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}

State Equation for Ideal Gas

pv = RT/MW

Perfect Gas

c_p = constant

k = c_p / c_v

Isentropic Compression

T₂ / T₁  =  (p₂ / p₁)^(k-1)/k

T₂ / T₁  =  (V₁ / V₂)^(k-1)

p₂ / p₁  =  (V₁ / V₂)^k

Combustion -- Flame Temperature

h_reactants = h_products 

Combustion -- HHV

HHV = h_reactants - h_products    

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

T_t / T  =  (1 + M²(k - 1)/2)

p_t / p  =  (1 + M²(k - 1)/2)^k/(k-1)

h_t  =  (h + v²/2)

T_t  =  (T + v²/(2c_p))

Thrust

Thrust =  vm + (p - p_a)A

Cycle Efficiency

Cycle Efficiency =  Net Work/Heat

Carnot Cycle Efficiency

Carnot Cycle Efficiency =  1 - T_{heat rejection} / T_{heat 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 - (COR^k - 1)/(k*CR^(k-1) *(COR - 1)) 

Fuel Cell

Fuel Cell Efficiency =  - (G_out - G_in)/HHV

Heat Rate

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

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].

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 [/]

h_reactants  -- 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]