View source for 2-Methoxy-6-polyprenyl-1,4-benzoquinol methylase

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).<ref name=McNown>{{cite journal|last=McNown|first=J.S.|title=Mechanics of manifold flow. Transactions of ASCE,|year=1954|volume=119|pages=1103~1142}}</ref><ref name=Acrivos>{{cite journal|last=Acrivos|first=A.|coauthors= Babcock, B.D.,  Pigford,R.L.|title=Flow distributions in manifolds. Chemical Engineering Science,|year=1959|volume=10|pages=112~124|doi=10.1016/0009-2509(59)80030-0|url=http://dx.doi.org/10.1016/0009-2509(59)80030-0}}</ref><ref>{{cite journal|last=Pigford|first=R.L.|coauthors= Ashraf, M., Mlron, Y.D.|title=Flow distribution in piping manifolds. Ind. Eng. Chem. Fundam.|year=1983|volume=22|pages=463–471|doi=10.1021/i100012a019
|url=http://pubs.acs.org/doi/abs/10.1021/i100012a019}}</ref> A key question is the uniformity of the flow distribution and pressure drop. 
 
[[File:Manifold arrangement.jpg|thumbnail|default|Fig. 1. Manifold arrangement for flow distribution]]

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|thumbnail|default|Fig. 2. Control volume]]
{{NumBlk|:|<math>\Delta\,P+\tfrac{\rho f}{2\,D}\,W^2\Delta\,X+\tfrac{\rho}2\Delta\,W^2\,=\,0</math>|{{EquationRef|Eq.1}}}}
where 
:<math>W\,</math> is the velocity, 
:<math>P\,</math> is the pressure,
:<math>\rho</math> is the density,
:<math>D\,</math> is the hydraulic diameter,
:<math>f\,</math> is the frictional coefficient,
:<math>X\,</math> is the axial coordinate in the manifold,
''∆X''&nbsp;=&nbsp;''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.<ref>{{cite journal|last=Tondeur|first=D.|coauthors=Fan, Y. Commenge, J.M., Luo., L.|title=Uniform flows in rectangular lattice networks. Chemical Engineering Science,|year=2011|volume=66|pages=5301–5312|doi=10.1016/j.ces.2011.07.027|url=http://dx.doi.org/10.1016/j.ces.2011.07.027}}</ref><ref>{{cite journal|last=Commenge|first=J.M.|coauthors=Saber, M.,  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|url=http://dx.doi.org/10.1016/j.ces.2011.07.027}}</ref><ref name=Kee>{{cite journal|last=Kee|first=R.J.|coauthors=Korada, P., Walters, Pavol, K.M.|title=A generalized model of the flow distribution in channel networks of planar fuel cells|journal=J Power Sources|year=2002|volume=109|pages=148–159|doi=10.1016/S0378-7753(02)00090-3|url=http://dx.doi.org/10.1016/S0378-7753(02)00090-3}}</ref>  A generalized model of the flow distribution in channel networks of planar fuel cells.<ref name=Kee /> 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 ''Q''&nbsp;=&nbsp;''∆P/R''. ''f''&nbsp;=&nbsp;64/''Re'' for laminar flow where ''Re'' is the [[Reynolds number]]. The frictional resistance, <math>\,R\,=\tfrac{\,128\mu\,L}{\pi\,d^4}</math> using [[Poiseuille's law]]. Since they have same diameter and length in Fig. 3, their resistances are same, ''R''<sub>2</sub> = ''R''<sub>3</sub>. Thus the velocities should be equal in two outlets or the flow rates should be equal according to the assumptions. Obviously this disobeys our observation. Our observations shows that the greater the velocity (or momentum), the more fluid fraction through the straight direction. Only under very slow laminar flow, Q<sub>2</sub> may be equal to Q<sub>3</sub>.  
[[File:T-junction and corresponding network.jpg|thumbnail|default|Fig. 3. T-junction and corresponding network]]
The question raised from the experiments by McNown<ref name=McNown /> and by Acrivos et al.<ref name=Acrivos />  Their experimental results showed a pressure rise after T-junction due to flow branching. This phenomenon was explained by Wang.<ref name=Wang2008>{{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=Int. J. of Hydrogen Energy|year=2008|volume=33|issue=21|pages=6339–6350|doi=10.1016/j.ijhydene.2008.08.020|url=http://dx.doi.org/10.1016/j.ijhydene.2008.08.020}}</ref><ref name=Wang2010>{{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=Int. J. of Hydrogen Energy|year=2010|volume=35|issue=11|pages=5498–5509|doi=10.1016/j.ijhydene.2010.02.131|url=http://dx.doi.org/10.1016/j.ijhydene.2010.02.131}}</ref><ref name=Wang2011>{{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|url=http://dx.doi.org/10.1016/j.cej.2011.02.050}}</ref> 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|thumbnail|default|Fig. 4. Velocity profile along a manifold]]
Thus, mass, momentum and energy conservations must be employed together for description of flow in manifolds.<ref>{{cite journal|last=Bajura|first=R.A.|title=A model for flow distribution in manifolds|journal=J. of Eng. for power, Transaction of ASME|year=1971|volume=93|pages=7–12}}</ref><ref>{{cite journal|last=Bajura|first=R.A.|coauthors=Jones Jr., E.H.|title=Flow distribution manifolds.|journal=J. Fluid Eng., Transaction of ASME|year=1976|volume=98|pages=654–65}}</ref><ref>{{cite journal|last=Bassiouny|first=M.K.|coauthors=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|url=http://dx.doi.org/10.1016/0009-2509(84)80176-1}}</ref><ref>{{cite journal|last=Bassiouny|first=M.K.|coauthors=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|url=http://dx.doi.org/10.1016/0009-2509(84)80177-3}}</ref><ref>{{cite journal|last=Wang|first=J.Y.|coauthors=Gao, Z.L., Gan, G.H., 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|url=http://dx.doi.org/10.1016/S1385-8947(00)00263-1}}</ref>  Wang<ref name=Wang2008 /><ref name=Wang2010 /><ref name=Wang2011 /> 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|:|<math>\frac{1}{\rho}\frac{\,d\,P}{\,d\,X}+\tfrac{\,f}{2\,D}\,W^2+\left(\frac {2-\beta}{2}\right)\frac{\,d\,W^2}{\,d\,X} \,=\,0</math>|{{EquationRef|Eq.2a}}}}
or to a discrete equation:
{{NumBlk|:|<math>\Delta\,P+\tfrac{\rho f}{2\,D}\,W^2\Delta\,X+\rho\left(\frac {2-\beta}{2}\right)\Delta\,W^2   = 0</math>|{{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<ref name=Kee /> when β=0. Moreover, {{EquationNote|Eq.2}} can be simplified to Acrivos et al.’s model<ref name=Acrivos /> after substituting Blasius’ equation, <math>\,f\,=\,0.3164\,/\,Re^{0.25}\,=\,f_0\,W^{-0.25}</math>. 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 combing flow:
{{NumBlk|:|<math>\frac{1}{\rho}\frac{\,d\,P}{\,d\,X}-\tfrac{\,f}{2\,D}\,W^2+\left(\frac {2-\beta}{2}\right)\frac{\,d\,W^2}{\,d\,X} \,=\,0</math>|{{EquationRef|Eq.3a}}}}
or to a discrete equation:
{{NumBlk|:|<math>\Delta\,P-\tfrac{\rho f}{2\,D}\,W^2\Delta\,X+\rho\left(\frac {2-\beta}{2}\right)\Delta\,W^2   = 0</math>|{{EquationRef|Eq.3b}}}}

The Governing equation of the U-type flow:
{{NumBlk|:|<math>\frac{1}{\rho}\frac{\,d\left(\,P-P_e\right)}{\,d\,X}+\tfrac12\,\left[\frac{\,f}{\,D}+\frac{f_e}{D_e}\left(\frac{F}{F_e}\right)^2\right]\,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</math>|{{EquationRef|Eq.4a}}}}
or to a discrete equation:
{{NumBlk|:|<math>\Delta\left(\,P-P_e\right)+\frac{\rho}{2}\,\left[\frac{\,f}{\,D}+\frac{f_e}{D_e}\left(\frac{F}{F_e}\right)^2\right]\,W^2\Delta\,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</math>|{{EquationRef|Eq.4b}}}}

The Governing equation of the Z-type flow:
{{NumBlk|:|<math>\frac{1}{\rho}\frac{\,d\left(\,P-P_e\right)}{\,d\,X}+\tfrac12\,\left[\frac{\,f}{\,D}-\left(\,1-\frac{W_0}{W}\right)\frac{f_e}{D_e}\left(\frac{F}{F_e}\right)^2\right]\,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</math>|{{EquationRef|Eq.5a}}}}
or to a discrete equation:
{{NumBlk|:|<math>\Delta\left(\,P-P_e\right)+\frac{\rho}{2}\,\left[\frac{\,f}{\,D}-\left(\,1-\frac{W_0}{W}\right)\frac{f_e}{D_e}\left(\frac{F}{F_e}\right)^2\right]\,W^2\Delta\,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</math>|{{EquationRef|Eq.5b}}}}

[[File:Manifold Configuration01.jpg|thumbnail|default|Fig. 5. Different configurations]]
{{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.<ref name=Wang2008 /> Wang<ref name=Wang2008 /><ref name=Wang2010 /><ref name=Wang2011 /> 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<ref>{{cite journal|last=Wang|first=J.Y.|coauthors=Wang, H.L.|title=Flow field designs of bipolar plates in PEM fuel cells: theory and applications, Fuel Cells,|year=2012|volume=12|issue=6|pages=989–1003|doi=10.1002/fuce.201200074|url=http://onlinelibrary.wiley.com/doi/10.1002/fuce.201200074/abstract}}</ref><ref>{{cite journal|last=Wang|first=J.Y.|coauthors=Wang, H.L.|title=Discrete approach for flow-field designs of parallel channel configurations in fuel cells|journal=Int. J. of Hydrogen Energy|year=2012|volume=37|issue=14|pages=10881–10897|doi=10.1016/j.ijhydene.2012.04.034|url=http://dx.doi.org/10.1016/j.ijhydene.2012.04.034}}</ref> 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.

==See also==
{{commons category|Plate and frame heat exchangers}}
*[[Plate heat exchanger]]
*[[Fuel Cells]]

==References==
{{reflist}}

<!-- This will add a notice to the bottom of the page and won't blank it! The new template which says that your draft is waiting for a review will appear at the bottom; simply ignore the old (grey) drafted templates and the old (red) decline templates. A bot will update your article submission. Until then, please don't change anything in this text box and press "Save page". -->



[[Category:Fluid dynamics]]