Laminar flows continued transition

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. 

A link between laminar and turbulent lifted flames has been demonstrated based on the observation of a continuous transition from laminar to turbulent lifted flames, as shown in Figure 4.3.13 [56]. The flame attached to the nozzle lifted off in the laminar regime, experienced the transition by the jet breakup characteristics, and became turbulent lifted flames as the nozzle flow became turbulent. Subsequently, the liftoff height increased linearly and finally blowout (BO) occurred. This continuous transition suggested that tribrachial flames observed in laminar lifted flames could play an important role in the stabilization of turbulent lifted flames. Recent measurements supported the existence of tribrachial structure at turbulent lifted edges [57], with the OH zone indicating that the diffusion reaction zone is surrounded by the rich and lean reaction zones. 

In many cases, a region of laniinar flow will have to be allowed for over the initial part of the body. The calculation bf this portion of the flow can be accomplished by using the same equations as for turbulent flow with E set equal to 0. This calculation can be started using initial conditions of the type discussed in Chapter 3 for purely laminar flows. From the point where transition is assumed to occur, the calculation can be continued using the equations presented above to describe E. 

Rollin et al. have published results for the continuous emulsion polymerization of styrene in a tubular reactor the particular concern is with the transition between laminar flow and turbulent flow in such a reactor. Chiang and Thompson have studied the factors which affect the stability of a continuous emulsion polymerization reactor. Birtwistle, Blackley, and Jeffershave examined a modification of a model proposed by Brooks, in order to ascertain whether it permits the possibility of either periodic fluctuations in rate of polymerization in the vicinity of the steady state, or sustained fluctuations in rate of polymerization in any physically attainable state. The conclusion is reached that neither of these possibilities is realizable in reaction systems which conform to the model considered, nor are sustained oscillations possible in reaction systems which conform to certain variants of the model. 

Forced convection cooling can be divided into laminar flow and turbulent flow. The transition from laminar to turbulent flow in air usually occurs at a velocity of 180 ft/min (180 Ifm). In laminar (or sfreamline) flow, the fluid particles follow a smooth, continuous path where the velocity vectors of the particles are always parallel and never intersect. The heat is transferred by molecular conduction in the fluid and by the solid-fluid interface. Tm-bulent flow, characterized by the irregular motion of fluid particles, has eddies in the fluid in which the particles are continuously mixed and rearranged. The heat is transferred in turbulent flow from the eddies back and forth across the streamlines. The greater heat transfer occurs for turbulent flow. 

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. 

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 ... 

During recent years experimental work continued actively upon the macroscopic aspects of thermal transfer. Much work has been done with fluidized beds. Jakob (D5, J2) made some progress in an attempt to correlate the thermal transport to fluidized beds with transfer to plane surfaces. This contribution supplements work by Bartholomew (B3) and Wamsley (Wl) upon fluidized beds and by Schuler (S10) upon transport in fixed-bed reactors. The influence of thermal convection upon laminar boundary layers and their transition to turbulent boundary layers was considered by Merk and Prins (M5). Monaghan (M7) made available a useful approach to the estimation of thermal transport associated with the supersonic flow of a compressible fluid. Monaghan s approximation of Crocco s more general solution (C9) of the momentum and thermal transport in laminar compressible boundary flow permits a rather satisfactory evaluation of the transport from supersonic compressible flow without the need for a detailed iterative solution of the boundary transport for each specific situation. None of these references bears directly on the problem of turbulence in thermal transport and for that reason they have not been treated in detail. 

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). 

Measurements of kinetic parameters of liquid-phase reactions can be performed in apparata without phase transition (rapid-mixing method [66], stopped-flow method [67], etc.) or in apparata with phase transition of the gaseous components (laminar jet absorber [68], stirred cell reactor [69], etc.). In experiments without phase transition, the studied gas is dissolved physically in a liquid and subsequently mixed with the liquid absorbent to be examined, in a way that ensures a perfect mixing. Afterwards, the reaction conversion is determined via the temperature evolution in the reactor (rapid mixing) or with an indicator (stopped flow). The reaction kinetics can then be deduced from the conversion. In experiments with phase transition, additionally, the phase equilibrium and mass transport must be taken into account as the gaseous component must penetrate into the liquid phase before it reacts. In the laminar jet absorber, a liquid jet of a very small diameter passes continuously through a chamber filled with the gas to be examined. In order to determine the reaction rate constant at a certain temperature, the jet length and diameter as well as the amount of gas absorbed per time unit must be known. 

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... 

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... 

Since the speed near the surface in a laminar boundary layer has a lower velocity than its turbulent counterpart, the laminar boundary layer is more likely to separate. When this occurs, the laminar boundary layer leaves the surface and usually undergoes a transition to a turbulent flow away from the surface. This process takes place over a certain distance that is inversely related to Re, but if it happens quickly enough, the flow may reattach as a turbulent boundary layer and continue along the surface. To compute when the separation will occur, we can solve the Navier-Stokes equations or apply one of the several separation criteria to the solutions of the boundary layer equations. 

These equations are based on experimental data for 18 different offset strip fin geometries, and they represent the data continuously in the laminar, transition, and turbulent flow regions, as shown in Fig. 11.13c. The development of accurate power-law correlations for a variety of enhancement configurations is possible when large databases are available. 

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. 

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