Diesel Cycle

Introduction

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  Diesel Cycle

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  Diesel Cycle

Ratio of specific heats (heat capacity ratio) is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{ C_p }{ C_v } }

pV diagram

1. Isentropic (adiabatic) expansion
2. Isochoric cooling (Qout): Heat rejection. Power stroke ends, heat rejection starts.
3. Isobaric compression: Exhaust
4. Isobaric expansion: Intake
5. Isentropic (adiabatic) compression
6. Isobaric heating (Qin): Combustion of fuel (heat is added in a constant pressure;)

Engine displacement is the cylinder volume swept by all of the pistons of a piston engine, excluding the combustion chambers. A combustion chamber is part of an internal combustion engine in which the fuel/air mix is burned.

Only air is compressed, and then diesel fuel is injected directly into that hot, high-pressure air.

• Cylinder pressure: ~30–80 bar
• Injection pressure: ~1,000–2,500+ bar

Diesel Cycle and Ideal Gas

There are three (3) different processes:

1. Isentropic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dU = nC_v dT = -pdV}
2. Isobaric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q = \Delta U + p \Delta V}
3. Isochoric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta Q = m c_v \Delta T}

where the specific heat capacity at constant volume is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_v = \frac{dQ/dT}{m}} .

For a closed system, the total change in energy of a system is the sum of the work done and the heat added Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dU = \delta W + \delta Q} , and the reversible work done on a system by changing the volume is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta W = - p dV} . Furthermore, for any transformation of an ideal gas, it is always true that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dU = nC_v dT} .

If the system is reversible and adiabatic (isentropic) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta Q = 0} , which gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dU = \delta W + \delta Q = -pdV + 0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dU = nC_v dT = -pdV }

For the constant pressure (isobaric process) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta p = 0} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = \int p dV = p \Delta V} , and by applying the ideal gas law, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = n R \Delta T} .

For the Isochoric process Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta V=0} , and thus we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dU = dQ = mc_v dT} which gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta Q = m c_v\Delta T} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle }

Real gas: Air

Basic ideal gas model (good first approximation). Air is usually approximated as an ideal gas with:

• Gas constant:
  • R≈287 J/(kg\cdotpK)
  • R≈287J/(kg\cdotpK)
• Equation of state:
• p=ρRT

This works well at:

• pressures near atmospheric
• temperatures roughly 200–500 K

“Generalized” ideal gas → temperature-dependent properties. To go beyond the simple model, you allow properties like heat capacity to vary with temperature:

Heat capacity. For air, a common approximation is a polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_p(T)=a+bT. } Typical coefficients (for dry air, ~200–1000 K range):

• a≈1005
• b≈0.1

More accurate forms come from NASA polynomials: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{c_p}R=a_1+a_2T+a_3T^2+a_4T^3+a_5T^4 } Which are widely used in CFD and thermodynamics.

Compressibility factor Z (real gas correction). If you want a generalized ideal gas, you often introduce Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=Z\rho RT} . Where

• Z=1 → ideal gas
• Z≠1 → real gas behavior

For air:

• At normal conditions: Z≈1
• At high pressure: use virial expansion

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=1+\frac{B(T)}V+\frac{C(T)}{V^2}+ \cdots \approx 1+ \frac{B(T)}{RT}p. }

Mixture-based formulation (more fundamental). Air is a mixture mainly of:

• N₂ (~78%)
• O₂ (~21%)
• Ar (~1%)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} R &=\sum_i y_i R_i \\ c_p & = \sum_i y_ i c_{p,i}(T) \end{align} }

Realistic Diesel Cycle

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  The size of the combustion chamber of MB W211.

Ratio of specific heats γ

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{ \sum y_1 c_{p,i} }{ \sum y_1 ( c_{p,i} -R_u ) } }

Caption text

Condition	γ (approx.)
Stoichiometric	~1.30–1.33
Moderate lean	~1.33–1.37
Very lean (diesel)	~1.37–1.40

Injection pressures

• Older systems: 200–500 bar
• Modern common-rail: 1,000–2,500 bar
• Latest systems: up to ~3,000 bar

• Air pressure in cylinder: ~30–80 bar, which is the pressure of the combustion chamber.
• Temperature: ~700–1000 K

Diesel-air Mixture

Only diesel is ejected into the cylinder. Clean air comes in during the intake stroke.

The heat capacity ratio (known as the adiabatic index) for a diesel-air mixture is typically around 1.4.

Air-fuel ratio

14.5:1	Near-stoichiometric; good combustion efficiency but higher emissions.
16:1	Balanced performance; good power output and efficiency.
18:1	Lean burn; improved fuel economy but potential for higher NOx emissions.

Diesel is a complicated compound, but it’s commonly approximated as a hydrocarbon like C₁₂H₂₃ (or C₁₂H₂₆). Air is about 21% O₂ and 79% N₂. Without nitrogen, the stoichiometric ratio is about

C₁₂​H₂₃​+17.75O₂​→12CO₂​+11.5H₂​O

and by including nitrogen, we get

C₁₂​H₂₃​+17.75( O₂ + 3.76 N₂ )​→12CO₂​+11.5H₂​O + 66.74 N₂.

The molar masses

• Fuel: 167 g/mol
• Air: 137.28 g/mol

And the total air needed is 17.72 x 137.28 = 2436 g, which gives air-to-fuel-ratio to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AFR = \frac{2436}{167} = 14.6:1 }

MB data

Motor: OM648.

Mercedes Benz W211 (2003)

• Engine displacement: 3222 cm³ = 0.003222 m³
• Bore x Stroke: 88.0 x 88.4 mm³
• Compression Ratio: 18.0

Bore x stroke gives V = 6xπ(8.8/2)² x 8.84 cm³ = 3225.95cm³, which is rather close.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1 V_1^\gamma = \text{constant}_1 }