DELFT UNIVERSITY OF TECHNOLOGY

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1 DELFT UNIVERSITY OF TECHNOLOGY REPORT -3 Stability analysis of the numerical density-enthalpy model Ibrahim, F J Vermolen, C Vui ISSN Reports of the Department of Applied Mathematical Analysis Delft

2 Copyright by Department of Applied Mathematical Analysis, Delft, The Netherlands No part of the Journal may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission from Department of Applied Mathematical Analysis, Delft University of Technology, The Netherlands

3 STABILITY ANALYSIS OF THE NUMERICAL DENSITY-ENTHALPY MODEL IBRAHIM, F J VERMOLEN, AND C VUI Abstract In 5, we numerically solved a fluid system by using the numerical density-enthalpy model which consists of mass and energy conservation laws, Darcy s Law and other thermodynamics relations In the current report, the convergence behavior of this model is investigated We transform the original model to two-equation system Which is further approximated by a linear model The eigenvalues of the linear model are used to estimate convergence of the original model Introduction In the density-enthalpy model, we solve a thermodynamic multi-phase flow system by considering density and enthalpy as state variables and compute rest of the system variables as a post processing step We refer to,,3,, for more detail about the usage of numerical density-enthalpy phase diagrams in short, ρ-h diagrams and merits of this approach Here, enthalpy h is actually the specific enthalpy with units J/ g However, we will use ρ and s as our state variables in this report, where s represents the total enthalpy with units J/m 3 In Figure P x 6 T ρ ρ s x s x 7 Figure Partially negative total-enthalpy values corresponding to left pressure and right temperature, two ρ, s phase diagrams are shown for P and T However, we observe that these values are valid for a certain temperature values To mae this point clear, s is plotted as a function of T at constant X G in Figure From this graph and other experiments, we conclude that currently available ρ, s or equivalently ρ, h diagrams are valid approximately for 75 T 36 The author is indebted to HEC, Paistan and NUFFIC, The Netherlands for their financial and logistic support

4 x 7 st 8 6 s T Figure A plot of total enthalpy s as a function of T at constant X G Two Equations Approach In 5, we numerically solved a fluid flow system in a porous medium The mathematical model for the one-dimensional system is given by the following equations t + ρv =, x, t >, mass conservation, t + sv T λ = q, x, t >, energy conservation, v + =, µ x, t >, Darcy s law, 3 T = T ρ, h, x, t >, thermodynamical relation, P = P ρ, h, x, t >, thermodynamical relation, 5 s = ρh, x, t >, total enthalpy, 6 X G = X G ρ, h, x, t >, thermodynamical relation, 7 where the permeability, dynamic viscosity µ, and heat diffusivity λ are assumed to be constants and q is a heat source The initial and boundary conditions are given as follows T x, = T x, x, X G x, = X G, x, x, ρv =, x Γ, t > zero mass flux, 8 λ T + sv =, x Γ, t > zero energy flux 9 This system is solved and discussed in 5 We give the numerical solution results for this system in Figure 3 with a reduced resolution for fast printing Later on this figure is used for comparison with other simulation results We transform the model to two equations in a specific format This approach helps in analyzing system stability Transformation to two equations Consider the mass equation and substitute v by its value as given by the Darcy s law, we obtain t + µ ρ =

5 a ρ b X G 3 c Iterations x m t sec x m t sec τ x 7 d s e T x 5 f P x m t sec x m t sec x m t sec Figure 3 The original model =, x = The plots of a ρ, b X G, c Newton iterations/timestep d s, e T, and f P Now, using the value of, ie, into the above equation, we realize t µ = +, ρ By maing similar substitutions of v, written as t s µ + + = T, and, the energy equation can be λ T + T = Hence the many-equation system given by equations to 6 is written in the following two-equation format t = D + D, t = D + D, 3 where D ij are given by D = µ ρ, D = µ s T, D = µ ρ, The boundary conditions are given by µ s + µ ρ λ 3 D = µ s T + T + T =, = 5

6 3 Numerical solution algorithm To verify that the two approaches many-equations versus two-equations model are indeed equivalent, we solve the system given by equation and 3 by Standard Galerin Algorithm, as follows 3 The mass equation We start the solution algorithm by considering the transformed mass equation ie, equation and write down its linearized wea form t φd µ ρ + Apply the product rule to the second integral t φd µ ρ + φ + µ φd = ρ + dφ d = dx The boundary term vanishes see equation By using Euler Bacward time integration, the above equation is written as ρ τ ρ τ φdx + ρ τ τ τ dφ µ dx dx + ρ τ τ τ dφ dx = µ dx For brevity, we use a different convention for, the following τ i for ρτ, s τ, for ρτ,, s τ,, for ρτ, i, s τ, i, T, and T terms such as The convention used for, T T, and is analogous The linearization about ρ and s is given by the following equation where we omit the index τ for brevity, except for explicit terms and use the notation δρ = ρ + ρ and δs = s + s ρ ρ τ + δρφdx + ρ µ + δρ + + ρ δρ dφ + ρ dx dx + ρ µ + δρ + + ρ δs dφ + ρ dx dx = 6 We use central difference approximations for the density and enthalpy derivatives, given by the following expressions i P i = ρ i, s i = ɛ ρ P ρ i + ɛ ρ, s i P ρ i ɛ ρ, s i, = ɛ P ρ i + ɛ, s i P ρ i, s i + P ρ i ɛ, s i, ρ Pi = P ρ i + ɛ ρ, s i + ɛ s P ρ i + ɛ ρ, s i ɛ s ɛ ρ ɛ s P ρ i ɛ ρ, s i + ɛ s + P ρ i ɛ ρ, s i ɛ s,

7 where ɛ ρ and ɛ s are suitable small numbers in our case, ɛ ρ = and ɛ s = and P i The approximations for i from Taylor series expansion about ρ, s leads to The expression + T + = ρ+ ρ P are analogous The approximation for + s+ s P T is defined in a similar way Using these values in equation 6, we get ρ ρ τ + δρφdx + ρ µ + δρ + ρ δρ P + P δs + ρ + ρ µ + δρ + ρ δρ P + P δs + ρ δρ dφ dx dx Now, we rearrange these terms into explicit and implicit parts δs dφ dx = dx δρφdx + P δρ + ρ µ δρ + δρ ρ + P dφ δρ + ρ δρ dx dx + ρ P P δs dφ δs + ρ µ δs + ρ dx dx + ρ ρ τ φdx + ρ dφ + ρ dx = µ dx We apply the Standard Galerin discretization by using approximations, δρ N j= δρ jφ j, δs N j= δs jφ j and choosing φ φ i N δρ j φ i φ j dx + µ j= N δρ j j= + φ j + ρ P φ j + N δs j ρ µ j= + ρ ρ τ φ i dx + µ dφi dx dx P φ j + ρ ρ φ j + ρ P φ j + ρ + ρ P φ j + ρ dφ j dx dφ j dx dφi dx dx dφi dx = dx The equivalent matrix form of the above equation is given by S δρ + S δs + f = 7 5

8 The element matrices are defined as S e = x + µ i + µ i + µ i ρ i ρ i i i P i P i i i i i ρ P i i ρ P i i + µ i i i + µ x ρ i ρ i ρ i P i P i i + ρ i i ρ P i i ρ P i i, where i S e = µ i + µ x = ρ i ρ i x ρ i ρ i and i P i ρ P i i = s i s i x ρ P i i ρ P i i i + ρ i i + µ i, ρ P i i ρ i P i ρ P i i ρ P i i f i = ρ ρ τ φ i dx + µ ρ f i ρ τ i e ρ i + ρτ i µ = x + i ρ i i + ρ i i ρ i ρ i + ρ dφi dx dx, i + ρ i i 3 The energy equation As a next step, we treat the transformed energy equation equation and write down its wea formulation t φdφ s µ + φd T λ + T φd = Applying the product rule to second and third integral in the above equation, we have t φdφ + s µ + dφ T dx d + T dφ dx d + s µ + T φ λ + T = The boundary terms vanish by applying the boundary conditions equation 5 For the time integration, we use Euler Bacward formula s τ s τ φdx + s τ τ τ dφ µ dx dx + s τ τ τ dφ µ dx dx T τ τ dφ dx dx T τ τ dφ dx = dx 6

9 Using linearization about ρ and s s + δs s τ φdx + s µ + δs δρ + s P + P δs + s µ + δs δρ + s P + P δs T δρ + T + T δs + T δρ dφ dx dx T δρ + T + T δs + T δs dφ dx = dx + s + s Rearranging this equation so that the terms containing δρ come first, then the terms having δs, and lastly the explicit terms s δρ P δρ µ + s + s δρ P dφ dx dx δρ T + T δρ + T δρ dφ dx dx + δs φdx + δs µ + s δs P + δs + s δs P δs dφ + s dx dx T + T δs dφ T dx dx δs dφ dx dx + s s τ φdx + s µ + dφ dx dx T + T dφ dx = dx Applying the approximation for δρ and δs as defined in the case of mass equation, we have N δρ j s P µ φ j + s φ j + s P φ dφi j dx dx j= + N j= j= δρ j T + T N δs j φ i φ j dx + µ dφ i φ j + φ j + s P φ j + s N T δs j + T j= + s s τ φ i dx + s µ T + T dφi dx = dx dx dx N j= δρ j T N δs j φ j + s P j= φ j dφi dx dx φ j dφ i 7 φ j N dx dx T δs j j= dφi + s dx dx dφ i dφ j dx dx dx dφ i dφ j dx dx dx δρ dφ dx dx δs dφ dx dx

10 The equivalent matrix form is given by S δρ + S δs + f = 8 The element matrices are defined as S ij s = P µ T or S e = s i x µ + i s i µ s i i i T i T i φ j + s + T + s i P i P i T i T i i s P i i s P i i + φ j + s P φ j dφ i T φ j dx dx λ x i T i + i µ + T i T i T i dφi dx dx dφ i dφ j dx dx dx P i s P i i s i T i T i Similarly, the element matrix for S is computed as S ij = φ i φ j dx + µ φ j + s P φ j + φ j + s P φ j + s φ j dφi dx dx T + T dφ i T φ j dx dx dφ i dφ j dx dx dx or S e = x + i µ + i µ i + i i i i T i T i T i + T i x x µ i i i i T i T i s i i + i µ + i µ i + s i i s i P i s i P i s P i i s i P i T i T i T i T i The element vector, containing the explicit terms, is given by f i = s s τ φ i dx + s + s µ T + T dφi dx =, dx 8 s P i i s P i i s P i i s P i i dφi dx dx s P i i s P i i

11 or f e = x + i µ i s i sτ i s i sτ i s i T i i + T i + s i + i µ i s i i i + s i i T i + T i 33 Comparison of numerical results from two approaches Equations 7 and 8 can be written in the following matrix form S S δρ f = 9 S S δs f or G + = G J F, where J is the Jacobian matrix Furthermore S S J = f, F =, G ρ = S S f Equation is solved by a direct method Gaussian elimination Here, we give a comparison between the two-equation approach and the original 6-equation system In Figure, the relative difference of density, total enthalpy, and temperature are provided The number of Newton iteration per time step is also presented From these results, we conclude that the two-equation model is an equivalent representation of the system given by equations to 6 Approximation by a linear system We approximate the two-equation model by a linear system in the following way As a first step, the constants a, b, c, and d are computed from {T, X G 8 T 36, X G } Their value is given by a b A = = c d D D = D D µ ρ µ s T These constants are used in the approximate system, given as t = a + b, t = c + d s µ ρ µ s T We compute the eigenvalues of A to determine the stability of this linear system Let λ be an eigenvalue of A, then it is computed as A λi =, where I is a unity matrix of Hence, we solve The solution is given by λ = a λd λ bc =, λ a + dλ + ad bc = a + d ± a + d ad bc 9

12 x 5 a ρ x 5 b s x m t sec x m t sec x 6 c T 3 d Iterations 5 x m t sec 6 τ Figure Comparison of two-equation model with the original system The solution plots are a ρ t sec, b s, and d Newton iterations/timestep j s j s j, c T j ρ j ρ j j T j T j, for j and We show that ad = bc in the following expressions Here we mae use of the fact that P = P T, ie, = T T and = T T Similarly ad = µ ρ µ s T, = µ ρs µ ρ T, = µ ρs µ ρ T T T bc = µ ρ µ s T, Comparing expressions and, we have = µ ρs µ ρ T, = µ ρs µ ρ T T T ad = bc

13 } or equiv- Therefore, the eigen values of A are given by {, µ ρ + µ s T alently { {, } for a + d =, λ = {, a + d} for a + d It is difficult to find a + d analytically We numerically computed this value for the entire ρ, s-diagram, and it is given by 73 < a + d < 3 for X G and 8 T 36 Hence, the original system is unconditionally stable because λ = {, a positive value} Possibility of one state variable One of the two eigenvalues is zero for the entire phase diagram, hence the approximated linear system can be reformulated such that only one variable is sufficient to describe system dynamics This can be achieved by diagonalization of A Such conclusion can only be drawn for a linear system However, we checed the possibility of one state variable, experimentally Using the following initial conditions 9 for x, 5, T x, = x 9 for x 5, 95, 9 for x 95,, X G x, =, = / s N = x = /N ɛ r = 6 = 5 m, µ = 5 5 P a s, λ = 5 W/m/, t max = 3 s time step, mesh size, spatial step, error tolerance on ρ and h, process time Figure 5 shows the relative difference between the initial and steady state value of x Relative difference of hx, and hx, x Figure 5 Relative difference between initial and steady-state h for original system h, when the above initial conditions are used by the original 6-equation model We do not observe a significant relative difference between the two values

14 In an another experiment, we tae only one equation ie, the mass equation and ignore the energy equation In other words, the original system is approximated by one equation only The simulation results are comparable to the original model and they are given in Figure x Original vs one equation model, T 8 x Original vs one equation model, X G x x Figure 6 Comparison of variables from the original system and one-equation model at steady-state left Relative difference in T and right relative difference in X G Gibbs Phase Rule Gibbs Phase Rule is given by the following relation where F = number of degrees of freedom, F = C + Φ, C = number of component or substances, Φ = number of phases in thermodynamic equilibrium with each other For our system, C = because the only substance here is Propane, Φ =, for a two phase flow Therefore, the results we obtained are consistent with Gibbs Phase Rule ie, one equation is sufficient to solve the system for a two phase flow 5 Conclusions The original system can be transformed to two-equation model Which can further be approximated by a linear two-equation system The eigenvalues of the linear system suggest that the original nonlinear system is stable for the given range of T and X G We also conclude that the system obeys Gibbs Phase Rule, at least for a two-phase flow References A R J Arendsen, A I van Berel, A B M Heesin, and G F Versteeg Dynamic modelling of thermal processes with phase transitions by means of a density-enthalpy phase diagram 7th World Congress of Chemical Engineering, Glasgow, Scotland, 5 Ibrahim, F J Vermolen, and C Vui Application of the numerical density-enthalpy method to the multi-phase flow through a porous medium Procedia Computer S, cience 78-79, Amsterdam, The Netherlands, 3 A R J Arendsen and G F Versteeg Dynamic modelling of refrigeration cycles using density and enthalpy as state variables 7th International Congress of Chemical and Process Engineering, Prague, The Czech Republic, 6 A R J Arendsen and G F Versteeg Dynamic Thermodynamics with Internal Energy, Volume, and Amount of Moles as States: Application to Liquefied Gas Tan Ind Eng Chem Res 9, 8,

15 5 M Fabbri and V R Voller The Phase-Field Method in the Sharp-Interface Limit: A comparison between Model Potentials Journal of Computational Physics, A Abouhafç Finite Element Modeling Of Thermal Processes With Phase Transitions Master Thesis, 7, Delft University of Technology 7 Ibrahim, C Vui, FJ Vermolen, D Hegen Numerical Methods for Industrial Flow Problems Delft University of Technology, Report 8-3, 8 8 E Javierre, C Vui, F J Vermolen, S van der Zwaag A comparison of numerical models for one-dimensional Stefan problems J Comp Appl Math, 6, 9, H Emmerich The Diffuse Interface Approach in Materials Science, Thermodynamic Concepts and Applications of Phase-Field Models Springer, Berlin, 3 JH Brusche, A Segal, and C Vui An efficient numerical method for solid-liquid transitions in optical rewritable recording International Journal for Numerical Methods in Engineering, 77, pp 7-78, 9 OC Zieniewicz, RL Taylor, & JZ Zhu The Finite Element Method, Its Basis & Fundamentals 6e, Butterworth-Heinemann, 5 A Hoffmann Computational Fluid Dynamics, Vol I e, EES, Wichita, USA, 3 J van an, A Segal, F Vermolen Numerical Methods in Scientific Computing VSSD, R J Leveque Finite Volume Methods for Hyperbolic Problems Cambridge University Press, USA, 5 JN Reddy An Introduction to the Finite Element Method e, McGraw-Hill, Ibrahim, C Vui, FJ Vermolen, D Hegen Numerical Methods for Industrial Flow Problems Delft University of Technology, Report 9-, 9 7 H S Udayumar, R Mittal, and Wei Shyy Computation of SolidLiquid Phase Fronts in the Sharp Interface Limit on Fixed Grids Journal of Computational Physics, 53, 999, H S Udayumar, R Mittal, and W Shyy Computation of SolidLiquid Phase Fronts in the Sharp Interface Limit on Fixed Grids Journal of Computational Physics 53, 999, S Chen, B Merriman, S Osher, P Smerea A simple level set method for solving Stefan problems Journal of Computational Physics, v35 n, p8-9, July 5, 997 S Osher, J A Sethian Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations Journal of Computational Physics, v79, pp-9, Nov 988 A Faghri, Y Zhang, J Howell Advanced Heat and Mass Transfer Global Digital Press, B Nedjar An enthalpy-based finite element method for nonlinear heat problems involving phase change Comput Struct G Comini, S DelGiudice, B W Lewis, O C Zieniewicz Finite element solution of nonlinear heat conduction problems with special reference to phase change Int J Numer Meth Engng, 8, pp 636, 97 G Segal, C Vui and F J Vermolen A conserving discretization for the free boundary in a two-dimensional Stefan problem J Comp Phys,, pp -, Ibrahim, F J Vermolen, C Vui Numerical Methods for Industrial Flow Problems Delft University of Technology, Report -3, 3