Ch 6 - Martini and Wine Glass Martini glass filled to 6cm...

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Martini and Wine Glass !" 2 22 R ha R #$ % 3 6 cap Vh a h & %$ 2 216 3 33 cap R h %# % Martini glass filled to 6cm depth contains ~ 216 & /3 of wine. How deep must you fill wine glass to ensure that it contains the same amount of wine? ( 4 ) Volume of a spherical cap Use Pythagorean Theorem 2 aR h h 3 2 216 () 7 0 h fh h # % Substitute into formula (R=7) 1 Peng-Robinson Equation of State ! Ideal gas law: PV i =RT ! Many gases not “ideal”: Reservoirs ! High temperatures ! High Pressures ! Peng-Robinson accounts for non-ideality ! At P= 50 bar and T=473K…. 2 iii RT a P VbV b Vb # 0.457 2.3 6 0.0778 24.7 c c c c RT aE P RT b P %% 2 39325 2.3 6 () 5 0 0 24.7 49.4 611 i ii i E fV VV V $ % # Methane 2
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Falling Parachutist ! Newton’s Law: ! Force Balance: ! Plugging in gravity and drag: ! Integrate: ! What is “c” if: g = 9.8 m/s2, m = 68.1 kg, v =40 m/s, t = 10s dv Fm am dt %% !" (/ ) () 1 cmt gm vt e c # %# DU dv FFm dt $% dv mg cv m dt #% 0 40 1 38 . 667 ) ( 146843 . 0 % # # % # c e c c f 3 Roots of Nonlinear Equations: f(x) = 0 ! Find the root of the quadratic equation ! Many problems aren’t so simple ! Martini and wine glass ! Peng-Robinson Equation of State ! Falling Parachutist ! Use a “numerical” method to solve ! Approximate technique a ac b b x c bx ax x f 2 4 ; 0 ) ( 2 2 # ' # % % $ $ % no “analytical” solution!!! 4
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Root Finding Techniques ! Bracketing Methods ! Graphical ! Bisection ! False Position ! Open Methods ! Fixed Point Iteration ! Newton-Raphson ! Secant Method ! Polynomials ! Muller’s Method ! Bairstows Method 3 2 216 () 7 0 33 h fh h %# # % 2 39325 2.3 6 ( ) 50 0 24.7 49.4 611 i ii i E fV VV V $ % #$ # !" 0 40 1 38 . 667 ) ( 146843 . 0 % # # % # c e c c f 5 Graphical Methods • Find “c”, but how? • Maybe I could just plot points? c = 4; f(c) = 34.115 c = 8; f(c) = 17.653 c = 12; f(c) = 6.067 c = 16; f(c) = -2.269 c = 20; f(c) = -8.401 0 40 1 38 . 667 ) ( 146843 . 0 % # # % # c e c c f Nonlinear Equation for drag coefficient root 6
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Bisection (Interval Halving) ! We saw from the graph ! The root: 12 < c < 16 ! Why f(c) went from being positive to negative? ! The value of “c”, that gave f(c)= 0 had to be between those numbers !
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Ch 6 - Martini and Wine Glass Martini glass filled to 6cm...

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