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Introduction 1. Similarity 1.1. Mechanism and mathematical description

Similarity and modeling Prof. Christo Boyadjiev. Contents. Introduction 1. Similarity 1.1. Mechanism and mathematical description 1.2. Generalized variables 1.3. Qualitative analysis 1.4. Generalized individual case (process) and similarity 1.5. Similarity criteria

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Introduction 1. Similarity 1.1. Mechanism and mathematical description

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  1. Similarity and modeling Prof. Christo Boyadjiev Contents • Introduction • 1. Similarity • 1.1. Mechanism and mathematical description • 1.2. Generalized variables • 1.3. Qualitative analysis • 1.4. Generalized individual case (process) and similarity • 1.5. Similarity criteria • 2. Modeling • 2.1. Similarity and physical modeling • 2.2. Incompatible criteria • 2.3. Mathematical modeling • 2.4. Model equation structures • 2.5. Scale-up and scale effect • 2.6. Parameters identification and incorrectness of the inverse problem • 2.7. Some specific problems • Conclusions

  2. INTRODUCTION Modeling and simulation – An united approach for quantitative description of the processes and systems and its optimal design, control or renovation. Modeling - the creation of a model on the bases of hypothesis of physical mechanism, mathematical structure, model parameter identification using experimental data and analysis of the model adequacy. Simulation – the numerical (mathematical) experiment for quantitative description of the process on the bases of an algorithm and computer program for solution of the model equations.

  3. SIMILARITY • 1.1. Mechanism and mathematical description The model equations are a full correspondence between different physical effect in the process and the mathematical operators in the mathematical description. Model of the physical absorption in falling film Convection–diffusion equation - mass balance between convective longitudinal and cross transfer and diffusive transfer : . For the vertical film with constant thickness h , , .

  4. 1.2. Generalized variables Every variable have a characteristic scale (very often the maximal value increase in a characteristic interval): 0 ≤ x ≤ l , h – δ ≤ y ≤ h , 0 ≤ u ≤ u0 , c0 ≤ c ≤ c*, whereu0 and δ are interface velocity and thickness of the diffusion boundary layer: . The characteristic scales permit to obtain the generalized (dimensionless) variables X, Y, C: , ; , where .

  5. 1.3. Qualitative analysis Fo ~ 1 , C = C ( X,Y ) ; Fo < 1 (10 -1) , C = C0 + FoC1+ Fo2C2 + … ; Fo << 1 (10 -2), C = C0. 1.4. Generalized individual case (process) and similarity ; , . Generalized case – Fo have all possible values Generalized individual case (GIC) - Fo have a concrete value All processes in the general individual case are similar, i.e. every process is a model of the all others.

  6. 1.5. Similarity criteria The equality of the dimensionless parameters – in the GIC is similarity criterion Mass transfer kinetics , , Fo – similarity criteria; Pe, Sh is not similarity criterion Sh Pe-½=ψ(Fo)

  7. 2. MODELING 2.1. Similarity and physical modeling Let’s consider a real process (1) and its physical model (2). From the similarity condition follows: . If J2is an experimental value of the mass transfer rate for the model, the mass transfer coefficient value is . i.e. As a result the mass transfer rate for the real process is , . where

  8. 2.2. Incompatible criteria Let’s consider two phase flow in the boundary layer approximation with boundary conditions at the interphase (y=0 is flat face interphase). If ui(0) and δi (i = 1,2) are the scales of the flow the dimensionless equations are The parameters of the general individual case are: Rei(0)(i=1,2), α0, β0, i.e. It is seen that it is not possible to be satisfied these two conditions in the scale - up, i.e. the similarity criteria Rei (i =1,2) and β are incompatible.

  9. Let’s consider mass transfer with chemical reaction: where u0, c0 and l are the scales of the process. The dimensionless form of the equation is where The parameters of the general individual case are Pe0 and Bi0 , i. e. the similarity conditions are It is seen that the similarity criteria Pe and Bi are incompatible, i. e. the generalized individual case consist one process only and it is not possibility to obtain process for the physical model. The dimensionless model shows that similarity criteria express the balance between two physical effects. For example: Pe – convective and diffusive transfer, Bi – convective transfer and chemical reaction , Bi/Pe – chemical reaction and diffusive transfer.

  10. 2.3. Mathematical modeling The criteria incompatibility leads to impossibility to be used physical modeling for quantitative description of many processes. It is a main cause for the mathematical modeling development in the chemical engineering. If the rate of absorption with chemical reaction is for the Sherwood number is obtained If we have experimental data for the processes rate J for different values of the Pe and Bi numbers we will obtain an experimental dependence between Sherwood number and Peclet and Bio numbers: The introduction of a mathematical structure for the function ψ and an obtaining of the parameters in this structure (using experimental data) is equivalent to the mathematical modeling of the process, i.e. the equations with the similarity criteria are mathematical models of the process.

  11. 2.4. Model equation structure The model equation must be a function which approximation the experimental data. This function must be invariant with respect to the similarity transformations, i.e. do not change the type of this function when the model variables are replaced with the real process variables. Let’s consider model equations f(x1,…,xm)=0 . The function f is invariant with respect to the similarity transformations if it is homogenous function f(k1x1,…,kmxm)=0 , i.e. f(k1x1,…,kmxm)=φ(k1,…,km) f(x1,…,xm)=0 . If it use short record The problem is to obtain a function with this property. A differentiation of with respect to the first parameter k1lead to

  12. As a result i.e. If As a result and A repetition of this procedure for x2,…,xm lead to i.e. f is homogenous function if represents a power complex and as a result is invariant with respect to the similarity (metric) transformations. The result obtained explains the mathematical structure of the criteria models.

  13. 2.5. Scale-up and scale effect The scale-up on the base of the physical modeling. It is impossible if the similarity criteria are incompatibles. The scale effect is the problem of the mathematical modeling because the increase of the apparatuses size lead to the decrease of the process efficiency. This effect is result of the radial non-uniformity of the velocity in the column apparatuses which is none predicted in the model equations. The solution of the scale-up problem has two steps. The first is to decrease the radial non-uniformity of the velocity distribution by means of design modification of the column ends. The second is to give an account of the non-uniformity in the model equations. The scale-up of the column apparatuses uses very often diffusion type models. The behavior in these cases is very complicated but the processes include convective transport, diffusion transport and volume reaction, i.e. a convective – diffusion equation with volume reaction may be used as a mathematical structure of the model. The convective transfer is result of the laminar flow or large scale turbulent pulsations. The diffusion transport is molecular turbulent as a result of the small scale turbulent pulsations. The volume sources are result of the chemical, biochemical reactions or interphase mass transfer.

  14. Let’s consider gas absorption with chemical reaction in the liquid phase: where k is interphase mass transfer coefficient, χ – Henry’s number, k0 – chemical reaction rate constant. The velocities and concentration may be presented as where are average velocities and concentrations and - radial non -uniformities of the velocities and concentrations:

  15. As a result the average concentration model has the form: where , i.e. In many cases and The parameters Di, εi, k, k0, χare related with the process andαi, βi, γi– with the apparatuses, i.e. with the scale-up.

  16. 2.6. Parameter identification and incorrectness of the inverse problem. In the chemical engineering are used models of the continuum mechanics where the parameter must be obtained on the bases of experimental data. Let us consider a numerical model : y = f (x , b) , where f is an objective function, expressed analytically, numerically or through an operator (algorithm); x = (x1,…,xm) is a vector of independent variables, b = (b1,…,bj) - vector of parameters. The parameters of the model (1) should be determined by means of N experimental values of the objective function . This requires the introduction of a least square function: where yn = f ( xn, b) are the calculated values of the objective function of the model (1), while xn = ( x1n , … , xmn ) are the values of the independent variables from the different experimental conditions (regimes), n = (1,…,N) . The parameters of the model (1) can be determined upon the conditions imposed by the minimum of the function Q = (b1,…,bj) with respect to the parameters b = (b1,…,bj) .

  17. The determination of b faces many troubles due to the incorrectness of the problem. They are a result of the sensibility of the solution with respect to the experimental errors associated with the determination of . They can be avoided by applications of regularization methods that make the problem conditionally correct. Let us consider the one-parameter model :y = 1 – exp(–bx) , where y is an objective function, x is an independent variable and b is an parameter. In the fig. is shown a dependence of the objective function from the model parameter at a constant value of the independent variable. The relation between the objective function and the parameter in the fig. is typical for number models of heat and mass transfer processes. The fig. permits to obtain objective function y0 for a given parameter value b0, i.e. this is the direct problem solution. The inverse problem is an obtaining of the parameter value b0 if the experimental value of the objective function y0 is known. Objective function y for different values of the model parameter b at x = x0 = const.

  18. Let us consider the two -parameter model: y = 1 – b1 – exp(–b2x) where and are exact parameter values. The horizontals of the least square function Q – correct inverse problem The horizontals of the least square function Q– incorrect inverse problem The horizontals of the least square function Q– essentialy incorrect problem

  19. Let the iteration procedure starts with an initial approximation b(0) = (b1(0) , … , bJ(0)). The values of bi = (b1i , … , bJi), where i is iteration number, are result of the conditions imposed by the movement towards the anti-gradient of the functionQ (b) : Each iteration step is successful if two conditions are satisfied:

  20. 2.7. Some specific problems Similarity with out mathematical description - method of the dimensions The dimension of the left part of the equation of the physical processes must be equal to the dimension of the right part. Let’s consider the model equation z = f (x1 ,…, xm ; y1 ,… , yr) , where x1 , … , xm are primary quantities (length, mass time, …) and y1 , … , yr – secondary quantities (velocity, concentration, …). The secondary quantities may be expressed as a power complex The physical equation is invariant to the similarity transformations, i.e. it is possible to present as a power complex: The dimensions equality conditions lead to , … , i.e. n equations with n + r variables.

  21. To obtain dimensionless numbers must to put concrete values of r variables and to obtain another values, solving the linear system. • These dimensionless numbers are not similarity criteria because: • are not ratio between two physical effects; • are not uniquely obtained; • an error in the selection of the quantities x1,…,xm , y1,…,yr is fatal; • do not present an opportunity for to make the difference between similarity criteria and dimension numbers. Criteria models and autocorrelations On the figure is seen correlation between Reynolds and Nuselt numbers in the case of heat transfer in fluidized bed:

  22. The same experimental data are used for the correlation between Veber and Stanton numbers The same experimental data for u and h are shown on the next picture without correlation because they are obtained with the generator of random numbers. The autocorrelation is result of the common parameter (n) in Re and Nu, or ρuin St and We.

  23. CONCLUSION • The comparison analysis of the similarity and modeling shown: • the similarity and modeling are related in the cases of physical modeling for a quantitative description of the general individual cases; • the mathematical modeling permit a quantitative description of the general cases; • the mathematical technique of modeling is very simple but the similarity has very great sense. • The using of the modeling without an understanding (realization) of the similarity may to lead to reduce the similarity theory into a similarity of theory (Prof. P. G. Romankov). • In all cases the mathematical modeling must to finish with statistical analysis of the model adequacy, i.e. equality of the variances of model error and experimental error.

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