Diesel Cycle

Introduction

1

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  Diesel Cycle

• 

  Diesel Cycle

Ratio of specific heats (heat capacity ratio) is defined as $\gamma ={\frac {C_{p}}{C_{v}}}$

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

For a closed system, the total change in energy of a system is the sum of the work done and the heat added $dU=\delta W+\delta Q$, 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} .

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 }

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} , giving

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 }

Real gas: Air

1. 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 p=ρRT

This works well at:

pressures near atmospheric temperatures roughly 200–500 K 2. “Generalized” ideal gas → temperature-dependent properties

To go beyond the simple model, you allow properties like heat capacity to vary with temperature:

Heat capacity cp(T) c p ​

(T)

For air, a common approximation is a polynomial:

cp(T)=a+bT+cT2+dT3 c p ​

(T)=a+bT+cT 2 +dT 3

Typical coefficients (for dry air, ~200–1000 K range):

a≈1005 a≈1005 b≈0.1 b≈0.1 higher-order terms small depending on fit

More accurate forms come from NASA polynomials:

cpR=a1+a2T+a3T2+a4T3+a5T4 R c p ​

​

=a 1 ​

+a 2 ​

T+a 3 ​

T 2 +a 4 ​

T 3 +a 5 ​

T 4

These are widely used in CFD and thermodynamics.

3. Compressibility factor Z Z (real gas correction)

If you want a generalized ideal gas, you often introduce:

p=ZρRT p=ZρRT

Where:

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

For air:

At normal conditions: Z≈1 Z≈1 At high pressure: use virial expansion Virial equation: Z=1+B(T)V+C(T)V2+⋯ Z=1+ V B(T) ​

+ V 2 C(T) ​

+⋯

Usually truncated to:

Z≈1+B(T)RTp Z≈1+ RT B(T) ​

p 4. Mixture-based formulation (more fundamental)

Air is a mixture mainly of:

N₂ (~78%) O₂ (~21%) Ar (~1%)

You can model it as:

R=∑iyiRi R= i ∑ ​

y i ​

R i ​

cp=∑iyicp,i(T) c p ​

= i ∑ ​

y i ​

c p,i ​

(T)

This is the most physically grounded “generalized ideal gas” model.

5. When to use which model Simple engineering → constant R,cp R,c p ​

Moderate accuracy (aerodynamics, engines) → cp(T) c p ​

(T) High pressure / high temperature → include Z Z High fidelity (CFD, combustion) → mixture + NASA polynomials Bottom line

Yes—air is commonly treated as a generalized ideal gas by:

Keeping p=ρRT p=ρRT Allowing: cp=cp(T) c p ​

=c p ​

(T) possibly Z≠1 Z  =1

Realistic Diesel Cycle

• 

  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 }