Laminar flow special case

Topics that acquire special importance on the industrial scale are the quality of mixing in tanks and the residence time distribution in vessels where plug flow may be the goal. The information about agitation in tanks described for gas/liquid and slurry reactions is largely apphcable here. The relation between heat transfer and agitation also is discussed elsewhere in this Handbook. Residence time distribution is covered at length under Reactor Efficiency. A special case is that of laminar and related flow distributions characteristic of non-Newtonian fluids, which often occiu s in polymerization reactors. 

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). 

A special kind of tubular flow reactor has laminar flow. The specific rate of such a case is found and compared with plug flow in problem P3.09.15. 

Laminar flow In circular tubes with parabolic velocity distribution Is known as Poiseuille flow. This special case is found frequently in vacuum technology. Viscous flow will generally be found where the molecules mean free path is considerably shorter than the diameter of the pipe X d. 

Most of the relief sizing equations given in Chapters 6-8 yield the two-phase required relief rate, W. The two-phase mass flow capacity per unit area, G, is then needed in order to obtain the required relief area. Chapter 9 contains important background information about two-phase flow, and calculation methods for G. Some system types are special cases involving highly viscous (laminar) flow, solids and/or... 

For the special case of slow laminar shear flow in the direction of the 1-axis eq. (2.35) appears to reduce to ... 

Theoretical treatment of smooth laminar film flow on vertical surface, with and without gas flow, including inertia effects. Nusselt equations (N6, N7) are shown to be special cases of the present solutions. 

Burke and Schumann [1] calculated the form of the flame surface for the very special case of combustion in parallel concentric laminar flows of fuel and oxygen or air. In doing so they did not consider in detail the phenomena taking place in the flame zone. 

Separation of Unstable Systems The buoyancy of bubbles suspended in liquid can frequently be depended upon to cause the bubbles to rise to the surface and separate. This is a special case of gravity settling. The mixture is allowed to stand at rest or is moved along a flow path in laminar flow until the bubbles have surfaced. Table 14-30 shows the calculated rate of rise of air bubbles at atmospheric pressure in water at 20°C (68°F) as a function of diameter. It will be observed that the velocity of rise for 10-pm bubbles is very low, so that long separating times would be required for gas which is more finely dispersed. 

There are some special cases in FFF related to the two extreme limits of the cross-field driving forces. In the first case, the cross-field force is zero, and no transverse solute migration is caused by outer fields. However, because of the shear forces, transverse movements may occur even under conditions of laminar flow. This phenomenon is called the tubular pinch effect . In this case, these shear forces lead to axial separation of various solutes. Small [63] made use of this phenomenon and named it hydrodynamic chromatography (HC). If thin capillaries are used for flow transport, this technique is also called capillary hydrodynamic fractionation (CHDF). A simple interpretation of the ability to separate is that the centers of the solute particles cannot approach the channel walls closer than their lateral dimensions. This means that just by their size larger particles are located in streamlines of higher flow velocities than smaller ones and are eluted first (opposite to the solution sequence in the classical FFF mode). For details on hydrodynamic chromatography,see [64-66]. 

Calculation of the pressure drop from the prescribed field v on 5 , A, and is thus reduced to a quadrature whenever the classical solution for laminar flow through the cylinder is already known. A relationship comparable to the above was previously given for the special case of a circular cylinder by Brenner (B13) along with an example of its utility in applications. 

The expressions for frequencies of bubble collision in laminar and turbulent flow which derived in the previous paragraphs make it possible to And the kernels of coagulation K co, V) and then proceed to solve the kinetic equation (25.1). Because the solution, generally speaking, presents significant mathematical difficulties, we shall only consider some simple special cases. 

Heat transfer in square bundles is roughly estimated, so special experiments are needed for verification. By now there are no convincing methods of extrapolation of the data on temperature non-uniformity in the area around fuel pin to the square grid bundle. As a first approximation, it can be assumed that values increase in the same way as in case of laminar flow. Unfortunately, this cannot be adopted for the peripheral pins where temperature pattern non-uniformity is significant. Therefore, additional experimental studies are needed. 

Laminar flow conditions cease to exist at Rcmod = 2100. The calculation of the critical velocity corresponding to Rcmod = 2100 requires an iterative procedure. For known rheology (p, m, n, Xq) and pipe diameter (D), a value of the wall shear stress is assumed which, in turn, allows the calculation of Rp, from equation (3.9), and Q and Qp from equations (3.14b) and (3.14a) respectively. Thus, all quanties are then known and the value of Rcmod can be calculated. The procedure is terminated when the value of x has been found which makes RCjnod = 2100, as illustrated in example 3.4 for the special case of n = 1, i.e., for the Bingham plastic model, and in example 3.5 for a Herschel-Bulkley fluid. Detailed comparisons between the predictions of equation (3.34) and experimental data reveal an improvement in the predictions, though the values of the critical velocity obtained using the criterion Rqmr = 2100 are only 20-25% lower than those predicted by equation (3.34). Furthermore, the two... 

Detachment of Adherent Particles Situated in a Laminar Sublayer with Turbulent Flow Over the Surface. Let us examine the conditions for the detachment of adherent particles under conditions of turbulent flow across a dust-covered surface. We will first dwell on the simplest special case in which the particles are located in the laminar sublayer (see Fig. X. 1. b, Position III). The velocity increases within the boundary layer as we go from the laminar sublayer 2 to the buffer layer 3 (curve c). Such an increase in velocity, in accordance with Eq. (X.3), leads to an increase in drag and in the number of detached particles. Hence the thickness of the laminar sublayer is an important quantity, through which we can evaluate the conditions for detachment of adherent particles. 

In RIM, as in injection moulding, whether thermoplastic or thermoset, there is a transition zone between the nozzle (mix head) and the mould. In injection moulding, this transition zone includes the sprue/runner/gate system, and, in special cases, can also include a static mixer. The analogy to RIM is very apt. The transition zone for the RIM process also includes a sprue/runner/gate system, as well as an after mixer. One function of the RIM transition zone is to convert the mixing head liquid stream from a turbulent state to one which is in a laminar flow mode. If this does not occur, air traps (i.e. large subsurface voids in the finished part) can become a problem. 

For a straight circular pipe when the value of the Reynolds number is less than 2100, the flow is always laminar. When the value is over 4000, the flow will be turbulent, except in very special cases. In between, which is called the transition region, the flow can be viscous or turbulent, depending upon the apparatus details, which cannot be predicted. 

It should be noted that Eq. 3-77 reduces to the special cases of a) single-phase turbulent flow N = 0) and (Z ) single-phase laminar flow N = u = v = OL... 

Taylor dispersion is a special case of convection, where the dispersion is caused by a mean velocity gradient. It is most often referred to in the case of laminar pipe flow, where axial dispersion arises due to the parabolic velocity gradient in the pipe. 

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