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1、化工應用數學,授課教師：郭修伯,Lecture 6,Functions and definite integralsVectors,Chapter 5,Functions and definite integrals,There are many functions arising in engineering which cannot be integrated analytically in terms of elementary functions.The valuesof many integrals have been tabulated,much numerical work ca
2、n beavoided if the integral to be evaluated can be altered to a form that is tabulated.Ref.pp.153We are going to study some of these special functions.,Special functions,FunctionsDetermine a functional relationship between two or more variablesWe have studied many elementary functions such as polyno
3、mials,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.,The error f
4、unction,It occurs in the theory of probability,distribution of residence times,conduction of heat,and diffusion matter:,0,x,z,erf x,z:dummy variable,Proof in next slide,x and y are two independent Cartesian coordinates,in polar coordinates,Error between the volume determined by x-y and r-,The volume
5、 of has a base area which isless than 1/2R2 and a maximum height of e-R2,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
6、 error function“erfc x”,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
7、 integervalues of n,the gamma function can be replaced by a factorial.,(Fig.5.3 pp.147),More about the gamma function,Evaluate,Chapter 7,Vector analysis,It has been shown that a complex number consisted of a real part andan imaginary part.One symbol was used to represent a combinationof two other sy
8、mbols.It is much quicker to manipulate a single symbolthan 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)TensorsThe principal difference between tensors and
9、 matrices is the labelling andordering of the many distinct parts.,Tensors,Generalized as zm,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 thanone subscript,thus zmn is a tensor of second rank(i.e.m,n).,The symbol
10、ism for the general tensor consists of a main symbol suchas z with any number of associated indices.Each index is allowed totake any integer value up to the chosen dimensions of the system.Thenumber of indices associated with the tensor is the“rank”of the tensor.,Tensors of zero rank(a tensor has no
11、 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
12、:temperature,concentration and pressure which are all signed by a number which may vary with position but not depend upon direction.,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 dimension
13、s 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.,Tensors of second rank(a tensor has two indices),It has a magnitude and two di
14、rections 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:,The physical interpretation of the stress tensor,The first subscript denotes
15、 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”.,Geometrical applications,If A and B are two position vectors,find the equation of the straightline 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 cons
17、idering three separate scalar balances,one vector balance can be takenby 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:,Using vector ad
18、dition,Thus,OR with represents of the sum of the two streams must be a constantvector 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,Pon
19、chon-Savarit method(enthalpy-concentration diagram),x,y,z,M,R,N,B,A,P,The constant line OR cross the plane x=1 at point P,O,point A is:,point B is:,point P is:,Multiplication of vectors,Two different interactions(whats the difference?)Scalar or dot product:the calculation giving the work done by a f
20、orce during a displacementwork and hence energy are scalar quantities which arise from the multiplication of two vectorsif AB=0The vector A is zeroThe vector B is zero=90,A,B,Vector or cross product:n is the unit vector along the normal to the plane containing A and B and its positive direction is d
21、etermined as the right-hand screw rulethe magnitude of the vector product of A and B is equal to the area of the parallelogram formed by A and Bif 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=0The vector A
22、is zeroThe vector B is zero=0,A,B,Commutative law:,Distribution law:,Associative law:,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.,Scalar triple product,The magnitude of is the volume of the pa
23、rallelepiped with edges parallel to A,B,and C.,A,B,C,AB,Vector triple product,The vector is perpendicular to the plane of A and B.When the further vectorproduct with C is taken,the resulting vector must be perpendicular to and hence in the plane of A and B:,where m and n are scalar constants to be d
24、etermined.,Since this equation is validfor any vectors A,B,and CLet A=i,B=C=j:,Differentiation of vectors,If a vector r is a function of a scalar variable t,then when t varies by anincrement t,r will vary by an increment r.r is a variable associated with r but it needs not have either thesame magnit
25、ude of direction as r:,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 processcan be performed with respect to s thus:,is a unit vector in the direction of the tangent to the curve,is perp
26、endicular to the tangent.,The direction of is the normal to the curve,and the two vectors definedas the tangent and normal define what is called the“osculating plane”of the curve.,Temperature is a scalar quantity which can depend in general upon three coordinates defining position and a fourth indep
27、endent 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,A dependent variable such as temperature,having these properties,is called a“scalar point function”and the system of va
28、riables 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 transfe
29、r rate.,Scalar field and vector field,Hamiltons operator,It has been shown that the three partial derivatives of the temperaturewere vector gradients.If these three vector components are addedtogether,there results a single vector gradient:,which defines the operator for determining the complete vec
30、torgradient 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,More about the Hamiltons operator.,(vector)(vector),But T is the vector equilvalen
31、tof the generalized gradient,Physical meaning of T:,A variable position vector r to describe an isothermal surface:,Since dr lies on the isothermal planeandThus,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 AB=0The vec
32、tor A is zeroThe vector B is zero=90,dr,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.,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 an
33、d 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,countingvectors into the volume as negative,and vectors out of the volume as positive.,Ain,Aout,The flux leaving the one end must exceed the flux enterin
34、g 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,Another form of the vector product:,is the“curl”of a vector;,What is its physical meaning?,Assume a two-dimensional fluid element,Regarded
35、as the angular velocity of OA,direction:kThus,the angular velocity of OA is;similarily,the angular velocity of OB is,The angular velocityu of the fluid element is the average of the two angular velocities:,This value is called the“vorticity”of the fluid element,which is twice the angular velocity of
36、 the fluid element.This is the reason why it is called the“curl”operator.,We have dealt with the differentiation of vectors.We are going to review the integration of vectors.,Vector integration,Linear integralsVector area and surface integralsVolume integrals,An arbitrary path of integration can be
37、specified by defining a variableposition 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 AB=0The vector A is zeroTh
38、e vector B is zero=90,If a vector field A can be expressed as the gradient of a scalar field,the line integralof 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:,If the vector field is a force field and a
39、particle at a point r experiences a force f,then the work done in moving the particle a distance r from r is definedas 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 incrementsalong the path.As the inc
40、rements 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”fun
41、ction:gravitational potential function(potential energy).gravitational force fieldelectric potential function.electrostatic force fieldmagnetic potential function.magnetic force field,Surface:a vector by referece to its boundaryarea:the maximum projected area of the elementdirection:normal to this p
42、lane 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 flowacross the surface.,Volume:a scalar by referece to its boundar
43、y,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,S,Considering a surface S having element dS and curve C denotes the curve:,Stokes Theo
44、rem（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,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,Ain,Aout,The tubular element is“divergent”in th
45、e 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 givesthe net volumetric flow across the surface,The mass flow rate of a closed surface(volume),Gauss Divergence Theorem（連接面和體的關係）,Stokes Theor
46、em（連接線和面的關係）,Useful equations about Hamiltons operator.,A is to be differentiated,valid when the order of differentiation is notimportant in the second mixed derivative,Coordinates other than cartesian,Spherical polar coordinates(r,)Fig 7.15the edge of the increment element is general curved.If a,b,
47、c are unit vectors defined as point P:,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:,Coordinates other than cartesian,Cylindrical polar coordinates(r,z)Fig 7.17the edge of the increment element is general curved.If a,b,c
48、are unit vectors defined as point P:,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:,How can we use vectors in chemical engineering problems?Why the Hamiltons operator is important for chemical engineers?,Considering the st
49、udy of“fluid flow”,the heating effectdue to friction and mass transfer are ignored:,Newtonian fluid:coefficient of viscosity remains constant Independent variables:x,y,z and timeDependent variables:u,v,w,pressure,density,5 dependent variables 5 equations:(1)continuity equation(mass balance)(2)equati
50、on of state(density and pressure)(3)(5)Newtons 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?,Stokes Approximation(omit the inertia term,Re 1),dimensionless form,dimensionless
