1. 化工應用數學 授課教師： 郭修伯 Lecture 6 Functions and definite integrals Vectors

2. Chapter 5 Functions and definite integrals There are many functions arising in engineering which cannot be integrated analytically in terms of elementary functions. The values of many integrals have been tabulated, much numerical work can be avoided if the integral to be evaluated can be altered to a form that is tabulated. Ref. pp.153 We are going to study some of these special functions…..

3. Special functions Functions –Determine a functional relationship between two or more variables –We have studied many elementary functions such as polynomials, powers, logarithms, exponentials, trigonometric and hyperbolic functions. –Four kinds of Bessel functions are useful for expressing the solutions of a particular class of differential equations. –Legendre polynomials are solutions of a group of differential equations. Learn some more now….

4. The error function It occurs in the theory of probability, distribution of residence times, conduction of heat, and diffusion matter: 0x z erf x z: dummy variable Proof in next slide

5. x and y are two independent Cartesian coordinates in polar coordinates Error between the volume determined by x-y and r-  The volume of  has a base area which is less than 1/2R 2 and a maximum height of e -R2

6. More about error function Differentiation of the error function: Integration of the error function: The above equation is tabulated under the symbol “ ierf x” with (Therefore, ierf 0 = 0) Another related function is the complementary error function “erfc x”

7. The gamma function for positive values of n. t is a dummy variable since the value of the definite integral is independent of t. (N.B., if n is zero or a negative integer, the gamma function becomes infinite.) repeat The gamma function is thus a generalized factorial, for positive integer values of n, the gamma function can be replaced by a factorial. (Fig. 5.3 pp. 147)

8. More about the gamma function Evaluate

9. Chapter 7 Vector analysis It has been shown that a complex number consisted of a real part and an imaginary part. One symbol was used to represent a combination of two other symbols. It is much quicker to manipulate a single symbol than the corresponding elementary operations on the separate variables. This is the original idea of vector. Any number of variables can be grouped into a single symbol in two ways: (1) Matrices (2) Tensors The principal difference between tensors and matrices is the labelling and ordering of the many distinct parts.

10. Tensors Generalized as z m A tensor of first rank since one suffix m is needed to specify it. The notation of a tensor can be further generalized by using more than one subscript, thus z mn is a tensor of second rank (i.e. m, n). The symbolism for the general tensor consists of a main symbol such as z with any number of associated indices. Each index is allowed to take any integer value up to the chosen dimensions of the system. The number of indices associated with the tensor is the “rank” of the tensor.

11. Tensors of zero rank (a tensor has no index) It consists of one quantity independent of the number of dimensions of the system. The value of this quantity is independent of the complexity of the system and it possesses magnitude and is called a “scalar”. Examples: –energy, time, density, mass, specific heat, thermal conductivity, etc. –scalar point: temperature, concentration and pressure which are all signed by a number which may vary with position but not depend upon direction.

12. Tensors of first rank (a tensor has a single index) The tensor of first rank is alternatively names a “vector”. It consists of as many elements as the number of dimensions of the system. For practical purposes, this number is three and the tensor has three elements are normally called components. Vectors have both magnitude and direction. Examples: –force, velocity, momentum, angular velocity, etc.

13. Tensors of second rank (a tensor has two indices) It has a magnitude and two directions associated with it. The one tensor of second rank which occurs frequently in engineering is the stress tensor. In three dimensions, the stress tensor consists of nine quantities which can be arranged in a matrix form:

14. The physical interpretation of the stress tensor x z y p xx  xy  xz The first subscript denotes the plane and the second subscript denotes the direction of the force.  xy is read as “the shear force on the x facing plane acting in the y direction”.

15. Geometrical applications If A and B are two position vectors, find the equation of the straight line passing through the end points of A and B. A B C

16. Application of vector method for stagewise processes In any stagewise process, there is more than one property to be conserved and for the purpose of this example, it will be assumed that the three properties, enthalpy (H), total mass flow (M) and mass flow of one component (C) are conserved. In stead of considering three separate scalar balances, one vector balance can be taken by using a set of cartesian coordinates in the following manner: Using x to measure M, y to measure H and z to measure C Any process stream can be represented by a vector: M H C A second stream can be represented by:

17. Using vector addition, Thus, OR with represents of the sum of the two streams must be a constant vector for the three properties to be conserved within the system. To perform a calculation, when either of the streams OM or ON is determined, the other is obtained by subtraction from the constant OR. Example : when x = 1, Ponchon-Savarit method (enthalpy-concentration diagram) x y z M R N B AP The constant line OR cross the plane x = 1 at point P O point A is : point B is : point P is :

18. Multiplication of vectors Two different interactions (what’s the difference?) –Scalar or dot product : the calculation giving the work done by a force during a displacement work and hence energy are scalar quantities which arise from the multiplication of two vectors if A·B = 0 –The vector A is zero –The vector B is zero –  = 90 °  A B

19. –Vector or cross product : n is the unit vector along the normal to the plane containing A and B and its positive direction is determined as the right-hand screw rule the magnitude of the vector product of A and B is equal to the area of the parallelogram formed by A and B if there is a force F acting at a point P with position vector r relative to an origin O, the moment of a force F about O is defined by : if A  B = 0 –The vector A is zero –The vector B is zero –  = 0 °  A B

20. Commutative law : Distribution law : Associative law :

21. Unit vector relationships It is frequently useful to resolve vectors into components along the axial directions in terms of the unit vectors i, j, and k.

22. Scalar triple product The magnitude of is the volume of the parallelepiped with edges parallel to A, B, and C. A B C ABAB

23. Vector triple product The vector is perpendicular to the plane of A and B. When the further vector product with C is taken, the resulting vector must be perpendicular to and hence in the plane of A and B : A B C ABAB where m and n are scalar constants to be determined. Since this equation is valid for any vectors A, B, and C Let A = i, B = C = j:

24. Differentiation of vectors If a vector r is a function of a scalar variable t, then when t varies by an increment  t, r will vary by an increment  r.  r is a variable associated with r but it needs not have either the same magnitude of direction as r :

25. As t varies, the end point of the position vector r will trace out a curve in space. Taking s as a variable measuring length along this curve, the differentiation process can be performed with respect to s thus: is a unit vector in the direction of the tangent to the curve is perpendicular to the tangent. The direction of is the normal to the curve, and the two vectors defined as the tangent and normal define what is called the “osculating plane” of the curve.

26. Temperature is a scalar quantity which can depend in general upon three coordinates defining position and a fourth independent variable time. – is a “partial derivative”. – is the temperature gradient in the x direction and is a vector quantity. – is a scalar rate of change. Partial differentiation of vectors

27. A dependent variable such as temperature, having these properties, is called a “scalar point function” and the system of variables is frequently called a “scalar field”. –Other examples are concentration and pressure. There are other dependent variables which are vectorial in nature, and vary with position. These are “vector point functions” and they constitute “vector field”. –Examples are velocity, heat flow rate, and mass transfer rate. Scalar field and vector field

28. Hamilton’s operator It has been shown that the three partial derivatives of the temperature were vector gradients. If these three vector components are added together, there results a single vector gradient: which defines the operator  for determining the complete vector gradient of a scalar point function. The operator  is pronounced “del” or “nabla”. The vector  T is often written “grad T” for obvious reasons.  can operate upon any scalar quantity and yield a vector gradient. 應用於 scalar 的偏微

29. More about the Hamilton’s operator... (vector) · (vector) But  T is the vector equilvalent of the generalized gradient

30. Physical meaning of  T : A variable position vector r to describe an isothermal surface : Since dr lies on the isothermal plane… and Thus,  T must be perpendicular to dr. Since dr lies in any direction on the plane,  T must be perpendicular to the tangent plane at r. if A·B = 0 The vector A is zero The vector B is zero  = 90 ° dr TT  T is a vector in the direction of the most rapid change of T, and its magnitude is equal to this rate of change.

31. The operator  is of vector form, a scalar product can be obtained as : 應用於 vector 的偏微 application The equation of continuity : where  is the density and u is the velocity vector. Output - input : the net rate of mass flow from unit volume  A is the net flux of A per unit volume at the point considered, counting vectors into the volume as negative, and vectors out of the volume as positive.

32. A in A out The flux leaving the one end must exceed the flux entering at the other end. The tubular element is “divergent” in the direction of flow. Therefore, the operator  is frequently called the “divergence” : Divergence of a vector

33. Another form of the vector product : is the “curl” of a vector ; What is its physical meaning? Assume a two-dimensional fluid element u v  x  y O A B Regarded as the angular velocity of OA, direction : k Thus, the angular velocity of OA is ; similarily, the angular velocity of OB is

34. The angular velocityu of the fluid element is the average of the two angular velocities : u v  x  y O A B  This value is called the “vorticity” of the fluid element, which is twice the angular velocity of the fluid element. This is the reason why it is called the “curl” operator.

35. We have dealt with the differentiation of vectors. We are going to review the integration of vectors.

36. Vector integration Linear integrals Vector area and surface integrals Volume integrals

37. An arbitrary path of integration can be specified by defining a variable position vector r such that its end point sweeps out the curve between P and Q r P Q dr A vector A can be integrated between two fixed points along the curve r : Scalar product If the integration depends on P and Q but not upon the path r : if A·B = 0 The vector A is zero The vector B is zero  = 90 °

38. If a vector field A can be expressed as the gradient of a scalar field , the line integral of the vector A between any two points P and Q is independent of the path taken. If  is a single-valued function : and 假如與從 P 到 Q 的路徑無關，則有兩個性質： Example :

39. If the vector field is a force field and a particle at a point r experiences a force f, then the work done in moving the particle a distance  r from r is defined as the displacement times the component of force opposing the displacement : The total work done in moving the particle from P to Q is the sum of the increments along the path. As the increments tends to zero: When this work done is independent of the path, the force field is “conservative”. Such a force field can be represented by the gradient of a scalar function : Work, force and displacement When a scalar point function is used to represent a vector field, it is called a “potential” function : gravitational potential function (potential energy)……………….gravitational force field electric potential function ………………………………………..electrostatic force field magnetic potential function……………………………………….magnetic force field

40. Surface : a vector by referece to its boundary area : the maximum projected area of the element direction : normal to this plane of projection (right-hand screw rule) The surface integral is then : If A is a force field, the surface integral gives the total forace acting on the surface. If A is the velocity vector, the surface integral gives the net volumetric flow across the surface.

41. Volume : a scalar by referece to its boundary A B C Both the elements of length (dr) and surface (dS) are vectors, but the element of volume (d  ) is a scalar quantity. There is no multiplication for volume integrals. What are the relationships between them ? Stokes’ theorem

42. S Considering a surface S having element dS and curve C denotes the curve : Stokes’ Theorem （連接「線」和「面」的關係） If there is a vector field A, then the line integral of A taken round C is equal to the surface integral of  × A taken over S : Two-dimensional system

43. P Q How to make a line to a surface ? P and Q represent the same point! 你看到了一個「面」，你要如何去描述？ 從「線」著手 從「面」著手

44. A in A out The tubular element is “divergent” in the direction of flow. The net rate of mass flow from unit volume Gauss’ Divergence Theorem （連接「面」和「體」的關係） We also have : The surface integral of the velocity vector u gives the net volumetric flow across the surface The mass flow rate of a closed surface (volume)

45. Gauss’ Divergence Theorem （連接「面」和「體」的關係） Stokes’ Theorem （連接「線」和「面」的關係）

46. Useful equations about Hamilton’s operator... A is to be differentiated valid when the order of differentiation is not important in the second mixed derivative

47. Coordinates other than cartesian Spherical polar coordinates (r, ,  ) –Fig 7.15 –the edge of the increment element is general curved. –If a, b, c are unit vectors defined as point P :

48. The gradient of a scalar point function U : Assuming that the vector A can be resolved into components in terms of a, b, and c :

49. Coordinates other than cartesian Cylindrical polar coordinates (r, , z) –Fig 7.17 –the edge of the increment element is general curved. –If a, b, c are unit vectors defined as point P :

50. The gradient of a scalar point function U : Assuming that the vector A can be resolved into components in terms of a, b, and c :

51. How can we use vectors in chemical engineering problems? Why the Hamilton’s operator is important for chemical engineers?

52. Considering the study of “fluid flow”, the heating effect due to friction and mass transfer are ignored : Newtonian fluid: coefficient of viscosity remains constant Independent variables: x, y, z and time Dependent variables: u, v, w, pressure, density 5 dependent variables  5 equations : (1) continuity equation (mass balance) (2) equation of state (density and pressure) (3) ~ (5) Newton’s second law of motion to a fluid element (relating external forces, pressure force, viscous forces to the acceleration of fluid element) Navier - Stokes equation Solve together ?

53. Stokes’ Approximation (omit the inertia term, Re << 1) dimensionless form dimensionless groups dimensionless time dimensionless pressure coefficient Reynolds number incompressible not useful, usually u, not p, is given vorticity analogous to the heat and mass transfer equation

54. The ideal fluid Approximation (omit the viscous and inertia term, Re >> ) if steady state and vorticity = 0 Bernoulli’s equation : (1) laminar flow is steady (2) imcompressible (3) inviscid (4) irrotational incompressible The vorticity of any fluid element remains constant.

55. if a fluid motion starts from rest, the vorticity is zero and flow is irrotational Recall : if the curl of a vector is zero, the vector itself can be expressed as the gradient of a scalar point function (i.e. a potential function). An inviscid irrotational fluid where  is the velocity potential Can only be used in ideal fluid flow

56. Boundary layer theory (Prandtl) Assuming that ideal fluid flow existed everywhere except in a thin layer of fluid near any solid boundary : –within this thin layer, viscous effects are not negligible –velocity gradient normal to the boundary are quite large –velocity gradient parallel to the boundary are relatively small Boundary layer theory Navier-Stokes equation 2 assumptions, they are...

57. The thickness of the boundary layer at any point on a surface is small compared with the length of the surface to that point measured along the surface in the direction of flow. Viscous effects are confined to the boundary layer and ideal fluid flow exists outside it. flat two dimensional, steady-state x y continuity equation 01  Leave these to the transport phenomena

58. Heat transport –The rate of flow of heat per unit area at any point is proportional to the temperature gradient at that point –The constant of proportionality is the thermal conductivity –Divergence operator (  Q) represents the net flow of heat from unit volume –The total heat content of unit volume is  C p T (conservation law) = thermal diffusivity

59. Mass transport –The rate of mass transport by diffusion : –Divergence operator (  N) represents the net flow of mass from unit volume –Similarily N : molar flux density; D : diffusivity ; C : molar concentration

60. When bulk motion is involved : Heat transfer (scalar equation) Mass transfer (scalar equation) Momentum transfer (vector equation) They are very similar in vector form, but the momentum transfer is the ONLY vector equation, having two extra terms.