Flow over a sphere

This is known as Stokes flow, and Eq. (11-3) has been found be accurate for flow over a sphere for NRe < 0.1 and to within about 5% for NRe < 1. Note the similarity between Eq. (11-3) and the dimensionless Hagen-Poiseuille equation for laminar tube flow, i.e.,/ = 16/tVRe. 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

Flow over a sphere is an important geometry in catalyst spheres, liquid drops, gas bubbles, and small solid particles. In this case the characteristic length is the sphere diameter D, and... 

In the limit of slow flow over a sphere Sh/j = 2.0, and this corresponds to diffusion to or from a sphere surrounded by a stagnant fluid. When the sphere diameter is sufifrciently small. Re becomes sufficiently small that Sh/j = 2.0 in many common situations of flow around spheres. 

Let us apply the interpolation procedure to a case involving an electric field. It is well known that the efficiency of the granular bed filters can be significantly increased by applying an external electrostatic field across the filter. In this case, fine (<0.5-/rm) particles deposit on the surface of the bed because of Brownian motion as well as because of the electrostatically generated dust particle drift [51], The rate of deposition can be calculated easily for a laminar flow over a sphere in the absence of the electrostatic field [5]. The other limiting case, in which the motion of the particles is exclusively due to the electric field, could also be treated [52], When, however, the two effects act simultaneously, only numerical solutions to the problem could be obtained [51],... 

Davidson and Cullen (Dl), 1957 Consideration of the flow mechanics and diffusion in a liquid film flowing over a sphere. 

In retrospect, the effect of the change of variables has been to deform the velocity field from axisymmetric stagnation flow over a sphere to linear shear flow along a flat plate. The main advantage of the new coordinates is that the coefficients of the derivatives in (32) are independent of X, and consequently Duhamel s theorem can be applied. Thus, the following procedure can be used (1) solve equation (32) subject to a uniform surface concentration (2) extend this solution to one valid for an arbitrary, nonuniform surface concentration by applying Duhamel s theorem (3) select the surface concentration which satisfies (33a). 

Goren and O Neill (1971), who calculated the normal force exerted by creeping flow over a sphere near a plane surface. Although Ihe flow field in the present problem is somewhat more complex than that considered by Goren and O Neill, the curvature of the plane surface as well as a tangential velocity component have only secondary effects upon the normal force, so their result is reasonably applicable here. 

Assuming that the bubble in Prob. 9-27 moves through the liquid at a velocity of 4.5 m/s, estimate the time required to cool the bubble 0.3°C by calculating the heat-transfer coefficient for flow over a sphere and using this in a lumped-capacity analysis as described in Chap. 4. 

For flow over a sphere, Whitaker recommends the following comprehensive correlation ... 

Rimon, J. and Cheng, S. I., Numerical solution of a uniform flow over a sphere at intermediate Reynolds numbers, Phys. Fluid, Vol. 12, No. 5, pp. 949-959, 1969. 

The solution of Eq. (9.26) with conditions (9.27) can be obtained by the same method, as the solution of the problem on the Stokes flow over a sphere [51]. As a result, we will find the pressure and velocity distributions at the surface of the sphere, and the tangential stress at the sphere. Therefore the hydrodynamic force acting on the sphere, is equal to... 

As an example of unorthodox behavior, consider the large velocity (large Reynolds number) flow over a sphere as shown in Fig. 6.5. We might assume that the characteristic length scale in this problem is the sphere diameter d), that is, the Reynolds number, which is the ratio of fluid inertial to viscous terms, is postulated to be... 

The value of fc the species mass-transfer coefficient in the fluid film, may be obtained from a variety of correlations for the gas phase as fcjg and for the liquid phase as ku- The experimentally obtained correlations for ki popularly employed for a gas flowing over a sphere or in a packed bed are ... 

Similar correlations are available with regard to other physical situations such as fluid flow over a flat plate, a sphere or a cylinder. 

Overall mass-transfer rates at a sphere in forced flow, and mass-transfer rate distribution over a sphere as a function of the polar angle have been measured by Gibert, Angelino, and co-workers (G2, G4a) for a wide range of Reynolds numbers. The overall rate dependence on Re exhibited two distinct regimes with a sharp transition at Re = 1250. Local mass-transfer rates were deduced from measurements in which the sphere was progressively coated by an insulator, starting from the rear. 

Study of the eflSciency of packed columns in liquid-liquid extraction has shown that spontaneous interfacial turbulence or emulsification can increase mass-transfer rates by as much as three times when, for example, acetone is extracted from water to an organic solvent (84, 85). Another factor which may be important for flow over packing has been studied by Ratcliff and Reid (86). In the transfer of benzene into water, studied with a laminar spherical film of water flowing over a single sphere immersed in benzene, they found that in experiments where the interface was clean... 

Deals with smooth and rippling flow of a liquid over a sphere or series of spheres, especially with the problem of mixing in the liquid film at the junctions of spheres. 

Movement of the Huid may be generated by means external to the heat transfer process, us by fans, blowers, or pumps. It may also be created by density differences connected with the heat transfer process itself. The first mode is culled timet cniireeiirtn the second one natural or free t ttttveclion. Convection heal transfer may also be classified as heat transfer in iltni /fnn. or in interna flow (over cylinders, spheres, air foils, and similar objects). In ilie case of external flow, the heal transfer process is essentially concentrated in a thin fluid layer surrounding the object (boundary layer . 

For forced convective heat transfer over a sphere in a uniform flow, a frequently used empirical relation was proposed by Ranz and Marshall (1952) as... 

Kumati, M. and Nath. G.. "Non-Darcy Mixed Convection Flow over a Nonisother-mal Cylinder and Sphere Embedded in a Saturated Porous Medium". J. Heat Transfer. Vol. 112, pp.318-521, 1990. 

Water flows over a 3-mm-diameter sphere at 6 m/s. The free-stream temperature is 38°C, and the sphere is maintained at 93°C. Calculate the heat-transfer rate. 

Mears24 suggested that the fact that (4-6) correlated the data was fortuitous. He questioned the validity of Eq. (4-5) for the packed-bed trickle-bed reactor, since this equation was derived from the data taken for the flow over a string of spheres. He argued that the dependence of reactor performance on velocity in pilot-scale reactors is due to incomplete catalyst wetting at low flow rates. For a first-order reaction, he modified Eq. (4-4) as... 

When both gas and liquid flow downward over a string of spheres, it can simulate a trickle-bed reactor if the liquid flows downward in the form of a thin film. The hydrodynamics for this type of reactor are reasonably well known. Both the hydrodynamics of the liquid flow over a single sphere and the phenomena taking place at the junctions of two spheres have been extensively studied. Flow maldistribution encountered in the pilot-scale trickle-bed reactor is eliminated in this type of reactor. Furthermore, good estimations of the various mass-transfer resistances can be ascertained. The reactor is successfully used by Satterfield et al.st) for the catalytic hydrogenation of a-methyl styrene. Their experimental setup is shown in Fig. 5-9. 

Satterfield et al.80 studied the liquid holdup characteristics of flow over a string of spheres. Based on their data with 0.825-cm-diameter spherical porous catalyst pellets of palladium-on-alumina they proposed the following dimensionless relation for the dynamic liquid holdup ... 

Small Reynolds Number Flow, Re < 1. The slow viscous motion without interfacial mass transfer is described by the Hadamard (66)-Rybcynski (67) solution. For infinite liquid viscosity the result specializes to that of the Stokes flow over a rigid sphere. An approximate transient analysis to establish the internal motion has been performed (68), Some simplified heat and mass transfer analyses (69, 70) using the Hadamard-Rybcynski solution to describe the flow field also exist. These results are usually obtained through numerical integration since analytical solutions are usually difficult to obtain. 

The gas-phase slow viscous flow over a rigid sphere with interfacial mass injection or abstraction has been analyzed in Refs. 71, 72, 73, and 74, In general evaporation reduces the droplet drag significantly. 

In the absence of separation, the gas-phase free stream is described by the potential flow solution over a sphere (77) ... 

For a liquid droplet in a gaseous medium, the densities differ significantly. Whereas the linearized treatment can still be extended to the liquid-phase analysis, for the gas phase the conventional boundary layer analysis should be used (82). For high Reynolds number flow over a solid sphere, approximate solutions have been obtained using both the Blasius series and the momentum integral techniques (82). 

The average drag coefficients C j, for cross-flow over a smooth single circu lar cylinder and a sphere aie given in Fig, 7-17. The curves exhibit different behaviors in different ranges of Reynolds numbers ... 

Average drag coefficient for cross-flow over a smooth circular cylinder and a smooth sphere. From //. ScMichtift. Boyndary Layer Theory 7c. Copyright Q i979 The McGrow-Hill Companies. 

The average Nusselt numbers for cross flow over a cylinder and sphere are... 

In transverse flow, the buoyant motion is perpendicular to the forced motion. Transverse flow enhances fluid mixing and thus enhances heat transfer. An example is horizontal forced flow over a hot or cold cylinder or sphere. 

As the previous illustrations showed, the heat and mass transfer coefficients for simple flows over a body, such as those over flat or slightly curved plates, can be calculated exactly using the boundary layer equations. In flows where detachment occurs, for example around cylinders, spheres or other bodies, the heat and mass transfer coefficients are very difficult if not impossible to calculate and so can only be determined by experiments. In terms of practical applications the calculated or measured results have been described by empirical correlations of the type Nu = f(Re,Pr), some of which have already been discussed. These are summarised in the following along with some of the more frequently used correlations. All the correlations are also valid for mass transfer. This merely requires the Nusselt to be replaced by the Sherwood number and the Prandtl by the Schmidt number. 

Dandy and Dwyer [30] computed numerically the three-dimensional flow around a sphere in shear flow from the continuity and Navier-Stokes equations. The sphere was not allowed to move or rotate. The drag, lift, and heat flux of the sphere was determined. The drag and lift forces were computed over the surface of the sphere from (5.28) and (5.33), respectively. They examined the two contributions to the lift force, the pressure contribution and the viscous contribution. While the viscous contribution always was positive, the pressure contribution would change sign over the surface of the sphere. The pressure... 

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