Resistance ---------- RESIST (class) ^^^^^^^^^^^^^^ .. figure:: ../media/image699.png :figwidth: 0.80208in :align: center Local resistance or loss **Fall type** +-------------------------+--------------------------------------------------------------------------+--------+ | type label | Description | active | +=========================+==========================================================================+========+ | Resist quadr. xi | Dimensionless -value from loss equation | No | +-------------------------+--------------------------------------------------------------------------+--------+ | Resist quadr. C | -value from loss equation | No | +-------------------------+--------------------------------------------------------------------------+--------+ | Resist linear C | -value from loss equation | No | | | | | | | :math:`\Delta H=C Q` | | +-------------------------+--------------------------------------------------------------------------+--------+ | Resist quadr. Initial Q | Initial is specified, from loss equation is computed. | No | +-------------------------+--------------------------------------------------------------------------+--------+ | Resist 2-way quadr.xi | Dimensionless -value from loss equation -value depends on flow direction | No | +-------------------------+--------------------------------------------------------------------------+--------+ | Resistance polynomial | Resistance according user-defined polynomial | No | +-------------------------+--------------------------------------------------------------------------+--------+ .. _mathematical-model-29: Mathematical model """""""""""""""""" A resistance is modelled by the following equations: .. math:: H_{1}-H_{2}=a \xi Q_{1}\left|Q_{1}\right| in which: +---------------+------------------------------------------------+---------------------------+ | Variable | Description | Units | +===============+================================================+===========================+ | *H*\ :sub:`1` | Upstream head | [m] | +---------------+------------------------------------------------+---------------------------+ | *H*\ :sub:`2` | Downstream head | [m] | +---------------+------------------------------------------------+---------------------------+ | a | :math:`1 /\left(2 g A_{\mathrm{r}}^{2}\right)` | [s\ :sup:`2`/m\ :sup:`5`] | +---------------+------------------------------------------------+---------------------------+ | ξ | loss coefficient | [‑] | +---------------+------------------------------------------------+---------------------------+ | *Q*\ :sub:`1` | discharge through RESIST | [m\ :sup:`3`/s] | +---------------+------------------------------------------------+---------------------------+ | *A*\ :sub:`r` | discharge area resistance | [m\ :sup:`2`] | +---------------+------------------------------------------------+---------------------------+ The discharge area does not have to be equal to the area of the adjacent pipes. The area is calculated from the resistance diameter *D*\ :sub:`r` according to .. math:: A_{\mathrm{r}}=\pi D_{r}^{2} / 4 | **Tip**: | The RESIST can be used to model all kinds of resistant components without a relevant length, e.g.: - A number of bends - Entrance, exit losses - Filters - Heat exchangers, etc. Resist quadr. xi ^^^^^^^^^^^^^^^^ .. _hydraulic-specifications-30: Hydraulic specifications """""""""""""""""""""""" ================= ===== ==== ======= ======= ======= Description input unit range default remarks ================= ===== ==== ======= ======= ======= inner diam­eter real [m] (0-5] loss coeffi­cient real [-] [0-100] ================= ===== ==== ======= ======= ======= See also "Mathematical model" (:numref:`mathematical-model-29`). .. _component-specific-output-29: Component specific output """"""""""""""""""""""""" None .. _h-actions-27: H-actions """"""""" None .. _component-messages-26: Component messages """""""""""""""""" None Resist quadr. C ^^^^^^^^^^^^^^^ .. _hydraulic-specifications-31: Hydraulic specifications """""""""""""""""""""""" +---------------------------+-------+---------------------------+---------+---------+---------+ | description | input | Unit | range | default | remarks | +===========================+=======+===========================+=========+=========+=========+ | C-value (∆H=CQ\ :sup:`2`) | real | [s\ :sup:`2`/m\ :sup:`5`] | [0-100] | | | +---------------------------+-------+---------------------------+---------+---------+---------+ See also "Mathematical model" (:numref:`mathematical-model-29`). .. _component-specific-output-30: Component specific output """"""""""""""""""""""""" None .. _h-actions-28: H-actions """"""""" None .. _component-messages-27: Component messages """""""""""""""""" None Resist linear C ^^^^^^^^^^^^^^^ .. _hydraulic-specifications-32: Hydraulic specifications """""""""""""""""""""""" ============== ===== =============== ======= ======= ======= description input unit range default remarks ============== ===== =============== ======= ======= ======= linear C-value real [s/m\ :sup:`2`] [0-100] (­∆H=CQ) ============== ===== =============== ======= ======= ======= See also "Mathematical model" (:numref:`mathematical-model-29`). .. _component-specific-output-31: Component specific output """"""""""""""""""""""""" None .. _h-actions-29: H-actions """"""""" None .. _component-messages-28: Component messages """""""""""""""""" None Resist quadr. Initial Q ^^^^^^^^^^^^^^^^^^^^^^^ .. _hydraulic-specifications-33: Hydraulic specifications """""""""""""""""""""""" ================= ===== =============== ====== ======= ======= description input unit range default remarks ================= ===== =============== ====== ======= ======= initial discharge real [m\ :sup:`3`/s] (0-10] ================= ===== =============== ====== ======= ======= See also "Mathematical model" (:numref:`mathematical-model-29`). .. _component-specific-output-32: Component specific output """"""""""""""""""""""""" None .. _h-actions-30: H-actions """"""""" None .. _component-messages-29: Component messages """""""""""""""""" ================================== ==== =========== Message Type Explanation ================================== ==== =========== C‑value (resistance) = ... [s2/m5] Info ================================== ==== =========== Resist 2-way quadr.xi ^^^^^^^^^^^^^^^^^^^^^ .. _hydraulic-specifications-34: Hydraulic specifications """""""""""""""""""""""" ========================== ===== ==== ======= ======= ======= Description input unit range default remarks ========================== ===== ==== ======= ======= ======= inner diam­eter pos flow real [m] (0-5] inner diameter neg flow real [m] (0-5] loss coeff. pos flow (xi+) real [-] [0-100] loss coeff. new flow (xi-) real [-] [0-100] ========================== ===== ==== ======= ======= ======= .. _component-specific-output-33: Component specific output """"""""""""""""""""""""" None .. _h-actions-31: H-actions """"""""" None .. _component-messages-30: Component messages """""""""""""""""" None Resistance polynomial ^^^^^^^^^^^^^^^^^^^^^ .. figure:: ../media/image699.png :figwidth: 0.80208in :align: center Polynomial resistance **Fall type** +-----------------------+-------------------------------------------------+--------+ | type label | description | active | +=======================+=================================================+========+ | Resistance polynomial | Resistance described by second order polynomial | no | +-----------------------+-------------------------------------------------+--------+ .. _mathematical-model-30: Mathematical model """""""""""""""""" The resistance of a particular hydraulic object is mostly a non-linear function. If field measurements are carried out, the pressure head difference and related discharge can be shown in a chart, see figure below. [CHART] The function of this resistance can be described by a second order polynomial: - :math:`\Delta H=a+b Q+c Q|Q|` (1) in which: - *∆H =* pressure drop *H\ 1-H\ 2* [m] - *Q =* Discharge through resist [m\ :sup:`3`/s] - *a,b,c =* coefficients [-] - H\ *1 =* upstream pressure head [m] - H\ *2 =* downstream pressure head [m] The coefficients *a, b,* and *c* can be derived e.g. using EXCEL (insert trendline). The coefficient *a* is the static pressure head. In the figure below, the derived function together with the measured points are drawn. Note that the coefficients are based on SI-units. [CHART] Remark: The polynomial is valid for positive and negative flow. Note that the coefficient “A” is a constant head loss and is only valid for a positive flow direction. That means that with a negative flow, the headloss contribution “A” (defined as H1-H2) is the same as for positive flow. .. _hydraulic-specifications-35: Hydraulic specifications """""""""""""""""""""""" +---------------------------------+-------+------+-------------------------------+---------+---------+ | Description | input | unit | range | default | remarks | +---------------------------------+-------+------+-------------------------------+---------+---------+ | Coefficient a in ∆H=a+bQ+cQ|Q\| | real | [-] | (-10\ :sup:`8` -10\ :sup:`8`) | | | +---------------------------------+-------+------+-------------------------------+---------+---------+ | Coefficient b in ∆H=a+bQ+ cQ|Q | real | [-] | (-10\ :sup:`8` -10\ :sup:`8`) | | | +---------------------------------+-------+------+-------------------------------+---------+---------+ | Coefficient c in ∆H=a+bQ+ cQ|Q | real | [-] | (-10\ :sup:`8` -10\ :sup:`8`) | | | +---------------------------------+-------+------+-------------------------------+---------+---------+ .. _component-specific-output-34: Component specific output """"""""""""""""""""""""" None .. _h-actions-32: H-actions """"""""" None .. _component-messages-31: Component messages """""""""""""""""" None .. _example-12: Example """"""" .. figure:: ../media/image705.png :figwidth: 3.1875in :align: center Schematic overview of the wanda model The level in the upstream reservoir varies from 50 m to -5 m; the level in the downstream reservoir remains constant at H= 0 m. Next figure shows the discharge and level as functions of time. .. figure:: ../media/image706.png :figwidth: 5.625in :align: center Pressure as function of time for a polynomial resist. The next chart shows the pressure head ΔH related to the discharge Q. The WANDA results are exactly the same as the theoretical polynomial. [CHART] .. include:: substitutions_liquid.rst