Laminar flows continued Reynolds number

Polystyrene can be easily prepared by emulsion or suspension techniques. Harkins (1 ), Smith and Ewart(2) and Garden ( ) have described the mechanisms of emulsTon polymerization in batch reactors, and the results have been extended to a series of continuous stirred tank reactors (CSTR)( o Much information on continuous emulsion reactors Ts documented in the patent literature, with such innovations as use of a seed latex (5), use of pulsatile flow to reduce plugging of the tube ( ), and turbulent flow to reduce plugging (7 ). Feldon (8) discusses the tubular polymerization of SBR rubber wTth laminar flow (at Reynolds numbers of 660). There have been recent studies on continuous stirred tank reactors utilizing Smith-Ewart kinetics in a single CSTR ( ) as well as predictions of particle size distribution (10). Continuous tubular reactors have been examined for non-polymeric reactions (1 1 ) and polymeric reactions (12.1 31 The objective of this study was to develop a model for the continuous emulsion polymerization of styrene in a tubular reactor, and to verify the model with experimental data. 

At velocities greater than the critical, the fluid velocity profile in the conduit is uniform across the conduit diameter except for a thin layer of fluid at the conduit wall. This boundary layer continues to move in laminar flow. In connection with flow measurement, most flowmeters have constant coefficients under turbulent flow conditions. Some flowmeters have the advantage of constant coefficients over Reynolds Number ranges encompassing both turbulent and laminar flows. See also Fluid and Fluid Flow and Reynolds Number. 

Figure 2 shows an example of a static mixer and a schematic representation of how such structures operate—see, for example, [1] and [2]. The open intersecting channels divide the main fluid stream into a number of substreams. In addition to the lateral displacement caused by the obliquity of the channels, a fraction of each substream shears off into the adjacent channel at every intersection. This continuous division and recombination of the substreams causes transition from laminar to turbulent flow at Reynolds numbers (based on channel hydraulic diameter) as low as 20Q-300 and results in... 

These relations are valid for laminar flow of the liquid. Laminar flow continues until around a Reynolds number Re = pVz,nvfiR/p) of 2100. For a given Q, AP increases linearly with L it increases drastically as R is reduced. 

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

Continuous Flat Surface Boundaiy layers on continuous surfaces drawn through a stagnant fluid are shown in Fig. 6-48. Figure 6-48 7 shows the continuous flat surface (Saldadis, AIChE J., 7, 26—28, 221-225, 467-472 [1961]). The critical Reynolds number for transition to turbulent flow may be greater than the 500,000 value for the finite flat-plate case discussed previously (Tsou, Sparrow, and Kurtz, J. FluidMech., 26,145—161 [1966]). For a laminar boundary layer, the thickness is given by... 

Continuous Cylindrical Surface The continuous surface shown in Fig. 6-48b is apphcable, for example, for a wire drawn through a stagnant fluid (Sakiadis, AIChE ]., 7, 26-28, 221-225, 467-472 [1961]). The critical-length Reynolds number for transition is Re = 200,000. The laminar boundary laver thickness, total drag, and entrainment flow rate may be obtained from Fig. 6-49 the drag and entrainment rate are obtained from the momentum area 0 and displacement area A evaluated at x = L. 

Coefficient A and exponent a can be evaluated readily from data on Re and T. The dimensionless groups are presented on a single plot in Figure 15. The plot of the function = f (Re) is constructed from three separate sections. These sections of the curve correspond to the three regimes of flow. The laminar regime is expressed by a section of straight line having a slope P = 135 with respect to the x-axis. This section corresponds to the critical Reynolds number, Re < 0.2. This means that the exponent a in equation 53 is equal to 1. At this a value, the continuous-phase density term, p, in equation 46 vanishes. 

Pierce proposes and illustrates good agreement between the test data and the correlation for a smooth continuous curve for the Colburn factor over the entire range of Reynolds numbers for the laminar, transition, and turbulent flow regimes inside smooth tubes ... 

It is well known in fluid flow studies that below a certain critical value of the Reynolds number the flow will be mainly laminar in nature, while above this value, turbulence plays an increasingly important part. The same is true of film flow, though it must be remembered that in thin films a large part of the total film thickness continues to be occupied by the relatively nonturbulent laminar sublayer, even at large flow rates (N e ARecr J- Hence, the transition from laminar to turbulent flow cannot be expected to be so sharply marked as in the case of pipe flow (D12). Nevertheless, it is of value to subdivide film flow into laminar and turbulent regimes depending on whether (Ar6 5 Ar u). 

The flowing emulsion was assumed homogeneous, so that the continuity equations could be used. Additional assumptions were the fluid is an incompressible Newtonian with constant properties the flow is laminar at the maximum experimental Reynolds number of 210 and less there is negligible viscous heating flow is at steady... 

Rcout - 253.3 this indicates continued wavy laminar flow The flection of water evaporated is (Remin - Re ut) / Rfiin = 0.726. Since the averagip heat transfer coefficient can be related to the inlet and exit Reynolds numbers as... 

Agitated dispersions at low impeller speeds or high continuous phase viscosities are in a state of laminar or transition flow. At low impeller Reynolds number, (NRe)T < 15, the flow is laminar around the impeller... 

In aU methods there is Hquid flow with unbounded and strongly confined flow. In the unbounded flow, any droplets are surrounded by a large amount of flowing liquid (the confining walls of the apparatus are far away from most droplets), while the forces can be either frictional (mostly viscous) or inertial. Viscous forces cause shear stresses to act on the interface between the droplets and the continuous phase (primarily in the direction of the interface). The shear stresses can be generated by either laminar flow (LV) or turbulent flow (TV) this depends on the Reynolds number R, ... 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

However, the curve of the sphere drag coefficient has some marked differences from the friction factor plot. It does not continue smoothly to higher and higher Reynolds numbers, as does the / curve instead, it takes a sharp drop at an of about 300,000. Also it does not show the upward jump that characterizes the laminar-turbulent transition in pipe flow. Both differences are due to the different shapes of the two systems. In a pipe all the fluid is in a confined area, and the change from laminar to turbulent flow affects all the fluid (except for a very thin film at the wall). Around a sphere the fluid extends in all directions to infinity (actually the fluid is not infinite, but if the distance to the nearest obstruction is 100 sphere diameters, we may consider it so), and no matter how fast the sphere is moving relative to the fluid, the entire fluid cannot be set in turbulent flow by the sphere. Thus, there cannot be the sudden laminar-turbulent transition for the entire flow, which causes the jump in Fig. 6.10. The flow very near the sphere, however, can make the sudden switch, and the switch is the cause of the sudden drop in Q at =300,(300. This sudden drop in drag coefficient is discussed in Sec. 11.6. Leaving until Chaps. 10 and 11 the reasons why the curves in Fig. 6.22 have the shapes they do, for now we simply accept the curves as correct representations of experimental facts and show how to use them to solve various problems. 

One concludes from (12-17a) and (12-17c) that neither 4> nor Vp is a function of the Reynolds number because Re does not appear in either equation. Consequently, dynamic pressure and its gradient in the x direction are not functions of the Reynolds number because Re does not appear in the dimensionless potential flow equation of motion, given by (12-16), from which /dx is calculated. In summary, two-dimensional momentum boundary layer problems in the laminar flow regime (1) focus on the component of the equation of motion in the primary flow direction, (2) use the equation of continuity to calculate the other velocity component transverse to the primary flow direction, (3) use potential flow theory far from a fluid-solid interface to calculate the important component of the dynamic pressure gradient, and (4) impose this pressme gradient across the momentum boundary layer. The following set of dimensionless equations must be solved for Vp, IP, u, and v in sequential order. The first three equations below are solved separately, but the last two equations are coupled ... 

Based on the dimensionless equation of continuity [i.e., eq. (12-18dimensionless equation of motion [i.e., eq. (12-18e)] in the laminar flow regime, one concludes that both v and v are functions of dimensionless spatial coordinates x and y, as well as the Reynolds number and the geometiy of the flow configuration. Unfortunately, the previous set of equations does not reveal the specific dependence of v and v on Re. 

Fluid flow in small devices acts differently from those in macroscopic scale. The Reynolds number (Re) is the most often mentioned dimensionless number in fluid mechanics. The Re number, defined by pf/L/p, represents the ratio of inertial forces to viscous ones. In most circumstances involved in micro- and nanofluidics, the Re number is at least one order of magnitude smaller than unity, ruling out any turbulence flows in micro-/nanochannels. Inertial force plays an insignificant role in microfluidics, and as systems continue to scale down, it will become even less important. For such small Re number flows, the convective term (pu Vu) of Navier-Stokes equations can be dropped. Without this nonlinear convection, simple micro-/ nanofluidic systems have laminar, deterministic flow patterns. They have parabolic velocity... 

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