**Simple Finite Heat Release Model**

Introduction
We now account for finite burn duration, in which combustion occurs at
, and continues until .
Figure 6 is a representation of this cycle. The peak pressure will not
be as high as the Otto cycle which has a "delta" function heat release. The finite heat release model
assumes that the heat input Qin is delivered to the cylinder over a finite crank angle duration.

Heat Release Model
Link to: __Heat Release Fraction Applet__

Link to: __Simple Heat Release Applet__

####
Derivation of Pressure versus Crank Angle for Finite Heat Release

The differential first law for this model for a small crank angle change, ,
is:
Using the following definitions,heat
release , and ,
results in :

The ideal gas equation is PV = mRT, so

and

The first law now becomes

Further reducing the equation:

Using R = c_{p}-c_{v} and k = c_{p}/c_{v},
to define ,
the energy equation after rearrangement becomes:
or
If we know the pressure, P, volume, V, ,
the heat released gradient,
, we can compute the change in pressure, .
Thus explicitly solving the equation for pressure as a function of crank
angle. Alternatively, we can use experimental data for the pressure, P
and the volume, V, to determine the heat release term by solving for
.

First, the volume, V and
, have to be defined. From the __slider-crank
model__, we have a definition for cylinder volume, V. Both terms
are only dependent on engine geometry.

So taking the derivative with respect to the crank angle, ,
results in :

For heat release term, ,
the Wiebe function for the burn fraction is used.

Where:
f = the fraction of heat added
= the crank angle
_{0}
= angle of the start of the heat addition
=
the duration of the heat addition (length of burn)
a = usually 5
n = usually 3
To view the burn fraction, f, as a function of the crank angle, click
here: Heat Release Applet.

At the beginning of combustion, f = 0, and at the end the fraction is
almost 1.

The heat release, ,
over the crank angle change, ,
is:

Where Q_{in} is the overall
heat input.

Taking the derivative of the heat release function, f, with respect
to crank angle, gives the following definition of
.

If
, dF = 0.
So now with
and
defined, the pressure as a function of the crank angle can be solved.

The following applet plots the pressure, work and temperature as a function
of the crank angle: __Simple Heat Release Applet__.
The effect
of __heat transfer__
to the cylinder wall can also be included.