# Formulas – Power and processes

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Formula hydro power

The power available from a certain hydropower resource may be determined by calculating the hydraulic head and the flow rate of the water.

Power from falling water (P):

P = ηρQgh

P = power (W)
η = efficiency factor of the turbine
ρ = density of water (Kg/m3)
Q = the water flow (m3/sec)
g = the acceleration due to gravity (m/sec2)
h = the height difference between inlet and outlet (m)

Formula for wind power
Wind power density:

$Power = \dfrac{1}{2}\rho v^2 (W/m^2)$

Power = 0,5 * 1,225 * v3 ,

The available effect of the wind, which passes perpendicularly through a circular surface, is:

P = ½ v3 r2
P = Wind power (W).
ρ = Density of dry air = 1.225 kg/m3 (at 15° C and athmosferic pressure at sea level)
v = Wind speed (m/s)π = 3.14159
r = Rotor radius (m).

Formula for wave power
If the water depth is larger than half the wavelength, the wave energy flux is:

$P = \dfrac{\rho g^2}{64\pi}H^2_{mo}\approx (0,5\dfrac{kw}{m^3s})H^2_{mo}T_e$

P = Wave energy flux per unit of wave-crest length (kW)
Hm0 = Significant wave height (m)
Te = Wave energy period (s)
ρ = Water density
G = Acceleration by gravity.

Wave energy and wave-energy flux
The mean energy density waves on the water surface is:

$E = \dfrac{1}{2}\rho gH^2_{mo}$

E = The sum of kinetic and potential energy density per unit horizontal area. The potential energy density is equal to the kinetic energy, both contributing half to the wave energy density E.

Formula for tidal energy
The energy available from a kinetic system is:

$P= \dfrac{\rho AV^3}{2}C_P$

CP = the turbine power coefficient
P = the power generated (W)
ρ = the density of the water (if seawater =1027 kg/m³)
A = the sweep area of the turbine (in m²)
V = the velocity of the flow

A ducted turbine are capable of 3 to 4 times the power of the same turbine rotor in open flow.

Formula for gas turbine
Thermal efficiency (Joule process):

$\eta_0 = 1 - \dfrac{T_4-T_1}{T_3-T_2}$

η0 = (Theoretical/ideal thermal efficiency)
T1 = Temperature at beginning of compression
T2= Temperature after isentropic compression
T3 = Temperature after combustion at beginning of expansion
T4 = Temperature after isentropic expansion

$\eta_i = 1 - \dfrac{W_i}{Q_{appl.}}$

ηi = Indicated thermal efficiency
Wi= Indicated work
Qappl.= Applied energy

ηi always is less than η0 , the actual temperature raise in the compressor is larger and the actual temperature drop in the turbine is less than for a theoretical process (adiabatic).

The work (W) during compression or expansion can be expressed as the difference in energy (enthalpy) before and after the change (compression /expansion).

$W = c_p \Delta t$

W = Work (J)
cp = Specific heat Capacity (at constant pressure) (J/KgoK)
Δt = Temperature difference (before and after compression/expansion) (oK)

To quantify the losses due to friction in the turbine and the compressor, the efficiency factors ηc (compressor) and ηt (turbine) are established.

$\eta_C = \dfrac{Temperature-increase (adiabatic-compression)}{Temperature-increase(actual-compression)}$

$\eta_C = \dfrac{Temperature drop (actual-expansion)}{Temperature drop (adiabatic-expansion)}$

Available work for the ideal (theoretical) process is expressed:
W0 = Wt0 – WC0

Wt0 = Work turbine
WC0 = Work compressor

The indicated work is then:
Wi = Wti – WCi = Wt0ηT – WCiηC

Some of the indicated work (Wi) will not be available at the shaft since it is used to overcome mechanical losses and for operation of auxiliary systems (injection pumps lubrication pumps etc).
The remaining work is available as effective work at the output shaft (We)

ηe = ηi ηm

ηe = Effectiv (actual) efficiency factor
ηm = Mechanical effieciency factor

Fuel consumption and power
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$\eta_e = \dfrac{1}{f_eh_g}$

ηe = Effectiv (actual) efficiency factor
fe = Effective specific fuel consumption (Kg/J)
hg = Higher heating Value for the fuel (J/Kg)

$f_e = \dfrac{1}{h_g\eta_e}(kg/J)$

$m_f = f_eP_e = \dfrac{m_a}{\lambda(\dfrac{L}{B})_r}(kg/s)$

$P_e = \dfrac{m_a}{f_e\lambda(\dfrac{L}{B})_r}=\dfrac{m_a\eta_eh_g}{f_e\lambda(\dfrac{L}{B})_r}$

Pe= Actual power (at the shaft) (W)
mf = Fuel consumption (mass, fuel)
f= Effective specific fuel consumption (Kg/J)
ma = Air consumtion (Kg)
λ = Air index (related to air consumption)

$(\dfrac{L}{B})_r$ = Theoretical Air- fuel relationship, related to chemical reaction equation (“reaction equivalent”) (Kg/Kg)

Formula for the Otto cycle
1-2 Adiabatic compression
2-3 Combustion (temperasture increase) at constant volume
3-4 Adiabatic expansion
4-1 Gas exchange (heat removal) at constant volume

Ideal (theoretical) thermal efficiency
$\eta_0= \dfrac{w_0}{Q_t}= \dfrac{Q_t-Q_b}{Q_t}=1- \dfrac{Q_b}{Q_t} = 1- \dfrac{c_v(T_4-T_1)m}{c_v(T_3-T_2)m} = 1-\dfrac{(T_4-T_1)}{(T_3-T_2)}$

cv = Specific heat capacity for heating at constant volume (J/KgoK)
m = Charge (Kg)

Since:
PV/T = Constant
and during the adiabatic expansion (no heat transfer)
PVK = Constant
and during polytropic process
Pen = Constant

κ = adiabatic factor
n = Polytropic factor

The ideal (theoretical) thermal efficiency may then be expressed:

$\eta_0=1- \dfrac{V_2}{V_1}$

V1 = Cylinder volume above the piston top at BDC
V2 = Cylinder volume above the piston top at TDC

$\varepsilon_n= \dfrac{V_1}{V_2}$ = Nominal compression ratio

$\eta_0=1- \dfrac{1}{\varepsilon^{\varkappa-1}_n}$

Formula for the diesel cycle
1-2 Adiabatic compression
2-3 Combustion (temperasture increase) at constant pressure
3-4 Adiabatic expansions
4-1 Gas exchange (heat removal) at constant

Ideal (theoretical) thermal efficiency:
$\eta_0=1- \dfrac{1}{\varepsilon^{\varkappa-1}_n} \times \dfrac{\rho^\varkappa-1}{\varkappa(\rho-1)}$

κ = Cp/Cv = Adiabatic factor
εn =V1/V2 = Nominal compression ratio
ρ = V3/V2 = Volume differential during combustion