Data are reported for heat transfer from water to melting ice spheres and for mass transfer in the case of dissolving spheres of pivalic acid suspended in water agitated in a stirred vessel.The transport coefficients are found to depend on agitator power input but not on agitator design, in agreement with the Kolmogoroff theory. These experimental results are used with others in the literature to develop a correlation involving Nusselt and Prandtl or Schmidt numbers together with a dimensionless group involving agitation power. The correlation is essentially independent of solid-liquid density ratio in the ranqe 0.8 to 1.25, and in this range the gravity group also appears to be unimportant.Particles or drops suspended in agitated liquids are employed in various chemical processes, including crystallization, solvent extraction, polymerization, slurry catalysis, and direct-contact heat exchange. In many cases the size of the process equipment is directly related to the rate of mass or heat transfer between the particles and the liquid. Though numerous studies of the transport rates in such systems have been reported, the large number of variables involved has made it difficult to develop a full understanding of the physics of the processes of interest. Evidently the physical properties of both phases are important, as well as the nature of the turbulence in the liquid, as determined by the type of agitation and the geometry of the system.Perhaps the most widely used method of estimating transport rates in such systems is that based on the work of Harriott (12). Hariott suggested that the transport rates in a stirred tank can be estimated as a multiple of the transport which would result if the particles fell through a stagnant medium at their terminal velocity. There are, however, no reliable methods of predicting this multiple. KOLMOGORBFF'S THEORYSeveral investigators (8, 21,27) have suggested that KolmogoroFs theory of isotropic turbulence may be applied to a stirred liquid in turbulent motion and, therefore, that the transport results can be correlated with the agitation power per unit volume. However, no satisfactory correlation of transport data based upon these ideas has been developed.Kolmogoroff's theory postulates that turbuleiice results in a continuous inertial transfer of kinetic energy from the larger eddies to smaller and smaller eddies, finally resulting in viscous dissipation of the energy by the smallest eddies in the dissipation range. In a stirred tank, the impeller continually creates eddies described by a certain size and frequency distribution and also by a certain geometric orientation. The largest eddies are of the order of the size of the container and constitute the bulk flow. These large eddies transfer energy to smaller and smaller eddies through inertial interaction. As the kinetic energy is transferred from large to small eddies, the geometric orientation is lost. The large eddies interact to produce more random smaller ones and eventually, if sufficient interaction takes place, all ...

Mass transfer to spheres suspended in on agitated liquid has been studied both experimentally and theoretically. Finite-difference solutions ore obtained for mass transfer from a sphere to a fluid flowing post it in steady viscous flow. The effects of a transpiration velocity a t the surface of the sphere and of a continuously changing sphere diameter are included. A normolized presentation of these effects is quite insensitive to the bulk flow Peclet number. When these theoretical corrections for transpiring ond shrinking spheres are applied to the mass transfer data for ice spheres that are melting in an agitated brine bath, the corrected moss transfer coefficients ore brought into agreement with a generalized correlation published elsewhere. This agreement suggests that the theoretical results apply, with reasonable accuracy, to a shrinking and transpiring sphere that is suspended in a turbulent liquid.Rates of mass and heat transport to small particles suspended in agitated liquids are important in many chemical processes that involve dissolution, crystallization, and heterogeneous chemical reaction. The transport process often proceeds with a continuously changing particle size and is therefore transient in nature. Furthermore, there is often a finite radial fluid velocity at the particle surface, and this transpiration velocity can affect the transport process. In much of the literature the effects of transpiration and of a changing particle size have been assumed to have a negligible influence upon the transport rate to suspended particles. This assumption is probably acceptable in many situations, but in others it may not have been valid.The purpose of this work is to present a theoretical analysis of these effects. The equation of continuity for mass or heat diffusion is solved numerically for a sphere in a steady unidirectional viscous fluid flow. Solutions are obtained which include the effects of transpiration at the particle surface and a continuously changing particle radius. Generalized graphs of these results are then compared with experimental results for transport to particles freely suspended in a turbulent liquid. THEORYThe theoretical analysis to be presented involves a numerical solution of the partial differential equation describing the transport of mass from a sphere: The model further assumes the velocity profile existing around the sphere to be that which occurs with steady Stokes flow past a sphere of constant radius, thus The last term in the first of these equations accounts for the effect of a transpiration velocity at the particle surface (10). For the case of a sphere of changing radius, this velocity profile corresponds to the steady state profile at the instantaneous sphere radius, and thus it neglects transient effects in momentum transfer. This should be valid if the momentum diffusivity is much greater than the diffusivity for the transport process under consideration, that is, for a large Schmidt or Prandtl number. Equation ( 1 ) is not in a convenient form for numerical sol...

Finite difference approximations to partial derivatives are generally based on Taylor series, which are polynomial expressions for the unknown variable as a function of the grid locations. In many problems, approximate analytical solutions are known that incorporate the physics of the process. It is proposed that such expressions be used to derive finite difference equations. Increased accuracy is anticipated, particularly when the solutions are highly non-linear, singular, or discontinuous.Reservoir simulation is such a problem. Flow in petroleum reservoirs results from injection and productions from wells, which are relatively small sources and sinks. Near singularities in the pressure around the wells result. The immiscibility of the fluids causes an oil bank to form in front of displacing water, and near discontinuities in the saturations occur. This paper investigates the utility and accuracy of finite difference equations for reservoir pressures based on two new functional forms: ln(r) and 1/r, where r is the distance to the well. The ln(r) form is based on pressures from line sources, and thus is effective at representing straight line wells. The 1/r form is based on pressures from point sources. The sum of many points represent more complex wells. Both are found to greatly increase the accuracy of the simulated reservoir pressures relative to solutions based on the polynomial approach.

The simulation of reservoir performance is particularly difficult, compared with the simulation of similar processes, because of the extreme behavior of reservoir pressures and saturations. In particular, the reservoir pressures are very non-linear and exhibit near-singularities at the wells as a result of injection and production. Finite difference methods, which are used almost universally for reservoir simulation, are troublesome with such highly non-linear solutions. Accuracy is reduced and the solutions become more time consuming. This paper describes an attempt to eliminate these nonlinearities from the finite difference solutions by incorporating analytical solutions in the pressure solution. The reservoir pressures are the sum of analytical functions and traditional finite difference solutions. The analytical functions are based on well test equations which describe the pressures around the wells in an infinite, homogeneous system. Although heterogeneities and reservoir geometries make these analytical functions grossly inaccurate for predicting actual reservoir pressures, they do exhibit nonlinearities much like the actual solutions. As a result, the finite difference components become much more linear, improving both the accuracy and the speed of the solution. Several other advantages of this new technology are also demonstrated:–Empirical well equations, which relate cell pressures to well pressures, are unnecessary.–The technique is applicable to Cartesian grids as well as other grid types.–Wells need not be centered in the cells for best results.–Accurate simulations of early-time, pressure transients make well test simulations practical. Introduction Despite the similarity of reservoir simulation with other engineering problems such as laminar fluid flow, conductive heat transfer, and convective/diffusive mass transfer, reservoir simulation is uniquely difficult. These difficulties must be due, at least in part, to the large non-linearities inherent in the solutions. For example, the reservoir pressures develop near-singularities at the wells as illustrated in Figure 1. Unlike the other engineering problems, reservoir simulation results depend on flow through very small areas, the well bores. The wells act much as line sources resulting in the very sharp pressure gradients near them. The accuracy of finite difference solutions deteriorates when the solutions are highly non-linear, as can be easily demonstrated through the Taylor's series derivation of the finite difference approximations to the partial differential terms. The solution speed is also affected. This work investigates the elimination of the near-singularities in the finite difference equations, through the combination of analytical functions. As illustrated in Figure 1, an analytical function which accurately represents the pressure gradients around the well bores is added to the finite difference solution. The analytical solution may not accurately represent the actual reservoir pressure, particularly at large distances from the wells, but it does reduce the non-linear behavior of the finite difference component of the solution. Aftab has tried a similar approach to eliminate both the pressure singularities and the saturation discontinuities. Well Equations This new treatment of reservoir pressures eliminates the need for well equations. In the past, well equations have been used to relate well bore pressures to the well cell pressures. This approach follows Peaceman's classical work of 1978.

Pseudo-relative peraeability functions were Pseudo-relative peraeability functions were generated by autoaatically aatching the pressures and oil production voluaes obtained froa two-diaensional production voluaes obtained froa two-diaensional cross-sectional siitulations. The 2-D siaulations used actual reservoir fluid and rock properties, but incorporated hypothesized stratifications and dip angles. The resulting pseudo-relative permeability curves were correlated as a function of the water breakthrough tiae and the aaxiaua pressure differences observed in the corresponding siaulations. The resulting suite of relative peraeability curves then offered a siaple aethod of aatching historically observed pressures and production volumes in three-diaensional siaulations of the reservoirs. Siailar possibilities exist for altering pseudofunctions to obtain history aatches in areal aodels. pseudofunctions to obtain history aatches in areal aodels Introduction Despite the advent of ever larger and faster coaputers the use of pseudo-relative peraeabilities reaain a coaaon practice in reservoir siaulation. Pseudo-relative peraeabilities are applied to finite Pseudo-relative peraeabilities are applied to finite difference reservoir siaulation aodels to coapensate for intra-cell variations in rock properties and fluid saturations or coapositions. To coapensate for these difficulties through grid refineaent can be difficult because 1) reservoirs comaonly exhibit large and frequent variations in reservoir properties, e.g. peraeability and porosity. as a result of properties, e.g. peraeability and porosity. as a result of geological stratification, and because 2) very sharp saturation gradients can occur in the reservoir as one phase displaces another. Very saall finite difference grid blocks aay be required to represent these changes accurately. Achieving adequate accuracies can tax even the new super coaputers. Moreover, to represent these intricacies in a aodel through saall finite difference grid blocks can be very expensive. The use of pseudo-relative permeability functions offers a compromise to permeability functions offers a compromise to account for these errors without the increased expense of a larger model. Several methods have been proposed for the develop aent of pseudo-relative peraeabilities which would account for these intra-cell variations. Coats et al. suggested the assuaption of capilliary equilibrium within each cell to account for saturation variations. This method has received general acceptance. The assuaption of vertical equilibrium is justified when vertical communication is good, displaceitent velocities are saall, and phase density differences are large. Hearn has suggested a method for deteraining pseudo-relative per. eabilities in the other extreae pseudo-relative per.eabilities in the other extreae when convection forces doainate and vertical communication is small. He develops pseudo-relative permeability functions by assuming that the invading permeability functions by assuming that the invading fluid aoves along isolated strata in a piston-like fashion. Hawthorne has shown that Hearn's model can be modified to include the influence of capillary pressure. More recently, Simon and Koederitz have aodified Hearn's aethod to account for continuing oil production from each layer after water breakthrough. Jacks et al have suggested the use of "dynamic" pseudo-functions which involve detailed simulations pseudo-functions which involve detailed simulations of sections of the reservoir. Their procedure is suggested when neither of the above analytical techniques are applicable, which is often the case. Their method derives the pseudo-relative permeabilities from detailed vertical permeabilities from detailed vertical cross-sectional simulation studies of segments of the reservoir under conditions to be expected during the full scale simulations. The pseudos are derived by combining each column of cells in the vertical cross-sectional model to act as a single cell in the larger model. Average saturations and pressure gradients are calculated for the entire column of cells. The total flux from the column for each phase is also calculated. Then Darcy's Law is applied using these fluxes and pressure gradients to obtain the effective pseudo-relative permeabilities. This procedure was improved by Kyte and Berry to procedure was improved by Kyte and Berry to include an analogous pseudo-capillary pressure deteraination and to allow for differences in cell sizes between the vertical cross sections and the larger model. P. 445