The pressure-volume (pV) diagram and how work is produced in an ICE

The internal combustion engine is a heat engine. It’s working principle is based on the variation of pressure and volume inside the engine’s cylinders. All heat engines are characterized by a pressure-volume diagram, also known as pV diagram, which basically shows the variation of the pressure in the cylinder function of its volume, for a complete engine cycle.

Also, the work produced by the internal combustion engine is directly dependent on the variation of the pressure and volume inside the cylinder.

By the end of this tutorial, the reader should be able to:

• understand the meaning of the pV diagram
• how a pV diagram is drawn for a 4 stroke internal combustion engine
• when the intake and exhaust valves are actuated during the engine cycle
• when the ignition / injection is produced during the engine cycle
• how the work is produced by the internal combustion engine
• what’s the difference between indicated and brake work
• what is the mechanical efficiency of the engine

Let’s get started by looking at a pV diagram of a 4 stroke atmospheric internal combustion engine.

where:

S – piston stroke
V_c – clearance volume
V_d – displaced (swept) volume
p₀ – atmospheric pressure
W – work
TDC – top dead center
BDC – bottom dead center
IV – inlet valve
EV – exhaust valve
IVO – inlet valve opening
IVC – inlet valve closing
EVO – exhaust valve opening
EVC – exhaust valve closing
IGN (INJ) – ignition (injection)

The pressure-volume (pV) diagram is drawn by measuring the pressure inside the cylinder, and plotting its value against the angle of the crankshaft, over a complete engine cycle (720°).

Let’s see what’s happening in the cylinder during each piston stroke, how the pressure and volume are changing inside the cylinder.

Notice that the timing of the intake and exhaust valves have advance and delay, relative to the position of the piston. For example, the intake valve it’s opening during the exhaust stroke of the piston and it is closing during the compression stroke. In the same time, when the intake stroke is starting, the exhaust valve is still open for a short while. The opening of the exhaust valve is done before the power stroke has finished.

INTAKE (a-b)

The engine cycle starts in point a. The intake valve is already open and the piston moves from TDC towards BDC. The volume increases constantly as the piston travels the stroke length. The maximum volume is reached when the piston is at BDC. The pressure is below atmospheric pressure, during the whole stroke, because the piston movement is creating volume and the air is drawn inside the cylinder due to the vacuum effect.

COMPRESSION (b-c)

After the piston has passed BDC, the compression stroke begins. In this phase the volume starts to decrease and the pressure to increase. It takes a while until the pressure in the cylinder exceeds the atmospheric pressure so the intake valve is still open also after the piston passes BDC. As the piston goes towards TDC, the pressure increases gradually. Around 25° before TDC, the ignition is triggered and the pressure rises rapidly towards maximum pressure.

POWER (c-e)

After the ignition / injection event, the pressure in the cylinder rises sharply, until it hits the maximum values p_max. The value of the maximum pressure depends on the type of the engine, what fuel it’s used. For a typical passenger vehicle engine, the maximum cylinder pressure can be around 120 bar (gasoline) or 180 bar (diesel). The power stroke starts when the piston moves from TDC towards BDC. The high pressure in the cylinder is pushing the piston, therefore the volume rises and the pressure starts to drop gradually.

EXHAUST (e-a)

After the power stroke, the piston is again at the BDC. The volume in the cylinder is again at maximum value and the pressure around minimum (atmospheric pressure). The piston starts to move towards TDC and it’s pushing the burnt gases out of the cylinder.

As you can see, there is a continuous variation of the pressure and volume inside the engine’s cylinders. We’ll see that the work produced by the ICE is function of the pressure and volume changes.

Work W [J] is the product between the force F [N] which is pushing the piston and the displacement, which in our case is the stroke S [m].

\[W = F \cdot S \tag{1}\]

We know that pressure is force divided by area, therefore:

\[F = p \cdot A_p \tag{2}\]

where p [Pa] the pressure inside the cylinder and A_p [m²] is the piston’s area.

Replacing (2) in (1), gives:

\[W = p \cdot A_p \cdot S \tag{3}\]

We know that multiplying a distance with an area we get a volume, therefore:

\[W = p \cdot V \tag{4}\]

This is the instantaneous work produced in the cylinder for a certain pressure and volume. To determine the work for the complete engine cycle we need to integrate the instantaneous work:

\[W = \int F \cdot dx = \int p \cdot A_p \cdot dx \tag{5}\]

where x is the piston travel.

The product between the travel of the piston and the piston area gives the differential volume dV displaced by the piston:

\[dV = A_p \cdot dx \tag{6}\]

Replacing (6) in (5), gives the work produced in the cylinder for a complete cycle:

\[\bbox[#FFFF9D]{W = \int p \cdot dV} \tag{7}\]

Since the vast majority of the internal combustion engine have several cylinders, we are going to introduce a more appropriate parameter to quantify work, which is specific work w [J/kg].

\[w = \frac{W}{m} \tag{8}\]

where m [kg] is mass of air-fuel mixture inside the cylinders for a complete cycle.

We can define also the specific volume v [m³/kg] as:

\[v = \frac{V}{m} \tag{9}\]

The derivative of the specific volume will be:

\[dv = \frac{1}{m} \cdot dV \tag{10}\]

from which we can write:

\[dV = m \cdot dv \tag{11}\]

Replacing (7) in (8) gives:

\[w = \frac{1}{m} \int p \cdot dV \tag{12}\]

From (11) and (12) we get the mathematical expression of specific work for a complete engine cycle:

\[\bbox[#FFFF9D]{w = \int p \cdot dv}\]

The work produced inside the engine’s cylinders is called indicated specific work, w_i [J/kg]. What we get at the crankshaft is a brake specific work w_b [J/kg]. It is called “brake” because, when engines are tested on a test bench, they are connected to a braking device (hydraulic or electric), which is simulating the load.

To get the brake work we have to subtract from the indicated work all the losses of the engine. The losses are the internal frictions and the auxiliary devices which require power from the engine (oil pump, water pump, supercharger, air conditioner compressor, alternator, etc.). These losses have an equivalent friction specific work w_f [J/kg].

\[w_b = w_i – w_f\]

By looking at the indicated pressure-volume (pV) diagram above, we can see that there are two distinct areas:

• the upper area, formed during the compression and power strokes (+W)
• the lower area, formed during exhaust and intake strokes (-W), also named pumping work

Depending on the value of the intake pressure, the pumping work area can be negative or positive. For atmospheric engines, the pumping work is negative because it’s using energy from the engine to push exhaust gases out of the cylinders and draw fresh air during intake.

For gasoline atmospheric engines, due to intake air throttling, the pumping losses are higher, being maximum at idle speed. Diesel engines are more efficient than gasoline engines because there is no throttle on the intake, the load being controlled through fuel injection.

If we divide the brake specific torque to the indicated specific torque, we get the mechanical efficiency of the engine η_m [-]:

\[\bbox[#FFFF9D]{\eta_m = \frac{w_b}{w_i}}\]

For most of the engines, mechanical efficiency is around 80-85% at full load (wide open throttle) and it’s dropping to zero at idle, where all the engine torque is used to maintain idle speed and not for propulsion.

For any questions, observations and queries regarding this article, use the comment form below.

Don’t forget to Like, Share and Subscribe!