Variational recovery

The well-known inaccuracy of numerical differentiation precludes the direct calculation of pressure by the insertion of the computed velocity field into Equation (3.6). This problem is, however, very effectively resolved using the following variational recovery method Consider the discretized form of Equation (3.6) given as... 

In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

Continuous penalty method - to discretize the continuity and (r, z) components of the equation of motion, Equations (5.22) and (5.24), for the calculation of r,. and v. Pressure is computed via the variational recovery procedure (Chapter 3, Section 4). 

Step 2 - use the calculated pressures and find the velocity components by variational recovery. 

Note that the displacements are found using the working equations of the scheme stresses are found via the variational recovery method. 

STRESS. Applies the variational recovery method to calculate nodal values of pressure and, components of the stress. A mass lumping routine is called by STRESS to diagonalize the coefficient matrix in the equations to eliminate the... 

CALCULATION OF THE NODAL PRESSURE STRESS USING VARIATIONAL RECOVERY... 

A broad comparison of the main types of processes, the strength and quaUty of phosphoric acid, and the form and quaUty of by-product calcium sulfate are summarized in Table 7. Because the dihydrate process is the most widely used, the quaUty of its acid and calcium sulfate and its P2O3 recovery are taken as reference for performance comparisons. Illustrative flow diagrams of the principal variations in process types have been pubUshed (39). Numerous other variations in process details ar also used (40—42). The majority of plants use a dihydrate process and some of these have production capacity up to 2100 of P2O3 per day. 

Hardness is a measure of a material s resistance to deformation. In this article hardness is taken to be the measure of a material s resistance to indentation by a tool or indenter harder than itself This seems a relatively simple concept until mathematical analysis is attempted the elastic, plastic, and elastic recovery properties of a material are involved, making the relationship quite complex. Further complications are introduced by variations in elastic modulus and frictional coefficients. 

The purified acid is recovered from the loaded organic stream by contacting with water in another countercurrent extraction step. In place of water, an aqueous alkafl can be used to recover a purified phosphate salt solution. A small portion of the purified acid is typically used in a backwashing operation to contact the loaded organic phase and to improve the purity of the extract phase prior to recovery of the purified acid. Depending on the miscibility of the solvent with the acid, the purified acid and the raffinate may be stripped of residual solvent which is recycled to the extraction loop. The purified acid can be treated for removal of residual organic impurities, stripped of fluoride to low (10 ppm) levels, and concentrated to the desired P2 s Many variations of this basic scheme have been developed to improve the extraction of phosphate and rejection of impurities to the raffinate stream, and numerous patents have been granted on solvent extraction processes. 

Manufacture. The oldest method for producing Na2S is by the reduction of sodium sulfate with carbon in a refractory oven at 900—1000°C. Whereas this method is no longer used commercially in the United States, a variation is used to produce sodium sulfide captively during kraft pulp processing to replace lost sodium and sulfur values that were initiated into the system by merchant-suppHed sodium sulfide. In this method, sodium sulfate is added to the system in the recovery furnace, where it is reduced by carbon from the wood pulp to produce sodium sulfide. 

One more variation to the many methods proposed for sulfur extraction is the fire-flood method. It is a modem version of the Sickian method, by which a portion of the sulfur is burned to melt the remainder. It would be done in situ and is said to offer cost advantages, to work in almost any type of zone formation, and to produce better sweep efficiency than other systems. The recovery stream would be about 20 wt % sulfur as SO2 and 80 wt % elemental sulfur. The method was laboratory-tested in the late 1960s and patents were issued. However, it was not commercially exploited because sulfur prices dropped. 

Another variation of the Selectox process can be used with the Beavon process in tail gas treating. The hydrogenated Claus tail gas stream is sent to a Selectox reactor. Overall recoveries of up to 98.5% are possible. Use of Beavon/Selectox, however, typically costs more than use of Superclaus. 

Hot-Water Process. The hot-water process is the only successflil commercial process to be appHed to bitumen recovery from mined tar sands in North America as of 1997 (2). The process utilizes linear and nonlinear variations of bitumen density and water density, respectively, with temperature so that the bitumen that is heavier than water at room temperature becomes lighter than water at 80°C. Surface-active materials in tar sand also contribute to the process (2). The essentials of the hot-water process involve conditioning, separation, and scavenging (Fig. 9). 

Other variations of the fuming process under development by the Commonwealth Scientific and Industrial Research Organization in Clayton, Austraha, promise even greater efficiencies of metal recovery (5). 

FIG. 22-38 Tbe variation of adsorption density, oil-droplet contact angle, and oil-extraction recovery of bematite as a function of pH. To convert gram-moles per square centimeter to pound-moles per square foot, multiply hy 2.048. [From Raghavan and Fuerstenau, Am. Inst. Cbem. Eng. Symp. Ser., 71(150), 59... 

Due to the great variation in pressures, flux rates, materials of construction, heat recovery, burner configuration, etc., correlation of process heaters is difficult even with large amounts of data. For similar furnaces, heat absorption vs. cost gives the best correlation. It is again recommended that vendor help be obtained for estimating process furnaces, unless data on similar furnaces is available. Data can be found in References 24 and 25. 

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