By: Dr. Gerardo Ruiz UIC LPPD 03/18/2009

1. Arc length calculation in parametric curves, Temperature Collocation,Orthogonal Collocation in Composition Profiles, and Generalized Column Profiles Maps By: Dr. Gerardo Ruiz UIC LPPD 03/18/2009

2. Arc length calculation in parametric curves

3. Arc Length • If s is the arc length of a parametrized curve  defined by the end points of F(t), then dF/ds is a unit tangent to  in the direction of motion

4. Arc Length • The direction of Fis that of an infinitesimal chord of the parametrized curve  (F(t)) • t0, the direction of F, and hence the direction of F/t (dF/dt) approaches the direction of the tangent to  • If the scalar variable t is taken to be the length s of 

5. Arc Length • A point is moving with constant speed v (ds/dt) along a curve  • The distance covered from t0 and t1 is v(t1 - t0) • To obtain the distance along  t0tt1 by defining the integral of the speed with respect to time

6. Arc Length of a Circle • If the circle is parametrized in R2

7. Arc Length of a Circle • If the circle is parametrized in R2

8. Arc Length and Line Integral • Line integral is an integral where the function to be integrated is evaluated along a curve • The value of this function is the sum of values of the field at all points on the curve weighted by some scalar function on the curve (arc length) • This weighing distinguishes the line integral from simpler integrals

9. Temperature Collocation,Orthogonal collocation in composition profiles

10. Temperature Collocation • Zhang and Linninger 2004 • An affine mapping of the column height (h) into bubble-point temperature (BPT) • The mapping establishes a bijective relationship between h and BPT • The T transformation reduces the nonlinearity of the profile equations, because the use of T as an independent variable makes T-dependent expressions explicit. • BPT0and BPTpinch

11. Convectional continuous model Temperature collocation continuous model Temperature Collocation

12. Temperature CollocationConvectional T collocation

13. Temperature Collocation BP Isotherms Bubble Point Distance

14. Orthogonal Collocation • To approximate the differential equations with finite temperature elements and collocation in orthogonal polynomials. • Lagrange polynomials • ODE are converted in nonlinear algebraic equations • The variables are the unknown weights

15. Orthogonal Collocation • Discretization of derivatives of node j in element [i],

16. Orthogonal Collocation • Where,

17. Composition profiles using temperature orthogonal collocation

18. Column Profile Maps

19. Column Profile Maps • Tapp, Holland et al. 2004 • The difference point equation is used in a generalized column section • At steady state condition

20. Column Profile Maps, R=9, X  =[0.9; 0.05; 0.05]

21. Column Profile Maps • If the generalized reflux ratio go to , the difference point equation is mathematically identically with the RCE used in a generalized column section • At steady state condition

22. Column Profile Maps, R=+, X  =[0.9; 0.05; 0.05]

23. THANKS QUESTIONS?