2882:. 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 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.
462:
450:
40:
2851:
25:
2832:
2361:
438:. 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, Q
469:
Thus, mass, momentum and energy conservations must be employed together for description of flow in manifolds. 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
331:
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
457:
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. Because of inertial effects, the fluid will prefer to the straight direction. Thus the flow rate
2869:
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.
1078:
627:
20:
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
2387:
1561:
1903:
1877:
1219:
768:
152:
458:
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.
929:
478:
2827:{\displaystyle \Delta \left(\,P-P_{e}\right)+{\frac {\rho }{2}}\,\left\,W^{2}\Delta \,X+{\frac {\rho }{2}}\left\Delta \,W^{2}\,={\frac {\rho \,f_{e}}{\,2D_{e}}}\,W_{0}^{2}\left({\frac {F}{F_{e}}}\right)^{2}\Delta \,X}
1245:
2356:{\displaystyle {\frac {1}{\rho }}{\frac {\,d\left(\,P-P_{e}\right)}{\,d\,X}}+{\tfrac {1}{2}}\,\left\,W^{2}+\left\,W{\tfrac {\,dW}{\,dX}}\,={\frac {f_{e}}{\,2D_{e}}}\,W_{0}^{2}\left({\frac {F}{F_{e}}}\right)^{2}}
910:
332:
is usually described as the lattice network using analogy with the conventional electric circuit methods. A generalized model of the flow distribution in channel networks of planar fuel cells. Similar to
1587:
418:
32:
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
1104:
653:
470:
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:
21:
categorized into one of the following types: dividing, combining, Z-type and U-type manifolds (Fig. 1). A key question is the uniformity of the flow distribution and pressure drop.
52:
237:
306:
283:
260:
215:
192:
2870:
Their analytical solutions had been well-known challenges in this field for 50 years until 2008. Wang elaborated the most complete analytical solutions of
1073:{\displaystyle {\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}
622:{\displaystyle {\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}
798:
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
336:, 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
3064:
Kee, R.J.; Korada, P.; Walters, K.; Pavol, M. (2002). "A generalized model of the flow distribution in channel networks of planar fuel cells".
1556:{\displaystyle {\frac {1}{\rho }}{\frac {\,d\left(\,P-P_{e}\right)}{\,d\,X}}+{\tfrac {1}{2}}\,\left\,W^{2}+\left\,W{\tfrac {\,dW}{\,dX}}\,=\,0}
3305:
Wang, J.Y.; Gao, Z.L.; Gan, G.H.; Wu, D.D. (2001). "Analytical solution of flow coefficients for a uniformly distributed porous channel".
3138:
Wang, J.Y. (2010). "Pressure drop and flow distribution in parallel-channel of configurations of fuel cell stacks: Z-type arrangement".
3106:
Wang, J.Y. (2008). "Pressure drop and flow distribution in parallel-channel of configurations of fuel cell stacks: U-type arrangement".
837:
1872:{\displaystyle \Delta \left(\,P-P_{e}\right)+{\frac {\rho }{2}}\,\left\,W^{2}\Delta \,X+{\frac {\rho }{2}}\left\Delta \,W^{2}\,=\,0}
3278:
Bassiouny, M.K.; Martin, H. (1984). "Flow distribution and pressure drop in plate heat exchanges. Part II. Z-Type arrangement".
363:
3251:
Bassiouny, M.K.; Martin, H. (1984). "Flow distribution and pressure drop in plate heat exchanges. Part I. U-Type arrangement".
3332:
Wang, J.Y.; Wang, H.L. (2012). "Flow field designs of bipolar plates in PEM fuel cells: theory and applications, Fuel Cells".
3367:
Wang, J.Y.; Wang, H.L. (2012). "Discrete approach for flow-field designs of parallel channel configurations in fuel cells".
1214:{\displaystyle \Delta \,P-{\tfrac {\rho f}{2\,D}}\,W^{2}\Delta \,X+\rho \left({\frac {2-\beta }{2}}\right)\Delta \,W^{2}=0}
763:{\displaystyle \Delta \,P+{\tfrac {\rho f}{2\,D}}\,W^{2}\Delta \,X+\rho \left({\frac {2-\beta }{2}}\right)\Delta \,W^{2}=0}
33:
3032:
Commenge, J.M.; Saber, M.; Falk, L. (2011). "Methodology for multi-scale design of isothermal laminar flow networks".
3400:
147:{\displaystyle \Delta \,P+{\tfrac {\rho f}{2\,D}}\,W^{2}\Delta \,X+{\tfrac {\rho }{2}}\Delta \,W^{2}\,=\,0}
449:
3005:
Tondeur, D.; Fan, Y.; Commenge, J.M.; Luo, L. (2011). "Uniform flows in rectangular lattice networks".
2978:
Pigford, Robert L.; Ashraf, Muhammad; Miron, Yvon D. (1983). "Flow distribution in piping manifolds".
3073:
2891:
918:. Similarly, one can obtain the governing equations of the combining, U-type and Z-type arrangement.
421:
3349:
461:
3376:
3341:
3314:
3287:
3260:
3233:
3206:
3179:
3147:
3115:
3081:
3041:
3014:
2987:
2960:
2928:
39:
222:
2850:
357:
290:
267:
244:
199:
176:
3077:
3318:
3085:
3394:
3291:
3264:
2964:
2951:
Acrivos, A.; Babcock, B.D.; Pigford, R.L. (1959). "Flow distributions in manifolds".
3353:
24:
3380:
3151:
3119:
834:
can be simplified to
Acrivos et al.’s model after substituting Blasius’ equation,
424:. Since they have same diameter and length in Fig. 3, their resistances are same,
333:
3183:
3045:
3018:
2896:
3345:
2932:
2991:
3237:
3210:
3224:
Bajura, R.A.; Jones Jr., E.H. (1976). "Flow distribution manifolds".
822:
Bernoulli equation without the potential energy term when β=1 whilst
3197:
Bajura, R.A. (1971). "A model for flow distribution in manifolds".
2849:
460:
448:
38:
23:
905:{\displaystyle \,f\,=\,0.3164\,/\,Re^{0.25}\,=\,f_{0}\,W^{-0.25}}
36:. One obtains a governing equation of dividing flow as follows:
3170:
Wang, J.Y. (2011). "Theory of flow distribution in manifolds".
792:, the inertial effects are corrected by a momentum factor, β.
912:. Therefore, these main models are just a special case of
413:{\displaystyle \,R\,={\tfrac {\,128\mu \,L}{\pi \,d^{4}}}}
2921:
Transactions of the
American Society of Civil Engineers
2248:
1968:
1522:
1310:
1119:
971:
668:
520:
376:
111:
67:
2390:
1906:
1590:
1248:
1107:
932:
840:
656:
481:
366:
293:
270:
247:
225:
202:
179:
55:
2980:
Industrial & Engineering
Chemistry Fundamentals
2919:McNown, J.S. (1954). "Mechanics of manifold flow".
2826:
2355:
1871:
1555:
1213:
1072:
904:
762:
621:
412:
300:
277:
254:
231:
209:
186:
146:
28:Fig. 1. Manifold arrangement for flow distribution
3199:Journal of Engineering for Gas Turbines and Power
828:is simplified to Kee's model when β=0. Moreover,
921:The Governing equation of the combining flow:
8:
453:Fig. 3. T-junction and corresponding network
1895:The Governing equation of the Z-type flow:
1237:The Governing equation of the U-type flow:
3165:
3163:
3161:
3133:
3131:
3129:
3101:
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2723:
2718:
2704:
2692:
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2646:
2630:
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2610:
2595:
2575:
2568:
2559:
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2522:
2509:
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2477:
2471:
2464:
2449:
2443:
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2410:
2399:
2389:
2347:
2335:
2326:
2315:
2310:
2305:
2296:
2288:
2281:
2275:
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2260:
2251:
2247:
2243:
2232:
2220:
2211:
2208:
2187:
2181:
2174:
2158:
2147:
2138:
2123:
2104:
2099:
2088:
2076:
2067:
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2016:
2009:
1994:
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1985:
1979:
1967:
1957:
1953:
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1929:
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1917:
1907:
1905:
1865:
1861:
1855:
1850:
1836:
1824:
1815:
1812:
1796:
1785:
1776:
1761:
1741:
1734:
1725:
1720:
1709:
1697:
1688:
1675:
1665:
1659:
1649:
1643:
1640:
1634:
1624:
1610:
1599:
1589:
1549:
1545:
1534:
1525:
1521:
1517:
1506:
1494:
1485:
1482:
1466:
1455:
1446:
1431:
1412:
1407:
1396:
1384:
1375:
1362:
1352:
1346:
1336:
1330:
1327:
1321:
1309:
1299:
1295:
1282:
1271:
1262:
1259:
1249:
1247:
1199:
1194:
1169:
1155:
1146:
1141:
1133:
1118:
1111:
1106:
1066:
1062:
1055:
1051:
1043:
1038:
1034:
1031:
1009:
996:
991:
983:
974:
970:
960:
956:
950:
946:
943:
933:
931:
893:
888:
882:
877:
873:
867:
859:
854:
853:
849:
845:
841:
839:
748:
743:
718:
704:
695:
690:
682:
667:
660:
655:
615:
611:
604:
600:
592:
587:
583:
580:
558:
545:
540:
532:
523:
519:
509:
505:
499:
495:
492:
482:
480:
465:Fig. 4. Velocity profile along a manifold
400:
395:
386:
379:
375:
371:
367:
365:
297:
292:
274:
269:
251:
246:
224:
206:
201:
183:
178:
140:
136:
130:
125:
110:
103:
94:
89:
81:
66:
59:
54:
3369:International Journal of Hydrogen Energy
3140:International Journal of Hydrogen Energy
3108:International Journal of Hydrogen Energy
2914:
2912:
308:is the axial coordinate in the manifold,
2908:
3059:
3057:
3055:
7:
2381:
1897:
1581:
1239:
1098:
923:
647:
472:
46:
2817:
2715:
2565:
2391:
1847:
1731:
1591:
1191:
1152:
1108:
740:
701:
657:
122:
100:
56:
14:
2854:Fig. 5. Different configurations
812:
806:
800:
794:
3381:10.1016/j.ijhydene.2012.04.034
3152:10.1016/j.ijhydene.2010.02.131
3120:10.1016/j.ijhydene.2008.08.020
2878:
2872:
2865:
2859:
914:
830:
824:
818:
788:
285:is the frictional coefficient,
1:
3319:10.1016/S1385-8947(00)00263-1
3226:Journal of Fluids Engineering
3086:10.1016/S0378-7753(02)00090-3
360:. The frictional resistance,
3307:Chemical Engineering Journal
3292:10.1016/0009-2509(84)80177-3
3265:10.1016/0009-2509(84)80176-1
3034:Chemical Engineering Journal
3007:Chemical Engineering Science
2965:10.1016/0009-2509(59)80030-0
2953:Chemical Engineering Science
2379:or to a discrete equation:
1579:or to a discrete equation:
1096:or to a discrete equation:
645:or to a discrete equation:
327:is the number of ports and
3417:
262:is the hydraulic diameter,
3184:10.1016/j.cej.2011.02.050
3046:10.1016/j.cej.2011.07.060
3019:10.1016/j.ces.2011.07.027
352:for laminar flow where
34:Darcy–Weisbach equation
3346:10.1002/fuce.201200074
3172:Chemical Engineering J
2933:10.1061/TACEAT.0007058
2855:
2828:
2357:
1873:
1557:
1215:
1074:
906:
764:
623:
466:
454:
414:
302:
279:
256:
233:
211:
188:
148:
44:
43:Fig. 2. Control volume
29:
2853:
2829:
2358:
1874:
1558:
1216:
1075:
907:
765:
624:
464:
452:
415:
303:
280:
257:
234:
232:{\displaystyle \rho }
212:
189:
149:
42:
27:
2892:Plate heat exchanger
2388:
1904:
1588:
1246:
1105:
930:
838:
804:is limiting case of
654:
479:
364:
291:
268:
245:
223:
200:
177:
53:
3375:(14): 10881–10897.
3078:2002JPS...109..148K
2992:10.1021/i100012a019
2784:
2320:
301:{\displaystyle X\,}
278:{\displaystyle f\,}
255:{\displaystyle D\,}
210:{\displaystyle P\,}
187:{\displaystyle W\,}
2856:
2824:
2770:
2353:
2306:
2269:
1977:
1869:
1553:
1543:
1319:
1211:
1139:
1070:
989:
902:
760:
688:
619:
538:
467:
455:
410:
408:
298:
275:
252:
229:
207:
184:
144:
120:
87:
45:
30:
3238:10.1115/1.3448441
3211:10.1115/1.3445410
3146:(11): 5498–5509.
3114:(21): 6339–6350.
3013:(21): 5301–5312.
2848:
2847:
2805:
2767:
2698:
2668:
2583:
2537:
2515:
2486:
2454:
2432:
2377:
2376:
2341:
2303:
2268:
2226:
2196:
2082:
2060:
2031:
1999:
1976:
1962:
1915:
1893:
1892:
1830:
1749:
1703:
1681:
1654:
1632:
1577:
1576:
1542:
1500:
1390:
1368:
1341:
1318:
1304:
1257:
1235:
1234:
1185:
1138:
1094:
1093:
1060:
1025:
988:
965:
941:
816:is simplified to
784:
783:
734:
687:
643:
642:
609:
574:
537:
514:
490:
442:may be equal to Q
407:
168:
167:
119:
86:
18:flow in manifolds
3408:
3385:
3384:
3364:
3358:
3357:
3329:
3323:
3322:
3302:
3296:
3295:
3275:
3269:
3268:
3248:
3242:
3241:
3221:
3215:
3214:
3194:
3188:
3187:
3178:(3): 1331–1345.
3167:
3156:
3155:
3135:
3124:
3123:
3103:
3090:
3089:
3061:
3050:
3049:
3029:
3023:
3022:
3002:
2996:
2995:
2975:
2969:
2968:
2959:(1–2): 112–124.
2948:
2937:
2936:
2927:(1): 1103–1142.
2916:
2842:
2833:
2831:
2830:
2825:
2816:
2815:
2810:
2806:
2804:
2803:
2791:
2783:
2778:
2768:
2766:
2765:
2764:
2750:
2749:
2748:
2734:
2728:
2727:
2714:
2710:
2709:
2708:
2703:
2699:
2697:
2696:
2687:
2681:
2674:
2670:
2669:
2664:
2663:
2654:
2640:
2636:
2635:
2634:
2609:
2605:
2584:
2576:
2564:
2563:
2553:
2549:
2548:
2547:
2542:
2538:
2536:
2535:
2523:
2516:
2514:
2513:
2504:
2503:
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2492:
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2487:
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2481:
2472:
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2362:
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2339:
2327:
2319:
2314:
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2302:
2301:
2300:
2286:
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2276:
2270:
2267:
2258:
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2242:
2238:
2237:
2236:
2231:
2227:
2225:
2224:
2215:
2209:
2202:
2198:
2197:
2192:
2191:
2182:
2168:
2164:
2163:
2162:
2137:
2133:
2109:
2108:
2098:
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2092:
2087:
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2080:
2068:
2061:
2059:
2058:
2049:
2048:
2039:
2037:
2033:
2032:
2027:
2026:
2017:
2000:
1998:
1992:
1986:
1978:
1969:
1963:
1961:
1951:
1950:
1946:
1945:
1944:
1918:
1916:
1908:
1898:
1887:
1878:
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1813:
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1802:
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1775:
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1750:
1742:
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1708:
1704:
1702:
1701:
1689:
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1670:
1669:
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1647:
1641:
1633:
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1616:
1615:
1614:
1582:
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1554:
1544:
1541:
1532:
1523:
1516:
1512:
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1505:
1501:
1499:
1498:
1489:
1483:
1476:
1472:
1471:
1470:
1445:
1441:
1417:
1416:
1406:
1402:
1401:
1400:
1395:
1391:
1389:
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1376:
1369:
1367:
1366:
1357:
1356:
1347:
1342:
1340:
1334:
1328:
1320:
1311:
1305:
1303:
1293:
1292:
1288:
1287:
1286:
1260:
1258:
1250:
1240:
1229:
1220:
1218:
1217:
1212:
1204:
1203:
1190:
1186:
1181:
1170:
1151:
1150:
1140:
1137:
1128:
1120:
1099:
1088:
1079:
1077:
1076:
1071:
1061:
1059:
1049:
1048:
1047:
1032:
1030:
1026:
1021:
1010:
1001:
1000:
990:
987:
978:
972:
966:
964:
954:
944:
942:
934:
924:
911:
909:
908:
903:
901:
900:
887:
886:
872:
871:
858:
778:
769:
767:
766:
761:
753:
752:
739:
735:
730:
719:
700:
699:
689:
686:
677:
669:
648:
637:
628:
626:
625:
620:
610:
608:
598:
597:
596:
581:
579:
575:
570:
559:
550:
549:
539:
536:
527:
521:
515:
513:
503:
493:
491:
483:
473:
422:Poiseuille's law
419:
417:
416:
411:
409:
406:
405:
404:
390:
377:
348: = 64/
307:
305:
304:
299:
284:
282:
281:
276:
261:
259:
258:
253:
238:
236:
235:
230:
217:is the pressure,
216:
214:
213:
208:
194:is the velocity,
193:
191:
190:
185:
162:
153:
151:
150:
145:
135:
134:
121:
112:
99:
98:
88:
85:
76:
68:
47:
3416:
3415:
3411:
3410:
3409:
3407:
3406:
3405:
3391:
3390:
3389:
3388:
3366:
3365:
3361:
3340:(6): 989–1003.
3331:
3330:
3326:
3304:
3303:
3299:
3277:
3276:
3272:
3250:
3249:
3245:
3223:
3222:
3218:
3196:
3195:
3191:
3169:
3168:
3159:
3137:
3136:
3127:
3105:
3104:
3093:
3066:J Power Sources
3063:
3062:
3053:
3031:
3030:
3026:
3004:
3003:
2999:
2977:
2976:
2972:
2950:
2949:
2940:
2918:
2917:
2910:
2905:
2888:
2840:
2795:
2786:
2785:
2756:
2751:
2740:
2735:
2719:
2688:
2682:
2676:
2675:
2655:
2645:
2641:
2626:
2618:
2614:
2594:
2590:
2589:
2585:
2555:
2527:
2518:
2517:
2505:
2495:
2473:
2463:
2459:
2448:
2442:
2439:
2435:
2406:
2398:
2394:
2386:
2385:
2369:
2331:
2322:
2321:
2292:
2287:
2277:
2259:
2250:
2216:
2210:
2204:
2203:
2183:
2173:
2169:
2154:
2146:
2142:
2122:
2118:
2117:
2113:
2100:
2072:
2063:
2062:
2050:
2040:
2018:
2008:
2004:
1993:
1987:
1984:
1980:
1952:
1936:
1928:
1924:
1919:
1902:
1901:
1885:
1851:
1820:
1814:
1808:
1807:
1792:
1784:
1780:
1760:
1756:
1755:
1751:
1721:
1693:
1684:
1683:
1671:
1661:
1648:
1642:
1639:
1635:
1606:
1598:
1594:
1586:
1585:
1569:
1533:
1524:
1490:
1484:
1478:
1477:
1462:
1454:
1450:
1430:
1426:
1425:
1421:
1408:
1380:
1371:
1370:
1358:
1348:
1335:
1329:
1326:
1322:
1294:
1278:
1270:
1266:
1261:
1244:
1243:
1227:
1195:
1171:
1165:
1142:
1129:
1121:
1103:
1102:
1086:
1050:
1039:
1033:
1011:
1005:
992:
979:
973:
955:
945:
928:
927:
889:
878:
863:
836:
835:
776:
744:
720:
714:
691:
678:
670:
652:
651:
635:
599:
588:
582:
560:
554:
541:
528:
522:
504:
494:
477:
476:
445:
441:
437:
430:
396:
391:
378:
362:
361:
358:Reynolds number
289:
288:
266:
265:
243:
242:
239:is the density,
221:
220:
198:
197:
175:
174:
160:
126:
90:
77:
69:
51:
50:
12:
11:
5:
3414:
3412:
3404:
3403:
3401:Fluid dynamics
3393:
3392:
3387:
3386:
3359:
3324:
3297:
3286:(4): 701–704.
3280:Chem. Eng. Sci
3270:
3259:(4): 693–700.
3253:Chem. Eng. Sci
3243:
3232:(4): 654–665.
3216:
3189:
3157:
3125:
3091:
3072:(1): 148–159.
3051:
3040:(2): 334–340.
3024:
2997:
2986:(4): 463–471.
2970:
2938:
2907:
2906:
2904:
2901:
2900:
2899:
2894:
2887:
2884:
2846:
2845:
2836:
2834:
2823:
2819:
2814:
2809:
2802:
2798:
2794:
2789:
2782:
2777:
2773:
2763:
2759:
2755:
2747:
2743:
2738:
2732:
2726:
2722:
2717:
2713:
2707:
2702:
2695:
2691:
2686:
2679:
2673:
2667:
2662:
2658:
2652:
2649:
2644:
2639:
2633:
2629:
2625:
2622:
2617:
2613:
2608:
2604:
2601:
2598:
2593:
2588:
2582:
2579:
2574:
2571:
2567:
2562:
2558:
2552:
2546:
2541:
2534:
2530:
2526:
2521:
2512:
2508:
2502:
2498:
2491:
2485:
2480:
2476:
2470:
2467:
2462:
2458:
2452:
2446:
2438:
2431:
2428:
2423:
2419:
2413:
2409:
2405:
2402:
2397:
2393:
2375:
2374:
2365:
2363:
2350:
2345:
2338:
2334:
2330:
2325:
2318:
2313:
2309:
2299:
2295:
2291:
2284:
2280:
2274:
2266:
2263:
2257:
2254:
2246:
2241:
2235:
2230:
2223:
2219:
2214:
2207:
2201:
2195:
2190:
2186:
2180:
2177:
2172:
2167:
2161:
2157:
2153:
2150:
2145:
2141:
2136:
2132:
2129:
2126:
2121:
2116:
2112:
2107:
2103:
2097:
2091:
2086:
2079:
2075:
2071:
2066:
2057:
2053:
2047:
2043:
2036:
2030:
2025:
2021:
2015:
2012:
2007:
2003:
1997:
1991:
1983:
1975:
1972:
1966:
1960:
1956:
1949:
1943:
1939:
1935:
1932:
1927:
1923:
1914:
1911:
1891:
1890:
1881:
1879:
1868:
1864:
1858:
1854:
1849:
1845:
1839:
1834:
1827:
1823:
1818:
1811:
1805:
1799:
1795:
1791:
1788:
1783:
1779:
1774:
1770:
1767:
1764:
1759:
1754:
1748:
1745:
1740:
1737:
1733:
1728:
1724:
1718:
1712:
1707:
1700:
1696:
1692:
1687:
1678:
1674:
1668:
1664:
1658:
1652:
1646:
1638:
1631:
1628:
1623:
1619:
1613:
1609:
1605:
1602:
1597:
1593:
1575:
1574:
1565:
1563:
1552:
1548:
1540:
1537:
1531:
1528:
1520:
1515:
1509:
1504:
1497:
1493:
1488:
1481:
1475:
1469:
1465:
1461:
1458:
1453:
1449:
1444:
1440:
1437:
1434:
1429:
1424:
1420:
1415:
1411:
1405:
1399:
1394:
1387:
1383:
1379:
1374:
1365:
1361:
1355:
1351:
1345:
1339:
1333:
1325:
1317:
1314:
1308:
1302:
1298:
1291:
1285:
1281:
1277:
1274:
1269:
1265:
1256:
1253:
1233:
1232:
1223:
1221:
1210:
1207:
1202:
1198:
1193:
1189:
1184:
1180:
1177:
1174:
1168:
1164:
1161:
1158:
1154:
1149:
1145:
1136:
1132:
1127:
1124:
1117:
1114:
1110:
1092:
1091:
1082:
1080:
1069:
1065:
1058:
1054:
1046:
1042:
1037:
1029:
1024:
1020:
1017:
1014:
1008:
1004:
999:
995:
986:
982:
977:
969:
963:
959:
953:
949:
940:
937:
899:
896:
892:
885:
881:
876:
870:
866:
862:
857:
852:
848:
844:
810:when ∆X → 0.
782:
781:
772:
770:
759:
756:
751:
747:
742:
738:
733:
729:
726:
723:
717:
713:
710:
707:
703:
698:
694:
685:
681:
676:
673:
666:
663:
659:
641:
640:
631:
629:
618:
614:
607:
603:
595:
591:
586:
578:
573:
569:
566:
563:
557:
553:
548:
544:
535:
531:
526:
518:
512:
508:
502:
498:
489:
486:
443:
439:
435:
428:
403:
399:
394:
389:
385:
382:
374:
370:
310:
309:
296:
286:
273:
263:
250:
240:
228:
218:
205:
195:
182:
166:
165:
156:
154:
143:
139:
133:
129:
124:
118:
115:
109:
106:
102:
97:
93:
84:
80:
75:
72:
65:
62:
58:
13:
10:
9:
6:
4:
3:
2:
3413:
3402:
3399:
3398:
3396:
3382:
3378:
3374:
3370:
3363:
3360:
3355:
3351:
3347:
3343:
3339:
3335:
3328:
3325:
3320:
3316:
3312:
3308:
3301:
3298:
3293:
3289:
3285:
3281:
3274:
3271:
3266:
3262:
3258:
3254:
3247:
3244:
3239:
3235:
3231:
3227:
3220:
3217:
3212:
3208:
3204:
3200:
3193:
3190:
3185:
3181:
3177:
3173:
3166:
3164:
3162:
3158:
3153:
3149:
3145:
3141:
3134:
3132:
3130:
3126:
3121:
3117:
3113:
3109:
3102:
3100:
3098:
3096:
3092:
3087:
3083:
3079:
3075:
3071:
3067:
3060:
3058:
3056:
3052:
3047:
3043:
3039:
3035:
3028:
3025:
3020:
3016:
3012:
3008:
3001:
2998:
2993:
2989:
2985:
2981:
2974:
2971:
2966:
2962:
2958:
2954:
2947:
2945:
2943:
2939:
2934:
2930:
2926:
2922:
2915:
2913:
2909:
2902:
2898:
2895:
2893:
2890:
2889:
2885:
2883:
2881:
2880:
2875:
2874:
2868:
2867:
2862:
2861:
2852:
2844:
2837:
2835:
2821:
2812:
2807:
2800:
2796:
2792:
2787:
2780:
2775:
2771:
2761:
2757:
2753:
2745:
2741:
2736:
2730:
2724:
2720:
2711:
2705:
2700:
2693:
2689:
2684:
2677:
2671:
2665:
2660:
2656:
2650:
2647:
2642:
2637:
2631:
2627:
2623:
2620:
2615:
2611:
2606:
2602:
2599:
2596:
2591:
2586:
2580:
2577:
2572:
2569:
2560:
2556:
2550:
2544:
2539:
2532:
2528:
2524:
2519:
2510:
2506:
2500:
2496:
2489:
2483:
2478:
2474:
2468:
2465:
2460:
2456:
2450:
2444:
2436:
2429:
2426:
2421:
2417:
2411:
2407:
2403:
2400:
2395:
2384:
2383:
2380:
2373:
2366:
2364:
2348:
2343:
2336:
2332:
2328:
2323:
2316:
2311:
2307:
2297:
2293:
2289:
2282:
2278:
2272:
2264:
2261:
2255:
2252:
2244:
2239:
2233:
2228:
2221:
2217:
2212:
2205:
2199:
2193:
2188:
2184:
2178:
2175:
2170:
2165:
2159:
2155:
2151:
2148:
2143:
2139:
2134:
2130:
2127:
2124:
2119:
2114:
2110:
2105:
2101:
2095:
2089:
2084:
2077:
2073:
2069:
2064:
2055:
2051:
2045:
2041:
2034:
2028:
2023:
2019:
2013:
2010:
2005:
2001:
1995:
1989:
1981:
1973:
1970:
1964:
1958:
1954:
1947:
1941:
1937:
1933:
1930:
1925:
1921:
1912:
1909:
1900:
1899:
1896:
1889:
1882:
1880:
1866:
1862:
1856:
1852:
1843:
1837:
1832:
1825:
1821:
1816:
1809:
1803:
1797:
1793:
1789:
1786:
1781:
1777:
1772:
1768:
1765:
1762:
1757:
1752:
1746:
1743:
1738:
1735:
1726:
1722:
1716:
1710:
1705:
1698:
1694:
1690:
1685:
1676:
1672:
1666:
1662:
1656:
1650:
1644:
1636:
1629:
1626:
1621:
1617:
1611:
1607:
1603:
1600:
1595:
1584:
1583:
1580:
1573:
1566:
1564:
1550:
1546:
1538:
1535:
1529:
1526:
1518:
1513:
1507:
1502:
1495:
1491:
1486:
1479:
1473:
1467:
1463:
1459:
1456:
1451:
1447:
1442:
1438:
1435:
1432:
1427:
1422:
1418:
1413:
1409:
1403:
1397:
1392:
1385:
1381:
1377:
1372:
1363:
1359:
1353:
1349:
1343:
1337:
1331:
1323:
1315:
1312:
1306:
1300:
1296:
1289:
1283:
1279:
1275:
1272:
1267:
1263:
1254:
1251:
1242:
1241:
1238:
1231:
1224:
1222:
1208:
1205:
1200:
1196:
1187:
1182:
1178:
1175:
1172:
1166:
1162:
1159:
1156:
1147:
1143:
1134:
1130:
1125:
1122:
1115:
1112:
1101:
1100:
1097:
1090:
1083:
1081:
1067:
1063:
1056:
1052:
1044:
1040:
1035:
1027:
1022:
1018:
1015:
1012:
1006:
1002:
997:
993:
984:
980:
975:
967:
961:
957:
951:
947:
938:
935:
926:
925:
922:
919:
917:
916:
897:
894:
890:
883:
879:
874:
868:
864:
860:
855:
850:
846:
842:
833:
832:
827:
826:
821:
820:
815:
814:
809:
808:
803:
802:
797:
796:
791:
790:
780:
773:
771:
757:
754:
749:
745:
736:
731:
727:
724:
721:
715:
711:
708:
705:
696:
692:
683:
679:
674:
671:
664:
661:
650:
649:
646:
639:
632:
630:
616:
612:
605:
601:
593:
589:
584:
576:
571:
567:
564:
561:
555:
551:
546:
542:
533:
529:
524:
516:
510:
506:
500:
496:
487:
484:
475:
474:
471:
463:
459:
451:
447:
434:
427:
423:
401:
397:
392:
387:
383:
380:
372:
368:
359:
355:
351:
347:
343:
340: =
339:
335:
330:
326:
322:
318:
315: =
314:
294:
287:
271:
264:
248:
241:
226:
219:
203:
196:
180:
173:
172:
171:
164:
157:
155:
141:
137:
131:
127:
116:
113:
107:
104:
95:
91:
82:
78:
73:
70:
63:
60:
49:
48:
41:
37:
35:
26:
22:
19:
3372:
3368:
3362:
3337:
3333:
3327:
3310:
3306:
3300:
3283:
3279:
3273:
3256:
3252:
3246:
3229:
3225:
3219:
3202:
3198:
3192:
3175:
3171:
3143:
3139:
3111:
3107:
3069:
3065:
3037:
3033:
3027:
3010:
3006:
3000:
2983:
2979:
2973:
2956:
2952:
2924:
2920:
2877:
2871:
2864:
2858:
2857:
2838:
2378:
2367:
1894:
1883:
1578:
1567:
1236:
1225:
1095:
1084:
920:
913:
829:
823:
817:
811:
805:
799:
793:
787:
785:
774:
644:
633:
468:
456:
432:
425:
353:
349:
345:
341:
337:
328:
324:
320:
316:
312:
311:
169:
158:
31:
17:
15:
3334:Fuel Cells
3313:(1): 1–6.
2903:References
2897:Fuel Cells
2818:Δ
2737:ρ
2716:Δ
2651:−
2628:β
2624:−
2612:−
2603:β
2600:−
2578:ρ
2566:Δ
2469:−
2457:−
2427:ρ
2404:−
2392:Δ
2179:−
2156:β
2152:−
2140:−
2131:β
2128:−
2014:−
2002:−
1934:−
1913:ρ
1848:Δ
1794:β
1790:−
1778:−
1769:β
1766:−
1744:ρ
1732:Δ
1627:ρ
1604:−
1592:Δ
1464:β
1460:−
1448:−
1439:β
1436:−
1276:−
1255:ρ
1192:Δ
1179:β
1176:−
1163:ρ
1153:Δ
1123:ρ
1116:−
1109:Δ
1019:β
1016:−
968:−
939:ρ
895:−
741:Δ
728:β
725:−
712:ρ
702:Δ
672:ρ
658:Δ
568:β
565:−
488:ρ
393:π
384:μ
334:Ohm's law
227:ρ
123:Δ
114:ρ
101:Δ
71:ρ
57:Δ
3395:Category
3354:96529759
3205:: 7–12.
2886:See also
3074:Bibcode
356:is the
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