化工应用数学课件.pptx

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1、化工應用數學,授課教師： 郭修伯 助理教授,Lecture 3,應用數學方程式表達物理現象,化工應用數學授課教師： 郭修伯 助理教授Lecture 3應,建立數學模式,The conservation lawsmaterial balanceheat balanceenery balanceRate equationsthe relationship between flow rate and driving force in the field of fluid flowheat transferdiffusion of matter,建立數學模式The conservation laws,

2、建立數學模式,The conservation lawsmaterial balanceheat balanceenery balance(rate of) input - (rate of) output = (rate of) accumulation,建立數學模式The conservation laws,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 solven

3、t.The two streams are fed into a tank A where they are stirred 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 ha

4、s passed into the solvent phase.,water,toluene+benzoic acid,toluene+benzoic acid,water+benzoic acid,範例說明A single-stage mixer settl,簡化（理想化）,Rate equation for the extraction efficiency : y = mx,Material Balance : Input of benzoic acid = output of benzoic acid,Rc = Rx +Sy,Same method can be applied to

5、multi-stages.,簡化（理想化）S m3/s tolueneR m3/s to,隨時間變化,Funtion of time,隨時間變化Funtion of time,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 efficie

6、nt that the compositions of the two liquid 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.,非穩定狀態 (unsteady state)In unste,Material balance on

7、benzoic acid,Input - output = accumulation,單位時間的變化,t = 0, x = 0,Material balance on benzoic ac,Mathematical Models,Salt accumulation in a stirred tank,t = 0Tank contains 2 m3 of water,Q: Determine the salt concentration in the tankwhen the tank contains 4 m3 of brine,Mathematical ModelsSalt accumu,建

8、立數學模式,V and x are function of time tDuring t:balance of brinebalance of salt,建立數學模式V and x are function of,解數學方程式,Solvex = 20 - 20 (1 + 0.005 t)-2V = 2 + 0.01 t,解數學方程式Solve,Mathematical Models,Mixing,t = 0Tank 1 contains 150 g of chlorine dissolved in 20 l waterTank 2 contains 50 g of chlorine disso

9、lved in 10 l water,Q: Determine the amount of chlorine in each tank at any time t 0,Mathematical ModelsMixingPure,建立數學模式,Let xi(t) represents the number of grams of chlorine in tank i at time t. Tank 1: x1(t) = (rate in) - (rate out)Tank 2: x2(t) = (rate in) - (rate out)Mathematical model:,x1(t) = 3

10、 * 0 + 3 * x2/10 - 2 * x1/20 - 4 * x1/20,x2(t) = 4 * x1/20 - 3 * x2/10 - 1 * x2/10,建立數學模式Let xi(t) represents the,解數學方程式,How to solve?Using MatricesX = AX ; X(0) = X0 where x1(t)=120e-t/10+30e-3t/5x2(t)=80e-t/10-30e-3t/5,解數學方程式How to solve?,Mathematical Models,Mass-Spring SystemSuppose that the uppe

11、r weight is pulled down one unit and the lower weight is raised one unit, then both weights are released from rest simultaneously at time t = 0.,Q: Determine the positions of the weights relative totheir equilibruim positions at any time t 0,Mathematical ModelsMass-Spring,建立數學模式,Equation of motionwe

12、ight 1: weight 2: Mathematical model:,m1 y1”(t) = - k1 y1 +k2 (y2 - y1),m2 y2”(t) = - k2 (y2 - y1) - k3 y2,建立數學模式Equation of motionm1 y1”,解數學方程式,How to solve?y1(t)=-1/5 cos (2t) + 6/5 cos (3t)y2(t)=-2/5 cos (2t) - 3/5 cos (3t),解數學方程式How to solve?,隨位置變化,Funciotn of position,隨位置變化Funciotn of position,

13、Mathematical Models,Radial heat transfer through a cylindrical conductor,Temperature at a is ToTemperature at b is T1,Q: Determine the temperature distributionas a function of r at steady state,r,r +dr,a,b,Mathematical ModelsRadial heat,建立數學模式,Considering the element with thickness rAssuming the hea

14、t flow rate per unit area = QRadial heat fluxA homogeneous second order O.D.E.,where k is the thermal conductivity,建立數學模式Considering the element,解數學方程式,Solve,解數學方程式Solve,流場 (Flow systems) - Eulerian,The analysis of a flow system may proceed from either of two different points of view:Eulerian method

15、the analyst takes a position fixed in space and a small volume element likewise fixed in spacethe laws of conservation of mass, energy, etc., are applied to this stationary systemIn a steady-state condition:the object of the analysis is to determine the properties of the fluid as a function of posit

16、ion.,流場 (Flow systems) - EulerianTh,流場 (Flow systems) - Lagrangian,the analyst takes a position astride a small volume element which moves with the fluid.In a steady state condition:the objective of the analysis is to determine the properties of the fluid comprising the moving volume element as a fu

17、nction of time which has elapsed since the volume element first entered the system.The properties of the fluid are determined solely by the 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

18、 steady state condition:both the elapsed time and the absolute time affect the properties of the fluid comprising the element.,流場 (Flow systems) - Lagrangian,Eulerian 範例,A fluid is flowing at a steady state. Let x denote the distance from theentrance to an arbitrary position measured along the centr

19、e 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.Apply the law of conservation of mass to an infinitesimal element of volume fixed in space and of length dx.,Eulerian 範例A fl

20、uid is flowing,If Vx and are essentially constant across the area A,The rate of input of mass is:,The rate of mass output is:,Rate of input - rate of output = rate of accumulation,0,Equation of continuity,xdx, A, Vx+d, A+dA, Vx+dVxI,Lagrangian 範例,Consider a similar system. An infinitesimal volume el

21、ement 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 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 vel

22、ocity of the element relative to the stationary wall is Vx.Apply the law of conservation of mass to the volume element.,Lagrangian 範例Consider a simila,t integral,x,The elapsed time :,The difference between the relative velocity of the forward face and the relativevelocity of the trailing face is the

23、 change rate of the length of the element:,Mass balance of the element at steady-state,和Eulerian結果一樣,xa, A, Vxt integralxThe elap,獨立參數 (independent variable),These are quantities describing the system which can be varied by choice during a paticular experiment independently of one another.Examples:t

24、imecoordinates,獨立參數 (independent variable)The,非獨立參數 (dependent variable),These are properties of the system which change when the independent variables are altered in value. There is no direct control over a dependent variable during an experiment.The relationship between independent and depend vari

25、ables is one cause and effect; the independent variable measures the cause and the depend variable measures the effect of a particular action.Examples:temperatureconcentrationefficiency,非獨立參數 (dependent variable)Thes,變數 (Parameter),It consists mainly of the charateristics properties of the apparatus

26、 and the physical properties of the materials.It contains all properties which remain constant during an individual experiment. However, a different constant value can be taken by a property during different experiments.Examplesoverall dimensions of the apparatusflow rateheat transfer coefficientthe

27、rmal conductivitydensityinitial or boundary values of the depent variables,變數 (Parameter)It consists main,各符號之間的關係,A dependent variable is usually differentiated with respect to an independent variable, and occasionally with respect to a parameter.When a single independent variable is involved in th

28、e problem, it gives rise to ordinary differential equations.When more than one independent variable is needed to describe a system, the usual result is a partial differential equation.,各符號之間的關係A dependent variable i,邊界條件 (Boundary conditions),There is usually a restriction on the range of values whi

29、ch the independent variable can take and this range describes the scope of the problem. Special conditions are placed on the dependent variable at these end points of the range of the independent varible. These are natually called “boundary conditions”.,邊界條件 (Boundary conditions)Ther,常見的邊界條件,熱傳 (hea

30、t 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/dx = h (T-T0).k is the thermal conductivity;

31、h is the heat transfer coefficient; and T0 is the temperature of the surrendings.,常見的邊界條件熱傳 (heat transfer),邊界值與起始值(Boundary value and initial value),Specifying conditions on a solution and its derivative at the ends of an interval (boundary value problem) is quite different from specifying the valu

32、e of a solution and its derivative at a given point (initial value problem). Boundary 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.,邊界值與起始值(Boundary value and

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