Numerical Calculation of the Tee Local Resistance Coefficient

The local head loss of tee could be calculated with the determination of local resistance coefficient by CFD simulation and test. Based on the mesh-independent feature identified, the flow field inner tee was numerically simulated by the standard k ε turbulent model and SIMPLEC algorithms, which has revealed the mainstream was obliged to turn to the opposite side of tee junction, and a rise in pressure drop between upstream and downstream was caused as a result. Furthermore, the frictional resistance coefficient was calculated for eliminating the frictional head loss of model, which decreased from 0.0207 to 0.0133 when the inlet velocity increased from 1 m/s to 12 m/s. Additionally, the local resistance coefficients of tee at flow conditions were attained, and the quadratic polynomial between the local resistance coefficient and flux ratio was presented due to the influence of branch on mainstream. Through the test, the simulation result has been compared and the effectiveness of simulation has been verified.


INTRODUCTION
As a kind of pipe fittings, tee is widely used in the petrochemical industry.When fluid flow through the tee, the fluid velocity in the tee is redistributed for fluid particles turbulently fluctuate, and local energy loss is caused during energy transfer.Determining the local head loss is the key of hydraulic calculation of pipeline.In general, the local resistance coefficient of the tee is regarded as a constant for convenience, and this will cause larger calculation error because local resistance coefficient changes at different flow patterns for the same tee.To precisely compute the local head loss, the coefficient is needed to be studied.Wu Shengmin et al. [1] obtained the local resistance coefficient of DN50 standard tee at different velocity through experiments.Jorgen et al. [2] found that the local resistance coefficient was related with the flow pattern and the flux ratio through the test and simulation of tee.Additionally, other previous works [3][4][5] also drew similar conclusion, and the methods of simulation and experiment were used to study the flow characteristic of Y-type tube [6,7].
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In this paper, the internal flow field of tee is simulated with the Computational Fluid Dynamics method (CFD) to investigate the cause of local head loss for tee, and the pressure drops at different velocities are gotten to calculate the local resistance coefficient for tee.

Fluid Flow Model
To simulate the flow field, the conservation equations for mass and momentum in combination with transport equations for turbulence model are computed.
Assuming that the pipe flow is three-dimensional, incompressible flow, the equation for conservation of mass can be written as (1) where ρ is the fluid density and uj is the average velocity component.

Momentum equation is written as
where p is the static pressure and the stress tensor is expressed as where is the Kronecker delta and the turbulent viscosity, is eddy diffusivity expressed in the high-Reynolds number form as (4) where k and ε are respectively the kinetic energy and dissipation rate of turbulence, they can be written as (5) (6) As above equations are not closed, standard k -ε turbulent model [8] is applied for turbulence modeling.The k and ε are obtained from the following transport equations (7) where Pk is the generation of turbulence kinetic energy defined as (8) The model constants , , , , have the values: =0.09, =1.44, =1.92, =1.0, =1.3.

CFD Numerical Method
Based on the finite volume method, the flow equations are spatially discretized with the convection term of second order upstream scheme, and other terms are discretized with the central difference scheme.To obtain a converged solution, the SIMPLEC algorithms is used to separately solve the momentum equations and the continuity equation, additionally, at the end of each solver iteration, the residual for each of the conserved variables is computed and the convergence is judged if the maximal residual decreases to 10-5.

CFD Mesh Model
The structure schematic drawing of tee is shown as Fig. (1a).The inner diameters D1 of the main inlet pipe and outflow pipe are the same as 101.6 mm, and the inner diameter D2 of branch pipe is 50.8 mm.As the hypothesis of full developed flow in the tee, the length of each straight pipe should be more than 3 times of the diameter, and L=1800mm, L1=720mm, L2=500mm, L3=360mm.The fluid is liquid with the density of 1000 kg/m3, and the viscosity is 0.001Pa•s.To establish the CFD mesh model, the flow zone is divided into several subdomain for meshing.The region around tee junctions is meshed with tetrahedrons element for its good adaptability to the unstructured model, and hexahedron elements are generated in other zones to reduce the number of elements.As the energy loss is mostly caused in junctions and the simulation should be independent of mesh, the tetrahedrons element is much finer and the number of element is determined with checking computations.The mesh model shown in  with v1, v2, and one pressure outlet at outlet with a specified static pressure p0.The wall is static and a no-slip condition is specified.
Considering that the total head loss from the inlet 1 to outlet includes frictional head loss and local head loss of tee, the straight pipe model is built as Fig. (2) to compare energy loss with the tee model.

SIMULATION RESULT AND ANALYSIS
When the inlet velocity v1 is equal to 2m/s, partial streamlines are shown in Fig. (3).It shows that the branch with the inlet velocity of 0.1 m/s is on the upper part of flow channel due to the jacking role of the mainstream, and it flows to the pipe center gradually during the mixing of two streams.However, the branch with the inlet velocity of 1m/s forces the mainstream to significantly turn to the opposite side of junction after it flows into the junction, and a vortex is formed at the downstream of junction as a result Fig.When the velocity of branch is higher, junction of tee has a larger effect on the surrounding pressure, which is characterized by the increasing of pressure in the upstream and decreasing of pressure in the downstream of junction.Additionally, a negative pressure zone usually exists at the same side of junction.Fig. (6) shows the cross-sectional area-weighted average pressures along the main stream.There is a maximal average pressure value in 0.9m where the center axis of the branch locates in, and the average pressure increase slightly at the range of about 1 time the pipe diameter in the upstream.However, the pressure in the downstream decreases significantly, and it is affected by the junction more greatly as the branch velocity increases.With the branch velocity of 1.0 m/s, downstream becomes stable when the distance from the axis of branch is larger than 3 times the pipe diameter.

HEAD LOSS OF TEE
When the average pressures at upstream and downstream of junction is determined, the local head loss hj can be calculated with the pressure drop,  where λ is frictional resistance coefficient, L and d are length and diameter of straight pipe respectively.Δp is the pressure drop between the inlet and outlet, and λ is attained as followed (13) Fig. (7) shows the frictional resistance coefficients calculated at different velocities.When the inlet velocity increases from 1 m/s to 12 m/s, the frictional resistance coefficient λ declines from 0.0207 to 0.0133, following the power function relationship, which can be written as (14) Through Eq. ( 14), the frictional resistance coefficient for different segment is determined at different flow conditions, and the simulation result is shown in Table 1.It is noted that Re number is calculated according to v3 and D1, and the pressure p3 of outlet is 0.
The pressure drop p1-p3 is also tested for the tee shown in Fig. (8), of which the size is the same as the CFD model, and the data is listed in Table 2.It shows that the simulation is in good agreement with the test as the maximum relative deviation is less than 0.087.This proves that the CFD numerical method adopted is effective.

CONCLUSION
In summary, we have simulated the flow field inner tee for revealing the cause of local head loss.The flow characteristic of tee junction is that the branch forces the mainstream to turn to the opposite side of the junction, and the pressure drop between upstream and downstream increases.As the pressure drop includes the frictional head loss and local head loss, the frictional resistance coefficient is determined through the simulation of straight pipe, and the coefficient decreases from 0.0207 to 0.0133 when the inlet velocity increases from 1 m/s to 12 m/s, which follows the power function relationship.
By eliminating the frictional head loss, the local resistance coefficient of tee is attained at different flow conditions, and a relationship of quadratic polynomial between the coefficient and the flux ratio of branch to mainstream is obtained, which objectively reflected the influence of branch on mainstream.The present work also compares the simulation and test, and there is a good agreement between them with the maximum relative deviation of 0.087, which proves that the simulation is effective.
Fig. (1b) includes 25763 tetrahedrons elements and 56592 hexahedron elements.The flow boundary conditions include velocity inlet at inlet 1 and inlet 2, where the velocity vectors are defined Fig. (1).Tee model.

Fig. ( 5 )
Fig.(5) shows the static pressure filed on the symmetry.When the velocity of branch is higher, junction of tee has a larger effect on the surrounding pressure, which is characterized by the increasing of pressure in the upstream and decreasing of pressure in the downstream of junction.Additionally, a negative pressure zone usually exists at the same side of junction.
Fig. (4).Velocity filed on the symmetry plane (m/s).(a) v 1 =2m/s, v 2 =0.1m/s resistance coefficient ςj is calculated(10)    where p3 and p1 are fluid pressures on outlet and inlet 1. ρ denotes the fluid density, and g is gravitational acceleration.hf1 and hf3 are upstream frictional head loss and downstream frictional head loss, respectively, they can be written as(11)    where v3 is the outlet velocity, λ1 and λ1 are frictional resistance coefficients based on velocity v1 and v1, respectively, which are determined by the simulation of the straight pipe flow.

Fig. ( 6 )
Fig. (6).Cross-sectional area-weighted average pressures along the main steam.The frictional head loss hf is calculated with the pressure drop Δp of straight pipe shown in Fig. (2),
Fig. (10) shows the relationship curve of local resistance coefficient and flux ratio of branch to mainstream.The coefficient increases monotonically with increasing ratio, and they nearly follows the quadratic polynomial, which can be written as (15) 0205v 3 −0.178R 2 = 0.9989 m/s) λ ζ j = −4.2527β 2 + 3.521β + 0.0087 R 2 = 0.9999