PUMPSS
------

PUMPSS (class)
^^^^^^^^^^^^^^

.. figure:: ../media/image682.png
	:figwidth: 0.84375in
	:align: center
	
	Schematic of the Wanda Piston Pump component

Single cylinder, single-acting pump with mechanically driven piston

**Supplier type**

+-----------------------------+----------------------------------------------------------------------+--------+
| type label                  | description                                                          | active |
+=============================+======================================================================+========+
| Pump Simplex, single-acting | A single cylinder, with a mechanically driven single-acting piston.. | Yes    |
+-----------------------------+----------------------------------------------------------------------+--------+

A single cylinder reciprocating pump model can be created by connecting the cylinder to the node shared by two check valves in series See Figure 1.

.. figure:: ../media/image683.png
	:figwidth: 1.16667in
	:align: center
	
	Simplex, single-acting reciprocating pump model

A multi-cylinder pump model can be created by placing multiple instances of these cylinders and check valves in series

.. _mathematical-model-26:

Mathematical model
""""""""""""""""""

Figure 2 shows the geometry of the one cylinder piston pump and its motion.

.. figure:: ../media/image684.png
	:figwidth: 3.36458in
	:align: center
	
	Geometry of piston motion

The crank angle is defined as:

.. math::
	:label: eq_piston_crank_angle

	\phi=\omega t+c

where *Φ* is the crank angle relative to the top dead centre (radians), *ω* is the constant pump speed (radians/sec), *t* is time and *c* is the crankshaft offset angle (radians). 
For a variable pump speed equation :eq:`eq_piston_crank_angle` can be rewritten as:

.. math::

	\phi^{n}=\phi^{n-1}+\omega^{n} \Delta t

where *n* denotes the values of the present time-step, *n-1* denotes the values of the previous time-step, *Δt* is the time-step length and *x* is the piston position relative to the top dead centre, *e* is the crank radius and *r* is the connecting rod length.

.. math::

	x=e(1-\operatorname{Cos} \phi)+r\left(1-\sqrt{1-\left(\frac{e}{r} \operatorname{Sin} \phi\right)^{2}}\right)

The piston position may also be normalized [-1,+1] as:

.. math::

	x_{n o r m}=\frac{x-e}{e}

The cylinder volume *V* and piston flow rate *Q\ p* (derived from the piston position) is given by:

.. math::

	V=V_{T D C}+A_{p}\left[e(1-\operatorname{Cos} \phi)+r\left(1-\sqrt{1-\left(\frac{e}{r} \operatorname{Sin} \phi\right)^{2}}\right)\right]

.. math::

	Q_{p}=-\omega A_{p} e \operatorname{Sin} \phi\left[1+\frac{e \operatorname{Cos} \phi}{r \sqrt{1-\left(\frac{e}{r} \operatorname{Sin} \phi\right)^{2}}}\right]

where *V\ TDC* is the cylinder volume at top dead centre and *A\ p* is the piston area.

In the steady state there is no piston movement, hence no discharge is entering or leaving the cylinder. This can be written as a residual :

.. math::

	Q=0

During motion the following continuity equation applies:

.. math::
	:label: eq_piston_continuity_equation

	Q=Q_{p}-\frac{d P}{d t} \frac{V}{K}

where *Q* is the discharge out of the cylinder, *Q\ P* is the volumetric discharge of the piston, *P* is the pressure inside the cylinder and *K* is the bulk modulus of the fluid inside the cylinder. 
Dividing :eq:`eq_piston_continuity_equation` by fluid density and gravity acceleration and integrating fully implicitly in time gives:

.. math::

	F^{n}=H^{n}-H^{n-1}+\left(Q^{n}-Q_{p}^{n}\right) \frac{K \Delta t}{\rho g V^{n}}=0

Where *H* is the pressure head within the cylinder.

The piston pump model does not account for cavitation occurring within the pump cylinder.

PUMPSS (Piston Pump)
^^^^^^^^^^^^^^^^^^^^

.. _hydraulic-specifications-27:

Hydraulic specifications
""""""""""""""""""""""""

+-----------------------+-------+-------------+--------+---------+---------------------------+
| Description           | input | unit        | Range  | default | remarks                   |
+=======================+=======+=============+========+=========+===========================+
| Initial motor speed   | Real  | rad/s       |        |         |                           |
+-----------------------+-------+-------------+--------+---------+---------------------------+
| Initial phase         | Real  | radians     | 0 - 2π | 0       |                           |
+-----------------------+-------+-------------+--------+---------+---------------------------+
| Cylinder volume       | Real  | m\ :sup:`3` |        |         | volume at top dead centre |
+-----------------------+-------+-------------+--------+---------+---------------------------+
| Piston area           | Real  | m\ :sup:`2` |        |         |                           |
+-----------------------+-------+-------------+--------+---------+---------------------------+
| Crank radius          | Real  | m           |        |         |                           |
+-----------------------+-------+-------------+--------+---------+---------------------------+
| Connecting rod length | Real  | m           |        |         |                           |
+-----------------------+-------+-------------+--------+---------+---------------------------+

.. _component-specific-output-26:

Component specific output
"""""""""""""""""""""""""

Motor speed [rad/s]

Piston position (normalized) [-]

Crank shaft angle [radians]

.. _h-actions-24:

H-actions
"""""""""

**Example**:

To simulate a duplex, triplex or any other multicylinder pump, multiple instances of the PUMPSS component together with both checkvalves need to be placed parallel, with each cylinder having a unique phase angle.

The example below shows a triplex piston pump, with initial phase angle of respectively 0°, 120° and 240°.

.. figure:: ../media/image694.png
	:figwidth: 3.89583in
	:align: center
	
	Schematic overview of Wanda model

.. figure:: ../media/image695.png
	:figwidth: 3.09375in
	:align: center
	
	Discharge of the modeled triplex piston pump.

.. _component-messages-23:

Component messages
""""""""""""""""""

+------------------------------------------------------------+-------+------------------------+
| Message                                                    | Type  | Explanation            |
+============================================================+=======+========================+
| Time-step should be smaller than reciprocating pump period | Error | Time-step is too large |
+------------------------------------------------------------+-------+------------------------+

.. include:: substitutions_liquid.rst