Process of whiskey maturation & Current Efforts to improve Comparison Goals Modeling nonlinear Diffusion

Transcription

1 Brian England

2 Objective Process of whiskey maturation & Current Efforts to improve Comparison Goals Modeling nonlinear Diffusion Cylindrical Coordinates Initial and Boundary Conditions Methodologies and Computational Results Finite Difference Finite Volume Function Space Final Comparison and Conclusion

3 Maturation of Whiskey Driven by two processes Diffusion of Oak barrel goodness Short time scales Chemical reaction Long time scales Current Efforts Diffusion Wooden inserts, tastes more processed and is lacking Chemical Reactions Pressure vessels / burners to shift % yields and reaction rates Much better and they re improving on this front

4 Comparison Goal General Comparisons Accuracy Stability Computational Efficiency I will primarily deal with Computational Efficiency Validate a methodology for utilization in future work

5 Modeling Nonlinear Diffusion What makes the Diffusion Nonlinear? Changes in differential operators Best coordinate system Ellipsoidal Much more Complex Cylindrical Coordinates Easier & exact solution are readily available

6 Cylindrical Coordinates Differential Operator Adjustments Gradient Divergence Curl Laplacian U 1 U U U,, r r z F 1 rfr 1F F r r r z 1Fz F Fr Fz 1 rf F r F,, r z z r r r 1 ru 1 U U U r r r r z z

7 Initial and Boundary Conditions First Comparisons Boundary Conditions Chosen to ensure an equilibrium U( t,1,, z) cos sin z U( t, r,, 1) cosr Initial Condition U(0, r,, z) 0

8 Finite Difference Approach General PDE t r r r r z U 1 U U 1 U U Centered Difference Spatial Discretizations U r U U U i1, j, k i, j, k i1, j, k Forward Euler in Time r U f U r U U t f U r t U r n1 n (, ) ( ) (, ) U U i1, j, k i1, j, k r

9 Stability The discretization n n n U U n1 n Ui 1U i U i1 U U ( t) t x x Requirement for Stability t x For our PDE, the largest linearized coefficient is the most restrictive 4 1 U 1 r t r min r

10 Good news (->Stability) Can get away with ~ dr = 0.05 & dt = Wall Clock time = Seconds Tolerance set to Flux/function values of 0.01

11 Finite Volume Method More work apriori! Approach PDE with Gauss Law & Integral Form Our PDE in Integral Form V U dv K U dv t V The Integral Form allows us to represent the average value of our function U 1 U dv UdV V UdV V t t t V t V V V

12 Final Exact Formulation Utilizing Gauss Law V FdV F ds S Making appropriate substitutions V KU dv K U ds U 6 K ds We obtain the final form U S t V i 1 S i i

13 Finite Volume Approximations Difference in time K Net Flux n 1 n U U t V Differences on the boundaries U r U n i1 U r n i Calculation of Fluxes S ( r dr / )( U U ) ( r dr / )( U U ) Net Radial Sr U U U Net Azimuthal r j1 j i1 r i1 i i i1 S U U U Net Axial Z j1 j i1

14 Cell Volume & Surface Integrals Volume 1 V rout rin z rzr Axial Surface Radial Surfaces 1 SZ rout rin rr S rz r Azimuthal Surfaces S r z

15 Items of Note Method is fully conservative Only approximations lie in the derivative at the cell surfaces When proper substitutions of the volume / surfaces is performed, it s almost identical to finite difference What benefits are there? We re still approximating derivatives Leads to order of accuracies Computations is about the same

16 Results Same scenario as finite difference Stability Required dt = /6 th the dt There may have been corner issues Wall Clock 359 Seconds 10 times as long Even if the dt was matched, this method would still be slower For now

17 Function Space Methods Rely on Eigen function products Coefficients are solved via inner products Integrals are approximated by function evaluations at collocation points Our Problem - Diffusion Equation in Cylindrical Coordinates Choice of Eigenfunctions Radial - Bessel functions of the first kind Azimuthal Trigonometric Axial Trigonometric Temporal - Exponential

18 Exact Solution Fundamental Technique Principle of Superposition Handle a separate problem for each boundary and initial state for a steady state solution and transient U( t, r,,1) f ( r, ) top U( t,1,, z) f (, z) Lat U( t, r,, 1) f ( r, ) bot U(0, r,, z) f ( r,, z) Lat

19 General Solution Step by step Separation of Variables First Time Which Gives Separating and solving T() t e t U( t, r,, z) T( t) ( r,, z) Tt () t ( r,, z) T( t) ( r,, z) r r r r z Letting ( r,, z) R( r) ( ) Z( z) 1 R( r) 1 R( r) 1 ( ) 1 Z( z) rr( r) r R( r) r r ( ) Z( z) z

20 General Solution You can show that our ODE s 1 ( ) ( ) l 1 Z( z) Z() z z m Provides R( r) R( r) r r ( m) r l R( r) 0 r r General Solution depends on our BC-s

21 Solution Primary Case - Initial Conditions with homogenous BC=0 lmnt U ( t, r,, z) Almne cos mz Jl m r S in l l0 m0 n0 l0 m0 n0 l = is an integer greater than or equal to 0. k ln,, is the n th root of the Bessel function of the first kind of order L m is an integer greater than or equal to 0. Finally lmnt Blmne cos mz Jl m r Cos l k m lmn l, n

22 Constants The constants are solved via A lmn U (0, r,, z)cos mz Jl m r sin l rdrddz cos mz Jl m r sin l rdrddz B lmn U (0, r,, z)cos mz Jl m r cos l rdrddz cos mz Jl m r cos l rdrddz

23 Collocation Points Azimuthal Fourier j Evenly distributed Axial fourier Chebyshev nodal locations Radial Bessel / N j 1 cos N z Roots of m th order Bessel function Best used in tabular form to save on computations x j

24 Collocation Methods Numerous Integration techniques Quadrature at collocation points for A and B Standard Gaussian Quadrature Radial and Axial Weights Polynomial Approximations Azimuthal Weights Trigonometric Approximation We ll use nodal locations, i,j,k, to calculate weights f ( r,, z) rdrddz w f ( r,, z ) i1 j1 k1 ijk i j k

25 A Separation of Variables lmn Through some integration and appropriate initial conditions we can state B lmn l Z ( z)cos mz dz ( )sin l d R ( r) J m r rdr J l We can now approach each integral with our approximation m r rdr l Z ( z)cos mz dz ( )cos l d R ( r) J m r rdr J l m r rdr

26 Weight Calculation (Axial) A bit of matrix manipulation and brute force With fixed nodes 1 inverse and done! c1 1 1 x1 x... xn 1 x N c N N N N 1 x1 x... xn 1 xn cn 1 ( N 1) N 1 N 1 N 1 N 1 1 x1 x... xn 1 x N c N N n

27 Weight Calculation (Azimuthal) Trigonometric Gaussian Quadrature Maximal Trigonometric Degree of Exactness New Quadrature Methodologies Trigonometric Orthogonal Systems and Quadrature Formulae with Maximal Trigonometric Degree of Exactness Gradimir V. Milovanovic Easier assumption Requires proper initial conditions w1 1 1 sin x1 sin x... sin xn 1 sin x N w N N N N 1 sin x1 sin x... sin xn 1 sin xn wn 1 ( N 1) N 1 N 1 N 1 N 1 1 sin x1 sin x... sin xn 1 sin x N w N N sin( n ) d wif ( i) 1 n1 i1

28 Weight Calculation (Radial) Requires Bessel functions 1 0 J ( r) dr w f ( r ) n i n i i1 m mn x m! ( m n 1) m! ( m n 1) m n * J 0 n() r dr J N dr 0 m0 m0 J ( x ) J ( x )... J ( x ) J ( x ) w J J x J x J x J x w J J ( x ) J ( x )... J ( x ) J ( x ) w J J x J x J x J x w J * N1 0 N 1 1 * 1( 1) 1( )... 1( N1) 1( N) * N 1 1 N 1 N 1 N 1 N 1 N N 1 N 1 * N ( 1) N ( )... N ( N 1) N ( N ) N N m

29 Conclusion Function Space methods Very time consuming apriori Didn t get to finish Expectation of fast convergence Approximation lies only with initial / boundary conditions Exact solution for all time after (for the given approx)