v -- Velocity [m/s]

A -- Cross Sectional Area [m2]

F -- Force [N]

p -- Pressure [N/m2]

Q -- Heat [kW]

W -- Work [kW]]

g -- Gravitational Acceleration [m/s2]

h -- Height [m]

k -- Kappa [/]

hreactants
-- Reactants Enthalpy [kJ/kg]

h
products
-- Products Enthalpy [kJ/kg]

HHV -- Higher Heating Value [Btu/lbm]

vs -- Sonic Velocity [m/s]

M -- Mach Number [/]

Cp -- Specific Heat [kJ/kmol*K]

cp -- Specific Heat [kJ/kg*K]

MW -- Molecular Weight [kg/kmol]

R -- Universal Gas Constant [kJ/kmol*K]

Gas Constant = R/MW [kJ/kg*K]

H -- Enthalpy [kJ/kmol]

h -- Enthalpy [kJ/kg]

T -- Temperature [K]

S -- Entropy [kJ/kmol*K]

s -- Entropy [kJ/kg*K]

p -- Pressure [atm]

U -- Internal Energy [kJ/kmol]

u -- Internal Energy [kJ/kg]

V -- Volume [m3]

v -- Specific Volume [m3/kg]

G -- Gibbs Free Energy [kJ/kmol]

g -- Gibbs Free Energy [kJ/kg]