The flow in manifolds is extensively encountered in many industrial processes when it is necessary to distribute a large fluid stream into several parallel streams and then to collect them into one discharge stream, such as fuel cells, plate heat exchanger, radial flow reactor, and irrigation. Manifolds can usually be categorized into one of the following types: dividing, combining, Z-type and U-type manifolds (Fig. 1).[1][2][3] A key question is the uniformity of the flow distribution and pressure drop.
Traditionally, most of theoretical models are based on Bernoulli equation after taking the frictional losses into account using a control volume (Fig. 2). The frictional loss is described using the Darcy–Weisbach equation. One obtains a governing equation of dividing flow as follows:
(Eq.1)
where
is the velocity,
is the pressure,
is the density,
is the hydraulic diameter,
is the frictional coefficient,
is the axial coordinate in the manifold,
∆X = L/n. The n is the number of ports and L the length of the manifold (Fig. 2). This is fundamental of manifold and network models. Thus, a T-junction (Fig. 3) can be represented by two Bernoulli equations according to two flow outlets. A flow in manifold can be represented by a channel network model. A multi-scale parallel channel networks is usually described as the lattice network using analogy with the conventional electric circuit methods.[4][5][6] A generalized model of the flow distribution in channel networks of planar fuel cells.[6] Similar to Ohm's law, the pressure drop is assumed to be proportional to the flow rates. The relationship of pressure drop, flow rate and flow resistance is described as Q2 = ∆P/R. f = 64/Re for laminar flow where Re is the Reynolds number. The frictional resistance, using Poiseuille's law. Since they have same diameter and length in Fig. 3, their resistances are same, R2 = R3. Thus the velocities should be equal in two outlets or the flow rates should be equal according to the assumptions. Obviously this disobeys our observations. Our observations show that the greater the velocity (or momentum), the more fluid fraction through the straight direction. Only under very slow laminar flow, Q2 may be equal to Q3.
The question raised from the experiments by McNown[1] and by Acrivos et al.[2] Their experimental results showed a pressure rise after T-junction due to flow branching. This phenomenon was explained by Wang.[7][8][9] Because of inertial effects, the fluid will prefer to the straight direction. Thus the flow rate of the straight pipe is greater than that of the vertical one. Furthermore, because the lower energy fluid in the boundary layer branches through the channels the higher energy fluid in the pipe centre remains in the pipe as shown in Fig. 4.
Thus, mass, momentum and energy conservations must be employed together for description of flow in manifolds.[10][11][12][13][14] Wang[7][8][9] recently carried out a series of studies of flow distribution in manifold systems. He unified main models into one theoretical framework and developed the most generalised model, based on the same control volume in Fig. 2. The governing equations can be obtained for the dividing, combining, U-type and Z-type arrangements.
The Governing equation of the dividing flow:
(Eq.2a)
or to a discrete equation:
(Eq.2b)
In Eq.2, the inertial effects are corrected by a momentum factor, β. Eq.2b is a fundamental equation for most of discrete models. The equation can be solved by recurrence and iteration method for a manifold. It is clear that Eq.2a is limiting case of Eq.2b when ∆X → 0. Eq.2a is simplified to Eq.1 Bernoulli equation without the potential energy term when β=1 whilst Eq.2 is simplified to Kee's model[6] when β=0. Moreover, Eq.2 can be simplified to Acrivos et al.’s model[2] after substituting Blasius’ equation, . Therefore, these main models are just a special case of Eq.2.
Similarly, one can obtain the governing equations of the combining, U-type and Z-type arrangement.
The Governing equation of the combining flow:
(Eq.3a)
or to a discrete equation:
(Eq.3b)
The Governing equation of the U-type flow:
(Eq.4a)
or to a discrete equation:
(Eq.4b)
The Governing equation of the Z-type flow:
(Eq.5a)
or to a discrete equation:
(Eq.5b)
Eq.2 - Eq.5 are second order nonlinear ordinary differential equations for dividing, combining, U-type and Z-type manifolds, respectively. The second term in the left hand represents a frictional contribution known as the frictional term, and the third term does the momentum contribution as the momentum term. Their analytical solutions had been well-known challenges in this field for 50 years until 2008.[7] Wang[7][8][9] elaborated the most complete analytical solutions of Eq.2 - Eq.5. The present models have been extended into more complex configurations, such as single serpentine, multiple serpentine and straight parallel layout configurations, as shown in Fig. 5. Wang[15][16] also established a direct, quantitative and systematic relationship between flow distribution, pressure drop, configurations, structures and flow conditions and developed an effective design procedures, measurements, criteria with characteristic parameters and guidelines on how to ensure uniformity of flow distribution as a powerful design tool.
^ abMcNown, J.S. (1954). "Mechanics of manifold flow". Transactions of the American Society of Civil Engineers. 119 (1): 1103–1142. doi:10.1061/TACEAT.0007058.
^ abcAcrivos, A.; Babcock, B.D.; Pigford, R.L. (1959). "Flow distributions in manifolds". Chemical Engineering Science. 10 (1–2): 112–124. doi:10.1016/0009-2509(59)80030-0.
^Pigford, Robert L.; Ashraf, Muhammad; Miron, Yvon D. (1983). "Flow distribution in piping manifolds". Industrial & Engineering Chemistry Fundamentals. 22 (4): 463–471. doi:10.1021/i100012a019.
^Tondeur, D.; Fan, Y.; Commenge, J.M.; Luo, L. (2011). "Uniform flows in rectangular lattice networks". Chemical Engineering Science. 66 (21): 5301–5312. doi:10.1016/j.ces.2011.07.027.
^Commenge, J.M.; Saber, M.; Falk, L. (2011). "Methodology for multi-scale design of isothermal laminar flow networks". Chemical Engineering Journal. 173 (2): 334–340. doi:10.1016/j.cej.2011.07.060.
^ abcKee, R.J.; Korada, P.; Walters, K.; Pavol, M. (2002). "A generalized model of the flow distribution in channel networks of planar fuel cells". J Power Sources. 109 (1): 148–159. Bibcode:2002JPS...109..148K. doi:10.1016/S0378-7753(02)00090-3.
^ abcdWang, J.Y. (2008). "Pressure drop and flow distribution in parallel-channel of configurations of fuel cell stacks: U-type arrangement". International Journal of Hydrogen Energy. 33 (21): 6339–6350. doi:10.1016/j.ijhydene.2008.08.020.
^ abcWang, J.Y. (2010). "Pressure drop and flow distribution in parallel-channel of configurations of fuel cell stacks: Z-type arrangement". International Journal of Hydrogen Energy. 35 (11): 5498–5509. doi:10.1016/j.ijhydene.2010.02.131.
^ abcWang, J.Y. (2011). "Theory of flow distribution in manifolds". Chemical Engineering J. 168 (3): 1331–1345. doi:10.1016/j.cej.2011.02.050.
^Bajura, R.A. (1971). "A model for flow distribution in manifolds". Journal of Engineering for Gas Turbines and Power. 93: 7–12. doi:10.1115/1.3445410.
^Bajura, R.A.; Jones Jr., E.H. (1976). "Flow distribution manifolds". Journal of Fluids Engineering. 98 (4): 654–665. doi:10.1115/1.3448441.
^Bassiouny, M.K.; Martin, H. (1984). "Flow distribution and pressure drop in plate heat exchanges. Part I. U-Type arrangement". Chem. Eng. Sci. 39 (4): 693–700. doi:10.1016/0009-2509(84)80176-1.
^Bassiouny, M.K.; Martin, H. (1984). "Flow distribution and pressure drop in plate heat exchanges. Part II. Z-Type arrangement". Chem. Eng. Sci. 39 (4): 701–704. doi:10.1016/0009-2509(84)80177-3.
^Wang, J.Y.; Gao, Z.L.; Gan, G.H.; Wu, D.D. (2001). "Analytical solution of flow coefficients for a uniformly distributed porous channel". Chemical Engineering Journal. 84 (1): 1–6. doi:10.1016/S1385-8947(00)00263-1.
^Wang, J.Y.; Wang, H.L. (2012). "Flow field designs of bipolar plates in PEM fuel cells: theory and applications, Fuel Cells". Fuel Cells. 12 (6): 989–1003. doi:10.1002/fuce.201200074. S2CID 96529759.
^Wang, J.Y.; Wang, H.L. (2012). "Discrete approach for flow-field designs of parallel channel configurations in fuel cells". International Journal of Hydrogen Energy. 37 (14): 10881–10897. doi:10.1016/j.ijhydene.2012.04.034.
and 23 Related for: Flow distribution in manifolds information
flow reactor, and irrigation. Manifolds can usually be categorized into one of the following types: dividing, combining, Z-type and U-type manifolds (Fig...
exchanger include flowdistribution and pressure drop and heat transfer. The former is an issue of Flowdistributioninmanifolds. A layout configuration...
the normalized Ricci flow is not generally meaningful on noncompact manifolds. Let M {\displaystyle M} be a smooth closed manifold, and let g 0 {\displaystyle...
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distributionin the tangent bundle satisfying...
case of Riemannian manifolds. The article Levi-Civita connection discusses the more general case of a pseudo-Riemannian manifold and geodesic (general...
which converts to a single plane manifold around 3500 rpm for greater peak flow and horsepower. Older heat riser manifolds with 'wet runners' for carbureted...
manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature...
known as manifold learning, is any of various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with...
describe classical Hamiltonian mechanics. Poisson manifolds are further generalisations of symplectic manifolds, which arise by axiomatising the properties...
embedded CR manifolds, shows how to define these structures intrinsically, and then generalizes these to the abstract setting. Embedded CR manifolds are, first...
coordinates, there are cases where Ito calculus on manifolds is preferable. A theory of Ito calculus on manifolds was first developed by Laurent Schwartz through...
manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution...
isotropy has a few different meanings: Isotropic manifolds A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar...
which rotate, compress, and stretch around in an area preserving way. Even more revealing are groups, or manifolds of neighboring rays extending over significant...
systems of special kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results...
system's other components. In wind tunnels, rockets, and many flow applications, it is a chamber upstream on the fluid flow where the fluid initially resides...
^{-1}(\{b\}),b\in B} . Its space of leaves L is homeomorphic to B, in particular L is a Hausdorff manifold. If M → N is a covering map between manifolds, and F...
Flow measurement is the quantification of bulk fluid movement. Flow can be measured using devices called flowmeters in various ways. The common types of...
and distribution systems are an important part of the selection process. Gas cabinets may be distinguished by the types of valves used to control flow. Manual...
study of harmonic functions is the study of harmonic forms on Riemannian manifolds, and it is related to the study of cohomology. Also, it is possible to...
spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse...
without significantly altering the intake manifold design. More complex later designs use intake manifolds, and even cylinder heads, specially designed...