1、化工應用數學化工應用數學授課教師：授課教師：郭修伯郭修伯 助理教授助理教授Lecture 3應用數學方程式表達物理現象建立數學模式建立數學模式lThe conservation laws material balance heat balance enery balancelRate equations the relationship between flow rate and driving force in the field of fluid flow heat transfer diffusion of matter建立數學模式建立數學模式lThe conservation laws
2、 material balance heat balance enery balancel(rate of)input-(rate of)output=(rate of)accumulation範例說明範例說明A single-stage mixer settler is to be used for the continuous extractionof benzoic acid from toluene,using water as the extracting solvent.The two streams are fed into a tank A where they are sti
3、rred vigorously,and the mixture is then pumped into tank B where it is allowed to settleinto two layers.The upper toluene layer and the lower water layer areremoved separately,and the problem is to find what proportion of thebenzoic acid has passed into the solvent phase.watertoluene+benzoic acidtol
4、uene+benzoic acidwater+benzoic acid簡化（理想化）簡化（理想化）S m3/s tolueney kg/m3 benzoic acidR m3/s toluenex kg/m3 benzoic acidS m3/s waterR m3/s toluenec kg/m3 benzoic acidRate equation for the extraction efficiency:y=mxMaterial Balance:Input of benzoic acid=output of benzoic acidRc=Rx+SySame method can be a
5、pplied to multi-stages.隨時間變化隨時間變化Funtion of time非穩定狀態非穩定狀態(unsteady state)In unsteady state problems,time enters as a variable and someproperties of the system become functions of time.Similar to the previous example,but now assuming that the mixer isso efficient that the compositions of the two liq
6、uid streams are inequilibrium at all times.A stream leaving the stage is of the samecomposition as that phase in the stage.The state of the system at a general time t,wher x and y are now functions of time.S m3/s tolueney kg/m3 benzoic acidR m3/s toluenex kg/m3 benzoic acidS m3/s waterR m3/s toluene
7、c kg/m3 benzoic acidV1,xV2,yMaterial balance on benzoic acidS m3/s tolueney kg/m3 benzoic acidR m3/s toluenex kg/m3 benzoic acidS m3/s waterR m3/s toluenec kg/m3 benzoic acidV1,xV2,yInput-output=accumulationdtdxVdtdxVSyRxRc21)(單位時間的變化CmVVtmSRxmSRRc21)(lnt=0,x=0tmVVmSRmSRRcx21exp1Mathematical Modelsl
8、Salt accumulation in a stirred tankt=0Tank contains 2 m3 of waterQ:Determine the salt concentration in the tankwhen the tank contains 4 m3 of brineBrineconcentration 20 kg/m3feed rate 0.02 m3/sFlow0.01 m3/s建立數學模式建立數學模式lV and x are function of time tlDuring t:balance of brine balance of saltBrineconc
9、entration 20 kg/m3feed rate 0.02 m3/sBrine0.01 m3/sV m3x kg/m3tdtdVtt01.002.0VxtdtdxxtdtdVVtxt)(01.02002.0解數學方程式解數學方程式lSolvelx=20-20(1+0.005 t)-2lV=2+0.01 t0)0(2)0(01.04.001.0 xVxdtdxVdtdVxdtdVMathematical ModelslMixingPure water3 l/minMixture2 l/minMixture3 l/minMixture4 l/minMixture1 l/minTank 1Ta
10、nk 2t=0Tank 1 contains 150 g of chlorine dissolved in 20 l waterTank 2 contains 50 g of chlorine dissolved in 10 l waterQ:Determine the amount of chlorine in each tank at any time t 0建立數學模式建立數學模式lLet xi(t)represents the number of grams of chlorine in tank i at time t.lTank 1:x1(t)=(rate in)-(rate ou
11、t)lTank 2:x2(t)=(rate in)-(rate out)lMathematical model:x1(t)=3*0+3*x2/10-2*x1/20-4*x1/20 Pure water3 l/minMixture2 l/minMixture3 l/minMixture4 l/minMixture1 l/minTank 1Tank 2x2(t)=4*x1/20-3*x2/10-1*x2/10 50)0(150)0(525110310321212211xxxxdtdxxxdtdx解數學方程式解數學方程式lHow to solve?lUsing MatriceslX=AX;X(0)=
12、X0 where x1(t)=120e-t/10+30e-3t/5 x2(t)=80e-t/10-30e-3t/55015052511031030XandA50)0(150)0(525110310321212211xxxxdtdxxxdtdxMathematical ModelslMass-Spring System Suppose that the upper weight is pulled down one unit and the lower weight is raised one unit,then both weights are released from rest simul
13、taneously at time t=0.Q:Determine the positions of the weights relative totheir equilibruim positions at any time t 0k1=6k3=3k2=2m1=1m2=1y2y1建立數學模式建立數學模式lEquation of motionlweight 1:lweight 2:lMathematical model:m1 y1”(t)=-k1 y1+k2(y2-y1)0)0()0(1)0(1)0(52282121212211yyyyyyyyyyk1=6k3=3k2=2m1=1m2=1y2y
14、1m2 y2”(t)=-k2(y2-y1)-k3 y2 解數學方程式解數學方程式lHow to solve?y1(t)=-1/5 cos(2t)+6/5 cos(3t)y2(t)=-2/5 cos(2t)-3/5 cos(3t)0)0()0(1)0(1)0(52282121212211yyyyyyyyyy隨位置變化隨位置變化Funciotn of positionMathematical ModelslRadial heat transfer through a cylindrical conductorTemperature at a is ToTemperature at b is T1Q
15、:Determine the temperature distributionas a function of r at steady staterr+drab建立數學模式建立數學模式lConsidering the element with thickness rlAssuming the heat flow rate per unit area=QlRadial heat fluxlA homogeneous second order O.D.E.)(22rdrdQQrrrQdrdTkQwhere k is the thermal conductivity022drdTdrTdr解數學方程
16、式解數學方程式lSolve1022)()(0TbTTaTdrdTdrTdr)lnlnlnln)()(010abarTTTrT流場流場(Flow systems)-EulerianlThe analysis of a flow system may proceed from either of two different points of view:Eulerian methodlthe analyst takes a position fixed in space and a small volume element likewise fixed in spacelthe laws of c
17、onservation of mass,energy,etc.,are applied to this stationary systemlIn a steady-state condition:the object of the analysis is to determine the properties of the fluid as a function of position.流場流場(Flow systems)-Lagrangian the analyst takes a position astride a small volume element which moves wit
18、h the fluid.In a steady state condition:lthe objective of the analysis is to determine the properties of the fluid comprising the moving volume element as a function of time which has elapsed since the volume element first entered the system.lThe properties of the fluid are determined solely by the
19、elapsed time(i.e.the difference between the absolute time at which the element is examined and the absolute time at which the element entered the system).In a steady state condition:lboth the elapsed time and the absolute time affect the properties of the fluid comprising the element.Eulerian 範例範例A
20、fluid is flowing at a steady state.Let x denote the distance from theentrance to an arbitrary position measured along the centre line in thedirection of flow.Let Vx denote the velocity of the fluid in the x direction,A denote the area normal to the x direction,and denote thefluid density at point x.
21、Apply the law of conservation of mass to an infinitesimal element of volume fixed in space and of length dx.xdx,A,Vx+d,A+dA,Vx+dVxxdx,A,Vx+d,A+dA,Vx+dVxIf Vx and are essentially constant across the area A,The rate of input of mass is:wAVxThe rate of mass output is:dxdxdwwdxAVdxdAVdVVdAAdxxxxRate of
22、input-rate of output=rate of accumulation00)(dwAVdxEquation of continuityLagrangian 範例範例Consider a similar system.An infinitesimal volume element whichmoves with the fluid through the flow system.Let denote the elapsed time:=t-t0where t is the absolute time at which the element is observed andt0 is
23、the absolute time at which the element entered the system.At elapsed time,the volume of the element is Aa,the density is,and the velocity of the element relative to the stationary wall is Vx.Apply the law of conservation of mass to the volume element.xa,A,Vx2adxdVVxx2adxdVVxx0)(aAdtdxa,A,Vx2adxdVVxx
24、2adxdVVxxt integralconstmaAxThe elapsed time:00ttVdxxxThe difference between the relative velocity of the forward face and the relativevelocity of the trailing face is the change rate of the length of the element:adxdVdtadx)(Mass balance of the element at steady-state xVdtdxxxVdVaad)(0)(aAd0)(xAVd和E
25、ulerian結果一樣獨立參數獨立參數(independent variable)lThese are quantities describing the system which can be varied by choice during a paticular experiment independently of one another.lExamples:time coordinates非獨立參數非獨立參數(dependent variable)lThese are properties of the system which change when the independent
26、variables are altered in value.There is no direct control over a dependent variable during an experiment.lThe relationship between independent and depend variables is one cause and effect;the independent variable measures the cause and the depend variable measures the effect of a particular action.l
27、Examples:temperature concentration efficiency變數變數(Parameter)lIt consists mainly of the charateristics properties of the apparatus and the physical properties of the materials.lIt contains all properties which remain constant during an individual experiment.However,a different constant value can be t
28、aken by a property during different experiments.lExamples overall dimensions of the apparatus flow rate heat transfer coefficient thermal conductivity density initial or boundary values of the depent variables各符號之間的關係各符號之間的關係lA dependent variable is usually differentiated with respect to an independ
29、ent variable,and occasionally with respect to a parameter.lWhen a single independent variable is involved in the problem,it gives rise to ordinary differential equations.lWhen more than one independent variable is needed to describe a system,the usual result is a partial differential equation.邊界條件邊界
30、條件(Boundary conditions)lThere is usually a restriction on the range of values which the independent variable can take and this range describes the scope of the problem.lSpecial conditions are placed on the dependent variable at these end points of the range of the independent varible.These are natua
31、lly called“boundary conditions”.常見的邊界條件常見的邊界條件l熱傳熱傳(heat transfer)Boundary at a fixed temperature,T=T0.Constant hear flow rate through the boundary,dT/dx=A.Boundary thermally insulated,dT/dx=0.Boundary cools to the surroundings through a film resistance described by a heat transfer coefficient,k dT/
32、dx=h(T-T0).lk is the thermal conductivity;h is the heat transfer coefficient;and T0 is the temperature of the surrendings.邊界值與起始值邊界值與起始值(Boundary value and initial value)lSpecifying conditions on a solution and its derivative at the ends of an interval(boundary value problem)is quite different from
33、specifying the value of a solution and its derivative at a given point(initial value problem).lBoundary value problems usually do not have unique solutions,and it is this lack of uniqueness which makes certain boundary value problems important in solving P.D.E.of physics and engineering.心得心得l如何建立一個應
34、用數學問題？如何建立一個應用數學問題？由假設，將問題簡化。確定所要探討的目標，找出非獨立參數。例如溫度、濃度等。找出獨立參數，使得非獨立參數可經由獨立參數表示。例如位置、時間等。找出可將獨立參數 及非獨立參數的關係經由數學式表示出的變數。例如氣體流速、熱傳係數等。選定一個特殊點，應用非獨立參數來描述該系統的狀態。增加微量非獨立參數。應用泰勒展開式來表示該微量增加後，該系統的狀態。應用守恆定律或速率方程式來顯示增加的微量。將增加的微量取極限值，建立該模型方程式。將邊界條件確定。l树立质量法制观念、提高全员质量意识。树立质量法制观念、提高全员质量意识。22.11.1622.11.16Wednesd
35、ay,November 16,2022l人生得意须尽欢，莫使金樽空对月。人生得意须尽欢，莫使金樽空对月。18:40:1518:40:1518:4011/16/2022 6:40:15 PMl安全象只弓，不拉它就松，要想保安全，常把弓弦绷。安全象只弓，不拉它就松，要想保安全，常把弓弦绷。22.11.1618:40:1518:40Nov-2216-Nov-22l加强交通建设管理，确保工程建设质量。加强交通建设管理，确保工程建设质量。18:40:1518:40:1518:40Wednesday,November 16,2022l安全在于心细，事故出在麻痹。安全在于心细，事故出在麻痹。22.11.16
36、22.11.1618:40:1518:40:15November 16,2022l踏实肯干，努力奋斗。踏实肯干，努力奋斗。2022年年11月月16日日下午下午6时时40分分22.11.1622.11.16l追求至善凭技术开拓市场，凭管理增创效益，凭服务树立形象。追求至善凭技术开拓市场，凭管理增创效益，凭服务树立形象。2022年年11月月16日星期三日星期三下午下午6时时40分分15秒秒18:40:1522.11.16l严格把控质量关，让生产更加有保障。严格把控质量关，让生产更加有保障。2022年年11月月下午下午6时时40分分22.11.1618:40November 16,2022l作业标准
37、记得牢，驾轻就熟除烦恼。作业标准记得牢，驾轻就熟除烦恼。2022年年11月月16日星期三日星期三18时时40分分15秒秒18:40:1516 November 2022l好的事情马上就会到来，一切都是最好的安排。好的事情马上就会到来，一切都是最好的安排。下午下午6时时40分分15秒秒下午下午6时时40分分18:40:1522.11.16l一马当先，全员举绩，梅开二度，业绩保底。一马当先，全员举绩，梅开二度，业绩保底。22.11.1622.11.1618:4018:40:1518:40:15Nov-22l牢记安全之责，善谋安全之策，力务安全之实。牢记安全之责，善谋安全之策，力务安全之实。2022
38、年年11月月16日星期三日星期三18时时40分分15秒秒Wednesday,November 16,2022l相信相信得力量。相信相信得力量。22.11.162022年年11月月16日星期三日星期三18时时40分分15秒秒22.11.16谢谢大家！谢谢大家！l踏实，奋斗，坚持，专业，努力成就未来。踏实，奋斗，坚持，专业，努力成就未来。22.11.1622.11.16Wednesday,November 16,2022l弄虚作假要不得，踏实肯干第一名。弄虚作假要不得，踏实肯干第一名。18:40:1518:40:1518:4011/16/2022 6:40:15 PMl安全象只弓，不拉它就松，要想
39、保安全，常把弓弦绷。安全象只弓，不拉它就松，要想保安全，常把弓弦绷。22.11.1618:40:1518:40Nov-2216-Nov-22l重于泰山，轻于鸿毛。重于泰山，轻于鸿毛。18:40:1518:40:1518:40Wednesday,November 16,2022l不可麻痹大意，要防微杜渐。不可麻痹大意，要防微杜渐。22.11.1622.11.1618:40:1518:40:15November 16,2022l加强自身建设，增强个人的休养。加强自身建设，增强个人的休养。2022年年11月月16日日下午下午6时时40分分22.11.1622.11.16l追求卓越，让自己更好，向上而
40、生。追求卓越，让自己更好，向上而生。2022年年11月月16日星期三日星期三下午下午6时时40分分15秒秒18:40:1522.11.16l严格把控质量关，让生产更加有保障。严格把控质量关，让生产更加有保障。2022年年11月月下午下午6时时40分分22.11.1618:40November 16,2022l重规矩，严要求，少危险。重规矩，严要求，少危险。2022年年11月月16日星期三日星期三18时时40分分15秒秒18:40:1516 November 2022l好的事情马上就会到来，一切都是最好的安排。好的事情马上就会到来，一切都是最好的安排。下午下午6时时40分分15秒秒下午下午6时时40分分18:40:1522.11.16l每天都是美好的一天，新的一天开启。每天都是美好的一天，新的一天开启。22.11.1622.11.1618:4018:40:1518:40:15Nov-22l务实，奋斗，成就，成功。务实，奋斗，成就，成功。2022年年11月月16日星期三日星期三18时时40分分15秒秒Wednesday,November 16,2022l抓住每一次机会不能轻易流失，这样我们才能真正强大。抓住每一次机会不能轻易流失，这样我们才能真正强大。22.11.162022年年11月月16日星期三日星期三18时时40分分15秒秒22.11.16谢谢大家！谢谢大家！
