m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Internal combustion engine design

Derive ideal cycles, displacement, indicated and brake performance, mixture budgets and cooling loads.

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1. Geometry and four-stroke timing

Definitions & inputs. B bore,S stroke,nc cylinders,Vd swept volume,Vc clearance volume,N rev/s.

  1. Swept and clearance volumes set compression ratio.

    Vd=ncπB2S/4,r=(Vd/nc+Vc)/VcV_d=n_c\pi B^2S/4,\quad r=(V_d/n_c+V_c)/V_c

  2. Intake, compression, expansion and exhaust occupy two crank revolutions.

    fcycle=N/2f_{cycle}=N/2

  3. Mean piston speed counts two strokes per revolution.

    Uˉp=2SN\bar U_p=2SN

Interpretation. Piston acceleration, rod ratio, valve timing and breathing need more detailed kinematics and gas dynamics.

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2. Otto cycle efficiency

Definitions & inputs. r compression ratio,γ heat capacity ratio,Qin andQout cycle heat.

  1. Isentropic relations connect temperature to volume ratio.

    T2/T1=rγ−1,T3/T4=rγ−1T_2/T_1=r^{\gamma-1},\quad T_3/T_4=r^{\gamma-1}

  2. Use the first law and constant cv heat transfers.

    η=1−Qout/Qin=1−(T4−T1)/(T3−T2)\eta=1-Q_{out}/Q_{in}=1-(T_4-T_1)/(T_3-T_2)

  3. Eliminate temperatures using the matched isentropic ratios.

    ηOtto=1−r1−γ\eta_{Otto}=1-r^{1-\gamma}

Interpretation. This efficiency is not a real brake efficiency or a recommendation to increase compression without knock analysis.

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3. Diesel cycle and performance metrics

Definitions & inputs. rc cutoff ratio,Vd total displacement,imep andbmep indicated/brake mean effective pressures,τ shaft torque.

  1. Substitute constant-pressure heat addition into the ideal cycle heat ratio.

    ηD=1−rcγ−1γrγ−1(rc−1)\eta_D=1-\frac{r_c^\gamma-1}{\gamma r^{\gamma-1}(r_c-1)}

  2. Brake power equals torque speed or four-stroke brake work times cycles per second.

    Pb=τω=bmep VdN/2P_b=\tau\omega=\mathrm{bmep}\,V_dN/2

  3. Separate indicated power from mechanical and pumping/friction accounting under a consistent convention.

    ηm=Pb/Pi,Pf=Pi−Pb\eta_m=P_b/P_i,\quad P_f=P_i-P_b

Interpretation. A high mean effective pressure does not guarantee thermal efficiency or durability.

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4. Fuel, mixture and thermal balance

Definitions & inputs. fuel mass flow rate fuel mass rate,LHV lower heating value,AFR air fuel ratio,λ excess air ratio.

  1. Define brake thermal efficiency and brake-specific fuel consumption with consistent units.

    ηb=Pb/(m˙fLHV),BSFC=m˙f/Pb\eta_b=P_b/(\dot m_fLHV),\quad\mathrm{BSFC}=\dot m_f/P_b

  2. Compare mixture with the fuel-specific stoichiometric ratio.

    λ=AFR/AFRstoich\lambda=AFR/AFR_{stoich}

  3. Account for all outgoing energy rather than equating ideal cycle efficiency with measured output.

    Q˙fuel=Pb+Q˙cool+Q˙exhaust+Q˙other\dot Q_{fuel}=P_b+\dot Q_{cool}+\dot Q_{exhaust}+\dot Q_{other}

Interpretation. Spark timing, injection, turbulence, heat transfer, catalyst temperature and control connect this subject to chemical kinetics, fluids and embedded systems.

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Graphical worked example

Calorically perfect air-standard Otto cycle, γ=1.4; not measured brake efficiency. X axis: Compression ratio (dimensionless). Y axis: Ideal Otto thermal efficiency (dimensionless).
Calorically perfect air-standard Otto cycle, γ=1.4; not measured brake efficiency. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Single-cylinder displacement

Definitions & inputs. B.08m,S.09m.

  1. Choose the governing model and isolate the requested quantity.

    Vd=πB2S/4V_d=\pi B^2S/4

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    π(.08)2(.09)/4\pi(.08)^2(.09)/4

  3. Evaluate the expression; the result uses the units shown.

    Result=452.3893 cm3\mathrm{Result}=452.3893\ \mathrm{cm^3}

Interpretation. Cubic meters converted to cubic centimeters.

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Example 02. Four-cylinder displacement

Definitions & inputs. Same bore stroke,nc4.

  1. Choose the governing model and isolate the requested quantity.

    Vd,total=4VdV_{d,total}=4V_d

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4π(.08)2(.09)/44\pi(.08)^2(.09)/4

  3. Evaluate the expression; the result uses the units shown.

    Result=1.809557 L\mathrm{Result}=1.809557\ \mathrm L

Interpretation. Liters conversion.

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Example 03. Compression ratio

Definitions & inputs. Vs500cm³,Vc50cm³.

  1. Choose the governing model and isolate the requested quantity.

    r=(Vs+Vc)/Vcr=(V_s+V_c)/V_c

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    550/50550/50

  3. Evaluate the expression; the result uses the units shown.

    Result=11 \mathrm{Result}=11\ {}

Interpretation. Per-cylinder volumes.

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Example 04. Clearance volume

Definitions & inputs. Vs500cm³,r10.

  1. Choose the governing model and isolate the requested quantity.

    Vc=Vs/(r−1)V_c=V_s/(r-1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    500/9500/9

  3. Evaluate the expression; the result uses the units shown.

    Result=55.55556 cm3\mathrm{Result}=55.55556\ \mathrm{cm^3}

Interpretation. Positive clearance.

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Example 05. Revolutions per second

Definitions & inputs. n3000rpm.

  1. Choose the governing model and isolate the requested quantity.

    N=n/60N=n/60

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    3000/603000/60

  3. Evaluate the expression; the result uses the units shown.

    Result=50 s−1\mathrm{Result}=50\ \mathrm{s^{-1}}

Interpretation. Crank speed.

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Example 06. Four-stroke cycle rate

Definitions & inputs. N50rev/s.

  1. Choose the governing model and isolate the requested quantity.

    f=N/2f=N/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    50/250/2

  3. Evaluate the expression; the result uses the units shown.

    Result=25 s−1\mathrm{Result}=25\ \mathrm{s^{-1}}

Interpretation. Per-cylinder firing cycle frequency.

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Example 07. Mean piston speed

Definitions & inputs. S.09m,N50/s.

  1. Choose the governing model and isolate the requested quantity.

    Up=2SNU_p=2SN

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2(.09)(50)2(.09)(50)

  3. Evaluate the expression; the result uses the units shown.

    Result=9 m/s\mathrm{Result}=9\ \mathrm{m/s}

Interpretation. Mean absolute piston speed.

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Example 08. Ideal Otto efficiency

Definitions & inputs. r10,γ1.4.

  1. Choose the governing model and isolate the requested quantity.

    η=1−r1−γ\eta=1-r^{1-\gamma}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1−10−.41-10^{-.4}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6018928 \mathrm{Result}=0.6018928\ {}

Interpretation. Air-standard upper benchmark under these constraints.

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Example 09. Compressed temperature

Definitions & inputs. T1300K,r10,γ1.4.

  1. Choose the governing model and isolate the requested quantity.

    T2=T1rγ−1T_2=T_1r^{\gamma-1}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    300(10.4)300(10^{.4})

  3. Evaluate the expression; the result uses the units shown.

    Result=753.5659 K\mathrm{Result}=753.5659\ \mathrm K

Interpretation. Constant gamma approximation.

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Example 10. Compressed pressure

Definitions & inputs. p1100kPa,r10,γ1.4.

  1. Choose the governing model and isolate the requested quantity.

    p2=p1rγp_2=p_1r^\gamma

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    100(101.4)100(10^{1.4})

  3. Evaluate the expression; the result uses the units shown.

    Result=2511.886 kPa\mathrm{Result}=2511.886\ \mathrm{kPa}

Interpretation. Ideal adiabatic compression.

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Example 11. Ideal Diesel efficiency

Definitions & inputs. r18,rc2,γ1.4.

  1. Choose the governing model and isolate the requested quantity.

    η=1−(rcγ−1)/[γrγ−1(rc−1)]\eta=1-(r_c^\gamma-1)/[\gamma r^{\gamma-1}(r_c-1)]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1−(21.4−1)/[1.4(18.4)(1)]1-(2^{1.4}-1)/[1.4(18^{.4})(1)]

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6315775 \mathrm{Result}=0.6315775\ {}

Interpretation. Constant-pressure heat addition.

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Example 12. Brake power

Definitions & inputs. Torque100Nm,speed3000rpm.

  1. Choose the governing model and isolate the requested quantity.

    P=τ2πn/60P=\tau 2\pi n/60

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    100(2π)(3000)/60100(2\pi)(3000)/60

  3. Evaluate the expression; the result uses the units shown.

    Result=31415.93 W\mathrm{Result}=31415.93\ \mathrm W

Interpretation. Shaft output.

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Example 13. BMEP

Definitions & inputs. Torque100Nm,totalVd.002m³,four stroke.

  1. Choose the governing model and isolate the requested quantity.

    bmep=4πτ/Vd\mathrm{bmep}=4\pi\tau/V_d

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4π(100)/.0024\pi(100)/.002

  3. Evaluate the expression; the result uses the units shown.

    Result=628318.5 Pa\mathrm{Result}=628318.5\ \mathrm{Pa}

Interpretation. Brake work per cycle divided by displacement.

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Example 14. Mechanical efficiency

Definitions & inputs. Pb30kW,Pi36kW.

  1. Choose the governing model and isolate the requested quantity.

    ηm=Pb/Pi\eta_m=P_b/P_i

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    30/3630/36

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8333333 \mathrm{Result}=0.8333333\ {}

Interpretation. Specified indicated/brake convention.

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Example 15. Friction budget

Definitions & inputs. Pi36kW,Pb30kW.

  1. Choose the governing model and isolate the requested quantity.

    Pf=Pi−PbP_f=P_i-P_b

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    36−3036-30

  3. Evaluate the expression; the result uses the units shown.

    Result=6 kW\mathrm{Result}=6\ \mathrm{kW}

Interpretation. Aggregate loss difference.

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Example 16. Fuel mass flow

Definitions & inputs. Pb30kW,ηb.3,LHV44MJ/kg.

  1. Choose the governing model and isolate the requested quantity.

    m˙f=Pb/(ηbLHV)\dot m_f=P_b/(\eta_bLHV)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    30000/[.3(44×106)]30000/[.3(44\times10^6)]

  3. Evaluate the expression; the result uses the units shown.

    Result=0.002272727 kg/s\mathrm{Result}=0.002272727\ \mathrm{kg/s}

Interpretation. Steady averaged fuel rate.

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Example 17. BSFC

Definitions & inputs. ηb.3,LHV44MJ/kg.

  1. Choose the governing model and isolate the requested quantity.

    BSFC=3.6×109/(ηbLHV)\mathrm{BSFC}=3.6\times10^9/(\eta_bLHV)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    3.6×109/[.3(44×106)]3.6\times10^9/[.3(44\times10^6)]

  3. Evaluate the expression; the result uses the units shown.

    Result=272.7273 g/kWh\mathrm{Result}=272.7273\ \mathrm{g/kWh}

Interpretation. Unit conversion included.

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Example 18. Excess-air ratio

Definitions & inputs. AFR16,stoichiometric14.7for assumed fuel.

  1. Choose the governing model and isolate the requested quantity.

    λ=AFR/AFRs\lambda=AFR/AFR_s

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    16/14.716/14.7

  3. Evaluate the expression; the result uses the units shown.

    Result=1.088435 \mathrm{Result}=1.088435\ {}

Interpretation. Fuel-specific ratio.

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Example 19. Air mass flow

Definitions & inputs. Fuel.002kg/s,AFR14.7.

  1. Choose the governing model and isolate the requested quantity.

    m˙a=AFRm˙f\dot m_a=AFR\dot m_f

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    14.7(.002)14.7(.002)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0294 kg/s\mathrm{Result}=0.0294\ \mathrm{kg/s}

Interpretation. Mixture mass balance.

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Example 20. Cooling load

Definitions & inputs. Fuel power100kW,brake30,exhaust40,other5kW.

  1. Choose the governing model and isolate the requested quantity.

    Qc=100−30−40−5Q_c=100-30-40-5

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2525

  3. Evaluate the expression; the result uses the units shown.

    Result=25 kW\mathrm{Result}=25\ \mathrm{kW}

Interpretation. Complete assumed energy budget.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.