Flow distribution in manifolds

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, or to collect them into one discharge stream, such as in fuel cells, heat exchangers, radial flow reactors, hydronics, fire protection, and irrigation. Manifolds can usually be categorized into one of the following types: dividing, combining, Z-type and U-type manifolds (Fig. 1).{{cite journal|last=McNown|first=J.S.|title=Mechanics of manifold flow |journal=Transactions of the American Society of Civil Engineers |year=1954|volume=119|issue=1|pages=1103–1142|doi=10.1061/TACEAT.0007058 }}{{cite journal |doi=10.1016/0009-2509(59)80030-0|title=Flow distributions in manifolds|journal=Chemical Engineering Science|volume=10|issue=1–2|pages=112–124|year=1959|last1=Acrivos|first1=A.|last2=Babcock|first2=B.D.|last3=Pigford|first3=R.L.|bibcode=1959ChEnS..10..112A }}{{cite journal |doi=10.1021/i100012a019|title=Flow distribution in piping manifolds|journal=Industrial & Engineering Chemistry Fundamentals|volume=22|issue=4|pages=463–471|year=1983|last1=Pigford|first1=Robert L.|last2=Ashraf|first2=Muhammad|last3=Miron|first3=Yvon D.}} A key question is the uniformity of the flow distribution and pressure drop.

File:Manifold arrangement.jpg

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:

File:Control volume of manifold.jpg

{{NumBlk|:|$\backslash Delta\backslash ,P+\backslash tfrac\{\backslash rho\; f\}\{2\backslash ,D\}\backslash ,W^2\backslash Delta\backslash ,X+\backslash tfrac\{\backslash rho\}2\backslash Delta\backslash ,W^2\backslash ,=\backslash ,0$|{{EquationRef|Eq.1}}}}

where

:$W\backslash ,$ is the velocity,

:$P\backslash ,$ is the pressure,

:$\backslash rho$ is the density,

:$D\backslash ,$ is the hydraulic diameter,

:$f\backslash ,$ is the frictional coefficient,

:$X\backslash ,$ 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.{{cite journal|last=Tondeur|first=D. |author2=Fan, Y. |author3=Commenge, J.M. |author4=Luo, L.|title=Uniform flows in rectangular lattice networks|journal=Chemical Engineering Science |year=2011|volume=66|issue=21 |pages=5301–5312|doi=10.1016/j.ces.2011.07.027|bibcode=2011ChEnS..66.5301T }}{{cite journal|last=Commenge|first=J.M.|author2=Saber, M. |author3=Falk, L. |title=Methodology for multi-scale design of isothermal laminar flow networks.|journal=Chemical Engineering Journal|volume=173|issue=2|pages=334–340|doi=10.1016/j.cej.2011.07.060|year=2011|bibcode=2011ChEnJ.173..541C }}{{cite journal|last=Kee|first=R.J. |author2=Korada, P. |author3=Walters, K. |author4=Pavol, M. |title=A generalized model of the flow distribution in channel networks of planar fuel cells|journal=J Power Sources|year=2002|volume=109|issue=1 |pages=148–159|doi=10.1016/S0378-7753(02)00090-3|bibcode=2002JPS...109..148K }} A generalized model of the flow distribution in channel networks of planar fuel cells. 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, $\backslash ,R\backslash ,=\backslash tfrac\{\backslash ,128\backslash mu\backslash ,L\}\{\backslash pi\backslash ,d^4\}$ 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.

File:T-junction and corresponding network.jpg

The question raised from the experiments by McNown and by Acrivos et al. Their experimental results showed a pressure rise after T-junction due to flow branching. This phenomenon was explained by Wang.{{cite journal|last=Wang|first=J.Y.|title=Pressure drop and flow distribution in parallel-channel of configurations of fuel cell stacks: U-type arrangement|journal=International Journal of Hydrogen Energy|year=2008|volume=33|issue=21|pages=6339–6350|doi=10.1016/j.ijhydene.2008.08.020}}{{cite journal|last=Wang|first=J.Y.|title=Pressure drop and flow distribution in parallel-channel of configurations of fuel cell stacks: Z-type arrangement|journal=International Journal of Hydrogen Energy|year=2010|volume=35|issue=11|pages=5498–5509|doi=10.1016/j.ijhydene.2010.02.131}}{{cite journal|last=Wang|first=J.Y.|title=Theory of flow distribution in manifolds|journal=Chemical Engineering J|year=2011|volume=168|issue=3|pages=1331–1345|doi=10.1016/j.cej.2011.02.050|bibcode=2011ChEnJ.168.1331W }} 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.

File:Velocity profile in manifold.jpg

Thus, mass, momentum and energy conservations must be employed together for description of flow in manifolds.{{cite journal|last=Bajura|first=R.A.|title=A model for flow distribution in manifolds.|journal= Journal of Engineering for Gas Turbines and Power|year=1971|volume=93|pages=7–12|doi=10.1115/1.3445410}}{{cite journal|last=Bajura|first=R.A.|author2=Jones Jr., E.H.|title=Flow distribution manifolds.|journal= Journal of Fluids Engineering|year=1976|volume=98|issue=4|pages=654–665|doi=10.1115/1.3448441}}{{cite journal|last=Bassiouny|first=M.K.|author2=Martin, H.|title=Flow distribution and pressure drop in plate heat exchanges. Part I. U-Type arrangement.|journal=Chem. Eng. Sci.|year=1984|volume=39|issue=4|pages=693–700|doi=10.1016/0009-2509(84)80176-1}}{{cite journal|last=Bassiouny|first=M.K.|author2=Martin, H.|title=Flow distribution and pressure drop in plate heat exchanges. Part II. Z-Type arrangement.|journal=Chem. Eng. Sci.|year=1984|volume=39|issue=4|pages=701–704|doi=10.1016/0009-2509(84)80177-3}}{{cite journal|last=Wang|first=J.Y.|author2=Gao, Z.L. |author3=Gan, G.H. |author4=Wu, D.D. |title=Analytical solution of flow coefficients for a uniformly distributed porous channel.|journal=Chemical Engineering Journal|year=2001|volume=84|issue=1|pages=1–6|doi=10.1016/S1385-8947(00)00263-1|bibcode=2001ChEnJ..84....1W }} Wang 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:

{{NumBlk|:|$\backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash ,d\backslash ,P\}\{\backslash ,d\backslash ,X\}+\backslash tfrac\{\backslash ,f\}\{2\backslash ,D\}\backslash ,W^2+\backslash left(\backslash frac\; \{2-\backslash beta\}\{2\}\backslash right)\backslash frac\{\backslash ,d\backslash ,W^2\}\{\backslash ,d\backslash ,X\}\; \backslash ,=\backslash ,0$|{{EquationRef|Eq.2a}}}}

or to a discrete equation:

{{NumBlk|:|$\backslash Delta\backslash ,P+\backslash tfrac\{\backslash rho\; f\}\{2\backslash ,D\}\backslash ,W^2\backslash Delta\backslash ,X+\backslash rho\backslash left(\backslash frac\; \{2-\backslash beta\}\{2\}\backslash right)\backslash Delta\backslash ,W^2\; =\; 0$|{{EquationRef|Eq.2b}}}}

In {{EquationNote|Eq.2}}, the inertial effects are corrected by a momentum factor, β. {{EquationNote|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 {{EquationNote|Eq.2a}} is limiting case of {{EquationNote|Eq.2b}} when ∆X → 0. {{EquationNote|Eq.2a}} is simplified to {{EquationNote|Eq.1}} Bernoulli equation without the potential energy term when β=1 whilst {{EquationNote|Eq.2}} is simplified to Kee's model when β=0. Moreover, {{EquationNote|Eq.2}} can be simplified to Acrivos et al.’s model after substituting Blasius’ equation, $\backslash ,f\backslash ,=\backslash ,0.3164\backslash ,/\backslash ,Re^\{0.25\}\backslash ,=\backslash ,f\_0\backslash ,W^\{-0.25\}$. Therefore, these main models are just a special case of {{EquationNote|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:

{{NumBlk|:|$\backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash ,d\backslash ,P\}\{\backslash ,d\backslash ,X\}-\backslash tfrac\{\backslash ,f\}\{2\backslash ,D\}\backslash ,W^2+\backslash left(\backslash frac\; \{2-\backslash beta\}\{2\}\backslash right)\backslash frac\{\backslash ,d\backslash ,W^2\}\{\backslash ,d\backslash ,X\}\; \backslash ,=\backslash ,0$|{{EquationRef|Eq.3a}}}}

or to a discrete equation:

{{NumBlk|:|$\backslash Delta\backslash ,P-\backslash tfrac\{\backslash rho\; f\}\{2\backslash ,D\}\backslash ,W^2\backslash Delta\backslash ,X+\backslash rho\backslash left(\backslash frac\; \{2-\backslash beta\}\{2\}\backslash right)\backslash Delta\backslash ,W^2\; =\; 0$|{{EquationRef|Eq.3b}}}}

The Governing equation of the U-type flow:

{{NumBlk|:|$\backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash ,d\backslash left(\backslash ,P-P\_e\backslash right)\}\{\backslash ,d\backslash ,X\}+\backslash tfrac12\backslash ,\backslash left[\backslash frac\{\backslash ,f\}\{\backslash ,D\}+\backslash frac\{f\_e\}\{D\_e\}\backslash left(\backslash frac\{F\}\{F\_e\}\backslash right)^2\backslash right]\backslash ,W^2$

+\left[\left(\,2-\beta\right)\,-\left(\,2-\beta_e\right)\left(\frac{\,F}{F_e}\right)^2\right]\,W\tfrac{\,dW}{\,dX}\,=\,0|{{EquationRef|Eq.4a}}}}

or to a discrete equation:

{{NumBlk|:|$\backslash Delta\backslash left(\backslash ,P-P\_e\backslash right)+\backslash frac\{\backslash rho\}\{2\}\backslash ,\backslash left[\backslash frac\{\backslash ,f\}\{\backslash ,D\}+\backslash frac\{f\_e\}\{D\_e\}\backslash left(\backslash frac\{F\}\{F\_e\}\backslash right)^2\backslash right]\backslash ,W^2\backslash Delta\backslash ,X$

+\frac{\rho}{2}\left[\left(\,2-\beta\right)\,-\left(\,2-\beta_e\right)\left(\frac{\,F}{F_e}\right)^2\right]\Delta\,W^2\,=\,0|{{EquationRef|Eq.4b}}}}

The Governing equation of the Z-type flow:

{{NumBlk|:|$\backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash ,d\backslash left(\backslash ,P-P\_e\backslash right)\}\{\backslash ,d\backslash ,X\}+\backslash tfrac12\backslash ,\backslash left[\backslash frac\{\backslash ,f\}\{\backslash ,D\}-\backslash left(\backslash ,1-\backslash frac\{W\_0\}\{W\}\backslash right)\backslash frac\{f\_e\}\{D\_e\}\backslash left(\backslash frac\{F\}\{F\_e\}\backslash right)^2\backslash right]\backslash ,W^2$

+\left[\left(\,2-\beta\right)\,-\left(\,2-\beta_e\right)\left(\,1-\frac{W_0}{W}\right)\left(\frac{\,F}{F_e}\right)^2\right]\,W\tfrac{\,dW}{\,dX}\,=\frac{f_e}{\,2D_e}\,W_0^2\left(\frac{F}{F_e}\right)^2|{{EquationRef|Eq.5a}}}}

or to a discrete equation:

{{NumBlk|:|$\backslash Delta\backslash left(\backslash ,P-P\_e\backslash right)+\backslash frac\{\backslash rho\}\{2\}\backslash ,\backslash left[\backslash frac\{\backslash ,f\}\{\backslash ,D\}-\backslash left(\backslash ,1-\backslash frac\{W\_0\}\{W\}\backslash right)\backslash frac\{f\_e\}\{D\_e\}\backslash left(\backslash frac\{F\}\{F\_e\}\backslash right)^2\backslash right]\backslash ,W^2\backslash Delta\backslash ,X$

+\frac{\rho}{2}\left[\left(\,2-\beta\right)\,-\left(\,2-\beta_e\right)\left(\,1-\frac{W_0}{W}\right)\left(\frac{\,F}{F_e}\right)^2\right]\Delta\,W^2\,=\frac{\rho\,f_e}{\,2D_e}\,W_0^2\left(\frac{F}{F_e}\right)^2\Delta\,X|{{EquationRef|Eq.5b}}}}

File:Manifold Configuration01.jpg

{{EquationNote|Eq.2}} - {{EquationNote|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. Wang elaborated the most complete analytical solutions of {{EquationNote|Eq.2}} - {{EquationNote|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{{cite journal|last=Wang|first=J.Y.|author2=Wang, H.L.|title=Flow field designs of bipolar plates in PEM fuel cells: theory and applications, Fuel Cells|journal=Fuel Cells|year=2012|volume=12|issue=6|pages=989–1003|doi=10.1002/fuce.201200074|s2cid=96529759 }}{{cite journal|last=Wang|first=J.Y.|author2=Wang, H.L.|title=Discrete approach for flow-field designs of parallel channel configurations in fuel cells|journal=International Journal of Hydrogen Energy|year=2012|volume=37|issue=14|pages=10881–10897|doi=10.1016/j.ijhydene.2012.04.034|bibcode=2012IJHE...3710881W }} 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.