Dynamic air valve with water release ------------------------------------ DYNAIRVA (class) ^^^^^^^^^^^^^^^^ .. figure:: ../media/image1245.svg :figwidth: 100% :align: center Dynamic air valve **Supplier type** +-----------------------------+-------------------------------------------------------------------------------+--------+ | type label | description | active | +=============================+===============================================================================+========+ | Dynamic air valve | Detailed air valve model with automated level effect. Water is released from | No | | | the air valve to dampen the surge during the transition from open to closed | | +-----------------------------+-------------------------------------------------------------------------------+--------+ .. _mathematical-model-dyn-airva: Mathematical model """""""""""""""""" :numref:`fig-dyn-air-valve-1` shows the dynamic air valve which is modelled. Air can enter the system when the stem (yellow part) is in the open position (Left figure of :numref:`fig-dyn-air-valve-1`). The stem will open when the pressure difference over the air valve (difference between atmospheric pressure and the pressure in the pipeline) is larger than the pressure required for opening the air valve (opening pressure). It will then start to move towards its full open position when this pressure difference increases. When the pressure difference is equal or greater than the full lift pressure the valve is fully open. In reality air can also enter via the piston at the top of the air valve (red part). This is neglected since the air flow through this part is small compared to the flow through the in/outlet. Air is also released via the opening of the stem. The stem starts to move to its fully open position when the pressure difference over the air valve (difference between the pressure in the pipeline and atmospheric pressure) is larger than the opening pressure. When this pressure difference reaches its fully lift pressure the stem is in its full open position. Air is then expelled from the air valve and the water level rises. When the water level reaches the air valve, air and water will be released. When all air is expelled the stem will start to move towards its fully closed position. :numref:`tab-modes-of-air-valve` shows an overview of the different stages of the air valve for different pressure differences. In the following sections the equation for the closed and open stage will be derived. .. _fig-dyn-air-valve-1: .. figure:: ../media/image1246.svg :figwidth: 100% :align: center Dynamic air valve .. _tab-modes-of-air-valve: .. table:: Different modes of the air valve based on the pressure inside the system. ======================================================= ============================================================================================ Situation Result ======================================================= ============================================================================================ :math:`P_{air} - P_{atm} < -P_{fyllyopen}` Stem fully open air inlet :math:`-P_{fullyopen} < P_{air} - P_{atm} < -P_{lift}` Stem between fully open and closed, air inlet :math:`-P_{lift} < P_{air} - P_{atm} < P_{lift}` Air valve closed :math:`P_{lift} < P_{air} - P_{atm} < P_{fyllyopen}` Stem between fully open and closed, air outlet :math:`P_{air} - P_{atm} > P_{fyllyopen}` Stem fully open air outlet and possible water outlet depending on the water level ======================================================= ============================================================================================ **Closed state** In the closed stage air will be trapped in the air valves, the amount will depend on the history of the system. This air can be compressed or expanded, depending on the pressure changes in the system. This is governed by the ideal gas law: .. math:: :label: eq-dynairva-ideal-gas-law P_{air} V_{air}^{k} = C with ============================= ========================================================= ======================= variable Description Units ============================= ========================================================= ======================= :math:`P_{air}` Absolute air pressure on fluid level Pa :math:`V_{air}` Air volume m\ :sup:`3` :math:`k` Laplacer coefficient \- :math:`C` Constant J ============================= ========================================================= ======================= This will result in a change of air volume and this change in volume will result in a discharge in the connected system. This is governed by: .. math:: :label: eq-dynairva-mass-balance Q_1 - Q_2 + Q_{airce} = 0 with ============================= ========================================================= ======================= variable Description Units ============================= ========================================================= ======================= :math:`Q_{1}` Discharge at connection point 1 m\ :sup:`3`/s :math:`Q_{2}` Discharge at connection point 2 m\ :sup:`3`/s :math:`Q_{airce}` Air volume change due to compression and expansion m\ :sup:`3`/s ============================= ========================================================= ======================= During the closed state the change of volume of air is due to compression and expansion of the air. This is given by: .. math:: :label: eq-dynairva-discharge Q_{airce} = \left( \left( \frac{P_{old}}{P_{new}} \right)^{1/\lambda} - 1 \right) \frac{V_{old}}{dt} with ============================= ========================================================= ======================= variable Description Units ============================= ========================================================= ======================= :math:`P_{old}` Air pressure at previous time step Pa :math:`P_{new}` Air pressure at present time step Pa :math:`V_{old}` Air volume at previous time step m\ :sup:`3` :math:`dt` Time step s ============================= ========================================================= ======================= The air pressure is given by: .. math:: :label: eq-dynairva-air-pressure P = P_{atm} + \rho g (H_i - w_i) with ============================= ========================================================= ======================= variable Description Units ============================= ========================================================= ======================= :math:`P` Absolute air pressure on fluid level Pa :math:`P_{atm}` Atmospheric pressure Pa :math:`\rho` Denisty of the fluid kg/m\ :sup:`3` :math:`H_i` Head at the :math:`i^{th}` connection point m :math:`w_i` Fluid level at the :math:`i^{th}` connection point m ============================= ========================================================= ======================= The equations :math:numref:`eq-dynairva-ideal-gas-law`, :math:numref:`eq-dynairva-mass-balance`, :math:numref:`eq-dynairva-discharge` and :math:numref:`eq-dynairva-air-pressure` govern the behavior of the air valve in closed state. **Open stage** When the air valve is open air can either enter or exit the system. The actual direction depends upon the pressure inside the system. :numref:`tab-modes-of-air-valve` shows an overview of the different possibilities. The air in- or outflow can be defined by either coefficients or by a characteristic. **Vent capacity (defined by coefficients)** The following formuleas are based on :cite:p:`Wylie1978`: 1. Subsonic air flow .. math:: \begin{eqnarray} Q_{air} = C_{in} A_{in} \sqrt{7RT_0} \sqrt{ \left(\frac{P}{P_{atm}}\right)^{1.4286} - \left( \frac{P}{P_{atm}} \right)^{1.714}} & P_{atm} > P > 0.53 P_{atm} \end{eqnarray} with ============================= ========================================================= ======================= variable Description Units ============================= ========================================================= ======================= :math:`A_{in}` Inlet area m\ :sup:`2` :math:`C_{in}` Inlet discharge coefficient \- :math:`R` Gas constant J/(kg K) :math:`T_0` Ambient temperature K ============================= ========================================================= ======================= 2. Critical flow in .. math:: \begin{eqnarray} Q_{air} = C_{in} A_{in} \sqrt{7RT_0} 0.259 & \textrm{ when } P_{atm} > P > 0.53 P_{atm} \end{eqnarray} 3. Subsonic air flow out .. math:: \begin{eqnarray} Q_{air} = -C_{out} A_{out} \sqrt{7RT_0} \left( \frac{P}{P_{atm}} \right)^{\frac{k+1}{2k}} \sqrt{ \left( \frac{P}{P_{atm}} \right)^{1.4286} - \left( \frac{P_{atm}}{P} \right)^{1.714} } & \mathrm{ when } \frac{P_{atm}}{0.53} > P > P_{atm} \end{eqnarray} ============================= ========================================================= ======================= variable Description Units ============================= ========================================================= ======================= :math:`A_{out}` Inlet area m\ :sup:`2` :math:`C_{out}` Inlet discharge coefficient \- ============================= ========================================================= ======================= 4. Critical flow out .. math:: \begin{eqnarray} Q_{air} = -0.259 C_{out} A_{out} \sqrt{7RT_0} \left( \frac{P}{P_{atm}} \right)^{\frac{k+1}{2k}} & \textrm{ when } \frac{P_{atm}}{0.53} < P \\ \end{eqnarray} **Vent capacity (defined by characteristic)** The capacity of the vent can also be defined by the air valve characteristic (see :numref:`fig-dynairva-characteristic`). In this case, you can define both the characteristic for inflow and outflow. The characteristic should be supplied as pressure against discharge. **Note**: Often the characteristic supplied by the manufacturer will have the axes reversed from this definition. See also the example characteristic in this manual. .. _fig-dynairva-characteristic: .. figure:: ../media/image1247.svg :figwidth: 100% :align: center Example of an air valave characteristic **Level effect** The air entering or leaving the pipeline will change the water level in the system. This is taken into account via the so called level effect. This is automatically included based upon the pipe profile of pipes directly connecting to the air valve. There are 4 possible cases: 1. No pipes are connected to the air valve, only the level effect for the stand pipe as specified by the user is taken into account. A warning is given when the water level drops below the bottom of the standpipe. The calculations will continue assuming that the standpipe continues with the same diameter below this point. 2. One pipe is connected to the air valve. For the side the pipe is connected the complete level area effect is taken into account. For the other side only the standpipe is taken into account. 3. Two pipes are connected to the air valve. For both sides the complete level area table is calculated based on the profile of the connected pipes. 4. More than 1 pipe is connected to one side of the air valve. In this case one pipe is selected and a message is given which pipe is selected. A connection pipe (stand pipe) between the air valve and the pipeline can be included by specifying its diameter and height see :numref:`fig-dynairva-example-system`. For the level effect an engineering approach is used to calculate the volume in bends. Since the most important result is the expelling of the last air, this approximation is acceptable. The dynamic behaviour of the water levels in both legs depends on the actual water levels relative to the bottom elevation. If we denote the local pipe bottom elevation with b, then the following states are feasible: .. _tab-states-of-the-pipe: .. table:: States of the pipe. =========== =========== ============================================== ============================================================================================ :math:`W_1` :math:`W_2` Description Applied equations =========== =========== ============================================== ============================================================================================ :math:`>b` :math:`>b` both levels are identical :math:`w_1 = w_2` :math:`Q_1 - Q_2 = A_1(w_1) \frac{dw_1}{dt} + A_2(w_2)\frac{dw_2}{dt}` :math:`=b` :math:` 0` :math:`Q_1 - Q_2 = A_2(w_2)\frac{dw_2}{dt}` :math:`