Linking Correlations Spanning Adjacent Applicability Domains

Transcription

1 Ian David Lockhart Bogle and Michael Fairweather (Editors), Proceedings of the 22nd European Symposium on Computer Aided Process Engineering, June 2012, London Elsevier B.V. All rights reserved. Linking Correlations Spanning Adjacent Applicability Domains Tareg M. Alsoudani and I. D. L. Bogle Centre for Process Systems Engineering, University College London, Dept. of Chemical Engineering, Torrington Place, London, WC1E 7JE, U.K. Abstract Discontinuities between distinct regions, described by different equation sets, cause difficulties for solvers. We present a new algorithm that eliminates integrator discontinuities through two steps. First, it determines the optimum switch point between two empirical correlations spanning two or more domains. The optimum switch point is determined by searching for a jump point that minimizes a discontinuity between results obtained through calculating a parameter using two or more different methods at an intersection or proimity of their validity domains. Second, it links the two adjacent discontinuous domains with an interpolating polynomial. When applied, this approach should eliminate the need for conventional integrators to discretize and then link discontinuities through the conventional approach that generates interpolating polynomials based on state variables. The new approach tackles each discontinuity at the level of the constitutive equations rather than at the level of state variables, hence eliminating errors associated with polynomials generated at state variable level. Keywords: discontinuity, domains overlap. 1. Introduction and Literature Survey Literature is rich in empirical correlations, spanning different domains, to calculate certain parameters. Each of these correlations spans a certain validity domain with a specified accuracy. Theoretically, parameters calculated through different correlations should have eact or very close values at the intersection points of the correlations' applicability domains. However, this is not the case with most empirical formula; resulting in large discontinuities when solving spatio-temporal problems. A typical eample, illustrating the discussion, would be the calculation of heat transfer coefficient at the intersections between laminar and turbulent regimes. A process can be thought of as a comple system that is described by, mostly, continuous mathematical functions (algebraic or differential). The solution of these mathematical functions, usually through integration, brings an insight into the behaviour of the process under study. However, the continuity of these mathematical functions is sometimes broken by internal or eternal influences. A rapid phase change or flow reversal are eamples of an internally generated discontinuity in a ODE/DEA system whereas switching a pump on or off can be considered as an eternal influence that raises a discontinuity. Regardless of the form or source of discontinuity, it needs to be resolved either before starting to integrate the ODE/DAE system (if possible) or whenever it is encountered during the evolution of the integration process. Eternal influences affecting model continuity are usually resolved through the use of the modelling language, especially modern modelling languages. Once a DAE/ODE solver encounters an eternal

2 2 T. Alsoudani and I.D.L. Bogle discontinuity, it reinitializes state variables based on the new conditions post discontinuity. Eamples of such cases include changes in process boundary conditions, changes in process network through flow rerouting or pump switching, relief valve triggering, etc. The literature in handling internal discontinuities can be classified into two types. The first type (Type I) handles discontinuities generically through detection and resolution algorithms that are usually embedded within the solver. The solver usually detects the discontinuity through observing a jump in one or more of the state variables although the origin of the discontinuity might not necessarily be the state variable. The solver, then, tries to bridge between the two discontinuous states either through discretizing the discontinuous domain or through generating an interpolating polynomial in the discontinuous variable as illustrated in the figure below (Borst, 2008). For a comprehensive review of state event detection algorithms, the reader may refer to (Park and Barton, 1996), (Mao and Petzold, 2002) and (Archibald, et al, 2008). (Javey, 1988) reports three methods for resolving discontinuities once detected: changing equations to those post discontinuity conditions, halving the step in an effort to find a resolution at a finer grid, and reinitializing the ODE/DAE system to conditions post discontinuity. The second type (Type II) of discontinuity handling techniques deals with detection and resolution of discontinuities through structuring the ODE/DAE system in a way that minimizes, and hopefully eliminates, discontinuities. The literature is etremely sparse on this type of discontinuity detection and resolution. The first encountered attempt into resolving such discontinuity was by Brackbill et al. (1992). Borst (2008) emphasized that the use of regulating functions derived from the physics of the problem (Type II) will better eliminate discontinuities than the sole use of discretization techniques. 2. Methodology In this work, we try to resolve type II internal discontinuities arising from the use of logical epressions such as (1a) by devising discontinuity detection and resolution algorithms. Such epressions usually arise when simulating adjacent domains defined by different empirical correlations. The modeler usually starts with an epression of the form (1a.) and then transforms it into form (1b.) during model construction. Assuming a < b, c < d and b c in epression (1b.) will satisfy (the last condition enforces overlapping or touching domains), g c g b to maintain continuity in dimension. We are then free to select any g that satisfies g [ b, c]. However, an arbitrary choice will f ( ) f 1 = f 2 ( ), [ a,b] ( ), [ c,d] 1a. 1b. If (<g) then (Domain I) f() = () Elseif ( g) then (Domain II) f() = ()

3 Linking Correlations Spanning Adjacent Applicability Domains 3 probably result in selecting g points that increase or create discontinuities as illustrated in (2a) and (2b), respectively. Nevertheless, this freedom in selecting g facilitates transforming the search for the best g into an optimization problem of the form: min. () () s.t. [ b, c] The solution of this optimization problem will result in g that minimizes the discontinuity between non-intersecting and (2a); or eliminates the discontinuity between intersecting and (2b). a c g ' g b d 2a. 2b. For multiple functions f i (i=1 n) spanning multiple adjacent domains [a i, b i ] and satisfying a i b i-1 (i=2 n), there eist n-1 intersecting g i points that need to be located by applying the minimization algorithm at each intersection interval. Once g i points are identified results can be reflected directly in the logical epression (1b). We can then rely on type I discontinuity handlers to resolve any discontinuity in a state variable resulting from (2a) type of discontinuity in the state variable s respective constitutive equation. However, a better solution would be to eliminate the discontinuity at the constitutive equation level. This is achievable through linking and by an interpolating polynomial at the location of the discontinuity. Any interpolating polynomial would satisfy zero order smoothness. However, 2 nd order smoothness (usually required in integration routines) is only achieved through a few of the readily available interpolating functions such as cubic splines and cubic hermite interpolating polynomials. In subsequent discussion, we will demonstrate the concept using hermite interpolating polynomials although any 2 nd order interpolating polynomial would fit. Three points are usually enough to construct an interpolating polynomial. However, to ensure better closure of the interpolating polynomial with functions and, a fourpoint interpolating polynomial is selected as illustrated in (3a.) and (3b.). The points are a c g g ' b d f ( ) Rela tive to { p=0.0 p =0.1 p =0.2 p =0.3 p =0.4 } p =0.4 p =0.5 p =0.3 p =0.2 p =0.1 p =0.0 f ( ) p=0.0 p=0.1 p=0.2 p=0.3 p=0.4 p=0.5 Rela tive to 3a. Three point interpolation 3b. Four point interpolation

4 4 T. Alsoudani and I.D.L. Bogle If (<g-1.5h) then (Domain I) f() = () elseif (g-1.5h g-1.5h) f() = Interpolating polynomial Elseif (>g) then (Domain II) f() = () 4. equally separated by a distance h and equally distributed between the right and left sides of the g point location. Two of the points lie eactly on the respective functions and and located at distances g-1.5g and g+1.5h, respectively. The other two points are located at distances g-0.5h and g+0.5h. To ensure proper closure of the interpolating polynomial with and, interpolating points values of the latter locations will correspond to the values of their respective functions after adding p (g)- (g) to the lower valued function and subtracting p (g)- (g) from the higher valued one, where p ranges from 0 to 0.5. The p factor is a tuning parameter that can be used to increase smoothness at the link points between the interpolating polynomial and the respective functions as illustrated in (3a.) and (3b.). The logical epression in (1b) then converts to (4) and the algorithm terminates. The 1D detection algorithm can be epanded to cover 2D functions f(,y) through treating each dimension separately. However, the 2D discontinuity resolution algorithm differs because the independent variables and y move freely in a 2D space based on evolution of simulation run. This simulation-dependent movement makes it hard to determine where the discontinuity would be hit during a simulation run and hence it is hard to fi the coordinates of the interpolating polynomial. Thus, we recommend tracking the movement trajectory of the independent variables and y through a vector v i at each simulation integration step and etrapolating the trajectory to the discontinuity plane in order to continuously update the coordinates of the interpolating polynomial points as in (5a). P o v 1 P1 v 2 P 2 v n f (, y) P n B v p A P n 1 P n 2 C y 5a. 5b. 3. An Eample To test the theory, we used a simplified model of the paraffins isomerization reactor that was part of the patent published by Minkkinen (1993). The model is constructed using the gproms modeling language (PSE,2011). We tested the effect of transition from Laminar to Turbulent flow regimes on the wall heat transfer coefficient. For Laminar

5 Linking Correlations Spanning Adjacent Applicability Domains 5 flow, we used the simplified constant heat-flu equation of Nu d = For turbulent flow we used the Gnielinski correlation (Keith, 2000): / where: ^ 2 A plot of the Nu d versus Re and Pr is illustrated in (5b.). Since minimum Re for Gnielinski correlation is 2300, this value was taken as the separation value between Laminar and Turbulent flow regimes. We started the simulation at a feed velocity corresponding to a Reynolds number o030. Then, during the course of the simulation, we ramped the velocity to cross the 2300 laminar boundary. Unlike other simulation packages that detect discontinuities only at the state variable level, gproms is smart enough to detect discontinuities at the constitutive equation level. However, gproms still performs reinitialization once a discontinuity is detected. This reinitialization is a form of discretization that we discussed earlier (Type I). To apply the two dimensional solution we proposed, we needed to link gproms to a C++ code through gproms Foreign Object Interface. gproms passes the required parameters to compute Nu d and the C++ code determines whether it needs to calculate Nu d using Laminar, Turbulent, or the interpolating polynomial equation. Once the simulation is run with the C++ code, gproms previously reported discontinuity disappeared and hence gproms reinitialization was not performed. The evolution of the simulation vector is superimposed on the Nu d mesh. Simulation results also demonstrate a noticeable decline in overall simulation computations due to elimination of model reinitialization; although it is not conclusive. To conclude, we were able to demonstrate an algorithm that detects and resolves discontinuities resulting from the use of correlations describing behaviors of adjacent domains. The solution to the 1D detection and resolution problem is unique for a fied set of correlations irrespective of the model. Thus the algorithm can be eecuted prior to starting the simulation run. For the 2D case, the solution is dependent on the evolution of the independent variables trajectory. We think the 2D solution can be further etended to cover higher dimension correlations. Also this approach is best embedded inside the simulator since it allows a simulation compiler to automatically generate the code containing the interpolating polynomial from that written by the modeler. References 1. Archibald, R., A. Gelb and J. Yoon, "Determining the Locations and Discontinuities in the Derivatives of Functions", Applied Numerical Mathematics, vol. 58,, pp , Borst, R., "Challenges in computational materials science: Multiple scales, multi-physics and evolving discontinuities", Computational Materials Science, vol. 43,, pp. 1-15, Brackbill, J.U., D.B. Kothe and C. Zemach, "A Continuum Method for Modeling Surface Tension", Journal of Computational Physics, vol. 100,, pp , gproms Modelling Language, Copyright Process Systems Enterprise Ltd. 5. Kreith, Frank, CRC Handbook of Thermal Engineering, CRC Press, Mao, G. and L.R. Petzold, "Efficient Integration over Discontinuities for Differential- Algebraic Systems", Computers and Mathematics with Applics, vol. 43,, pp , Minkkinen, A., L. Mank, and S. Jullian, "Process for the Isomerization of of C5/C6 Normal Paraffins with Recycling of Normal Paraffins", U. S. Patent 5,233,120 (1993). 8. Park, T. and P.I. Barton, "State Event Location in Differential-Algebraic Models", ACM Transactions on Modeling & Computer Simulation, vol. 6, issue 2, pp , 1996.