Transitional flow defined

In the first two cases the Navier-Stokes equation can be applied, in the second case with modified boundary conditions. The computationally most difficult case is the transition flow regime, which, however, might be encountered in micro-reactor systems. Clearly, the defined ranges of Knudsen numbers are not rigid rather they vary from case to case. However, the numbers given above are guidelines applicable to many situations encoimtered in practice. 

Dl) uses the adjusted sphere, i.e., the sphere with the same value of C in free-moleeule flow. The justifieation for this approach is that it gives the correct result in the limits of high and low Kn, and is therefore likely to be a good approximation in slip and transition flow as well. Figure 10.14 shows values for the diameter of the adjusted sphere for spheroids and cylinders, taken from Dahneke s tabulation. For cubes the adjusted sphere diameter to be used in defining Kn in Eq. (10-50) is 1.43 times the length of a side. 

The Reynolds number at observed transition location (defined as a location where the intermittency factor is about 0.1 i.e. the flow is 10 % of time turbulent and rest of the time it is laminar) for zero pressure gradient flat plate boundary layer is of the order of 3.5 X 10 . This corresponds to Re = 950. The distance between the point of instability and the point of transition depends on the degree of amplification and the kind of disturbance present with the oncoming flow. This calls for a study of local and total amplification of disturbances. The following description is as developed in Arnal (1984) for two-dimensional incompressible flows. 

Viscous flow is a bit complex and can be further divided into laminar flow, turbulent flow, and transition flow. At a low gas flow velocity, the flow is laminar where layered, parallel flow lines may be visnalized, no perpendicular velocity is present, and mixing of the gas is by diffnsion only. In this flow, the velocity is zero at the gas-wall interface and gradually increases as one moves away from the interface, reaching a maximnm at the center when gas is flowing inside a pipe. Viscons flow behavior can be defined by the so-called Reynolds number. Re, given below taking gas flow inside a pipe ... 

A whole range of cations and anions in different combinations have been explored. The results are surprising. Measurements of coalescence rates for a range of typical electrolytes as a function of electrolyte concentration are shown in Fig. (3.5). There is a correlation between valency of the salt and transition concentration, defined as 50% bubble coalescence, with more highly charged salt effective at lower concentration. The effect is independent of gas flow rate. All the results scale with Debye length (ionic strength). Some salts and acids have no effect at all on bubble coalescence, a situation summarised in Table 3.1. 

The Knudsen number (Kn) is used to determine the different regimes of the gas flow. These regimes can be divided into continuous flow, transitional flow and free molecular flow. This division is based on the understanding that the flow behaviour differs within each of the flow regimes. The Knudsen number is defined as... 

Reynold s number It describes the nature of hydraulic regime such as laminar flow, transitional flow or turbulent flow. It is defined as the ratio of the product of hydraulic diameter and mass flow velocity to that of fluid viscosity. Mass velocity is the product of cross-flow velocity and fluid density. Laminar flow exists for Reynold s numbers below 2000 whereas turbulent is characterized by Reynold s numbers greater than 4000. 

Critical Reynolds Number. The Reynolds number, defined as umDhlv, is widely adopted to identify flow status such as laminar, turbulent, and transition flows. A great number of experimental investigations have been performed to ascertain the critical Reynolds number at which laminar flow transits to turbulent flow. It has been found that the transition from laminar flow to fully developed turbulent flow occurs in the range of 2300 Re < 104 for circular ducts [41]. Correspondingly, flow in this region is termed transition flow. More conservatively, the lower end of the critical Reynolds number is set at 2100 in most applications. 

Interestingly, in tube flow the Reynolds number clearly indicates the range of a given type of flow. For example, for Reynolds numbers up to 2100 the flow is laminar from 2100 to about 4000 we have transition flow and from 4000 on up, the flow is turbulent. Actually, it is possible to extend laminar flow beyond 2100 if done in carefully controlled experiments. This, however, is not the usual situation found in nature. Furthermore, it should be mentioned that the boundary between transition and turbulent flow is not always clearly defined. The ranges given above are considered to hold for most situations that would be encountered. 

As of this writing, it has not been possible to use the seismic data which defines the volume of the reservoir to also determine the joint stmcture. Extended flow testing is the most direct measure of the efficiency and sustainabiUty of energy recovery from the reservoir. The use of chemical tracers in the circulating fluid can also provide valuable supporting data with regard to the multiplicity of flow paths and the transit time of fluid within the reservoir (37). 

Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

F r d ic Current. The double layer is a leaky capacitor because Faradaic current flows around it. This leaky nature can be represented by a voltage-dependent resistance placed in parallel and called the charge-transfer resistance. Basically, the electrochemical reaction at the electrode surface consists of four thermodynamically defined states, two each on either side of a transition state. These are (11) (/) oxidized species beyond the diffuse double layer and n electrons in the electrode and (2) oxidized species within the outer Helmholtz plane and n electrons in the electrode, on one side of the transition state and (J) reduced species within the outer Helmholtz plane and (4) reduced species beyond the diffuse double layer, on the other. 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

This is the basis for establishing the condition or type of fluid flow in a pipe. Reynolds numbers below 2000 to 2100 are usually considered to define laminar or thscous flow numbers from 2000 to 3000-4000 to define a transition region of peculiar flow, and numbers above 4000 to define a state of turbulent flow. Reference to Figure 2-3 and Figure 2-11 will identify these regions, and the friction factors associated with them [2]. 

In order to predict Lhe transition point from stable streamline to stable turbulent flow, it is necessary to define a modified Reynolds number, though it is not clear that the same sharp transition in flow regime always occurs. Particular attention will be paid to flow in pipes of circular cross-section, but the methods are applicable to other geometries (annuli, between flat plates, and so on) as in the case of Newtonian fluids, and the methods described earlier for flow between plates, through an annulus or down a surface can be adapted to take account of non-Newtonian characteristics of the fluid. 

The critical value of the Reynolds number (Remit) for the transition from laminar to turbulent flow may be calculated from the Ryan and Johnson001 stability parameter, defined earlier by equation 3.56. For a power-law fluid, this becomes ... 

Equation 5.2 is found to hold well for non-Newtonian shear-thinning suspensions as well, provided that the liquid flow is turbulent. However, for laminar flow of the liquid, equation 5.2 considerably overpredicts the liquid hold-up e/,. The extent of overprediction increases as the degree of shear-thinning increases and as the liquid Reynolds number becomes progressively less. A modified parameter X has therefore been defined 16 171 for a power-law fluid (Chapter 3) in such a way that it reduces to X both at the superficial velocity uL equal to the transitional velocity (m )f from streamline to turbulent flow and when the liquid exhibits Newtonian properties. The parameter X is defined by the relation... 

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