The problem of natural convection in a cavity of small aspect ratio with differentially heated end walls is considered. It is shown by use of matched asymptotic expansions that the flow consists of two distinct regimes : a parallel flow in the core region and a second, non-parallel flow near the ends of the cavity. A solution valid at all orders in the aspect ratio A is found for the core region, while the first several terms of the appropriate asymptotic expansion are obtained for the end regions. Parametric limits of validity for the parallel flow structure are discussed. Asymptotic expressions for the Nusselt number and the single free parameter of the parallel flow solution, valid in the limit as A -+ 0, are derived.
The efficiency of solvent extraction of bitumen from mined Athabasca oil sands has been studied in laboratory-scale, batch, stirred vessels. Solvent type, solids concentration, stirrer speed and contact time have been identified as important parameters. A theoretical analysis of the extraction process provides a basis by which the various parameters are assessed in terms of relative overall mass transfer coefficients. However, because the size distribution of the oil sand aggregates is difficult to reproduce, the analysis is limited. Greater reproducibility, and similar quantitative results were obtained for the dissolution of pure bitumen from the bottom of the stirred vessels.It is shown that highly aromatic solvents such as benzene and toluene can dissolve bitumen 3-5 times faster than an essentially alphatic solvent such as kerosene. There is evidence, however, which indicates that solvent properties other than aromaticity are also important.The data indicate that a minimum stirrer speed must be exceeded before the extraction of oil sand or pure bitumen will proceed at a reasonable rate. Above this minimum rate, the mass transfer coefficient increases linearly with stirrer speed. The solids concentration is also important since the data suggest that a higher efficiency is obtained at higher solids concentrations in the range of solid loadings studied. any schemes for the separation of bitumen from
Numerical solutions of the full Navier-Stokes equations are obtained for the problem of natural convection in closed cavities of small aspect ratio with differentially heated end walls. These solutions cover the parameter range Pr = 6.983, 10 6 Gr d 2 x lo4 and 0.05 4 A d 1. A comparison with the asymptotic theory of part 1 shows excellent agreement between the analytical and numerical solutions provided that A 5 0.1 and Gr2A3Pr2 5 lo5. In addition, the numerical solutions demonstrate the transition between the shallow-cavity limit of part 1 and the boundary-layer limit; A fixed, Gr -+ co.
An analytic procedure for calculating vertical fracture extent in symmetrical trilayered formations was extended to multilayered, asymmetrical formations using a semianalytic technique. The fracture extents computed by this method were compared with those calculated with the finite-element method. It was found that even for modulus variations between layers as large as a factor of 5, the semianalytic procedure gave exactly the same results as the finite-element solution in a fraction of the computation time and with significantly less manual data manipulation. It is recommended that the analytic and numerical procedures be used in a complementary manner to calculate fracture-width profiles in layered formations.
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