T-JUNCTION ---------- .. figure:: ../media/image517.png :figwidth: 5.33333in :align: center T-Junction +------------+-----------------------------------------------------------------------------------------------------------+--------+ | type label | description | active | +============+===========================================================================================================+========+ | T-straight | Three-node junction with the combined flow in the straight leg. Both straight legs have the same diameter | no | +------------+-----------------------------------------------------------------------------------------------------------+--------+ | T-branch | Three-node junction with the combined flow in the branch. Both straight legs have the same diameter | no | +------------+-----------------------------------------------------------------------------------------------------------+--------+ Both components can be used for all kind of flow regimes in a 90 degree T‑junction. The difference between the two T-junctions has to do with the positive flow definition. The arrows in the symbol specify the positive direction of the flow (and the velocity). The two different symbols allow the user to express his design (normal) flow scenario with positive flows. If he expects in the steady state the combined (total) flow in the straight part he may prefer the left symbol. A combined flow, coming into or out of the branch, gives preference to the right symbol. .. _mathematical-model-16: Mathematical model ^^^^^^^^^^^^^^^^^^ Positive flow definition """""""""""""""""""""""" The definition for the positive direction of flow (and velocity) is important to understand the results in the property window. Each T-junction type has his own definition. For the **T-straight** component the positive flow definition is defined in the left figure below. .. figure:: ../media/image518.png :figwidth: 3.33333in :align: center T-straight For the **T-branch** component (right figure) the positive flow definition for the first connection point is opposite to that of the T-straight component; for the two other connection points the positive flow definitions are the same as for the T-straight component. .. figure:: ../media/image519.png :figwidth: 3.33333in :align: center T-branch Flow regimes """""""""""" In the figures below all different flow regimes which can occur are listed and explained with a scheme with corresponding actual flow arrows. It is called straight flow if the total flow (in some literature also called: combined flow) is in one of the straight legs, branch flow means that the total flow is in the branch. .. figure:: ../media/image520.png :figwidth: 1.30208in :align: center Positive straight flow .. figure:: ../media/image522.png :figwidth: 1.30208in :align: center Negative straight flow .. figure:: ../media/image524.png :figwidth: 1.30208in :align: center Combining branch flow .. figure:: ../media/image525.png :figwidth: 1.30208in :align: center Dividing branch flow The sign of the discharge and velocity output (results) depends of the T-junction type chosen, but is independent of the flow regime definition. The example below will illustrate this. The main flow is from West to East with a supply from the South. That means we are dealing with a combin\ **ing** straight flow with the combin\ **ed** flow in leg 2. The solution of the calculation for the T-straight model (positive Q) as well as the T-branch model (negative Q) is shown in the figures below. .. figure:: ../media/image526.png :figwidth: 5.64583in :align: center Example solution with a T-straight model (positive Q) and a T-branch model (negative Q). Equations """"""""" The head loss over a 3-node component depends on the distribution of the discharges and the area of the connected legs. The T-junction supports two different ways for retrieving the resistance coefficient Xi (*ξ*). 1. the head loss functions according to Idelchik’s Handbook [ref. 1] 2. user specified table with Xi depending on discharge ratio. These table values belong to a fixed area ratio. Please note that the loss coefficients according to the handbooks apply to junctions with a certain minimum pipe length interval in between multiple junctions. The head loss in 3-node components is a function of the combined flow in the *combined* leg, which is either one leg (1 or 2) of the straight part or the branch (leg 3). Including the continuity equation (no production or loss of mass in the component) the general set of the three equations of the T-junction is: .. math:: f_{1}=Q_{1}-Q_{2}+Q_{3}=0 ΔH T-junction straight, combined flow in leg 2 .. math:: f_{2}=H_{1}-H_{2}-\xi_{12} \frac{Q_{2}\left|Q_{2}\right|}{2 g A_{2}^{2}} ΔH T-junction straight, combined flow in leg 1 .. math:: f_{2}=H_{1}-H_{2}-\xi_{12} \frac{Q_{1}\left|Q_{1}\right|}{2 g A_{1}^{2}} ΔH T-junction branch, combined flow in branch .. math:: f_{2}=H_{3}-H_{1}-\xi_{31} \frac{Q_{3}\left|Q_{3}\right|}{2 g A_{3}^{2}} Where: ===================================================== =============== Variable Units ===================================================== =============== Q :sub:`i` = total discharge in leg i [m\ :sup:`3`/s] H\ :sub:`i` = energy head in connect point i [m] A\ :sub:`i` = pipe area leg i (leg with total flow) [m\ :sup:`2`] ξ\ :sub:`ij` = loss coefficient between point i and j [-] ===================================================== =============== The subscripts (1), (2) and (3) correspond to the different legs. The subscript x at the discharge Q\ :sub:`x` refers always to the leg in which the combined (total) flow Q occurs. For positive straight dividing flow and negative straight combining flow, Q\ :sub:`1` will be taken into account. For positive straight combining flow and negative straight dividing flow, Q\ :sub:`2` will be taken into account. Resistance coefficients based on formulas """"""""""""""""""""""""""""""""""""""""" The resistance coefficients based of formulas are taken from the Idelchik Handbook with the following restriction to the section areas: .. math:: \begin{array}{l} A_{1}+A_{3}>A_{2} \\ A_{1}=A_{2} \end{array} For all different flow regimes the formulas used are described, and a table shows the *ξ*\ ‑values for various area and discharge ratios. Note that all indices used in the formulas are based on positive flow. **Straight flow combining** For the straight leg, the resistance coefficient is calculated according to: .. math:: \xi_{12} \approx 1.55 \frac{Q_{3}}{Q_{2}}-\left(\frac{Q_{3}}{Q_{2}}\right)^{2} The table below shows the *ξ\ 12* values according to the formula above for various values of the discharge ratio. .. _table_t12_combining_flow: .. table:: ξ\ :sub:`12` values for combining flow. ======================== ======================= Q\ :sub:`3`/Q\ :sub:`2` A\ :sub:`3`/A\ :sub:`2` ======================== ======================= Valid for all ranges 0.0 0.00 0.1 0.14 0.2 0.27 0.3 0.38 0.4 0.46 0.5 0.52 0.6 0.57 0.7 0.59 0.8 0.60 0.9 0.58 1.0 0.55 ======================== ======================= For the branch, the resistance coefficient ξ32 is calculated according to: .. math:: \xi_{32}=A\left[1+\left(\frac{Q_{3}}{Q_{2}} \frac{A_{2}}{A_{3}}\right)^{2}-2\left(1-\frac{Q_{3}}{Q_{2}}\right)^{2}\right] The coefficient *A* depends on the area and discharge ratios again. Table 3 shows the values. .. _table_t12_variable_A: .. table:: Values of the variable A. +-------------------------+---------+-----------------------------------------------+-------+ | A\ :sub:`3`/A\ :sub:`2` | ≤ 0.35 | > 0.35 | | +-------------------------+---------+-----------------------------------------------+-------+ | Q\ :sub:`3`/Q\ :sub:`2` | 0 – 1.0 | ≤ 0.4 | > 0.4 | +-------------------------+---------+-----------------------------------------------+-------+ | A | 1.0 | :math:`0.9\left(1-\frac{Q_{3}}{Q_{2}}\right)` | 0.55 | +-------------------------+---------+-----------------------------------------------+-------+ The table below shows the *ξ\ 32* values according to the formulas above for various values of the discharge and area ratios. .. _table_t32_combining_flow: .. table:: ξ\ :sub:`32` values for combining flow. +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | | A\ :sub:`3`/A\ :sub:`2` | | | | | | | | | | +=========================+=====+=========================+=======+=======+=======+=======+=======+=======+=======+=======+=======+ | | | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | Q\ :sub:`3`/Q\ :sub:`2` | 0.0 | ‑1.00 | ‑1.00 | ‑1.00 | ‑0.90 | ‑0.90 | ‑0.90 | ‑0.90 | ‑0.90 | ‑0.90 | ‑0.90 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.1 | 0.38 | ‑0.37 | ‑0.51 | ‑0.45 | ‑0.47 | ‑0.48 | ‑0.49 | ‑0.49 | ‑0.49 | ‑0.49 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.2 | 3.72 | 0.72 | 0.16 | ‑0.02 | ‑0.09 | ‑0.12 | ‑0.14 | ‑0.16 | ‑0.17 | ‑0.17 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.3 | 9.02 | 2.27 | 1.02 | 0.37 | 0.24 | 0.17 | 0.13 | 0.10 | 0.08 | 0.07 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.4 | 16.28 | 4.28 | 2.06 | 0.69 | 0.50 | 0.39 | 0.33 | 0.29 | 0.26 | 0.24 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.5 | 25.50 | 6.75 | 3.28 | 1.13 | 0.83 | 0.66 | 0.56 | 0.49 | 0.44 | 0.41 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.6 | 36.68 | 9.68 | 4.68 | 1.61 | 1.17 | 0.92 | 0.78 | 0.68 | 0.62 | 0.57 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.7 | 49.82 | 13.07 | 6.26 | 2.14 | 1.53 | 1.20 | 1.00 | 0.87 | 0.78 | 0.72 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.8 | 64.92 | 16.92 | 8.03 | 2.71 | 1.91 | 1.48 | 1.22 | 1.06 | 0.94 | 0.86 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 0.9 | 81.98 | 21.23 | 9.98 | 3.32 | 2.32 | 1.78 | 1.45 | 1.24 | 1.09 | 0.98 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ | | 1.0 | 101.00 | 26.00 | 12.11 | 3.99 | 2.75 | 2.08 | 1.67 | 1.41 | 1.23 | 1.10 | +-------------------------+-----+-------------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+ **Straight flow dividing** For the straight leg the resistance coefficient is given using the following formula: .. math:: \xi_{12}=\tau_{12} \frac{Q_{3}}{Q_{1}} where *τ*\ :sub:`12` depends on the area and discharge ratios as given in :numref:`table_t12_values_dividing_flow`. .. _table_t12_values_dividing_flow: .. table:: Values of :math:`\tau_{12}` +-------------------------+-------------------------------+-----------------------------------+-----------------------------------+ | A\ :sub:`3`/A\ :sub:`1` | ≤ 0.4 | 0.4 | | +-------------------------+-------------------------------+-----------------------------------+-----------------------------------+ | Q\ :sub:`3`/Q\ :sub:`1` | 0-1.0 | ≤ 0.5 | >0.5 | +-------------------------+-------------------------------+-----------------------------------+-----------------------------------+ | τ\ :sub:`12` | 0.4 (Q\ :sub:`3`/Q\ :sub:`1`) | 0.2(2Q\ :sub:`3`/Q\ :sub:`1` – 1) | 0.3(2Q\ :sub:`3`/Q\ :sub:`1` – 1) | +-------------------------+-------------------------------+-----------------------------------+-----------------------------------+ :numref:`table_xi12_values_dividing_flow` shows the *ξ\ 12* values according to the formula above for various values of discharge ratios. .. _table_xi12_values_dividing_flow: .. table:: Values of :math:`\xi_{12}` for dividing flow +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | | A\ :sub:`3`/A\ :sub:`1` | | | | | | | | | | +=========================+=====+=========================+======+======+======+=======+=======+=======+=======+=======+=======+ | | | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | Q\ :sub:`3`/Q\ :sub:`1` | 0.0 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.1 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.2 | 0.02 | 0.02 | 0.02 | 0.02 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.3 | 0.04 | 0.04 | 0.04 | 0.04 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.4 | 0.06 | 0.06 | 0.06 | 0.06 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.5 | 0.10 | 0.10 | 0.10 | 0.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.6 | 0.14 | 0.14 | 0.14 | 0.14 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.7 | 0.20 | 0.20 | 0.20 | 0.20 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.8 | 0.26 | 0.26 | 0.26 | 0.26 | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 0.9 | 0.32 | 0.32 | 0.32 | 0.32 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ | | 1.0 | 0.40 | 0.40 | 0.40 | 0.40 | 0.30 | 0.30 | 0.30 | 0.30 | 0.30 | 0.30 | +-------------------------+-----+-------------------------+------+------+------+-------+-------+-------+-------+-------+-------+ The resistance coefficient *ξ\ 13*\ for the branch is calculated using the following formula: For A\ :sub:`3`/A\ :sub:`1` ≤ 2/3: .. math:: \xi_{13}=A^{\prime}\left[1+\left(\frac{Q_{3}}{Q_{1}} \frac{A_{1}}{A_{3}}\right)^{2}\right] For A\ :sub:`3`/A\ :sub:`1` = 1 (up to velocity ratio to *v\ 3/v\ 1*\ ≈2.0): .. math:: \xi_{13}=A^{\prime}\left[1+0.3\left(\frac{Q_{3}}{Q_{1}} \frac{A_{1}}{A_{3}}\right)^{2}\right] The values of A’ are given in :numref:`table_a_dividing_flow`. .. _table_a_dividing_flow: .. table:: Values for :math:`A'` dividing flow. +-------------------------+-----------------------+--------+------------------------+-------+ | A\ :sub:`3`/A\ :sub:`1` | ≤ 0.35 | > 0.35 | | | +-------------------------+-----------------------+--------+------------------------+-------+ | Q\ :sub:`3`/Q\ :sub:`1` | ≤ 0.4 | > 0.4 | ≤ 0.6 | > 0.6 | +-------------------------+-----------------------+--------+------------------------+-------+ | A’ | 1.1 – 0.7 *Q\ 3/Q\ 1* | 0.85 | 1.0 – 0.65 *Q\ 3/Q\ 1* | 0.6 | +-------------------------+-----------------------+--------+------------------------+-------+ :numref:`table_xi13_dividing_flow` shows the *ξ\ 13* values according to the formulas above for various values of discharge and area ratio. .. _table_xi13_dividing_flow: .. table:: Values for :math:`\xi_{13}` +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | | A\ :sub:`3`/A\ :sub:`1` | | | | | | | | | | +=========================+=====+=========================+======+======+======+======+======+======+======+======+======+ | | | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | Q\ :sub:`3`/Q\ :sub:`1` | 0 | 1.10 | 1.10 | 1.10 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.1 | 1.34 | 1.11 | 1.06 | 0.95 | 0.95 | 0.94 | 0.94 | 0.94 | 0.94 | 0.94 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.2 | 2.11 | 1.25 | 1.09 | 0.94 | 0.91 | 0.90 | 0.89 | 0.89 | 0.88 | 0.88 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.3 | 3.29 | 1.49 | 1.16 | 0.94 | 0.89 | 0.87 | 0.85 | 0.84 | 0.83 | 0.83 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.4 | 4.76 | 1.80 | 1.26 | 0.96 | 0.88 | 0.84 | 0.81 | 0.80 | 0.78 | 0.78 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.5 | 7.23 | 2.44 | 1.56 | 0.99 | 0.88 | 0.82 | 0.78 | 0.75 | 0.74 | 0.73 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.6 | 10.03 | 3.15 | 1.87 | 1.02 | 0.87 | 0.79 | 0.74 | 0.71 | 0.69 | 0.68 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.7 | 13.35 | 3.97 | 2.24 | 1.15 | 0.95 | 0.85 | 0.78 | 0.74 | 0.71 | 0.69 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.8 | 17.17 | 4.93 | 2.66 | 1.32 | 1.06 | 0.92 | 0.84 | 0.78 | 0.74 | 0.72 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.9 | 21.51 | 6.01 | 3.14 | 1.51 | 1.18 | 1.00 | 0.90 | 0.83 | 0.78 | 0.75 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 1 | 26.35 | 7.23 | 3.68 | 1.73 | 1.32 | 1.10 | 0.97 | 0.88 | 0.82 | 0.78 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ **Branch flow combining** WANDA calculates the head loss coefficients for each leg (*ξ\ 31* and *ξ\ 32* ) from the following formula: .. math:: \xi_{31}=A\left\{1+\left(\frac{A_{3}}{A_{1}}\right)^{2}+3\left(\frac{A_{3}}{A_{1}}\right)^{2}\left[\left(\frac{Q_{1}}{Q_{3}}\right)^{2}-\left(\frac{Q_{1}}{Q_{3}}\right)\right]\right\} In which *A* is the same factor as used in the T-junction straight combining (table 3). :numref:`table_xi13_x32_values` shows the *ξ\ 31* values according to the formulas above for various values of discharge and area ratio (identical for *ξ\ 32* in leg 2). .. _table_xi13_x32_values: .. table:: Values for :math:`\xi_{13}` and :math:`\xi_{32}`. +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | | A\ :sub:`1`/A\ :sub:`3` | | | | | | | | | | +=========================+=====+=========================+=======+=======+======+======+======+======+======+======+======+ | | | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | Q\ :sub:`1`/Q\ :sub:`3` | 0.0 | 101.00 | 26.00 | 12.11 | 6.52 | 4.50 | 3.40 | 2.74 | 2.31 | 2.01 | 1.80 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.1 | 74.00 | 19.25 | 9.11 | 4.51 | 3.18 | 2.45 | 2.02 | 1.73 | 1.54 | 1.40 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.2 | 53.00 | 14.00 | 6.78 | 3.06 | 2.22 | 1.76 | 1.48 | 1.30 | 1.18 | 1.09 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.3 | 38.00 | 10.25 | 5.11 | 2.09 | 1.56 | 1.28 | 1.11 | 0.99 | 0.92 | 0.86 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.4 | 29.00 | 8.00 | 4.11 | 1.48 | 1.14 | 0.96 | 0.85 | 0.78 | 0.73 | 0.69 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.5 | 26.00 | 7.25 | 3.78 | 1.41 | 1.10 | 0.93 | 0.83 | 0.76 | 0.72 | 0.69 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.6 | 29.00 | 8.00 | 4.11 | 1.51 | 1.17 | 0.98 | 0.86 | 0.79 | 0.74 | 0.70 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.7 | 38.00 | 10.25 | 5.11 | 1.82 | 1.36 | 1.12 | 0.97 | 0.87 | 0.80 | 0.75 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.8 | 53.00 | 14.00 | 6.78 | 2.34 | 1.69 | 1.34 | 1.13 | 1.00 | 0.90 | 0.84 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 0.9 | 74.00 | 19.25 | 9.11 | 3.06 | 2.16 | 1.67 | 1.37 | 1.18 | 1.05 | 0.95 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ | | 1.0 | 101.00 | 26.00 | 12.11 | 3.99 | 2.75 | 2.08 | 1.67 | 1.41 | 1.23 | 1.10 | +-------------------------+-----+-------------------------+-------+-------+------+------+------+------+------+------+------+ **Branch flow dividing** WANDA calculates the head loss coefficients for each leg (*ξ\ 31* and *ξ\ 32* ) from the following formula: .. math:: \xi_{31}=1+k\left(\frac{Q_{1}}{Q_{3}} \frac{A_{3}}{A_{1}}\right)^{2} In which *k* is 0.3 valid for welded tees. :numref:`table_xi32_x32_dividing_branch` shows the *ξ\ 31* values according to the formulas above for various values of discharge and area ratio (identical for *ξ\ 32* in leg 2). .. _table_xi32_x32_dividing_branch: .. table:: ξ\ :sub:`31` and ξ\ :sub:`32` values for dividing branch flow +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | | A\ :sub:`1`/A\ :sub:`3` | | | | | | | | | | +=========================+=====+=========================+======+======+======+======+======+======+======+======+======+ | | | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | Q\ :sub:`1`/Q\ :sub:`3` | 0.0 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.1 | 1.30 | 1.08 | 1.03 | 1.02 | 1.01 | 1.01 | 1.01 | 1.00 | 1.00 | 1.00 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.2 | 2.20 | 1.30 | 1.13 | 1.08 | 1.05 | 1.03 | 1.02 | 1.02 | 1.01 | 1.01 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.3 | 3.70 | 1.68 | 1.30 | 1.17 | 1.11 | 1.08 | 1.06 | 1.04 | 1.03 | 1.03 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.4 | 5.80 | 2.20 | 1.53 | 1.30 | 1.19 | 1.13 | 1.10 | 1.08 | 1.06 | 1.05 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.5 | 8.50 | 2.88 | 1.83 | 1.47 | 1.30 | 1.21 | 1.15 | 1.12 | 1.09 | 1.08 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.6 | 11.80 | 3.70 | 2.20 | 1.68 | 1.43 | 1.30 | 1.22 | 1.17 | 1.13 | 1.11 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.7 | 15.70 | 4.68 | 2.63 | 1.92 | 1.59 | 1.41 | 1.30 | 1.23 | 1.18 | 1.15 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.8 | 20.20 | 5.80 | 3.13 | 2.20 | 1.77 | 1.53 | 1.39 | 1.30 | 1.24 | 1.19 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 0.9 | 25.30 | 7.08 | 3.70 | 2.52 | 1.97 | 1.67 | 1.50 | 1.38 | 1.30 | 1.24 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ | | 1.0 | 31.00 | 8.50 | 4.33 | 2.88 | 2.20 | 1.83 | 1.61 | 1.47 | 1.37 | 1.30 | +-------------------------+-----+-------------------------+------+------+------+------+------+------+------+------+------+ T-Junction properties ^^^^^^^^^^^^^^^^^^^^^ The input properties of both T-junction are not exactly the same. The differences are located in the sequence of the Xi tables and the description labels for input tables and output properties. The input and output properties for both types are specified below. Properties T-junction straight """""""""""""""""""""""""""""" Input properties T-junction straight +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Description | input | unit | range | default | Remarks | +==========================================+================+======+=======+=========+=================================+ | Diameter straight leg | Real | [mm] | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Diameter branch leg | Real | [mm] | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi method | Formula | | | Formula | | | | | | | | | | | Tables | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi tables valid for | Combining | | | | Only when Xi method = Tables | | | | | | | | | | Dividing | | | | | | | | | | | | | | All | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi straight flow combining, leg straight | Table | | | | if Xi tables = Combining or All | | | | | | | | | | (ξ\ :sub:`12`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi straight flow combining, leg branch | Table | | | | if Xi tables = Combining or All | | | | | | | | | | (ξ\ :sub:`32`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi straight flow dividing, | Table | | | | if Xi tables = Dividing or All | | | | | | | | | leg straight | (ξ\ :sub:`12`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi straight flow dividing, | Table | | | | if Xi tables = Dividing or All | | | | | | | | | leg branch | (ξ\ :sub:`13`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi branch flow combining, | Table | | | | if Xi tables = All | | | | | | | | | leg 1 | (ξ\ :sub:`13`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi branch flow combining, | Table | | | | if Xi tables = All | | | | | | | | | leg 2 | (ξ\ :sub:`23`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi branch flow dividing, | Table | | | | if Xi tables = All | | | | | | | | | leg 1 | (ξ\ :sub:`31`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi branch flow dividing, | Table | | | | if Xi tables = All | | | | | | | | | leg 2 | (ξ\ :sub:`32`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ With the tables for the Xi-values it is possible to choose user-defined loss coefficients, for instance for a certain kind of T-junction with rounded angles. The user has to specify a set of *loss coefficient values* related to the *discharge ratio* between the side and combined branch. If the user does not specify both combining and dividing Xi-tables, WANDA uses the formulas according to the Idelchik handbook for the unspecified flow regimes. Example of Xi table; straight flow combining, leg branch (ξ32) valid for Area ratio = 0.5 =================== ====== Discharge ratio [-] Xi [-] =================== ====== 0.0 0.90 0.2 0.12 0.4 0.39 0.6 0.92 0.8 1.48 1.0 2.08 =================== ====== Component specific output T-junction straight +-------------------------------+-----------------------------------------------------------------------------------------------------------+ | Loss coefficient straight [-] | The ξ\ :sub:`12` for positive flow and ξ\ :sub:`21` for negative flow | | | | | | In case of branch flow regime this is the ξ\ :sub:`31` coefficient | +===============================+===========================================================================================================+ | Loss coefficient branch [-] | The ξ\ :sub:`13` for positive flow and ξ\ :sub:`31` for negative flow | | | | | | In case of branch flow regime this is the ξ\ :sub:`32` coefficient | +-------------------------------+-----------------------------------------------------------------------------------------------------------+ | Head loss straight [m] | The ΔH\ :sub:`12` for positive flow and ΔH\ :sub:`21` for negative flow | | | | | | In case of branch flow regime this is the ΔH\ :sub:`13` for combining and ΔH\ :sub:`31` for dividing flow | +-------------------------------+-----------------------------------------------------------------------------------------------------------+ | Head loss branch [m] | The ΔH\ :sub:`13` for positive flow and ΔH\ :sub:`31` for negative flow | | | | | | In case of branch flow regime this is the ΔH\ :sub:`23` for combining and ΔH\ :sub:`32` for dividing flow | +-------------------------------+-----------------------------------------------------------------------------------------------------------+ Component messages T-junction straight ================ ==== ====================== ====================== Message Type Explanation ================ ==== ====================== ====================== Combining Info Positive straight flow Negative straight flow \ |image84| |image85| Dividing Info Positive straight flow Negative straight flow \ |image86| |image87| Branch combining Info |image88| Branch dividing info |image89| ================ ==== ====================== ====================== With the tables for the Xi-values it is possible to choose user-defined loss coefficients, for instance for a certain kind of T-junction with rounded angles. The user has to specify a set of *loss coefficients values* related to the *discharge ratio* between the side and combined branch. If the user does not specify both combining and dividing Xi-tables, WANDA uses the formulas according to the Idelchik handbook for the unspecified flow regimes. **NOTE:** The numbering of the branches in the literature is not always consistent with the WANDA definition as given under ‘Mathematical Model’ above. The Miller Handbook for instance indicates the side branch with *1*, the straight branch with *2* and the combined one as *3*. Furthermore please note that there is a discontinuity when the flow in one of the legs goes through zero. This might lead to unstable behaviour. Properties T-junction branch """""""""""""""""""""""""""" Input properties T-junction branch +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Description | input | unit | range | default | Remarks | +==========================================+================+======+=======+=========+=================================+ | Diameter straight leg | Real | [mm] | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Diameter branch leg | Real | [mm] | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi method | Formula | | | Formula | | | | | | | | | | | Tables | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi tables valid for | Combining | | | | Only when Xi method = Tables | | | | | | | | | | Dividing | | | | | | | | | | | | | | All | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi branch flow combining, | Table | | | | if Xi tables = Combining or All | | | | | | | | | leg 1 | (ξ\ :sub:`13`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi branch flow combining, | Table | | | | if Xi tables = Combining or All | | | | | | | | | leg 2 | (ξ\ :sub:`23`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi branch flow dividing, | Table | | | | if Xi tables = Dividing or All | | | | | | | | | leg 1 | (ξ\ :sub:`31`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi branch flow dividing, | Table | | | | if Xi tables = Dividing or All | | | | | | | | | leg 2 | (ξ\ :sub:`32`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi straight flow combining, leg straight | Table | | | | if Xi tables = All | | | | | | | | | | (ξ\ :sub:`12`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi straight flow combining, leg branch | Table | | | | if Xi tables = All | | | | | | | | | | (ξ\ :sub:`32`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi straight flow dividing, | Table | | | | if Xi tables = All | | | | | | | | | leg straight | (ξ\ :sub:`12`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ | Xi straight flow dividing, | Table | | | | if Xi tables = All | | | | | | | | | leg branch | (ξ\ :sub:`13`) | | | | | +------------------------------------------+----------------+------+-------+---------+---------------------------------+ Component specific output T-junction branch +------------------------+--------------------------------------------------------------------------------------+ | Loss coefficient 1 [-] | The ξ\ :sub:`31` for dividing flow and ξ\ :sub:`13` for combining flow | | | | | | In case of straight flow regime this is the ξ\ :sub:`12` or ξ\ :sub:`21` coefficient | +========================+======================================================================================+ | Loss coefficient 2 [-] | The ξ\ :sub:`32` for dividing flow and ξ\ :sub:`23` for combining flow | | | | | | In case of straight flow regime this is the ξ\ :sub:`32` or ξ\ :sub:`23` coefficient | +------------------------+--------------------------------------------------------------------------------------+ | Head loss 1 [m] | The ΔH\ :sub:`31` for dividing flow and ΔH\ :sub:`13` for combining flow | | | | | | In case of straight flow regime this is the ΔH\ :sub:`12` or ΔH\ :sub:`21` | +------------------------+--------------------------------------------------------------------------------------+ | Head loss 2 [m] | The ΔH\ :sub:`32` for dividing flow and ΔH\ :sub:`23` for combining flow | | | | | | In case of straight flow regime this is the ΔH\ :sub:`32` or ΔH\ :sub:`23` | +------------------------+--------------------------------------------------------------------------------------+ Component messages T-junction branch ================== ==== ====================== ====================== Message Type Explanation ================== ==== ====================== ====================== Branch combining Info |image90| Branch dividing info |image91| Combining straight Info Positive straight flow Negative straight flow \ |image92| |image93| Dividing straight Info Positive straight flow Negative straight flow \ |image94| |image95| ================== ==== ====================== ====================== With the tables for the Xi-values it is possible to choose user-defined loss coefficients, for instance for a certain kind of T-junction with rounded angles. The user has to specify a set of *loss coefficients values* related to the *discharge ratio* between the side and combined branch. If the user does not specify both combining and dividing Xi-tables, WANDA uses the formulas according to the Idelchik handbook for the unspecified flow regimes. **NOTE:** The numbering of the branches in the literature is not always consistent with the WANDA definition as given under ‘Mathematical Model’ above. The Miller Handbook for instance indicates the side branch with *1*, the straight branch with *2* and the combined one as *3*. Furthermore please note that there is a discontinuity when the flow in one of the legs goes through zero. This might lead to unstable behaviour. .. include:: substitutions_liquid.rst