Numerical solution of diffusion mass transfer model in adsorption systems. Prof. Nina Paula Gonçalves Salau, D.Sc.

Transcription

1 Numeical solution of diffusion mass tansfe model in adsoption systems Pof., D.Sc.

2 Agenda Mass Tansfe Mechanisms Diffusion Mass Tansfe Models Solving Diffusion Mass Tansfe Models Paamete Estimation 2

3 Mass Tansfe Mechanisms 3

4 Adsoption System Batch Opeation Stied-Tank Liquid Phase Spheical Adsobent Paticles 4

5 Mass Tansfe Mechanisms The oveall ate of adsoption ate on a poous solid includes the following thee simultaneous steps: Extenal Mass Tansfe Intapaticle Diffusion Adsoption on Active Site 5

6 Extenal Mass Tansfe It epesents the moviment of the adsobate (molecules/ions) fom bulk solution to the extenal suface of adsobent paticle. It is govened by extenal mass tansfe coefficient. 6

7 Intepaticle Diffusion It epesentes the moviment of the adsobate (molecules/ions) inside the paticle. It occus by: Effective poe volume diffusion (Fick s diffusion) Suface diffusion ombination of both mechanisms 7

8 Intepaticle Diffusion Effective poe volume diffusion It descibes the tanspot of the adsobate (molecules/ions) in the liquid phase inside of the paticle. It is epesented by the effective poe volume diffusion coefficient. 8

9 Intepaticle Diffusion Suface diffusion It is elative to the tanspot of the adsobate ove the suface of the adsobent paticles, fom sites of highe enegy to sites of lowe enegy. It is epesented by the suface diffusion coefficient. 9

10 Adsoption on Active Site It elative to the inteaction of the adsobate with the active sites of the adsobent. It can be consideed instantaneous and epesented by the isothem model equation. 10

11 Mass Tansfe Mechanisms 11

12 Diffusion Mass Tansfe Models 12

13 Diffusional Models They ae constucted on the basis in the thee consecutive steps: Extenal mass tansfe Intapaticle diffusion (effective poe volume diffusion, suface diffusion, o a combination of both mechanisms) Adsoption on an active site 13

14 Diffusional Models They epesent ealistically the adsoption kinetics. One of the most complete diffusional models is the poe volume and suface diffusion model (PVSDM). 14

15 PVSDM Assumptions Batch system adsoption occus at constant tempeatue. Mass tanspot by convection within the poes is negligible. 15

16 PVSDM Assumptions Intapaticle diffusion can occu by both poe volume diffusion and suface diffusion. Values of effective poe volume diffusion coefficient and suface diffusion coefficient ae constant. Adsoption ate on an active site is instantaneous. 16

17 PVSDM 17

18 PVSDM To solve this model, it is consideed that thee exists a local equilibium between the adsobate concentation of the poe solution and the mass of adsobate adsobed on the poe suface. This equilibium elationship is epesented by the adsoption isothem: 18

19 PVSDM In case of Redlich-Peteson isothem model equation: q k 1 a RP RP 19

20 PVSDM Fom chain ule: 20 q t t q RP RP a k t t q 1 q q RP RP a k q 1

21 PVSDM Also, if the volume is constant, it is evident that adsobate concentation fom bulk solution and the mass of adsobate pe mass of adsobent: 21

22 PVSDM Thee models can be deived fom the PVSDM model: Extenal mass tansfe model (EMTM) Poe volume diffusion model (PVDM) Suface diffusion model (SDM) 22

23 EMTM It assumes that the movement of solute fom the liquid phase to the adsobent is only due to extenal mass tansfe. Intapaticle diffusion is instantaneous. Thee is not a concentation gadient inside the paticle. Intapaticle diffusion esistance is consideed to be insignificant. 23

24 EMTM 24

25 EMTM 25

26 PVDM It is a simplification of the PVSDM model, used when the intapaticle diffusion is contolled only by poe volume diffusion. Suface diffusion coefficient is neglected. 26

27 PVDM 27

28 PVDM 28

29 SDM It is a simplification of the PVSDM model, used when the intapaticle diffusion is contolled only by suface diffusion. Effective poe volume diffusion coefficient is neglected. 29

30 SDM 30

31 SDM 31

32 PVDM & SDM If the extenal mass tansfe is negligible, the bounday condition can be eplaced by the adsoption isothem, o by othe bounday condition. PVDM: SDM: 32

33 HSDM Homogeneous suface diffusion model (HSDM) is anothe impotant diffusion model. It consides a dual mass tanspot mechanism acoss the hydodynamic bounday laye suounding the adsobent paticle and intapaticle esistance within the paticle in the fom of suface diffusion. 33

34 HSDM 34

35 HSDM In the same way of the othe models, HSDM equies a elation between the amount of adsobate adsobed on the adsobent and the amount of the adsobate in the bulk solution, which is given by the adsoption isothem. 35

36 Solving Diffusion Mass Tansfe Models 36

37 Solving Model Equations Numeical method of lines utilizes odinay diffeential equations (ODEs) fo the time deivative and finite diffeences on the spatial deivatives. In finite diffeence method, the deivatives in the patial diffeential equations (PDEs) ae appoximated by linea combinations of function values at the gid points. 37

38 Solving Model Equations Deivatives in the patial diffeential equation of diffusional tansfe models ae discetized into N + 1 points on the spatial deivatives (adius), whee N is the numbe of gid points. 38

39 Tanspot of adsobate molecules fom the bulk solution to the spheical paticle. Discetization (gid points) 39

40 Discetization (gid points) In this way, it has in diffeent points i whee i = 0 is the gid point at = 0 and i = N+1 is the gid point at = R. Gid points ae spaced equally with the step size given by: h N R 1 40

41 Finite Appoximation Functions Diffeence Fist-Ode Fomula Second-Ode Fomula Fowad Diffeence fo Fist Deivative f x = Δ f x = Δ ental Diffeence fo Fist Deivative - f x = Δ Backwad Diffeence fo Fist Deivative f x = Δ f x = Δ 41

42 Finite Appoximation Functions Diffeence Fist-Ode Fomula Second-Ode Fomula Fowad Diffeence fo Second Deivative ental Diffeence fo Second Deivative f x = Δ - f x f x = Δ = Δ Backwad Diffeence fo Second Deivative f x = Δ f x = Δ 42

43 Finite Appoximation Functions Tempoal vaiation of the adsobate concentation in bulk solution is ewitten in the discetized fom: d dt msk V F N 1 43

44 44 i RP i RP i i i i s p i i i p i RP i RP i i i s p i i p i RP i RP i p p a k d d h D h D a k d d h D h D a k d d dt d dt d Finite Appoximation Functions Secondode cental diffeence appoximation s of the fist and second deivative can be used to solve the PDEs.

45 Finite Appoximation Functions Isolating and simplifying: 45 RP p p i RP p RP p i RP p RP s p p i RP p RP s p i RP p i i i i i k a k a h D k D a D D k a D h dt d dt d A dt d

46 Finite Appoximation Functions Second-ode fowad diffeence appoximations of the fist deivative can be used to solve the bounday condition at = 0:

47 Second-ode backwad diffeence appoximations of the fist deivative can be used to solve the bounday condition at = R: N F N RP RP N N N N s p N N N p k a k d d h D h D Finite Appoximation Functions

48 . Bounday ondition Bounday condition at = R has no has no analytical solution, i. e. it is not possible to N 1 solve explicitly the equation to obtain. An implicit method to solve non linea equation is equied. The most common method fo solving nonlinea algebaic equations is the Newton s method. 48

49 Bounday ondition fsolve MatLab function that attemps to solve equations of the fom: Based on tust-egion modification of Newton method. 49

50 ODE s Integation Discetized PDEs yield ODE equation systems that ae vey stiff. To avoid a vey small time step, an implicit method can be used, as the backwad diffeence fomula (BDF). 50

51 ODE s Integation ode15s MatLab function that attemps to solve stiff diffeential equations and DAEs, vaiable ode method. Based on the numeical diffeentiation fomulas (NDFs). Optionally, it uses the backwad diffeentiation fomulas (BDF, also known as Gea method) that ae usually less efficient. 51

52 ase Study Mass of adsobate adsobed on each adial position ove time. 52

53 Paamete Estimation 53

54 Isothem Paametes In case of Redlich-Peteson isothem model equation: q k 1 a RP RP 54

55 Isothem Paametes The nonlinea least-squae objective function to be minimized is defined as the sum of the diffeences between the expeimental and model data of mass of adsobate adsobed on the poe suface: min f NE NY p q i, j qˆ i, j j1 i1 2 55

56 Model Paametes alculated mass tanspot paametes: Extenal mass tansfe coefficient (PVSDM, PVDM, SDM, HSDM) Molecula diffusivity (PVSDM, PVDM) Effective poe volume diffusion coefficient (PVSDM, PVDM) Totuosity facto (PVSDM, PVDM) 56

57 Model Paametes Estimated mass tanspot paametes: Suface diffusion coefficient (PVSDM, SDM, HSDM) Extenal mass tansfe coefficient (EMTM) 57

58 Model Paametes The nonlinea least-squae objective function to be minimized is defined as the sum of the diffeences between the expeimental and model data of adsobate concentation. 58

59 Paamete Estimation Nonlinea least-squae optimization can be used to estimate the isothem and mass tanspot paametes. 59

60 Paamete Estimation lsqnonlin MatLab function that solves non-linea least squaes poblems (nonlinea data-fitting) poblems with optional lowe and uppe bounds on the paametes. It implements two diffeent algoithms: Tust egion eflective Levenbeg-Maquadt 60

61 ase Study 61

62 ase Study 62

63 ase Study 63

64 Any questions? Moe infomation? 64

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67 EBA

68 ontact: Pof., D.Sc. 68