大学化工应用数学英文课件.pptx
《大学化工应用数学英文课件.pptx》由会员分享,可在线阅读,更多相关《大学化工应用数学英文课件.pptx(65页珍藏版)》请在淘文阁 - 分享文档赚钱的网站上搜索。
1、Differential equationAn equation relating a dependent variable to one or more independent variables by means of its differential coefficients with respect to the independent variables is called a“differential equation”.Ordinary differential equation-only one independent variable involved:xPartial di
2、fferential equation-more than one independent variable involved:x,y,z,http:/ and degreeThe order of a differential equation is equal to the order of the highest differential coefficient that it contains.The degree of a differential equation is the highest power of the highest order differential coef
3、ficient that the equation contains after it has been rationalized.3rd order O.D.E.1st degree O.D.E.http:/ or non-linearDifferential equations are said to be non-linear if any products exist between the dependent variable and its derivatives,or between the derivatives themselves.Product between two d
4、erivatives-non-linearProduct between the dependent variable themselves-non-linearhttp:/ order differential equationsNo general method of solutions of 1st O.D.E.s because of their different degrees of complexity.Possible to classify them as:exact equationsequations in which the variables can be separ
5、atedhomogenous equationsequations solvable by an integrating factorhttp:/ equationsExact?General solution:F(x,y)=CFor examplehttp:/ equationsIn the most simple first order differential equations,the independent variable and its differential can be separated from the dependent variable and its differ
6、ential by the equality sign,using nothing more than the normal processes of elementary algebra.For examplehttp:/ equationsHomogeneous/nearly homogeneous?A differential equation of the type,Such an equation can be solved by making the substitution u=y/x and thereafter integrating the transformed equa
7、tion.is termed a homogeneous differential equationof the first order.http:/ equation exampleLiquid benzene is to be chlorinated batchwise by sparging chlorine gas into a reaction kettle containing the benzene.If the reactor contains such an efficient agitator that all the chlorine which enters the r
8、eactor undergoes chemical reaction,and only the hydrogen chloride gas liberated escapes from the vessel,estimate how much chlorine must be added to give the maximum yield of monochlorbenzene.The reaction is assumed to take place isothermally at 55 C when the ratios of the specific reaction rate cons
9、tants are:k1=8 k2;k2=30 k3C6H6+Cl2 C6H5Cl+HClC6H5Cl+Cl2 C6H4Cl2+HClC6H4Cl2+Cl2 C6H3Cl3+HClhttp:/ a basis of 1 mole of benzene fed to the reactor and introducethe following variables to represent the stage of system at time,p=moles of chlorine presentq=moles of benzene presentr=moles of monochlorbenz
10、ene presents=moles of dichlorbenzene presentt=moles of trichlorbenzene presentThen q+r+s+t=1and the total amount of chlorine consumed is:y=r+2s+3tFrom the material balances:in-out=accumulationu=r/qhttp:/ solved by integrating factorThere exists a factor by which the equation can be multiplied so tha
11、t the one side becomes a complete differential equation.The factor is called“the integrating factor”.where P and Q are functions of x onlyAssuming the integrating factor R is a function of x only,thenis the integrating factorhttp:/ z=1/y3integral factorhttp:/ of 1st O.D.E.First order linear differen
12、tial equations occasionally arise in chemical engineering problems in the field of heat transfer,momentum transfer and mass transfer.http:/ O.D.E.in heat transfer An elevated horizontal cylindrical tank 1 m diameter and 2 m long is insulated withasbestos lagging of thickness l=4 cm,and is employed a
13、s a maturing vessel for abatch chemical process.Liquid at 95 C is charged into the tank and allowed tomature over 5 days.If the data below applies,calculated the final temperature of theliquid and give a plot of the liquid temperature as a function of time.Liquid film coefficient of heat transfer(h1
14、)=150 W/m2CThermal conductivity of asbestos(k)=0.2 W/m CSurface coefficient of heat transfer by convection and radiation(h2)=10 W/m2CDensity of liquid()=103 kg/m3Heat capacity of liquid(s)=2500 J/kgCAtmospheric temperature at time of charging=20 CAtmospheric temperature(t)t=10+10 cos(/12)Time in hou
15、rs()Heat loss through supports is negligible.The thermal capacity of the lagging can be ignored.http:/ of tank(A)=(x 1 x 2)+2(1/4 x 12)=2.5 m2TwTsRate of heat loss by liquid=h1 A(T-Tw)Rate of heat loss through lagging=kA/l(Tw-Ts)Rate of heat loss from the exposed surface of the lagging=h2 A(Ts-t)tAt
16、 steady state,the three rates are equal:Considering the thermal equilibrium of the liquid,input rate-output rate=accumulation rateB.C.=0,T=95http:/ O.D.E.Purpose:reduce to 1st O.D.E.Likely to be reduced equations:Non-linearEquations where the dependent variable does not occur explicitly.Equations wh
17、ere the independent variable does not occur explicitly.Homogeneous equations.LinearThe coefficients in the equation are constantThe coefficients are functions of the independent variable.http:/ 2nd O.D.E.-Equations where the dependent variables does not occur explicitlyThey are solved by differentia
18、tion followed by the p substitution.When the p substitution is made in this case,the second derivative of y is replaced by the first derivative of p thus eliminating y completely and producing a first O.D.E.in p and x.http:/ thereforeintegral factorerror functionhttp:/ 2nd O.D.E.-Equations where the
19、 independent variables does not occur explicitlyThey are solved by differentiation followed by the p substitution.When the p substitution is made in this case,the second derivative of y is replaced as Lethttp:/ thereforeSeparating the variableshttp:/ 2nd O.D.E.-Homogeneous equationsThe homogeneous 1
20、st O.D.E.was in the form:The corresponding dimensionless group containing the 2nd differential coefficient is In general,the dimensionless group containing the nth coefficient isThe second order homogenous differential equation can be expressed in a form analogous to ,viz.Assuming u=y/xAssuming x=et
21、If in this form,called homogeneous 2nd ODEhttp:/ by 2xyhomogeneousLetLetSingular solutionGeneral solutionhttp:/ graphite electrode 15 cm in diameter passes through a furnace wall into a watercooler which takes the form of a water sleeve.The length of the electrode betweenthe outside of the furnace w
22、all and its entry into the cooling jacket is 30 cm;and asa safety precaution the electrode in insulated thermally and electrically in this section,so that the outside furnace temperature of the insulation does not exceed 50 C.If the lagging is of uniform thickness and the mean overall coefficient of
23、 heat transferfrom the electrode to the surrounding atmosphere is taken to be 1.7 W/C m2 of surface of electrode;and the temperature of the electrode just outside the furnace is1500 C,estimate the duty of the water cooler if the temperature of the electrode atthe entrance to the cooler is to be 150
24、C.The following additional information is given.Surrounding temperature=20 CThermal conductivity of graphite kT=k0-T=152.6-0.056 T W/m CThe temperature of the electrode may be assumed uniform at any cross-section.xTT0http:/ sectional area of the electrode A=1/4 x 0.152=0.0177 m2A heat balance over t
25、he length of electrode x at distance x from the furnace isinput-output=accumulationwhereU=overall heat transfer coefficient from the electrode to the surroundingsD=electrode diameterhttp:/ factorhttp:/ differential equationsThey are frequently encountered in most chemical engineering fields of study
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 大学 化工 应用 数学 英文 课件
限制150内