Laminar flows continued

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. 

Some of the devices covered here handle the solids burden in a static or laminar-flowing bed. Other devices can be considered as continuously agitated kettles in their heat-transfer aspect. For the latter, unit-area performance rates are higher. 

Darrieus and Landau established that a planar laminar premixed flame is intrinsically unstable, and many studies have been devoted to this phenomenon, theoretically, numerically, and experimentally. The question is then whether a turbulent flame is the final state, saturated but continuously fluctuating, of an unstable laminar flame, similar to a turbulent inert flow, which is the product of loss of stability of a laminar flow. Indeed, should it exist, this kind of flame does constitute a clearly and simply well-posed problem, eventually free from any boundary conditions when the flame has been initiated in one point far from the walls. 

This regime is characterized by the presence of two continuous fluid phases and an interface which can easily be described. The term separated flows is frequently employed to describe these situations in both horizontal and vertical systems. Some flow patterns in Regime I are advantageous for transferring heat between the tube wall and the fluid mixture or for carrying out two-phase reactions. The special case of laminar-laminar flow is included in this regime, and two studies seem to be of interest, Byers and King (B7) and Bentwich and Sideman (B3). 

For a few highly idealized systems, the residence time distribution function can be determined a priori without the need for experimental work. These systems include our two idealized flow reactors—the plug flow reactor and the continuous stirred tank reactor—and the tubular laminar flow reactor. The F(t) and response curves for each of these three types of well-characterized flow patterns will be developed in turn. 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

Our treatment of Chemical Reaction Engineering begins in Chapters 1 and 2 and continues in Chapters 11-24. After an introduction (Chapter 11) surveying the field, the next five Chapters (12-16) are devoted to performance and design characteristics of four ideal reactor models (batch, CSTR, plug-flow, and laminar-flow), and to the characteristics of various types of ideal flow involved in continuous-flow reactors. Chapter 17 deals with comparisons and combinations of ideal reactors. Chapter 18 deals with ideal reactors for complex (multireaction) systems. Chapters 19 and 20 treat nonideal flow and reactor considerations taking this into account. Chapters 21-24 provide an introduction to reactors for multiphase systems, including fixed-bed catalytic reactors, fluidized-bed reactors, and reactors for gas-solid and gas-liquid reactions. 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Figure 8. Variation of the hydrodynamic boundary layer thickness (So, equation (26), continuous line), the diffusion layer thickness (<5,-, equation (34), dotted line) and the ensuing local flux (/, equation (32), dashed line) with respect to the distance from the leading edge (y) in the case of laminar flow parallel to an active plane (the surface is a sink for species i). Parameters /), = 10-9nrs, v= 10 3ms, c = lmolm-3, and v — 10-6 m2 s 1. Notice that c5, o (as required for the derivation of the flux equation (32)), and that the flux decreases when <5, increases...

Another variety of the continuous-coupling technique operates by transporting the coupling component suspension as a laminar flow upwards inside a vertical reaction tube. Portions of the diazonium compound, dissolved in an acidic aqueous medium, are added through appropriately located inlets in the walls of the reaction tube. The concentration of the added solution decreases as the reaction mixture flows upward and is designed to synchronize the uppermost inlet for the diazonium salt solution with the stoichiometric end point of the coupling reaction. 

The solid phase of a batch bed can either be fixed, moving or mixed, whereas continuous fuel beds can either be moving or mixed. A successful and comprehensive mathematical model needs to consider these three modes of fuel-bed movement. There is a close analogy between the fluid-solid dynamics, where we have proposed fixed, moving, and mixed, and the fluid dynamics where the corresponding terminology is stagnancy, laminar flow and turbulent flow, respectively. 

Example 2-6 Consider the situation where the reactants at constant density are fed continuously into a pipe of length L instead of a tank of volume V as in the batch reactor. The reactants react as they flow down the tube with a speed u, and we assume that they flow as a plug without mixing or developing the laminar flow profile. Show that the conversion of the reactants is exactly the same in these very different reactor configurations. 

Diffusive mixing in continuous laminar flow systems. Chem. Eng. Set. 21,... 

Laminar Flow A condition where fluid mass moves in a continuous, streamlined parallel path. 

All HVAC and laminar flow systems are to be in continuous operation when performing these tests. 

LAMINAR FLOW. A condition of fluid flow in a closed conduit in which the fluid panicles or "streams tend to move parallel to the flow axis and not mix. This behavior is characteristic of low flow rates and high viscosity fluid flows. As the flow rate increases (or viscosity significantly decreasesi. the streams continue to flow parallel until a velocity is reached where the streams waver and suddenly break into a diffused pattern. This point is called the critical velocity. See also Turbulent Flow. 

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. 

The profiles in Figure 3.37B represent the situation in which a potential is applied that requires equal concentrations of O and R at the electrode surface to satisfy the Nernst equation (i.e., E = Eq R). To fulfill this requirement, the electrode electrolyzes O to R at the rate required to maintain equal concentrations of O and R at the surface. If this potential is maintained, a continuous electrolysis of O to R is necessary to maintain surface concentrations because R diffuses away from the interface across the stagnant layer and is then swept away by the laminar flow. 

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