Study of the Possibility of Achieving the Same Per-Port Outflow in a Dividing Manifold

There have been several attempts to optimise fluid flow manifolds; these, however, have shown are limited and further investigation into the efficiency of these systems is needed. This work focuses on improving the distribution manifolds efficacy in outflow division, i.e. attaining the same flow rate per each exit port of the manifold. Water has been selected to be the working fluid. A numerical investigation utilising CFD (by ANSYS Fluent R16.2) analysis into two-dimensional, incompressible, and turbulent flow has been carried out to resolve the flow manifold problem using two turbulence modelling, Standard k-ε and RNG k-ε, approaches. Four values of flow rate have been considered, which are specified by the Reynolds numbers 101×10, 202×10, 303×10, and 404×10. These values correspond to the fluid inlet velocities 0.5, 1.0, 1.5, and 2.0 m/s, respectively. The manifold configuration is defined by the given area ratio (total cross-sectional area for laterals /header cross-sectional area). Three values of area ratio are considered; these are 0.703125, 0.84375, and 0.984375. The results indicate that the flow uniformity has a reverse proportional relationship with the fluid flow rate and area ratio for all manifold arrangements. However, there is no significant effect of the flow rate increase on flow mal-distribution. Also, the use of RNG k-ε model has shown higher values of the non-uniformity coefficient than those obtained by the Standard k-ε model. The outcomes of this analysis have been compared with experimental data and a good agreement among them has been found.


Sm
Source term: the mass added to the continuous phase from the dispersed second phase and any user-defined sources [10]. kg/(m 3  Generation of turbulence kinetic energy due to the mean velocity gradients [10]. W/m 3 Gb Generation of turbulence kinetic energy due to buoyancy [10]. W/m 3

YM
Contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate [10]. W/m 3

Introduction
Among all fluid-flow devices, manifolds are the most commonly encountered in practice, aside from valves, fittings, and pipes. Manifolds occupy major importance in numerous engineering applications including old conventional applications and modern sophisticated equipment. Even in very recent applications, the design of the participating manifolds has been treated in a casual manner. Furthermore, even when more attention has been given to manifold design, the numerical simulations have been incomplete owing to improper application of the boundary conditions. There is a clear need for an in-depth evaluation of the present status of manifold design and a determined research program to respond to the outcomes of such evaluation. A manifold is a chamber consisting of one fluid inlet and numerous fluid exits or, similarly, a chamber with many fluid inlets and a single fluid outlet. The former type may be designated as a distribution manifold while the latter is termed a collection manifold. Among all of the major design problems of fluid flow, the manifold problem still remains a primary one requiring a systematic solution method. The widespread applications involving manifolds have motivated a variety of solution methods, but these have generally been specific to the individual manifold belonging to that application. The history of solution methods that have been employed for the manifold problem is closely related to the availability and power of computational tools. Up until about 1980, one-dimensional models and corresponding algebraic solution methods were the standard. In order to enable such methods to be used as a design tool, it was necessary to make rather sweeping assumptions about the kind of the fluid flow style.
Many researchers have endeavored to attain an equal outflow distribution of fluid in dividing manifolds. Therefore, work of some of those will be reviewed. The performance investigation of flow distribution systems was performed analytically by Bajura [1] for both intake and exhaust manifolds. He employed a theoretical model for formulating dimensionless parameters describing the manifolds performance. The impacts of header configuration and Reynolds number variation on the flow allocation in a flow manifold were numerically studied by Kim et al. [2]. The manifold usage was for a liquid cooling unit utilized in an electronic pack. Three different designs of header shape were studied and the authors concluded that the triangular shape is the best one for flow division. Lu Hua [3] predicted the flow distribution in manifolds via computational modeling. He utilized many numerical methods to identify the difficulties related to the design of flow spreaders that are widely encountered in industrial applications, e.g. the manifold used in paper manufacturing system (Fig. 1). Weitbrecht et al. [4] accomplished an analytical and empirical investigation for the water flow in a dividing and collecting manifold used for solar heating (Fig. 2). The flow was analyzed in laminar conditions and the influence of frictional pressure and energy losses on the flow allocation were the main objective of the study. A numerical emulation of flow in manifolds is also submitted by Tong et al. [5] with the aid of the CFD software CFX-9.0. The work was achieved assuming a steady tow-dimensional and laminar flow. They focused their case studies on attaining the same flow rate division among the lateral tubes connecting the distribution and collecting manifolds. Andrew and Sparrow [7] studied numerically the impact of the outlet ports shape on the outflow rate equality in a dividing manifold with a circular cross-section header. The solving model selected for this aim was the realizable k-ε. They considered three geometries of the manifold exits and declared that the continual solo-slot shape is the best one for the flow uniformity. Andrew and Sparrow [8] also investigated the possibility of attainment the ideal performance of distribution manifold by two geometric methodologies. These methodologies include: (a) varying the area ratio and (b) raising the flow resistance in the lateral tubes via increasing their length-diameter ratio which eventually yields to more pressure drop (Figs. 3a & 3b). Tong et al. [9] carried out a numerical study for identifying the strategies of optimizing the manifold design to enhance its efficacy in outflow distribution. The difficulty encountered in the design was ascribed to the variations happening in fluid pressure along the header due to  Also, the distinction here is in the mechanism of varying the area ratio via changing the number of laterals rather than modifying their diameter or the header dimensions. Moreover, high turbulent flow limits have been selected (extended to more than 4.0×10 5 Reynolds number) in order to test and disclose the manifold outflow uniformity in high turbulence conditions.

Fluid Flow Governing Equations
The physical phenomenon of fluid flow in manifolds is governed by differential equations for both continuity and momentum and they will be indicated in the next subsidiary sections.

The Mass Conservation or Continuity Equation
The mass conservation equation of a flowing fluid is given as indicated below [10]: Equation (1) is the general equation and is applicable for incompressible and compressible flows. For special cases, i.e. problems with steady two-dimensional and incompressible flow (since the current flow problem is two-dimensional), the Eq. (1) will be reduced to a simplified form and given as follows [7],

Momentum Conservation Equations
In an inertial or non-accelerating reference frame, the conservation of momentum is given as following [10].
For steady incompressible and turbulent flows, the equation above is formulated as the Reynolds-averaged Navier-Stokes equations (RANS) and dismantled in general into three separate equations. The present study is two-dimensional, i.e. only two components of the velocity vector are considered (just x and y). Thus, the general momentum formula (Eq. 3) is simplified to form the components of RANS equations and written in two separate equations in x and y directions as following [7]: The effective viscosity ( eff) is defined as eff = + t. Where is the eddy or turbulent viscosity, it is not a property of fluid. It differs from an application to another and relies on the chosen turbulence model. It is also considered to be isotropic in most turbulence models [7]. In the current problem two models are selected, the standard k-ε and the RNG k-ε.

Transport Equations for the Standard k-ε Model
For this model, equations used to obtain k and ε are illustrated respectively below [11]:

Modelling the Turbulent Viscosity
The turbulent or eddy viscosity ( ) can be determined by incorporating k and ε together in a single formula as follows [10]:

Transport Equations for the RNG k-ε Model
Equations concerning the determination of k and ε are constructed respectively for the RNG model as follows [12,13]: For the high-Reynolds-number limit, both are taken nearly as 1.393 [10].

Modelling the Effective Viscosity
In the high-Reynolds-number limits, the values of the effective viscosity and the turbulent viscosity become approximately identical. Thus, the effective viscosity can be found from the Eq. (8) with Cμ = 0.0845. The Cμ value is derived using RNG theory [10].

RNG k-ε Model Constants
The constants C1ε, C2ε in the Eqs.

Non-Uniformity Flow Coefficient (Φ)
The manifold efficacy in flow distribution is evaluated by means of the dimensionless factors, Φ and . They are defined as follows [14]: The large value of Φ implies more flow mal-distribution, and thus the minimum value of non-uniformity coefficient gives the optimum design configuration for the manifold.
The header-tube area ratio (A.R) has been defined as the ratio of the total cross-sectional area for all exit ports to the header cross-sectional area, it can be determined as follows [14]: Kirkuk University Journal /Scientific Studies (KUJSS)

Grid Creation and Validation
The grid was generated for each manifold arrangement case using quadrilateral cells. An inflation was made for flow domains near wall boundaries to enhance the mesh quality and get solutions in the boundary layer regions. The bias factor of the layers inflation has been selected to be equal 40 for the horizontal walls and 20 for the vertical walls. The mesh generation is indicated in Fig. 5. The mesh quality was validated via knowing some of the main quality measures like warping factor, skewness, and orthogonal quality of the elements (cells) for each case of the manifold arrangement. Table 1 shows grid statistics for all manifold configurations.

The CFD Model and Simulation Approach
The manifold with the configuration indicated by the Fig. 6 has been prepared. Thereafter,

Problem Boundary Conditions
The boundary conditions considered in the simulating program are summarized in Table 2.

Effect of the Outlet Tube Number on the Flow Ratio
The      Figures (i.e. 10, 11, and 12) that the curves of Φ-Re have more linearity for the RNG k-ε model than the curves obtained by the standard k-ε. This is an advantage recorded for the first one, where the linearity feature is an important property for any graph which makes the extrapolation of figures more easily. Therefore, we can conclude that the RNG turbulence model is the best one for evaluating the flow problem presented here although it gives worse results for the flow uniformity.

Effect of the Fluid Inlet Flow Rate on the Outflow Uniformity
Another substantial observation can be found, that is when the fluid accelerates so that the Reynolds number becomes in the range of 3×10 5 to 4×10 5 (i.e. for high Reynolds numbers), the increase of Φ becomes slight in comparison with its value for the Re increase range of 1×10 5 to 2×10 5 . The explanation for this trend of the flow uniformity is related to the high viscous friction associated with the high turbulent flow which results in more increase of flow resistance in the laterals and hence the best chance for uniform flow distribution. This comparison is almost valid with all existing area ratios.

Effect of the Area Ratio on the Fluid Outflow Uniformity
The area ratio has a significant role

Present Findings versus Antecedent Works
Among many studies concerning the flow uniformity in manifolds, presentations of fluid distributers with the current manifold dimensions and flow ranges have not been found so far, therefore, comparisons are introduced with studies closer to the present work which were presented by Wang et al. [14,15]. Their manuscripts include the study of flow uniformity in a distribution manifold used in compact parallel flow heat exchangers. A rectangular cross section shape header with nine parallel tubes was used to construct the manifold with water as working fluid. The investigation has been done experimentally and numerically and the effect of area ratio and the fluid flow rate was included. Low turbulent flow limits were examined (Reynolds number ranges = 2640 up to nearly 11000) due to the small dimensions associated with the manifold. The comparison is achieved according to the dimensional and geometric similarity. The important ratio (area ratio) is approximately identical for the current manifold and that presented by the aforesaid researchers [15], and for average entry velocity of 0.83 m/s the two works can be compared. Fig. 15 indicates the variation of flow ratio with the laterals for fluid entry mean velocity of 0.83 m/s and nearly 0.80 area ratio for the two manifolds, the RNG solution model is considered for the current study.
It is clear from Fig. 15 that the values of flow ratio given by Wang et al. [15] are greater than those of the present study and the curve is less inclined especially when the fluid reaches the third tube, this is due to the small size of the selected manifold which increases the viscous flow resistance and static pressure drop at the laterals and hence more uniform distribution of the fluid through the laterals will occur. Also, the low turbulent limits taken have a substantial role in flow allocation, where at low turbulent flow the higher opportunity of uniform flow division will exist. However, a good conforming is found between the mentioned experimental work and the current numerical results particularly in the behavior of flow ratio curve. The two works exhibit roughly the same curve trend and the difference only in the values of flow ratio which is attributed to assumptions considered in the numerical study for solution facilitation that are not encountered in the empirical investigation. For instance, when the entry mean velocity is 0.20 m/s, the non-uniformity coefficient records 0.025 and 0.0207 for the current work and that presented by Wang et al. [15] respectively; while at a velocity of 0.83 m/s the Φ values are 0.02847 and 0.0332, respectively. This difference is ascribed to the nearly laminar flow at 0.20 m/s inlet velocity for the manifold examined by Wang et al. [15], thus the more uniform distribution of the fluid can occur and lower values of Φ result. But for the present manifold, the flow is more turbulent at the same 0.20 m/s inlet velocity (nearly 40400 Reynolds number) and hence Φ values will be larger. After the fluid velocity reaches 0.3 m/s, the experimental results of Φ have higher values than those obtained by the numerical procedure (current work), this is due to the aforementioned reason which is expounded by few assumptions that are made to get the numerical solution. Anyway, the two results have a good congruence as long as the curves have roughly the same trend.

Conclusions
The objective of this work is to attain equal outflow rate per each exit port of the manifold.
Therefore, the impact of the change of some hydraulic and geometric parameters has been tested to show their effects on the flow misdistribution through the manifold. From the previous discussions to the results in section five, we can deduce that the uniformity of fluid per-port outflow is affected by the inlet flow rate and it indicates a reverse relation, but the variations in fluid flow rate do not lead to a significant increase in the non-uniformity flow coefficient. This is almost observed with both turbulence models and for all existent area ratios. The study also indicates that the values of non-uniformity coefficient are higher with RNG k-ε model than those obtained utilizing the standard k-ε. This is valid with all manifold configurations and all selected flow rates. The area ratio (which identifies the manifold arrangement) has a serious influence on the fluid outflow uniformity, and its increase lowers the probability of obtaining the same per-port outflow. Nevertheless, the increase of area ratio through rising the laterals number may lead to identifying the largest outflow mal-distribution.