Constant-composition method

In the present work, a constant composition method has been used to investigate the growth of HAP from solutions of low supersaturation and in the presence of different background electrolytes. The influence of magnesium and strontium ions both on the rate of crystallization and upon the electrokinetic properties of the crystallite surfaces has also been investigated. 

A disadvantage of the conventional precipitation method in which the supersaturation was allowed to decrease during the reactions, was that different calcium phosphate phases could form and subsequently dissolve during the course of the reactions. In the present work, the constant composition method was used to investigate the influence of sodium chloride, potassium chloride, and potassium nitrate, as background electrolyte upon the rate of crystallization of HAP in solutions supersaturated only with respect to this phase. These experiments were made in solutions containing totaj... 

Because of the rearrangement, the constant-composition method requires calculating an enthalpy of a phase at the temperature of one stage but using the composition of a different phase and stage, As stated by Holland (8), this may be represented by a thermodynamic process which occurs at constant composition. The enthalpy per mole terms above are specifically ... 

Calculate the total liquid rates from the constant composition method, using the most recent set of compositions and temperatures. Total vapor rates are found from the material balance for each stage. 

Amjad, Z., Koutsoukos, P., Tomson, M.B., and Nancollas, G.H. The growth of hydroxyapatite from solution. A new constant composition method. J. Dental Res, (in press). 

There are basically two methods for recording phase diagrams the titration method and the constant composition method. Both have advantages and drawbacks. 

In the constant composition method a series of compositions covering the composition range to be studied are prepared in test tubes, which are sealed. The test tubes are shaken thoroughly and allowed to stand in a thermostatic bath. The test tubes containing turbid solutions are allowed to stand until they separate into two or more completely clear phases. The number of clear phases can be reported on the phase diagram, and the phase domains can be mapped. 

In this chapter, the 9 method is developed and applied to conventional distillation columns. In Sec. 2-1, the equations required to describe conventional distillation columns are presented. The formulation and application of the 9 method of convergence, the Kb method for computing temperatures, and the constant-composition method for solving the enthalpy balances for the total-flow rates are presented in Sec. 2-2. 

Determination of a Set of Improved Total-Flow Rates by Use of the Constant-Composition Method... 

In the constant-composition method, one of the total-flow rates (Vj or Lj) is eliminated from the enthalpy balance given by Eq. (2-1) for each stage by use of the component-material balances for the respective stage. The restatement of the enthalpy balances given by Eq. (2-1) in the form called the constant-composition method may be initiated by first observing that... 

This expression is developed in a manner analogous to that demonstrated above for Eq. (2-34). The above formulas are given the name constant-composition method because each of the summations appearing in Eqs. (2-34) through (2-37) may be represented by a thermodynamic process which occurs at constant... 

In summary, it was found that if the combination of the 0 method of convergence, the Kb method, and the constant-composition method were used, convergence of almost all problems involving conventional and complex columns as well as systems of distillation columns could be achieved. This combination of calculational procedures is also one of the fastest known methods.2,3... 

Although a proof of the convergence of the combination of the 0 method, Kb method, and constant-composition method cannot be constructed, it can be shown (and is shown below) that the 0 method constitutes an exact solution to certain total reflux problems. 

To illustrate the characteristics of the combination of these three calculational procedures, Example 2-7 was selected. The statement of this example is presented in Table 2-2. The temperature profiles obtained by use of the Kb method on the basis of the corrected compositions found by use of the 6 method [Eqs. (2-27) and (2-28)] are presented in Table 2-3. The constants a and b in the expression for Kb [Eq. (2-31)] were found by use of the K values for /-C4H10 at 510 and 960°R. The values of 0 and the calculated values of D at the end of each trial are also listed in this table. The vapor rates computed by use of the constant-composition method, the temperatures found by the Kp method, and the corrected compositions are presented in Table 2-4. The solution sets d, and bi are presented in Table 2-5. To satisfy the convergence criterion, twelve trials were required and 2.60 seconds of computer time on an AMDAHL 470 V/6 computer using a WATFIV compiler. 

Table 2-4 Vapor rates obtained by the constant-composition method for Example 2-7...

To obtain the convergence characteristics exhibited in Table 2-3 through 2-5, it cannot be overemphasized that the complete set of calculational procedures (the 9 method, the Kb method, and the constant-composition method) must be used. When the 6 method was introduced initially,11 other procedures were used for the determination of the temperatures and total-flow rates, and the convergence characteristics differed from those presented in Tables 2-3 through 2-5. 

To apply the 6 method to problems involving nonideal solutions, minor modifications of the equations presented above for ideal solutions are necessary. The development of the appropriate expressions for the component-material balances, the Kb method, and the constant-composition method is outlined in Probs. 2-6 through 2-10. 

The calculation procedures [the 0 method, Kb method, and constant composition method] developed in Chap. 2 for conventional distillation columns are applied to complex distillation columns in Sec. 3-1. For solving problems involving systems of columns interconnected by recycle streams, a variation of the theta method, called the capital 0 method of convergence is presented in Secs. 3-2 and 3-3. For the case where the terminal flow rates are specified, the capital 0 method is used to pick a set of corrected component-flow rates which satisfy the component-material balances enclosing each column and the specified values of the terminal rates simultaneously. For the case where other specifications are made in lieu of the terminal rates, sets of corrected terminal rates which satisfy the material and energy balances enclosing each column as well as the equilibrium relationships of the terminal streams are found by use of the capital 0 method of convergence as described in Chap. 7. 

The calculational procedure recommended by Nartker et al.12 consists of applying the 6 method of convergence (for conventional and complex distillation columns) one time to each column of the system in succession. The terminal rates so obtained for each column of the system are called the calculated values for the system, and denoted by the subscript ca. To initiate the calculational procedure for the system, the compositions for the minimum number of streams are selected as the independent variables for the system. For example, to initiate the calculational procedure for column 1 of the system shown in Fig. 3-9 the composition of the stream B2 is assumed. One complete column trial is made on column 1. A complete column trial consists of the application of the 6 method for single columns, the calculation of a new set of temperatures by use of the Kb method, and the calculation of a new set of total-flow rates for use of constant-composition method. These temperatures and total-flow rates are stored for use in the next trial calculation for the system. The set of bit s obtained by the... 

The enthalpy balances are stated in the form of the constant-composition method in a manner analogous to that demonstrated in Chap. 2 for conventional distillation columns with a finite number of stages. The balance enclosing the condenser-accumulator section may be stated in the form... 

Figure 6.10 Schematic of constant composition method (CC) experimental setup. (After Thomson and Nancollas 1978.)...

This equation is first order in [Ca ], [P04 ] and s. The average rate constant kf was determined to be 173 11 mof m sec The apparent Arrhenius activation energy was determined over a temperature range of 10 to 40°C to be 186 15 kJ mol Inskeep and Silvertooth (1988) speculated that the rate-limiting step is a surface process, specifically that it is the surface diffusion of Ca and PO4 ions and dehydration commensurate with binding at surface sites of incorporation. Using a constant-composition method and seeded solutions with Sha = 3.05-5.58 Koutsopoulos (2001) found hydroxylapatite formed without a precursor phase. The spiral growth model was found to best fit their kinetic data. 

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