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 *V _{t} [m^{3}]* is calculated function of the total number of cylinders

*n*and the volume of one cylinder

_{c}[-]*V*.

_{cyl}[m^{3}]The total volume of the cylinder is the sum between the displaced (swept) volume *V _{d} [m^{3}]* and the clearance volume

*V*.

_{c}[m^{3}]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.

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** *η _{v} [-]* is defined as the ratio between the actual (measured) volume of intake air

*V*drawn into the cylinder/engine and the theoretical volume of the engine/cylinder V

_{a}[m^{3}]_{d}[m

^{3}], during the intake engine cycle.

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 *m _{a} [kg]* and air density

*ρ*:

_{a}[kg/m^{3}]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.

where:

*N _{e} [rot/s]* – engine speed

*n*– number of crankshaft rotations for a complete engine cycle (for 4-stroke engine

_{r}[-]*n*)

_{r}= 2From 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 *p _{a} [Pa]* and temperature

*T*are measured in the intake manifold, the

_{a}[K]**intake air density**can be calculated as:

where:

*ρ _{a} [kg/m^{3}]* – intake air density

*p*– intake air pressure

_{a}[Pa]*T*– intake air temperature

_{a}[K]*R*– gas constant for dry air (equal to

_{a}[J/kgK]*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.

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).

The engine displacement was converted from *L* to *m ^{3}* and the engine speed from

*rpm*to

*rps*.

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.

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

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## Alok

it is a wonderful explanation- free from any jargon.

I wish to know if it is for any reason that equation 3 in denominator uses Vd instead of Vd+Vc as the air volume sucked in the engine.

## jeremie

I agree with flavio, air density decrease with low air temperature, strange !

## Anthony Stark

Air density increases with lower temperature. The reason is that the molecules have lower kinetic energy and they stay close to each other hence more molecules in the same volume => higher density.

## wilson

thanks a lot for the simplicity.

## Flávio Paoliello

We know that a higher air density is beneficial to volumetric efficiency of an engine, hence the use of intercooler. So I did not undersstand why air density appear in the denominetor of the volumetric eff formula. Thanks for explaining.

## Anthony Stark

The article explains step by step the final equation. Air density appears in the denominator because it’s used to calculate the intake air volume, see equation (4).

## Jordi

Thank god!

## Tim Edes

Excellent explanation…thank you !

## Ramniwas patidar

Very good sysmatic explanation in simple way. Thanks

## Fajrul

How to know the value of volumetric efficiency theoretically depends on engine speed (rpm)?