Farm Power Sources and I.C. Engines

Brake thermal efficiency and fuel consumption

Brake thermal efficiency is the ratio of brake power output to the rate of chemical energy supplied by the fuel, \eta_{th} = \dfrac{P_b}{\dot{m}_f \, CV}, where CV is the calorific value in kJ/kg. Since the brake power is set by the load, the fuel energy input rate is \dot{Q} = P_b/\eta_{th}, and dividing it by the calorific value gives the fuel mass flow rate. Converting mass flow to volume uses the density, \dot{V}_f = \dot{m}_f/\rho_f, and with \rho_f in kg/L the result is in litres per unit time. Brake specific fuel consumption (BSFC) is the fuel mass flow per unit brake power, usually given in g/(kW·h), and it is linked to efficiency by \eta_{th} = 3600/(\text{BSFC} \times CV) when BSFC is in kg/(kW·h) and CV in kJ/kg. A fuel with a lower calorific value needs a higher BSFC to deliver the same efficiency, so a blend such as B20 usually shows a lower efficiency than diesel when the BSFC rises. At equal brake power, the efficiency ratio between two fuels does not depend on the power level. For gas-fuelled dual-fuel operation the same energy logic applies with a specific gas consumption in m^3/(kW·h): the gas volume equals power times time times specific consumption. A biogas holder whose usable capacity is only a fraction of the plant size must be sized so that this usable fraction covers the requirement. The fuel equivalent power is the chemical power of the fuel burned, \dot{m}_f \times CV. It can be found from brake power divided by brake thermal efficiency, or from BSFC times brake power times CV, and when indicated power is known it is first converted to brake power through the mechanical efficiency.

Wind turbine power and Betz limit

A horizontal-axis wind rotor extracts power from the kinetic energy flux through its swept area, P = C_p \cdot \tfrac{1}{2}\rho A V^3, where A = \pi D^2/4 is the swept area, \rho the air density, V the upstream wind speed and C_p the power coefficient. The Betz limit states that no ideal rotor can have C_p above 16/27 \approx 0.593, because the air must leave the rotor with some residual velocity. Practical turbines stay well below this value, and design calculations often combine C_p with drivetrain efficiency. Momentum theory gives the axial thrust as the rate of momentum change, T = \dot{m}(V_1 - V_2), where the mass flow through the disc is \dot{m} = \rho A (V_1+V_2)/2. This gives T = \tfrac{1}{2}\rho A (V_1^2 - V_2^2), with V_1 the approach speed and V_2 the downstream speed. When a turbine drives a water pump, the power passes through transmission and pump efficiencies. The hydraulic power delivered is \rho g Q H, with Q the discharge and H the total head, so the calculation proceeds in sequence: wind power, then rotor shaft power, then transmitted power, then pump output. Because power depends on D^2 and V^3, the rotor diameter for a given shaft power follows from P = \eta_d C_p \tfrac{1}{2}\rho \tfrac{\pi D^2}{4} V^3. When the wind speed changes with the rotor fixed, the new power is found from the cube ratio, P_2 = P_1 (V_2/V_1)^3.

Solar panel efficiency and output power

Photovoltaic efficiency compares electrical output with the solar energy falling on the active cells, \eta = P_{out}/(G A_c), where G is irradiance in W/m^2 and A_c is the cell area. Panels have a frame and gaps, so the cell fraction of the panel area must be applied to the total panel area before the efficiency is used. At the maximum power point P_{max} = FF \cdot V_{oc} \cdot I_{sc}, where FF is the fill factor, V_{oc} the open-circuit voltage and I_{sc} the short-circuit current, and equivalently P_{max} = V_{mp} I_{mp}. Wiring governs the output of multiple panels. In series, voltages add while the current equals that of one panel; in parallel, currents add while the voltage stays the same. Thus two identical panels in series supply the current of a single panel at twice its voltage. Since the required area is A = P_{max}/(\eta G), area is inversely proportional to irradiance. A system that must deliver a given power over a range of irradiance is therefore sized at the lowest irradiance in that range.

Fuel properties (pour point)

Pour point is the lowest temperature at which a petroleum fuel or oil just flows under standard cooling conditions. It is set by the wax content: paraffinic components crystallise and form a network that stops the fluid moving. A fuel at or below its pour point cannot be pumped or drawn through filters and lines, so pour point is the key low-temperature fluidity property. Several related terms are easily confused with it. The cloud point is the temperature at which wax first appears as a haze and it lies above the pour point. The flash point is the lowest temperature at which vapour ignites in air and it concerns fire safety. The boiling point relates vapour pressure to surrounding pressure and governs volatility rather than flow. In a severe cold climate the fuel property that matters most for running an internal combustion engine is therefore the pour point, because it decides whether fuel can reach the injection or carburettor system at all. Heating value measures energy per unit mass and does not describe flow.

Solar altitude and zenith angles

The solar altitude angle \alpha is the angle between the sun's rays and the horizontal plane. The zenith angle \theta_z is measured from the vertical, so \theta_z = 90^\circ - \alpha; the zenith angle is the complement of the altitude and the two always sum to 90^\circ. The azimuth angle is measured in the horizontal plane and the hour angle describes the sun's position relative to solar noon, so neither is a complement of altitude. For a tilted flat-plate collector, the angle of incidence \theta of beam radiation on the surface normal is found from \cos\theta = \sin\delta\sin\phi\cos\beta - \sin\delta\cos\phi\sin\beta\cos\gamma + \cos\delta\cos\phi\cos\beta\cos\omega + \cos\delta\sin\phi\sin\beta\cos\gamma\cos\omega + \cos\delta\sin\beta\sin\gamma\sin\omega. Here \delta is declination, \phi latitude, \beta tilt, \gamma surface azimuth and \omega hour angle, which is 15 degrees per hour from solar noon. For a surface facing due south (\gamma = 0) the expression reduces to \cos\theta = \sin\delta\sin(\phi-\beta) + \cos\delta\cos\omega\cos(\phi-\beta). The declination follows from the day number n through \delta = 23.45\sin[360(284+n)/365] degrees.

Adiabatic compression work of ideal gas

For a reversible adiabatic process of an ideal gas, pv^{\gamma} = \text{constant} with \gamma = c_p/c_v, so the final specific volume follows from v_2 = v_1 (p_1/p_2)^{1/\gamma}. With no heat transfer the first law gives the work input equal to the change in internal energy, w = \Delta u = c_v \Delta T. Using pv = RT and c_v = R/(\gamma-1), the specific work of compression is w = \dfrac{p_2 v_2 - p_1 v_1}{\gamma - 1}, which is positive when work is done on the gas. This form avoids computing temperatures and is the usual route in exam problems. A polytropic process follows pv^n = \text{constant}, and for a reversible process the exponent is obtained from two states as n = \ln(p_2/p_1)/\ln(v_1/v_2). Setting n = \gamma recovers the adiabatic case and n = 1 gives isothermal compression. Because the exponent depends only on ratios, any consistent volume unit can be used.

Equivalence ratio and air-fuel ratio

The mixture strength of an engine is described relative to the stoichiometric air-fuel ratio. In the convention used for this topic, the relative air-fuel ratio is \lambda = (A/F)_{actual}/(A/F)_{stoich}, so \lambda < 1 means a rich mixture and \lambda > 1 a lean mixture. The stoichiometric ratio of a spark-ignition engine on gasoline is about 14.7:1 by mass, so the actual air-fuel ratio equals \lambda times this value. The equivalence ratio \phi = (F/A)_{actual}/(F/A)_{stoich} is the reciprocal of \lambda, and \phi > 1 means rich. Because texts use the same symbol in opposite senses, the direction must be read from the definition given. If a value is taken as an equivalence ratio, the actual air-fuel ratio is (A/F)_{stoich}/\phi rather than \phi (A/F)_{stoich}.

Engine volumetric efficiency and air intake

Volumetric efficiency is the ratio of the actual air volume drawn into the cylinder at intake conditions to the swept (displacement) volume, \eta_v = V_{actual}/V_s. In a four-stroke engine one intake stroke occurs every two crank revolutions, so the number of intakes per second for a cylinder is N/120 with N in rpm, while a two-stroke engine has N/60 intakes per second. The theoretical induced volume flow is the swept volume multiplied by the intake rate, and the actual air flow equals \eta_v times this value. Values below unity arise from the pressure drop across the inlet valve, residual gas, heating of the charge and valve timing. The actual air inducted into the cylinder is therefore found by applying the efficiency to the theoretical flow.