Internal Combustion Engine Formulas

Cheatsheet Content

1. Basic Engine Parameters Bore (B): Diameter of the cylinder. Stroke (L): Distance the piston travels in one direction. Crank Radius (a): $a = L/2$. Displacement Volume ($V_d$): Volume swept by the piston. For one cylinder: $V_{d,1} = \frac{\pi B^2}{4} L$ For $N_c$ cylinders: $V_d = N_c \frac{\pi B^2}{4} L$ Clearance Volume ($V_c$): Volume above the piston at TDC. Total Cylinder Volume ($V_t$): $V_t = V_d + V_c$. Compression Ratio (r): Ratio of total volume to clearance volume. $$ r = \frac{V_t}{V_c} = \frac{V_d + V_c}{V_c} = 1 + \frac{V_d}{V_c} $$ 2. Engine Performance Metrics 2.1 Power and Torque Indicated Power ($P_i$): Power generated within the cylinders. $$ P_i = \frac{IMEP \cdot V_d \cdot N}{n_R} $$ where $IMEP$ is Indicated Mean Effective Pressure, $N$ is engine speed (rpm), $n_R = 2$ for 4-stroke, $n_R = 1$ for 2-stroke. Brake Power ($P_b$): Power delivered at the crankshaft. $$ P_b = \frac{2 \pi N T_b}{60} $$ where $T_b$ is Brake Torque (Nm), $N$ is engine speed (rpm). Friction Power ($P_f$): Power lost due to friction. $P_f = P_i - P_b$. Brake Mean Effective Pressure (BMEP): Average pressure that would produce the brake torque. $$ BMEP = \frac{P_b \cdot n_R}{V_d \cdot N} $$ Indicated Mean Effective Pressure (IMEP): Average pressure exerted on the piston during the power stroke. $$ IMEP = \frac{\int P dV}{V_d} $$ Mechanical Efficiency ($\eta_m$): Ratio of brake power to indicated power. $$ \eta_m = \frac{P_b}{P_i} = \frac{BMEP}{IMEP} $$ 2.2 Efficiencies Thermal Efficiency ($\eta_{th}$): Ratio of work output to heat input. $$ \eta_{th} = \frac{W_{out}}{Q_{in}} $$ Brake Thermal Efficiency ($\eta_{b,th}$): $$ \eta_{b,th} = \frac{P_b}{\dot{m}_f \cdot Q_{HV}} $$ where $\dot{m}_f$ is mass flow rate of fuel, $Q_{HV}$ is heating value of fuel. Indicated Thermal Efficiency ($\eta_{i,th}$): $$ \eta_{i,th} = \frac{P_i}{\dot{m}_f \cdot Q_{HV}} $$ Volumetric Efficiency ($\eta_v$): Ratio of actual air intake to theoretical air intake. $$ \eta_v = \frac{\dot{m}_a}{\rho_a V_d N / n_R} = \frac{V_{actual}}{V_{swept}} $$ where $\dot{m}_a$ is mass flow rate of air, $\rho_a$ is air density. 3. Ideal Cycle Analysis 3.1 Otto Cycle (Spark Ignition Engines) Processes: 1-2 Isentropic compression, 2-3 Constant volume heat addition, 3-4 Isentropic expansion, 4-1 Constant volume heat rejection. Thermal Efficiency: $$ \eta_{Otto} = 1 - \frac{1}{r^{k-1}} $$ where $r$ is compression ratio, $k = c_p/c_v$ is specific heat ratio. 3.2 Diesel Cycle (Compression Ignition Engines) Processes: 1-2 Isentropic compression, 2-3 Constant pressure heat addition, 3-4 Isentropic expansion, 4-1 Constant volume heat rejection. Thermal Efficiency: $$ \eta_{Diesel} = 1 - \frac{1}{r^{k-1}} \left[ \frac{r_c^k - 1}{k(r_c - 1)} \right] $$ where $r_c = V_3/V_2$ is cutoff ratio. 3.3 Dual Cycle Combines constant volume and constant pressure heat addition. Thermal Efficiency: $$ \eta_{Dual} = 1 - \frac{1}{r^{k-1}} \left[ \frac{r_p r_c^k - 1}{(r_p - 1) + k r_p (r_c - 1)} \right] $$ where $r_p = P_3/P_2$ is pressure ratio during constant volume heat addition. 4. Air-Fuel Ratio and Combustion Air-Fuel Ratio (AFR): Mass of air per unit mass of fuel. $$ AFR = \frac{\dot{m}_a}{\dot{m}_f} $$ Fuel-Air Ratio (FAR): Mass of fuel per unit mass of air. $FAR = 1/AFR$. Stoichiometric AFR: Ideal AFR for complete combustion (e.g., $\approx 14.7:1$ for gasoline). Equivalence Ratio ($\phi$): $$ \phi = \frac{FAR_{actual}}{FAR_{stoich}} = \frac{AFR_{stoich}}{AFR_{actual}} $$ $\phi 1$ for rich mixture. Specific Fuel Consumption (SFC): Fuel consumed per unit power output. $$ SFC = \frac{\dot{m}_f}{P} $$ (Brake Specific Fuel Consumption, BSFC, uses $P_b$; Indicated Specific Fuel Consumption, ISFC, uses $P_i$). 5. Engine Kinematics Piston Position (s): Distance from TDC. $$ s = L - a \cos\theta - \sqrt{l^2 - (a \sin\theta)^2} $$ where $L$ is stroke, $a$ is crank radius, $l$ is connecting rod length, $\theta$ is crank angle from TDC. Piston Velocity (v): $$ v = \frac{ds}{dt} = \omega a \sin\theta \left( 1 + \frac{a \cos\theta}{\sqrt{l^2 - (a \sin\theta)^2}} \right) $$ where $\omega$ is angular velocity ($\omega = 2\pi N/60$). 6. Combustion Analysis Heat Release Rate: Often modeled using Wiebe function. $$ x_b(\theta) = 1 - \exp\left[ -a \left( \frac{\theta - \theta_s}{\Delta\theta} \right)^{m+1} \right] $$ where $x_b$ is fraction of fuel burned, $\theta$ is crank angle, $\theta_s$ is start of combustion, $\Delta\theta$ is combustion duration, $a$ and $m$ are Wiebe parameters.