University of Dunaújváros

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1 University of Dunaújváros
Institute of Informatics An explicit analytic solution of a coupled first order partial and ordinary differential equation system for a discontinuous initial-boundary value problem Dr. András Zachár associate professor

2 2. Literature review of the Plume-Entrainment model
Summary 1. Basic terminologies. 2. Literature review of the Plume-Entrainment model 3. Definition of the mathematical problem and the first step of the solution 4. Explicit analitic solution of the linear equation system 5. Existence and uniqueness of the solution 6. Improving the solution with an alternative series form 7. Comparison of the analitic and the numerical solutions Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 2

3 Basic terminologies W. Rodi [1], E. J. List [1]
Literature review Basic terminologies W. Rodi [1], E. J. List [1] Such a fluid motion where the primary source of kinetic energy and momentum flux is a pressure drop through an orifice is called a Jet. Institute of Informatics A fluid motion (flowing through an orifice) whose main source of kinetic energy and momentum flux is body forces called a Plume. University of Dunaújváros Flows whose motion is in transition from a jet to a plume is called a Forced Plume or a Buoyant Jet. Entrainment is the transport of fluid across an interface between two bodies of fluid by a shear induced turbulent flux. Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 3

4 A literature review of the Plume-Entrainment model
The applied simplifications of the physical reality W. F. Phillips, R. A. Pate [2] Institute of Informatics A small stream of water falling downward through a much larger tank of warmer water. University of Dunaújváros There are no horizontal temperature gradients in the tank except in the vicinity of the inlet fluid stream. The falling stream is small enough so that both axial conduction and thermal inertia can be neglected in the stream. Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 4

5 Tollmien [5], Schlichting [6], Bickley [7]:
Literature review The different material properties are constant (density, specific heat, thermal conductivity). Institute of Informatics Tollmien [5], Schlichting [6], Bickley [7]: University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 5

6 Heat loss of the storage tank has also been neglected.
Literature review Further simplification of the mathematical model G. F. Csordas et al. [3] Institute of Informatics Molecular diffusion of heat and momentum is neglected, the flow process is convection dominant. University of Dunaújváros Heat loss of the storage tank has also been neglected. Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 6

7 Literature review Hill [4] : cent=0.32 University of Dunaújváros
Institute of Informatics Hill [4] : cent=0.32 University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 7

8 Definition of the mathematical problem
Results Definition of the mathematical problem Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 8

9 Characteristics of equation (1).
Literature review Characteristics of equation (1). Institute of Informatics General form of the characteristic curve University of Dunaújváros General solution of equation (1). Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 9

10 Characteristic curves of equation (1)
Results Characteristic curves of equation (1) Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 10

11 Solution of equation (2):
Results Solution of equation (1) and (2) with the initial and boundary conditions Initial condition: Institute of Informatics Domain: University of Dunaújváros Solution of equation (1) : Solution of equation (2): Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 11

12 General form of the solution:
Results Boundary condition: Domain: Institute of Informatics General form of the solution: University of Dunaújváros Physical reason: Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 12

13 New form of the solution:
Results New form of the solution: Solution for the entire domain of the investigated problem: Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 13

14 First form of the solution:
Results First form of the solution: Institute of Informatics After some simplification: University of Dunaújváros Question: Satisfy: and into equation (2) Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 14

15 Physical meaning of the integral:
Results After substitution: Institute of Informatics Physical meaning of the integral: The harvested thermal energy from the storage tank into the fluid stream by the Entrainment process. University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 15

16 Boundary condition: Results University of Dunaújváros
Institute of Informatics Boundary condition: University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 16

17 Expanding into Taylor series to n-1 order:
Results Expanding into Taylor series to n-1 order: Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 17

18 is a Vandermonde type matrix:
Results Institute of Informatics Proof: Pull out the common factors from the second column to the last one: University of Dunaújváros where: is a Vandermonde type matrix: and expanding according to the first row we get another n-1 Vandermonde type matrix. Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 18

19 Applying the Lagrange formula:
Results Institute of Informatics Applying the Lagrange formula: University of Dunaújváros Decompose to partial fraction the following expression: Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 19

20 multiplying: using: we get: Results University of Dunaújváros
Institute of Informatics University of Dunaújváros using: we get: Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 20

21 Cramer rule Results University of Dunaújváros Institute of Informatics
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22 Results University of Dunaújváros Institute of Informatics
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23 Existence and uniqueness:
Results Existence and uniqueness: Institute of Informatics University of Dunaújváros Proof: Using the boundary characteristic curve the following estimate can be created: in the domain Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 23

24 for a fixed t a monotonically decreasing function of variable x
Results because the term for a fixed t a monotonically decreasing function of variable x Institute of Informatics and the value of this term along the boundary characteristic curve is University of Dunaújváros Using this estimate the maximum value of the function in the domain Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 24

25 Maximum value of the function
Results Maximum value of the function can be calculated by the following way along the boundary characteristic curve: Institute of Informatics University of Dunaújváros It is known from eq. (15): Using This ensures that is convergent in and the following function exist Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 25

26 Results Uniqueness: Unique solution of a first order quasi linear PDE can be obtained if the boundary curve defined by the condition is not a characteristic of eq. (1). Vvedensky [8]. Institute of Informatics It is enough to show that geometrically the condition is a vertical line in the (x,t) plane at a fixed value of x=L. This line intersects the characteristics of eq. (1). University of Dunaújváros where Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 26

27 Improving the solution:
Results Improving the solution: Institute of Informatics Satisfying the condition we get a linear equation system similar to eq. (15) where the vector b is the same vector what is in the right side of eq. (15). University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 27

28 Results University of Dunaújváros Institute of Informatics
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29 Results University of Dunaújváros Institute of Informatics
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30 Physical and geometrical parameters:
Numerical solution: Physical and geometrical parameters: Institute of Informatics p T University of Dunaújváros Applied numerical schems: For the time derivative: Explicit Euler For the spatial derivative: First order Up-Wind Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 30

31 Comparison of the results:
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 31

32 tini Application of the results: [s-1] 1 Results p
Institute of Informatics University of Dunaújváros 1 Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 32

33 tini t1 t2 1 1 1 Results University of Dunaújváros
Institute of Informatics University of Dunaújváros 1 1 1 Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 33

34 University of Dunaújváros
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 34

35 Thank you for your attention!
Institute of Informatics Thank you for your attention! University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 35

36 Irodalmi források: 1. University of Dunaújváros
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 36

37 Irodalmi források: 2. 3. University of Dunaújváros
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 37

38 Irodalmi források: 4. 5. 6. 7. 8. University of Dunaújváros
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 38

39 2. University of Dunaújváros Institute of Informatics
Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 39

40 University of Dunaújváros
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 40

41 3. University of Dunaújváros Institute of Informatics
Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 41

42 University of Dunaújváros
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 42

43 University of Dunaújváros
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 43

44 University of Dunaújváros
Institute of Informatics University of Dunaújváros Dátum Farkas Miklós Alkalmazott Analízis Szeminárium 44

45 Literature review University of Dunaújváros Institute of Informatics
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