《化工应用数学》PPT课件.ppt

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1、化工應用數學授課教師：郭修伯Lecture 6Functions and definite integralsVectorsChapter 5Functions and definite integralsThere 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 can beav

2、oided 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 functionsFunctionsDetermine a functional relationship between two or more variablesWe have studied many elementary functions such as polynomials,po

3、wers,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 functionIt

4、occurs in the theory of probability,distribution of residence times,conduction of heat,and diffusion matter:0 xzerf xz:dummy variableProof in next slidex and y are two independent Cartesian coordinatesin polar coordinatesError between the volume determined by x-y and r-The volume of has a base area

5、which isless than 1/2R2 and a maximum height of e-R2 More about error functionDifferentiation 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”Th

6、e gamma functionfor 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.)repeatThe gamma function is thus a generalized factorial,for positive integervalues of n,the gamma

7、 function can be replaced by a factorial.(Fig.5.3 pp.147)More about the gamma functionEvaluateChapter 7Vector analysisIt 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 symbols.It is much quicker to manipul

8、ate 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 matrices is the labelling andorderin

9、g of the many distinct parts.TensorsGeneralized as zmA 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 symbolism for the general tensor consists of a m

10、ain 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 index)It consists of one quantity independ

11、ent 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 whic

12、h 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 dimensions of the system.For practical purposes,this nu

13、mber 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 directions associated with it.The one tensor of se

14、cond 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 tensorxzypxxxyxzThe first subscript denotes the plane and the second subscript deno

15、tes the direction of the force.xy is read as“the shear force on the x facing plane acting in the y direction”.Geometrical applicationsIf A and B are two position vectors,find the equation of the straightline passing through the end points of A and B.ABCApplication of vector method for stagewise proc

16、essesIn 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 vecto

17、r 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 CAny process stream can be represented by a vector:MHCA second stream can be represented by:Using vector addition,Thus,OR with represents of the sum of the two s

18、treams 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,Ponchon-Savarit method(enthalpy-concentration diagram)xyzMR

19、NBAPThe constant line OR cross the plane x=1 at point POpoint A is:point B is:point P is:Multiplication of vectorsTwo different interactions(whats the difference?)Scalar or dot product:the calculation giving the work done by a force during a displacementwork and hence energy are scalar quantities wh

20、ich arise from the multiplication of two vectorsif AB=0The vector A is zeroThe vector B is zero=90ABVector 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 rulethe magnitude of the vector product o

21、f 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 is zeroThe vector B is zero=0ABCommutative law:Distribution law:Associative

22、 law:Unit vector relationshipsIt is frequently useful to resolve vectors into components along the axial directions in terms of the unit vectors i,j,and k.Scalar triple productThe magnitude of is the volume of the parallelepiped with edges parallel to A,B,and C.ABCABVector triple productThe vector i

23、s 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:ABCABwhere m and n are scalar constants to be determined.Since this equation is validfor any vectors A,B,and CLet A=i,B=C=j:Different

24、iation of vectorsIf 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 magnitude of direction as r:As t varies,the end point of the position vector r will trace out a

25、 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 curveis perpendicular to the tangent .The direction of is the normal to the curve,and the two vectors de

26、finedas 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 independent variable time.is a“partial derivative”.is the temperature gradient in the x direction

27、and is a vector quantity.is a scalar rate of change.Partial differentiation of vectorsA 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.Ther

28、e 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 fieldHamiltons operatorIt has been shown that the three partial

29、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 vectorgradient of a scalar point function.The operator is pronounced“del”or“nabla”.The vector T is oft

30、en 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 equilvalentof the generalized gradient Physical meaning of T:A variable position vector r to describe an isotherm

31、al 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 vector A is zeroThe vector B is zero=90drT T is a vector in the direction of the most rapid change of T,and i

32、ts 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applicationThe equation of continuity:where is the density and u is the velocity vector.Output-input:the net rate of mass flow from unit volumeA is the net flux of A per unit

33、 volume at the point considered,countingvectors into the volume as negative,and vectors out of the volume as positive.AinAoutThe 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 call

34、ed the“divergence”:Divergence of a vectorAnother form of the vector product:is the“curl”of a vector;What is its physical meaning?Assume a two-dimensional fluid element uv x yOABRegarded as the angular velocity of OA,direction:kThus,the angular velocity of OA is ;similarily,the angular velocity of OB

35、 is The angular velocityu of the fluid element is the average of the two angular velocities:uv x yOABThis 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.We have dealt with the different

36、iation of vectors.We are going to review the integration of vectors.Vector integrationLinear integralsVector area and surface integralsVolume integralsAn arbitrary path of integration can be specified by defining a variableposition vector r such that its end point sweeps out the curve between P and

37、QrPQdrA vector A can be integrated between two fixed points along the curve r:Scalar productIf the integration depends on P and Q but not upon the path r:if AB=0The vector A is zeroThe vector B is zero=90If a vector field A can be expressed as the gradient of a scalar field,the line integralof the v

38、ector 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 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

39、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 increments tends to zero:When this work done is independent of the path,the force field is“conservative”.Such a force field can

40、 be represented by the gradient of a scalar function:Work,force and displacementWhen a scalar point function is used to represent a vector field,it is called a“potential”function:gravitational potential function(potential energy).gravitational force fieldelectric potential function.electrostatic for

41、ce fieldmagnetic potential function.magnetic force fieldSurface:a vector by referece to its boundaryarea:the maximum projected area of the elementdirection: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

42、 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 boundaryABCBoth the elements of length(dr)and surface(dS)are vectors,but the element of volume(d)is a scalar quantity.There is no multipli

43、cation for volume integrals.What are the relationships between them?Stokes theorem SConsidering a surface S having element dS and curve C denotes the curve:Stokes Theorem（連接線和面的關係）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 ta

44、ken over S:Two-dimensional systemPQHow 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AinAoutThe tubular element is“divergent”in the direction of flow.The net rate of mass flow from unit volumeGauss Divergence Theorem（連接面和體的關係）連接面和體的關係）We also have:The surface i

45、ntegral of the velocity vector u givesthe net volumetric flow across the surfaceThe mass flow rate of a closed surface(volume)Gauss Divergence Theorem（連接面和體的關係）連接面和體的關係）Stokes Theorem（連接線和面的關係）連接線和面的關係）Useful equations about Hamiltons operator.A is to be differentiatedvalid when the order of differe

46、ntiation is notimportant in the second mixed derivativeCoordinates other than cartesianSpherical polar coordinates(r,)Fig 7.15the edge of the increment element is general curved.If a,b,c are unit vectors defined as point P:The gradient of a scalar point function U:Assuming that the vector A can be r

47、esolved into components in terms of a,b,and c:Coordinates other than cartesianCylindrical polar coordinates(r,z)Fig 7.17the edge of the increment element is general curved.If a,b,c are unit vectors defined as point P:The gradient of a scalar point function U:Assuming that the vector A can be resolve

48、d 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 study of“fluid flow”,the heating effectdue to friction and mass transfer are ignored:Newtonian fluid:coefficient of viscosity

49、 remains constant Independent variables:x,y,z and timeDependent variables:u,v,w,pressure,density5 dependent variables 5 equations:(1)continuity equation(mass balance)(2)equation of state(density and pressure)(3)(5)Newtons second law of motion to a fluid element(relating external forces,pressure forc

50、e,viscous forces to the acceleration of fluid element)Navier-Stokes equationSolve together?Stokes Approximation(omit the inertia term,Re )if steady state and vorticity=0Bernoullis equation:(1)laminar flow is steady(2)imcompressible(3)inviscid(4)irrotationalincompressibleThe vorticity of any fluid el