.. _sec-icengine:

================================
Internal Combustion Engine Model
================================

The internal combustion engine model in CanSen is included to enable
simulations of a reciprocating internal combustion engine. The
equation of motion for the piston follows from Heywood [#HEYW1988]_, Ch. 2.

Assuming the piston roughly appears as:

.. image:: /_piston/piston.*
   :width: 2in
   :align: center

where :math:`s` is distance from the crank axis to the piston pin axis,
:math:`\ell` is the connecting rod length, :math:`a` is the crank radius,
:math:`\theta` is the crank angle, with :math:`0^{\circ}` at the top of
the crank, :math:`L` is the stroke length, :math:`B` is the cylinder bore,
:math:`V_d` is the swept, or displacement, volume, :math:`V_c` is the
clearance volume, and TDC and BDC are top dead center and bottom dead center
respectively (i.e., the top and bottom of the stroke).

The compression ratio of the cylinder is defined as:

.. _compression-ratio:
.. math::
   r_c = \frac{V_d + V_c}{V_c}

The swept volume can either be specified directly, or can be calculated from
the cylinder bore and stroke length:

.. math::
   V_d = L * \pi * \frac{B^2}{4}

The initial volume of the cylinder (the volume at BDC) is:

.. math::
   V_0 = V_d + V_c = r_c * V_c = \frac{V_d}{r_c - 1} + V_d

==============================================================================

The distance from the crank center to the piston pin is given by:

.. math::
   s = a \cos(\theta) + \sqrt{\ell^2 - a^2 \sin^2(\theta)}

Cantera expects a moving wall to be given a velocity, so we find the piston
velocity by differentiating with respect to time:

.. math::
   \frac{ds}{dt} = -a \sin(\theta) \frac{d\theta}{dt} + \frac{a^2 \sin(\theta) \cos(\theta)\frac{d\theta}{dt}}{\sqrt{\ell^2 - a^2 \sin^2(\theta)}}

Defining :math:`\frac{d\theta}{dt}`, the angular velocity of the crank, as
:math:`\omega`, and using the definition of the stroke length :math:`L` and
the connecting rod length to crank radius ratio :math:`R`:

.. math::
   \begin{aligned}
      L &= 2a \\
      R &= \frac{\ell}{a}
   \end{aligned}

the equation for the velocity can be simplified to:

.. math::
   \frac{ds}{dt} = -\omega \frac{L}{2} \sin(\theta) \left[1 + \frac{\cos(\theta)}{\sqrt{R^2 - \sin^2(\theta)}}\right]

In CanSen, the angular velocity of the crank is input in revolutions per
minute, so it must be converted to radians per second:

.. math::
   \frac{rad}{s} = \text{RPM} * \frac{\pi}{30}

By default, the starting crank angle is :math:`180^{\circ}`, or BDC,
:math:`0^{\circ}` is TDC and the piston reaches BDC again at
:math:`-180^{\circ}`. The starting crank angle can be set with the
:ref:`DEG0 <DEG0>` keyword, so :math:`\theta` is calculated as a function of
time by:

.. math::
   \theta = \frac{\text{DEG0} * \pi}{180} - \omega t

.. [#HEYW1988] John B. Heywood. *Internal Combustion Engine Fundamentals.*
   New York: McGraw Hill, 1988. Print.