366化工应用数学

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 b

2、eavoided 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 functions determine a functional relationship between two or more variables we have studied many elementary functions such as polynom

3、ials, 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 erro

4、r function it occurs in the theory of probability, distribution of residence times, conduction of heat, and diffusion matter:dzexxz022erf0 xzerf x22zez: dummy variable1erfproof in next slidedyedxeiryrx0022x and y are two independent cartesian coordinatesdydxeiryxr0)(0222drdreirr210022in polar coordi

5、nateserror between the volume determined by x-y and r-the volume of has a base area which isless than 1/2r2 and a maximum height of e-r2 2221rer241412rei41,2irdzexxz022erf1erfmore about error functiondifferentiation of the error function:22erfxexdxddzexxz022erfintegration of the error function:cexxc

6、dxexxxxdxxx221erf2erferfthe above equation is tabulated under the symbol “ ierf x” with 1c(therefore, ierf 0 = 0)another related function is the complementary error function “erfc x”dzexxxz22erf1 erfcthe gamma functiondtetntn01)(for positive values of n.t is a dummy variable since the value of the d

7、efinite integral is independent of t.(n.b., if n is zero or a negative integer, the gamma function becomes infinite.) 1() 1() 1()(020101nndtetnetdtetntntntnrepeat)!1() 1 () 1)(2).(2)(1()(nnnnthe gamma function is thus a generalized factorial, for positive integervalues of n, the gamma function can b

8、e replaced by a factorial.(fig. 5.3 pp. 147)more about the gamma function02121dtett2xt xdxdt2001222221dxexdxexxxerf21evaluate213) 1() 1()(nnn2138152123252121232521225chapter 7vector analysisit has been shown that a complex number consisted of a real part andan imaginary part. one symbol was used to

9、represent a combinationof two other symbols. 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 p

10、rincipal difference between tensors and matrices is the labelling andordering of the many distinct parts.tensors21izziyxzgeneralized 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

11、 a tensor of second rank (i.e. m, n) .the symbolism 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”

12、 of the tensor.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, densit

13、y, 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.tensors of first rank (a tensor has a single index) the tensor of first rank is alternatively names a “

14、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.ten

15、sors 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 for

16、m:333231232221131211ttttttttttmnthe physical interpretation of the stress tensorxzypxxxyxzzzzyzxyzyyyxxzxyxxmnppptthe 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”.geometri

17、cal applicationsif a and b are two position vectors, find the equation of the straightline passing through the end points of a and b.abc)(abmbcmabmc) 1(application of vector method for stagewise processesin any stagewise process, there is more than one property to be conserved and for the purpose of

18、 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 takenby using a set of cartesian coordinates in the following manner:using

19、x to measure m, y to measure h and z to measure cany process stream can be represented by a vector:kcjhimom111mhckcjhimon222a second stream can be represented by: kccjhhimmonomor)()()(212121using vector addition, thus, or with represents of the sum of the two streams must be a constantvector for the

20、 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)xyzmrnbapthe constant line or cross t

21、he plane x = 1 at point po1111, 1mcmh2222, 1mcmh21212121, 1mmccmmhhpoint 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 force during a displacement work and hence energy are

22、scalar quantities which arise from the multiplication of two vectors if ab = 0 the vector a is zero the vector b is zero = 90abbabacos|ab 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 r

23、ule 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 zer

24、o = 0nsin|babaababbafrlcommutative law :abbaabbadistribution law :cabacba)(cabacba)(associative law :)(dcbadbcacbabca)(cbacba)(cbacba)()(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.10kkjjiiik

25、kjjijikikjkjikkjjii0zyxzyxzzyyxxzyxzyxbbbaaakjibababababakbjbibbkajaiaascalar triple productcbathe magnitude of is the volume of the parallelepiped with edges parallel to a, b, and c.cbaabcab,cbabacacbacbcbacbavector triple productcbathe vector is perpendicular to the plane of a and b. when the furt

26、her vectorproduct with c is taken, the resulting vector must be perpendicular to and hence in the plane of a and b :baabcabbanbmacba)(where m and n are scalar constants to be determined.0)(bncamccbacacnbcmbacabccba)()()(since this equation is validfor any vectors a, b, and clet a = i, b = c = j:1cba

27、bcacbaacbbcacba)()()()()()(differentiation 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 :dtdttrr0limzkyjxirkjirdtdzd

28、tdydtdxdtdbabababababadtddtddtddtddtddtd)()(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:kdsdzjdsdyidsdxdsdr1)()()()()()(|222222dsdzdydx

29、dsdzdsdydsdxdsdrdsdris a unit vector in the direction of the tangent to the curve22dsrdis perpendicular to the tangent .dsdr22dsrdthe 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 i

30、s 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.xtxtttpartial differentiation of vectors a d

31、ependent 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 posit

32、ion. 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 derivatives of the temperaturewere vector gradients. if these three vector

33、 components are addedtogether, there results a single vector gradient:tztkytjxtiwhich defines the operator for determining the complete vectorgradient 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 a

34、ny scalar quantity and yield a vector gradient.應用於 scalar 的偏微zkyjximore about the hamiltons operator .ztkytjxtidrtdr(vector) (vector)dzztdyytdxxtztkytjxtikdzjdyidxtdrttddrdrdtt but t is the vector equilvalentof the generalized gradient physical meaning of t :a variable position vector r to describe

35、an isothermal surface :czyxt),(0dt0dttdrsince 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 o

36、f 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 : zayaxakajaiazkyjxiazyxzyx)(應用於 vector 的偏微applicationthe equation of continuity :0)()()(twzvyuxwhere is the density and u is the velocity vector.0)(tuou

37、tput - input : the net rate of mass flow from unit volumea 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.zzyyxxbabababaainaout0 athe flux leaving the one end must exceed the flux entering at the ot

38、her end.the tubular element is “divergent” in the direction of flow.therefore, the operator is frequently called the “divergence” :aadivdivergence of a vectoranother form of the vector product :zyxaaazyxkjiazyxzyxbbbaaakjibais the “curl” of a vector ; acurla what is its physical meaning?assume a two

39、-dimensional fluid element uv x yxxvvyyuuoabregarded as the angular velocity of oa, direction : kthus, the angular velocity of oa is ; similarily, the angular velocity of ob is xvkyukyuxvkvuyxkji00uthe angular velocityu of the fluid element is the average of the two angular velocities :uv x yxxvvyyu

40、uoabkyuxv21yuxvkvuyxkji00uku2this 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 differentiation of vectors.we are going to review the integration of vectors

41、.vector integration linear integrals vector area and surface integrals volume 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 qrpqdra vector a can be integrated between two fixed points alon

42、g the curve r :)(dzadyadxazyqpxqpdrascalar productif the integration depends on p and q but not upon the path r :0)(dradradrdif ab = 0the vector a is zerothe vector b is zero = 900aif a vector field a can be expressed as the gradient of a scalar field , the line integralof the vector a between any t

43、wo points p and q is independent of the path taken.if is a single-valued function : yxaxyxxyayand0kyaxaxy0a假如與從p到q的路徑無關，則有兩個性質：a0aexample :qpdrfw0aif 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 d

44、efinedas the displacement times the component of force opposing the displacement :rfwthe 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:qpdrfwwhen this work done is independent of the path, the force field is “conservati

45、ve”.such a force field can be represented by the gradient of a scalar function :work, force and displacementawfwhen 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 p

46、otential function .electrostatic force 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)dsnds the surface integral is then :dsnadsai

47、f 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 boundaryabcboth the elements of length (dr) and surface (ds) are vectors,but

48、the element of volume (d) is a scalar quantity.there is no multiplication 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

49、line integral of a taken round c is equal to the surface integral of a taken over s :sscdsadsadratwo-dimensional systemjiayxaa jidrdydx kndxdyds kayaxaxycsxyyxdxdyyaxadyadxa)(pq)(dzadyadxazyqpxqpdrahow to make a line to a surface ?p and q represent the same point!scdsadra你看到了一個面，你要如何去描述？從線著手從面著手dsna

50、dsaainaoutthe tubular element is “divergent” in the direction of flow.uudivthe 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） aadivwe also have : the surface integral of the velocity vector u givesthe net volumetric flow across the surfacedsnudsudsudsuthe mass fl

51、ow 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） sscdsadsadradsadsauseful equations about hamiltons operator .)()(abbaabbabaabbaa is to be differentiateduuuaaauuuaaabaabbaabbabaabb)a(bababa)(ababab)(ba )(21aaaaa20u0aaaaa(2)valid wh

52、en the order of differentiation is notimportant in the second mixed derivativecoordinates 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 : cbarsinrrrddr0rdrcbasin11rrrthe gradient o

53、f a scalar point function u : cbaururruusin11arararrrarsin1)sin(sin1)(122assuming that the vector a can be resolved into components in terms of a, b, and c :cbaaaaarcbarra(rarrrararaara)1)(sinsin1)sin(sin122222222sin1sinsin11ururrurrrucoordinates other than cartesian cylindrical polar coordinates (r

54、, , z) fig 7.17 the edge of the increment element is general curved. if a, b, c are unit vectors defined as point p : cbarzrrddr0rdrcbazrr1the gradient of a scalar point function u : cbazuurruu1zrazarrarra)(1)(1assuming that the vector a can be resolved into components in terms of a, b, and c :cbaaz

55、raaacbarzrzarrarrazazaara)(1122222211zuurrurrruhow 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 o

56、f viscosity 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 ex

57、ternal forces, pressure force, viscous forces to the acceleration of fluid element)0)(tufuuuu21ptnavier - stokes equationsolve together ?stokes approximation (omit the inertia term , re )fuuuu21pt)(21aaaaa2pt121uu-uu2tpuuuu221 if steady state and vorticity = 0.21constp 2ubernoullis equation :(1) laminar flow is steady(2) imcompressible(3) inviscid(4) irrotationalincompressiblet u)(00)(utthe vorticity of any fluid element remains constant.0)(utif a fluid motion starts from rest, the vorticity is zero and flow i