化工應用數學

1. 化工應用數學 Solving Models Related to Partial Differentiation Equations 授課教師： 林佳璋

2. Solution of P.D.E Todetermine a particular relation between u, x, and y, expressed as u = f (x, y), that satisfies -the basic differential equation (P.D.E) -some particular conditions specified by practical problem linear If, however, one of the coefficients, say P, were a function of the dependent variable w, then the equation would be nonlinear. A “homogeneous” partial differential equation is one in which all terms contain derivatives of the same order.

3. Method of Solution of P.D.E No general formalized analytical procedure for the solution of an arbitrary partial differential equation is known. The solution of a P.D.E. is essentially a guessing game. The object of this game is to guess a form of the specialized solution which will reduce the P.D.E. to one or more total differential equations. Linear, homogeneous P.D.E.s with constant coefficients are generally easier to deal with.

4. Particular Solution of P.D.E • Compounding the independent variables into one variable • Superposition of solutions • If each of the functions v1, v2, …, vn, … is a solution of a linear, • homogeneous P.D.E., then the function • Method of images

5. Particular Solution of P.D.E Compounding the independent variables into one variable

6. Particular Solution of P.D.E This method is not very valuable.

7. Heat Transfer in a Flowing Fluid An infinitely wide flat plate is maintained at a constant temperature T0. The plate is immersed in an infinitely wide and thick stream of constant-density fluid originally at temperature T1. If the origin of coordinates is taken at the leading edge of the plate, a rough approximation to the true velocity distribution is: Turbulent heat transfer is assumed negligible, and molecular transport of heat is assumed important only in the y direction. The thermal conductivity of the fluid, k is assumed to be constant. It is desired to determine the temperature distribution within the fluid and the heat transfer coefficient between the fluid and the plate.

8. Heat Transfer in a Flowing Fluid y T1 T1 B.C. T = T1 at x = 0, y > 0 T = T1 at x > 0, y = T = T0 at x > 0, y = 0 dx dy x T0 Heat balance on a volume element of length dx and height dy situated in the fluid : Input energy rate: Output energy rate:

9. Heat Transfer in a Flowing Fluid Input - Output = Accumulation const. properties  = 0 at x = 0, y > 0 = 0 at x > 0, y =   = 1 at x > 0, y = 0 T = T1 at x = 0, y > 0 T = T1 at x > 0, y = T = T0 at x > 0, y = 0

10. Heat Transfer in a Flowing Fluid Assume Compounding the independent variables into one variable  = 0 at  =   = 1 at = 0 Replace y and x in the P.D.E by 

11. Heat Transfer in a Flowing Fluid In order to eliminate x and y, we choose n = 1/3  = 0 at  =   = 1 at = 0  = 1 at = 0

12. Heat Transfer in a Flowing Fluid local heat transfer coefficient  = 0 at y= 0

13. Particular Solution of P.D.E Superposition of solutions w.r.t x

14. Particular Solution of P.D.E Method of images

15. Heat Conduction in a Slab Suppose that a slab (depending indefinitely in the y and z directions) at an initial temperature T1 has its two faces suddenly cooled to T0. What is the relation between temperature, time after quenching, and position within the slab? Since the solid extends indefinitely in the y and z direction, heat flows only in the x direction. The heat- conduction equation: 2R x dx Boundary condition

16. Heat Conduction in a Slab dimensionless Assume separation of variables independent of t independent of x

17. Heat Conduction in a Slab when   0 when  = 0 superposition

18. Heat Conduction in a Slab A0, B0, A, B, and  have to be chosen to satisfy the boundary conditions. n is an integer B.C. The constant has to be determined. But no single value can satisfy the B.C.

19. Heat Conduction in a Slab More general format of the solution (by superposition) m=n orthogonal property

20. Heat Conduction in a Slab The representation of a function by means of an infinite series of sine functions is known as a “Fourier sine series”.

21. Orthogonal Functions Two functions m(x) and n(x) are said to be “orthogonal” with respect to the weighting function r(x) over interval a, b if: mn m=n sin(mx) and sin(nx) are orthogonal

22. Orthogonal Functions are orthogonal with respect to the weight function (i.e., unity) over the interval 0, 2R when m  n. and Each term is zero except when m = n. Back to our question, we had two O.D.E.s and the solutions are : where shows! These values of are called the “eigenvalues” of the equation, and the corresponding solutions, are called the “eigenfunctions”.

23. Sturm-Liouville Equation is a constant

24. Sturm-Liouville Equation If both boundary conditions are of the above three types, then the set of function n will be orthogonal with respect to the weighting function r(x).

25. Steady-state Heat Transfer with Axial Symmetry Assume dividing by fg and separate variables

26. Steady-state Heat Transfer with Axial Symmetry set

27. Steady-state Heat Transfer with Axial Symmetry Legendre’s equation of order l set Solved by the method of Frobenius and where Pl(m) is the “Legendre polynomial”

28. Steady-state Heat Transfer with Axial Symmetry superposition

29. Unsteady-state Heat Transfer to a Sphere A sphere, initially at a uniform temperature T0 is suddenly placed in a fluid medium whose temperature is maintained constant at a value T1. The heat-transfer coefficient between the medium and the sphere is constant at a value h. The sphere is isotropic, and the temperature variation of the physical properties of the material forming the sphere may be neglected. Derive the equation relating the temperature of the sphere to the radius r and time t. independent of  and  Boundary Condition

30. Unsteady-state Heat Transfer to a Sphere Assume Bessel’s equation

31. Unsteady-state Heat Transfer to a Sphere if  0 if = 0 if  0 if = 0

32. Unsteady-state Heat Transfer to a Sphere B.C. D = T1 B = C = 0

33. Unsteady-state Heat Transfer to a Sphere

34. Unsteady-state Heat Transfer to a Sphere B.C. superposition

35. Unsteady-state Heat Transfer to a Sphere m=n

36. Equations Involving Three Independent Variables The steady-state flow of heat in a cylinder is governed by Laplace’s equation in cylindrical polar coordinates: There are three independent variables r, , z. Assume separation of variables two independent variable P.D.E.

37. Equations Involving Three Independent Variables separation of variables Assume Bessel’s equation

38. Flow in a Packed Column In the study of flow distribution in a packed column, the liquid tends to aggregate at the walls. If the column is a cylinder of radius b m and the feed to the column is distributed within a central core of radius a m with velocity U0 m/s, determine the fractional amount of liquid on the walls as a function of distance from the inlet in terms of the parameters of the system. horizontal component of fluid velocity a U0 Material balance b z Input r U Output

39. Flow in a Packed Column B.C. at z = 0, if r < a, U = U0 at z = 0, if r > a, U = 0 at r = 0, U = finite at r = b, Assume Bessel’s equation

40. Flow in a Packed Column if   0 The solution of the Bessel’s equation if  = 0 general form An=? The fraction of total fluid in the wall layer is given by