Laminar flow definition

Pressure Driven Laminar Flow Definition of pressure driven laminar flow ... 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

As flow is, by definition, unpredictable, there is no single equation that defines the rate of turbulent flow as there is with laminar flow. However, there is a number that can be calculated in order to identify whether fluid flow is likely to be laminar or turbulent and this is called Reynold s number (Re). 

By definition, the laminar-flow reactor is segregated. Each radial element of fluid is assumed to slide past its adjacent elements with no mixing. Thus, eqn. (34) may be used to predict reactor conversion. The case of a first-order reaction has been analysed by Cleland and Wilhelm [43]. As we have seen previously... 

Bischoff and Levenspiel (B14) present some calculations using existing experimental data to check the above predictions about the radial coefficients. For turbulent flow in empty tubes, the data of Lynn et al. (L20) were numerically averaged across the tube, and fair agreement found with the data of Fig. 12. The same was done for the packed-bed data of Dorweiler and Fahien (D20) using velocity profile data of Schwartz and Smith (Sll), and then comparing with Fig. 11. Unfortunately, the scatter in the data precluded an accurate check of the predictions. In order to prove the relationships conclusively, more precise experimental work would be needed. Probably the best type of system for this would be one in laminar flow, since the radial and axial coefficients for the general dispersion model are definitely known each is the molecular diffusivity. 

A characteristic of micro channel reactors is their narrow residence-time distribution. This is important, for example, to obtain clean products. This property is not imaginable without the influence of dispersion. Just considering the laminar flow would deliver an extremely wide residence-time distribution. The near wall flow is close to stagnation because a fluid element at the wall of the channel is, by definition, fixed to the wall for an endlessly long time, in contrast to the fast core flow. The phenomenon that prevents such a behavior is the known dispersion effect and is demonstrated in Figure 3.88. 

Transition from laminar to turbulent flow within the condensed film can occur when the vapor is condensed on a tall surface or on a tall vertical bank of horizontal tubes [45] to [47]. It has been found that the film Reynolds number, based on the mean velocity in the film, um, and the hydraulic diameter, D, can be used to characterize the conditions under which transition from laminar flow occurs. The mean velocity in the film is given by definition as ... 

We have already likened the macroscopic transport of heat and momentum in turbulent flow to their molecular counterparts in laminar flow, so the definition in Eq. (5-60) is a natural consequence of this analogy. To analyze molecular-transport problems (see, for example. Ref. 7, p. 369) one normally introduces the concept of mean free path, or the average distance a particle travels between collisions. Prandtl introduced a similar concept for describing turbulent-flow phenomena. The Prandtl mixing length is the distance traveled, on the average, by the turbulent lumps of fluid in a direction normal to the mean flow. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters De defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, De = 2 Q. L/ nAPY . Equivalent mameters are not the same as hydraulic diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/IkDeH). Equivalent diameter De is not to be used in the friction factor and Reynolds number / 16/Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydrauhc diameter Dh used for turbulent flow. 

Figure 6 (a) Development of laminar flow in a rod bundle (side view) (b) definition of wall-to-wall distances in a rod bundle (top view). 

In laminar flow, /f reduces to a delta function. In turbulent flow, /f can be modeled using PDF methods. Thus, by definition, there is no mass transfer between phases. 

In a laminar flow at a definite shear rate, different parts of the polymer molecule move at different rates depending on whether they are in the zone of rapid or relatively slow flow, and as a result the polymer molecule is under the action of a couple of forces which makes it rotate in the flow. Rotation and translational movement of polymer molecules causes friction between their chain segments and the solvent molecules. This is manifested in an increase in viscosity of the solution compared to the viscosity of the pure solvent. 

Heat exchange in fully developed laminar flow of fluids in tubes of various cross-sections was studied in many papers (e.g., see [80, 253, 341]). In what follows, we present some definitive results for the limit Nusselt numbers corresponding to the region of heat stabilization in the flow in the case of high Peclet numbers (when the molecular heat transfer can be neglected). 

Two of the various types of laminar flow, and the effects such flows have on drops, are depicted in Figure 11.7. It also gives the definitions of the velocity gradient. 

When the definitions are inserted into Eq. (4) for laminar flow, one obtains for the pressure drop ... 

Thus, in laminar flow in a straight channel, the density of the fluid does not affect the pressure drop. To use the correlation for friction factor, or Fig. 8.1(a), one must know the density. This problem is avoided if one plots the product of friction factor and Reynolds number versus Reynolds number, since from their definition ... 

For circular pipes, Rh = R- The reader is cautioned that some definitions of Rh omit the factor of 2 shown in Equation 3.22 so that the result must be multiplied by 2 for use in equations such as 3.18 and 3.19. The use of Rh is not recommended for laminar flow, but alternatives are available in the literature. Also, the method of false transients applied to PDEs in Chapter 16 can be used to calculate laminar velocity profiles in ducts with noncircular cross sections. For turbulent, low-pressure gas flows in rectangular ducts, the American Society of Heating, Refrigerating and Air Conditioning Engineers recommends use of an equivalent diameter defined as... 

REYNOLDS NUMBER AND TRANSITION FROM LAMINAR TO TURBULENT PLOW. Reynolds studied the conditions under which one type of flow changes into the other and found that the critical velocity, at which laminar flow changes into turbulent flow, depends on four quantities the diameter of the tube and the viscosity, density, and average linear velocity the liquid. Furthermore, he found that these four factors can be combined into one group and that the change in kind of flow occurs at a definite value of the group. The grouping of variables so found was... 

This is the definition of the Reynolds number Nr, given in Eq. (3,9). This Reynolds number reduces to the Reynolds number for a newtonian fluid when n = 1, and it reproduces the linear portion of the logarithmic plot of / versus Nr, with a slope of — 1, for the laminar flow of newtonian fluids. 

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