Channel flow steady state

Macbeth, R. V., 1963b, Forced Convection Burnout in Simple, Uniformly Heated Channels A Detailed Analysis of World Data, European Atomic Energy Community Symp. on Two Phase Flow, Steady State Burnout and Hydrodynamic Instability, Stockholm, Sweden. (5)... 

Assuming a variable-area channel and steady-state flow, the integrals can be evaluated for the differential control volume (Fig. 16.3) as... 

In the FFF a field or gradient is applied in a direction, perpendicular to the axis of a narrow flow channel. At the same time a solvent is forced steadily through the channel forming a cross-sectional flow profile of parabolic shape. When a polymer sample is injected into the channel, a steady state is soon reached in which the field induced motion and the opposed diffusion are exactly balanced. The continuous size-distribution of the polymer will migrate with a continuous spectrum of velocities and will emerge at the end of the flow channel with a continuous time distribution. When processed through a detector and its associated electronics, the time distribution becomes an elution (retention) spectrum. 

Field-flow fractionation experiments are mainly performed in a thin ribbonlike channel with tapered inlet and outlet ends (see Fig. 1). This simple geometry is advantageous for the exact and simple calculation of separation characteristics in FFF Theories of infinite parallel plates are often used to describe the behavior of analytes because the cross-sectional aspect ratio of the channel is usually large and, thus, the end effects can be neglected. This means that the flow velocity and concentration profiles are not dependent on the coordinate y. It has been shown that, under suitable conditions, the analytes move along the channel as steady-state zones. Then, equilibrium concentration profiles of analytes can be easily calculated. 

The friction loss for a closed channel and steady-state single-phase flow was examined in Chapter 2. Using the Darcy factor, the head loss for an open launder can be expressed in terms of the hydraulic radius ... 

Liquid flows under steady-state conditions along an open channel of fixed inclination to the horizontal. On what factors will the depth of liquid in the channel depend Oblain a relationship between the variables using dimensional analysis. 

The behavior of liquid flow in micro-tubes and channels depends not only on the absolute value of the viscosity but also on its dependence on temperature. The nonlinear character of this dependence is a source of an important phenomenon - hydrodynamic thermal explosion, which is a sharp change of flow parameters at small temperature disturbances due to viscous dissipation. This is accompanied by radical changes of flow characteristics. Bastanjian et al. (1965) showed that under certain conditions the steady-state flow cannot exist, and an oscillatory regime begins. 

For adiabatic, steady-state, and developed gas-liquid two-phase flow in a smooth pipe, assuming immiscible and incompressible phases, the essential variables are pu, pG, Pl, Pg, cr, dh, g, 9, Uls, and Uas, where subscripts L and G represent liquid and gas (or vapor), respectively, p is the density, p is the viscosity, cr is the surface tension, dh is the channel hydraulic diameter, 9 is the channel angle of inclination with respect to the gravity force, or the contact angle, g is the acceleration due to gravity, and Uls and Ugs are the liquid and gas superficial velocities, respectively. The independent dimensionless parameters can be chosen as Ap/pu (where Ap = Pl-Pg), and... 

Assuming steady state in Eqs. (10.8-10.10) and (10.18-10.20), we obtain the system of equations, which determines steady regimes of the flow in the heated miero-channel. We introduce values of density p = pp.o, velocity , length = L, temperature r = Ti 0, pressure AP = Pl,o - Pg,oo and enthalpy /Jlg as characteristic scales. The dimensionless variables are defined as follows ... 

The steady states of such systems result from nonlinear hydrodynamic interactions with the gas flow field. For the convex flame, the flame surface area F can be determined from the relation fSl = b zv, where Sl is the laminar burning velocity, the cross-section area of the channel, and w is the propagation velocity at the leading point. 

In the articles cited above, the studies were restricted to steady-state flows, and steady-state solutions could be determined for the range of Reynolds numbers considered. Experimental work on flow and heat transfer in sinusoidally curved channels was conducted by Rush et al. [121]. Their results indicate heat-transfer enhancement and do not show evidence of a Nusselt number reduction in any range... 

Steady-state component and solvent mass balances can be written for single-pass operation by considering an incremental area element dA in the axial or flow direction for a feed channel, module, or membrane... 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

The HIPS resin was extruded at screw speeds of 30, 60, and 90 rpm at barrel temperatures of 200, 220, and 240 °C for Zones 1, 2, and 3, respectively. The screw temperatures in Zone 3 as a function of time at the screw speeds are shown in Fig. 10.20. Because the RTDs were positioned within 1 mm of the screw root surface, they were influenced by the temperature of the material flowing in the channels. Prior to the experiment, the screw was allowed to come to a steady-state temperature without rotation. Next, the screw speed was slowly increased to a speed of 30 rpm. The time for the screw to reach a steady state after changing the screw speed to 30 rpm was found to be about 10 minutes. The temperature of the T12 and T13 locations decreased with the introduction of the resin. This was caused by the flow of cooler solid resin that conducted energy out from the screw and into the solids. At sensor positions downstream from T13, the screw temperature increased at a screw speed of 30 rpm, indicating that the resin was mostly molten in these locations. These data suggest that the solid bed extended to somewhere between 15.3 and 16.5 diameters, that is, between T13 and T14. When the screw speed was increased to 60 rpm, the T12 and T13 sensors decreased in temperature, the T14 sensor was essentially constant, and the T15, T16, and T17 sensor temperatures increased. These data are consistent with solids moving further downstream with the increase in screw speed. For this case, the end of the solids bed was likely just upstream of the T14 sensor. If the solid bed were beyond this location, the T14 temperature would have decreased. Likewise, if the solid bed ended further upstream of the T14 sensor, the temperature would have increased. When the screw speed was increased to 90 rpm, the T12, T13, and T14 temperatures decreased while the T15, T16, and T17 temperatures increased. As before, the solids bed was conveyed further downstream with the increase in screw speed. At a screw speed of 90 rpm, the solid bed likely ended between the T14 and T15 sensor positions, that is, between 16.5 and 17.8 diameters. These RTDs were influenced by the cooler solid material because they were positioned within 1 mm of the screw root surface. 

The flow is steady state and fully developed in the down-channel direction. 

Johnson and Fierke Hammes have presented detailed accounts of how rapid reaction techniques allow one to analyze enzymic catalysis in terms of pre-steady-state events, single-turnover kinetics, substrate channeling, internal equilibria, and kinetic partitioning. See Chemical Kinetics Stopped-Flow Techniques... 

For steady-state, isothermal, single-phase, uniform, fully developed newtonian flow in straight pipes, the velocity is greatest at the center of the channel and symmetric about the axis of the pipe. Of those flowmeters that are dependent on the velocity profile, they are usually calibrated for this type of flow. Thus any disturbances in flow conditions can affect flowmeter readings. 

Albumin solutions (1, 2, and 5wt%) are continuously ultrafiltered through a flat plate filter with a channel height of 2 mm. Under cross-flow filtration with a transmembrane pressure of 0.5 MPa, steady-state filtrate fluxes (cm min ) are obtained as given in Table P9.2. 

The kinetic theory derivation of the thermal conductivity coefficient is very similar in spirit to the viscosity treatment just discussed. In the schematic shown in Fig. 12.2, we considered a fluid between two plates held at different temperatures. At steady state the fluid temperature varies linearly across the channel, and heat flows from the top, higher-temperature... 

The plug-flow problem is formulated as a steady-state problem—that is, nothing varies as a function of time. Consequently the surface composition at any point on the channel... 

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