Laminar equations describing

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

Thus in turbulent flow, the dispersion coefficient is independent of the diffusion coefficient, but in laminar flow, the dispersion coefficient depends inversely on the diffusion coefficient. This counterintuitive inverse dependence, the result of axial convection coupled with radial diffusion, is the foundation of the Goulay equation describing peak spreading in chromatography. We now return from this dispersion tangent back to diffusion and in particular, to mass transfer. 

The specific filter cake resistance, a [L-2], is defined with the equation describing the pressure loss, Ap, of the liquid in the porous filter cake at laminar flow ... 

This equation describes the temperature distribution in fully developed laminar plane duct flow when the wall heat flux is a constant. It can be written in terms of the specified wall heat flux. qw, by noting that when Eq. (4.77) is used to give the value of dT dy y = w in Eq. (4.71), the following is obtained ... 

Validation of the Global Rates Expressions. In order to validate the global rate expressions employed in the model, temperature and concentration profiles determined by probing the flames on a flat flame burner were studied. Attention was concentrated on Flames B and C. The experimental profiles were smoothed, and the stable species net reaction rates were determined using the laminar flat-flame equation described in detail by Fristrom and Westenberg (3) and summarized in Reference (8). 

The dispersion of a non-reactive solute in a circular tube of constant cross-section in which the flow is laminar is described by the convective-diffusion equation... 

Reynolds number is a fundamental parameter used to describe the fluid properties associated with an aerosol. Equations describing the resistance offered by a particle depend on whether the flow is laminar or turbulent, and the Reynolds number provides knowledge of the type of flow present. 

F.L. Schuyler T.P. Torda, AIAA J 4 (12), 2171-7 (1966) CA 66, 30631 (1967) Steady-state combustion of solid double-base and mono-proplnt rockets was investigated analytically. The nonlinear differential equations describing the motion of a laminar 2-dimensional, compressible, chemically reacting viscous fluid are intimately coupled with the 1-dimensional... 

These equations are identical to the equations describing mass transfer in monolithic reactors. For monolithic reactors it was shown [14] that when the reaction rate is very fast compared to the mass transfer rate in the fluid domain, the boundary condition of Eq. (1 5) becomes identical to the standard heat transfer boundary condition of constant wall temperature when the reaction rate is very slow compared to the mass transfer rate in the fluid domain, the boundary condition of Eq. (IS) becomes identical to the standard heat transfer boundary condition of constant heat flux. The influence of the boundary conditions on the mass transfer coefficient in case of laminar flow is discussed in the following section. 

Taking note of Eqs. 10.56 and 10.58, the final equation describing the fully established laminar velocity profile of a power-law fluid flowing through a rectangular duct is given by... 

This chapter addresses the three fundamental transport properties characteristic of Chemical Engineering heat transfer, momentum transfer, and mass transfer. The underlying physical properties that represent each of these phenomena are thermal conductivity, viscosity, and diffusivity and the equations describing them have a similar form. Heat flux through conduction is expressed as a temperature gradient with units of W m . Note that heat flux, mass flux, etc. are physical measures expressed with respect to a surface (m ). Momentum flux in laminar flow conditions is known as shear stress and has units of Pa (or N m ) it equals the product of viscosity and a velocity gradient. Finally, molar flux (or mass flux) equals the product of diffusivity and a concentration gradient with units of mol m s These phenomena are expressed mathematically as shown in Table 7.1. 

The modeling of cooling a fluid in a laminar pipe flow was considered in Example 10.3. Neglecting the axial heat conduction relative to the convection term, the following heat balance equation describes the temperature change inside the pipe... 

Application Channel electrodes have well-defined hydrodynamic properties [57], and the laminar flow convective-diffusion equation describing the mass... 

Conservation Equations The computational model consists of steady-state continuity equation and momentum equations describing the flow inside the laminar static mixer. 

As Figure 2 shows, the corrected pressure drop, AP, and the observed volumetric flow rate, Q, are used to determine an apparent viscosity, Uapp This calculation is based on the Hagen-Poiseuille equation describing steady laminar flow of a Newtonian fluid in a capillary ... 

The following equations describe transport of species, heat, and momentum in a cylindrical tubular reactor with laminar flow... 

This equation describes the rate of laminar flow of a liquid through a straight tube or pipe of length L and radius R. We assume that the tube is parallel to the z axis and we assume that the velocity depends only on r, the perpendicular distance from the center of the tube. Consider a portion of the fluid that is contained in an imaginary cylinder of radius r that is concentric with the tube walls, as depicted in Figure 10.5. 

This chapter has been written primarily as an introduction to the present state of the art of reactive flow modeling. It should guide the reader, after some effort, to appreciate the relevant literature, use or improve existing modeling codes, or write new ones. The basic hydrodynamic conservation equations and the expressions for the transport fluxes were presented and the finite-difference solution of the time-dependent equations for the one-dimensional laminar flame described in some detail. The most general transport flux model has been retained on the basis that it is easier to simplify than to construct. Conversion to more approximate formulations of transport fluxes, for example, those of Section 3.4, presents no difficulty where these are felt to be more appropriate. 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

Ca.rma.n-KozenyEfjua.tion, Flow through packed beds under laminar conditions can be described by the Carman-Kozeny equation in the... 

In configurations more complex than pipes, eg, flow around bodies or through nozzles, additional shearing stresses and velocity gradients must be accounted for. More general equations for some simple fluids in laminar flow are described in Reference 1. 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

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