On the adaptive time-stepping for large-scale parabolic problems: computer simulation of heat and mass transfer in vacuum freeze-drying

Transcription

1 On the adaptive time-stepping for large-scale parabolic problems: computer simulation of heat and mass transfer in vacuum freeze-drying Krassimir Georgiev, Nikola Kosturski, Svetozar Margenov Institute for Parallel Processing BAS Sofia Bulgaria Jiří Starý Institute of Geonics AS CR Ostrava Czech Republic SNA Liberec Page 1

2 Introduction The work is motivated by a process of the dehydrating frozen materials by sublimation under high vacuum. The apparatus consists of two containers, one (food camera) for the product to be dried and one (absorbent camera) filled by a material (zeolite granules) for the sorbtion of water molecules. Three phases: preparation of the source and activation (dehydration) of zeolites, self-freezing of the source, drying in conditions of an uniform sublimation of water steam. The main advantages for such kind of drying are a higher food quality of the dry due to the minimum loss of flavour and aroma, the minimal chances of the microbal to growth due to the water absence and no thermal and oxidizing processes. SNA Liberec Page 2

3 Heat and mass transfer problem Model of the absorbent camera cρ T t = LT + f(x, t, T) x Ω, t > LT = d i=1 ( k(x,t) T ) x i x i Notations: T(x, t) unknown distribution of the temperature, d dimension of the space (d = 2), Ω R d computational domain, k = k(x,t) > 0 heat conductivity, c = c(x, t) > 0 heat capacity, ρ > 0 material density, f(x, t, T) function responsible for the non-linear process of transfer of water molecules in the absorption container. Initial conditions: Boundary conditions: T(x, 0) = T 0 (x) x Ω T(x, t) = µ(x, t) x Γ Ω, t > 0 SNA Liberec Page 3

4 Numerical treatment of the problem M dt dt + KT = F Note: f(x, t, T) connects the system with heat sources in the food camera. The heat is supplied to the source material by radiation, conduction or combination of both types. Mass matrix: Stiffness matrix: Right hand side: [ ] N [ M = c ρ φ i φ j dx K = k φ i φ j dx F = f(x, t, T) φ i dx Ω i,j=1 Ω i,j=1 Ω ] N [ ] N i=1 The coarser finite element mesh Number of nodes: 6568 Number of elements: SNA Liberec Page 4

5 Solving the systems of linear equations After the variational formulation, the problem is discretized by the linear triangular finite elements in space (minimal angle of a triangle, maximal area of a single triangle) and the simplest finite differences in time. For the uniform discretization of the time interval (t j = t 0 + j τ, where τ is the time stepsize), we get the following simple algorithm: for j = 1,...,N: compute find T j : b = ( M τ (1 ϑ) K ) T j 1 + τ ϑ F j + τ (1 ϑ) F j 1 (M + τ ϑk) T j = b end for We restrict our attention to implicit methods with ϑ 1, 1. Particularly, we shall consider two 2 cases with ϑ= 1 and ϑ=1, which correspond to Crank-Nicholson (CN) and backward Euler (BE) 2 method, respectively. SNA Liberec Page 5

6 Adaptive time steps The time interval is divided by time steps t 0 < t 1 <... < t N 1 < t N and τ j = t j t j 1. for k = 1,2,... until stop: find T k+1 : ( M + τ k K ) T k = MT k 1 + τ k F k // for the BE steps is ϑ = 1 end for compute r = ( M τk K ) T k ( M 1 2 τk K ) T k τk F k 1 2 τk F k 1 compute η k = r / T k // we suppose T CN T BE r decide if η k <ε min then τ k+1 = 2τ k // in usual practice, we specify if η k >ε max then τ k+1 = τ k /2 if η k ε min, ε max or ( η k <ε min and η k 1 >ε max ) then τ k+1 = τ k and T k+1 = T k, stop if η k >ε max and η k 1 <ε min then τ k+1 = τ k /2 and T k+1 = T k 1, stop // ε min = 10 8, ε max = 10 7 SNA Liberec Page 6

7 How often to perform the ATS procedure? The adaptivity test depends on the input parameters ε min, ε max and is performed each #N NA time steps. The time interval is 5000s and the adaptive time stepping begins with τ = 5. The linear system is solved up to the relative residual accuracy The exact solution vector T ex means the solution of the linear system for the constant time stepsizes (no adaptivite time steps, τ = 5, Time = s, #It = 33067, #N TS = 999). ε min ε max #N NA Time [s] #It #It A #N TS τ T T ex 2 T 2 T T ex ScLion: Pentium 4 / 1.5GHz 256 L2 cache, 256 RAM, Scientific Linux 4.5, GNU C SNA Liberec Page 7

8 How to set the parameters ε min, ε max? No adaptivite time steps, τ = 5: Time = s, #It = 33067, #N TS = 999). ε min ε max #N NA Time [s] #It #It A #N TS τ T T ex 2 T 2 T T ex Impact for the practical usage For the given ε min, ε max : #N NA 3, 15 with the preference of smaller values. For the given ε PCG : ε min ε PCG, ε PCG 10 2 and then ε max = ε min 10. SNA Liberec Page 8

9 Numerical experiments in the full time interval The considered time interval is set to be s 8 days 19 hours 13 minutes. No adaptivite time steps, τ = 5: Time = s, #It = , #N TS = ). ε min ε max #N NA Time [s] #It #N TS τ T T ex 2 T 2 T T ex The temperature field at the end of time interval SNA Liberec Page 9

10 From the engineering point of view The water filling of the zeolites at the 1/3, 2/3 and 3/3 of the time interval. Red: 100 % Yellow: 75 % Green: 50 % Cyan: 25 % Blue: 0 % SNA Liberec Page 10

11 Conclusion The introduced work is devoted to mathematical modelling of the freeze-drying process under high vacuum. Especially, it concerns the computer simulation of heat and mass transfer in the absorbent camera. The origin of such simulation is in solving the time-dependent nonlinear partial differential equation of parabolic type and the core is the repeated solution of linear systems for different time steps. The existing program code used only the uniform discretization of the considered time interval, when the time stepsizes are constant. In an effort to optimize the whole simulation, we included the adaptive time stepping procedure in the code. This procedure is based on the local comparison of Crank-Nicholson and backward Euler time steps. Finally, we performed the numerical experiments to know the behaviour of the resulting program, to find the optimal (in some sense) parameters and to try it on a selected real-life problem. The tests confirmed its big usefulness for practice. SNA Liberec Page 11

12 Thank you for your attention SNA Liberec Page 12