## Simulink - Vehicle Acceleration 7

1. Introduction
This is a development of the previous model.
In this model the pressure drop across the throttle, the effect this has on the air mass flow rate and the cross section area opened by the throttle for the air to flow through are all considered. Again a single gear is assumed.
This model initially makes the assumption that the air flow is incompressible and that Bernoulli's equation can be applied. This simplification means the modeling is not excessively difficult, but is not very accurate as the flow should be considered as compressible, which is more complex to model.
Flow rates are 'per second', unless otherwise stated.
The block diagram for this model (fuel 4) is shown below:

2. The Model
Starting from near the top left, the output from the 'Product4' block is the air mass flow rate. This is multiplied by the 'Gain' of 0.068 (assuming a stoichiometric mixture) to give the fuel mass flow rate. 'Gain1' multiplies this by 46000000 to give the gross power produced by burning the fuel. 'Gain2' is an assumed efficiency of 30% to give the assumed net power.
The 'Product' block multiplies the assumed net power by 1/engine angular velocity to give the assumed engine torque. There is a 'Constant' of 200 added into this feedback loop to prevent division by zero and simulate the engine speed when letting out the clutch.

'Gain3' is the assumed overall gear ratio, 10, and 'Gain4' is 1/wheel radius to give forward thrust generated by the driving wheels. The summing point takes away the air resistance to give the net force available to accelerate the vehicle which goes into 'Gain5' which is 1/equivalent mass of the vehicle, giving the acceleration. This goes through the 'Integrator' to give the vehicle velocity.

The vehicle velocity is squared in 'Product1' and in 'Gain6' is multiplied by:

frontal area x drag coefficient x air density/2 This gives the air resistance opposing the forward motion.

The velocity is also fed back through 'Gain7', 1/wheel radius, to give the drive shaft angular velocity, then through 'Gain8', the overall gear ratio, to give the engine omega value. 'Constant1', value 1, is added to get the model started from rest. This is then fed back via the 'Math Function' reciprocal into the 'Product' block to obtain the net engine torque.

The engine omega is also multiplied by:

swept volume/(2 * 2 * 3.14159) NB - do NOT omit the brackets in expressions of this type. to give the air volume flow rate per second into the engine. This value is then multiplied (in 'Product4') by the calculated density, Ro2, of the inducted air, to give the air mass flow rate.

To apply Bernoulli the velocity in the inlet tract, V1, and across the throttle plate, V2 are required. As there is assumed no change in level, the equation can be written:

V12/2g + P1/Ro.g = v22/2g + P2/Ro.g which when re-arranged, assuming P1 = 100000N/m2, to give P2 reads: P2 = Ro.V12/2 + 100000 - Ro.V22/2 Ro is assumed to be 1.25, so Ro/2 = 0.625

The volume flow rate is also multiplied (in 'Product2') by 1/inlet tract open area to give the speed of the air past the throttle. This is squared in 'Product3' and multiplied by 1.25/2 ('Gain11').
The volume of air drawn in per second is also multiplied by 1/(full inlet tract cross section area) ('Gain14') to give the V1 value, which is squared in 'Product5'. This is multiplied by 1.25/2 in 'Gain 16'.
'Constant3', 100000, gives the assumed atmospheric pressure.
The two adjacent summing points are then used to evaluate P2.

At this point the assumption of incompressible flow is changed and it is assumed that the air density is proportional to the absolute pressure (P2) and P2 is multiplied by 1/100000 (in 'Gain12') and by 1.25 (in Gain13'). This gives the air density across the throttle plate, Ro2.

The air density, Ro2 is then multiplied in 'Product4' by the volume rate flow to give the mass flow rate.

The 'Signal Builder' source provides the throttle opening, which is multiplied in 'Gain10' by 0.97 * 0.05 * 0.06 to give the cross section area of inlet tract opened by the throttle. The 0.97 is to allow for a slight obstruction of the tract even when the throttle is fully open. This is inverted (a small constant added, 'Constant2' to avoid division by zero when starting) and this goes to 'Product2' to give the air velocity, V2, across the throttle plate.

3. Results
Running this model gives a 0 - 60 mph time of almost 9 seconds. Again because of the single gear ratio the engine revs at this speed are an unrealistically high 9000 rpm.

This model is however very sensitive to the constant value in 'Constant', the assumed rad/s at which the clutch is let out. Changing this to 100, gives a 0 - 60 time of about 6 seconds. However the airflow is exceeding the maximum after about 4.5 seconds, which should be limited for accuracy (the engine torque also rises above the maximum of 205 Nm). These effects may be due to the fact that the throttle cross section area is not accurate.

David Grieve, 4th November 2005.