How to calculate the volumetric efficiency of an internal combustion engine

For a thermal engine, the combustion process depends on the air-fuel ratio inside the cylinder. The more air we can get inside the combustion chamber, the more fuel we can burn, the higher the output engine torque and power.

Since air has mass, it has inertia. Also, the intake manifold, the valves and the throttle are acting as restrictions for the air flow into the cylinders. By volumetric efficiency we measure the capacity of the engine to fill the available geometric volume of the engine with air. It can be seen as a ratio between the volume of air drawn the cylinder (real) and the geometric volume of the cylinder (theoretical).

Most of the internal combustion engines used nowadays on road vehicles, have a fixed volumetric capacity (displacement), defined by the geometry of the cylinder and the crank mechanism. Strictly speaking, the total volume of an engine Vt [m3] is calculated function of the total number of cylinders nc [-] and the volume of one cylinder Vcyl [m3].

\[V_t=n_c \cdot V_{cyl} \tag{1}\]

The total volume of the cylinder is the sum between the displaced (swept) volume Vd [m3] and the clearance volume Vc [m3].

\[V_{cyl} = V_d + V_c \tag{2}\]

The clearance volume is very small in comparison with the displacement volume (e.g. ratio 1:12) so it can be neglected when calculating the volumetric efficiency of the engine.

Basic piston and cylinder geometry parameters of internal combustion engines

Image: Basic piston and cylinder geometry parameters of internal combustion engines

where:

IV – intake valve
EV – exhaust valve
TDC – top dead center
BDC – bottom dead center
B – cylinder bore
S – piston stroke
r – connecting rod length
a – crank radius (offset)
x – distance between the crank axis and the piston pin axis
θ – crank angle
Vd – displaced (swept) volume
Vc – clearance volume

The volumetric efficiency ηv [-] is defined as the ratio between the actual (measured) volume of intake air Va [m3] drawn into the cylinder/engine and the theoretical volume of the engine/cylinder Vd [m3], during the intake engine cycle.

\[\eta_v = \frac{V_a}{V_d} \tag{3}\]

The volumetric efficiency can be regarded also as the efficiency of the internal combustion engine to fill the cylinders with intake air. The higher the volumetric efficiency the higher the volume of intake air in the engine.

In case of indirect fuel injection engines (mainly gasoline) the intake air is mixed with fuel. Since the amount of fuel is relatively small (ratio 1:14.7), compared with the amount of air, we can neglect the fuel mass for volumetric efficiency calculation.

The actual intake air volume can be calculated function of air mass ma [kg] and air density ρa [kg/m3]:

\[V_a = \frac{m_a}{\rho_a} \tag{4}\]

Replacing (4) in (3) gives the volumetric efficiency equal to:

\[\eta_v = \frac{m_a}{\rho_a \cdot V_d} \tag{5}\]

Usually, on the engine dynamometer, intake air mass flow rate is measured [kg/s] instead of air mass [kg]. Therefore, we need to use air mass flow rate for volumetric efficiency calculation.

\[\dot{m}_a=\frac{m_a \cdot N_e}{n_r} \tag{6}\]

where:

Ne [rot/s] – engine speed
nr [-] – number of crankshaft rotations for a complete engine cycle (for 4-stroke engine nr = 2)

From equation (6), we can write the intake air mass as:

\[m_a=\frac{\dot{m}_a \cdot n_r}{N_e} \tag{7}\]

Replacing (7) in (5) gives the volumetric efficiency equal with:

\[\bbox[#FFFF9D]{\eta_v = \frac{\dot{m}_a \cdot n_r}{\rho_a \cdot V_d \cdot N_e}} \tag{8}\]

The volumetric efficiency is maximum 1.00 (or 100%). At this value, the engine is capable of drawing all of the theoretical volume of air available into the engine. There are special cases in which the engine is specifically designed for one operating point, for which the volumetric efficiency can be slightly higher than 100 %.

If intake air pressure pa [Pa] and temperature Ta [K] are measured in the intake manifold, the intake air density can be calculated as:

\[\rho_a = \frac{p_a}{R_a \cdot T_a} \tag{9}\]

where:

ρa [kg/m3] – intake air density
pa [Pa] – intake air pressure
Ta [K] – intake air temperature
Ra [J/kgK] – gas constant for dry air (equal to 286.9 J/kgK)

Example – How to calculate the volumetric efficiency

Let’s consider a compression ignition (diesel) engine with the following parameters:

\[ \begin{split}
V_d &= 3.8 \text{ L} \\
n_r &= 2 \\
p_a &= 1.5 \text{ bar} \\
T_a &= 40 \text{ } ^\circ \text{C} \\
R_a &= 286.9 \text{ J/kgK} \\
N_e &= 1000 \text{ rpm} \\
\dot{m}_a &= 0.0375 \text{ kg/s}
\end{split} \]

For the above engine parameters, calculate the volumetric efficiency.

Step 1. Calculate the intake air density using equation (9). Make sure that all measurement units match.

\[\rho_a = \frac{1.5 \cdot 10^5}{286.9 \cdot (40 + 273.15)}=1.67 \text{ kg/m}^3\]

The air intake pressure was converted from bar to Pa and the temperature from °C to K.

Step 2. Calculate the volumetric efficiency of the engine using equation (8).

\[\eta_v = \frac{0.0375 \cdot 2}{1.67 \cdot 3.8 \cdot 10^{-3} \cdot \frac{1000}{60}} = 0.7091081 = 70.91 \text{ %}\]

The engine displacement was converted from L to m3 and the engine speed from rpm to rps.

Volumetric efficiency function of intake air pressure and engine speed

Image: Volumetric efficiency function of intake air pressure and engine speed

The volumetric efficiency of an internal combustion engine depends on several factors like:

  • the geometry of the intake manifold
  • the intake air pressure
  • the intake air temperature
  • the intake air mass flow rate (which depends on engine speed)

Usually, engines are designed to have the maximum volumetric efficiency at medium/high engine speed and load.

You can also check your results using the calculator below.

Volumetric Efficiency Calculator

Vd [L]nr [-]pa [bar]Ta [˚C]Ra [J/kgK]Ne [rpm]ma [kg/s]
Air density, ρa [kg/m3] =
Volumetric efficiency, ηv [%] =

For any questions or observations regarding this tutorial please use the comment form below.

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