Residue curve equation derivation

Equation 9.14 is the membrane residue curve equation as identified by Peters et al. [14]. It is mathematically analogous to the residue curve equation derived in Chapter 2 for distillation processes. While the residue curve equation was derived... 

It is important to note that the residue curve equation (Equation 2.8) is simply a mass balance. In deriving it, there were no pre-defined notions regarding how the vapor phase is related to that of the liquid. The equation is merely providing a mathematical means of determining the composition of the residual liquid phase after an amount of material has transferred to the vapor phase. Solving the residue curve equation, however, requires knowledge about how the two compositions are related to each other. Such a relationship depends on the manner in which the boiling is conducted. 

The holdup effects can be neglected in a number of cases where this model approximates the column behavior accmately. This model provides a close approximation to the Rayleigh equation, and for complex systems (e.g., azeotropic systems) the synthesis procedures can be easily derived based on the simple distillation residue curve maps (trajectories of composition). However, note that this model involves an iterative solution of nonlinear plate-to-plate algebraic equations, which can be computationally less efficient than the rigorous model. 

Table 2.4 shows the SAS NLIN specifications and the computer output. You can choose one of the four iterative methods modified Gauss-Newton, Marquardt, gradient or steepest-descent, and multivariate secant or false position method (SAS, 1985). The Gauss-Newton iterative methods regress the residuals onto the partial derivatives of the model with respect to the parameters until the iterations converge. You also have to specify the model and starting values of the parameters to be estimated. It is optional to provide the partial derivatives of the model with respect to each parameter, b. Figure 2.9 shows the reaction rate versus substrate concentration curves predicted from the Michaelis-Menten equation with parameter values obtained by four different... 

Figure 21.5 Potential errors introduced by the assumption that Ar2+ is constant in the derivation of the linkage Eq. (21.30). The dependence of Ar2+ on [Mg2 1J was modeled by a polynomial fit to the solid gray data points in Fig. 21.4D. The polynomial was used in the integration of Eq. (21.30) to give expressions for In Kobs and 0 with [Mg2+]-dependent AT2 (in contrast to Eqs. (21.33) and (21.34), which assume AT2+ is constant). 9 is plotted for the calculated titration curve when the midpoint of the titration, [Mg2+]0, is 10 jiM (circles), 30 uA / (squares), or 100 li.M (diamonds). AT2+ (as used to calculate the displayed curves) at the titration midpoints (9 = 0.5) is 1.45, 2.30, and 2.73, respectively. The simulated data points have been fit to either a modified version of Eq. (21.34) that assumes the y-intercept of the curve has the value 6 = 0, 9 = 90 + (1 - 0o)([Mg2+]/[Mg2+]o)"/[l + ([Mg2+]/[Mg2+j0)"], or to an equation that allows a nonzero y-intercept, Eq. (21.35). The residuals of the fits are shown in the lower three panels closed symbols correspond to Eq. (21.34) and open symbols to Eq. (21.35). For the curve with a midpoint of [Mg2+]0 = 100 fiM, Eq. (21.35) could not be fit to the data because the value of Co became vanishingly small. The values of A IT. at the titration curve midpoints obtained from the modified Eq. (21.34) are 1.99, 2.39, and 2.73, in order ofincreasing [Mg2+]0. AT2+ obtained by fitting ofEq. (21.35) and application of Eq. (21.36) are 1.43 and 2.29 (10 and 30 fiM transition midpoints, respectively).

Here, Y is R. With the aid of equations 1-3, the best regression line is fitted using values of Y and K. The constants a , a, c, and Cq are constrained to make the calibration curve and its first derivative continuous at and K2. The boundaries K, and K2 are chosen to minimize the sum of the residuals squared. The constants, bo, bj, b2 and b governing that portion of the calibration curve with < K2 was obtained by non linear regression using the... 

Temperature Pulse Decay Technique. As described in Sec. 2.4 under Temperature Pulse Decay (TPD) Technique, local blood perfusion rate can be derived from the comparison between the dieoretically predicted and experimentally measured temperature decay of a thermistor bead probe. The details of the measurement mechanism have been described in that section. The temperature pulse decay technique has been used to measure the in vivo blood perfusion rates of different physical or physiological conditions in varimis tissues (Xu et al., 1991 1998). The advantages of this technique are that it is fast and induces little trauma. Using the Pennes bioheat transfer equation, the intrinsic thermal conductivity and blood perfusion rate can be simultaneously measured. In some of the applications, a two-parameter least-square residual fit was first performed to obtain the intrinsic therm conductivity of the tissue. This calculated value of thermal conductivity was then used to perform a one-parameter curve fit for the TPD measurements to obtain the local blood perfusion... 

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