【大学课件】化工应用数学(英文).ppt
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1、化工應用數學,http:/ equation,An 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: x,Pa
2、rtial differential equation - more than one independent variable involved: x, y, z, ,http:/ and degree,The 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
3、differential coefficient that the equation contains after it has been rationalized.,3rd order O.D.E. 1st degree O.D.E.,http:/ or non-linear,Differential equations are said to be non-linear if any products exist between the dependent variable and its derivatives, or between the derivatives themselves
4、.,Product between two derivatives - non-linear,Product between the dependent variable themselves - non-linear,http:/ order differential equations,No general method of solutions of 1st O.D.E.s because of their different degrees of complexity. Possible to classify them as: exact equations equations in
5、 which the variables can be separated homogenous equations equations solvable by an integrating factor,http:/ equations,Exact?,General solution: F (x,y) = C,For example,http:/ equations,In the most simple first order differential equations, the independent variable and its differential can be separa
6、ted from the dependent variable and its differential by the equality sign, using nothing more than the normal processes of elementary algebra.,For example,http:/ equations,Homogeneous/nearly homogeneous? A differential equation of the type, Such an equation can be solved by making the substitution u
7、 = y/x and thereafter integrating the transformed equation.,is termed a homogeneous differential equation of the first order.,http:/ equation example,Liquid benzene is to be chlorinated batchwise by sparging chlorine gas into a reaction kettle containing the benzene. If the reactor contains such an
8、efficient agitator that all the chlorine which enters the reactor 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 isotherma
9、lly at 55 C when the ratios of the specific reaction rate constants are:,k1 = 8 k2 ; k2 = 30 k3,C6H6+Cl2 C6H5Cl +HCl C6H5Cl+Cl2 C6H4Cl2 + HCl C6H4Cl2 + Cl2 C6H3Cl3 + HCl,http:/ a basis of 1 mole of benzene fed to the reactor and introduce the following variables to represent the stage of system at t
10、ime , p = moles of chlorine present q = moles of benzene present r = moles of monochlorbenzene present s = moles of dichlorbenzene present t = moles of trichlorbenzene present Then q + r + s + t = 1 and the total amount of chlorine consumed is: y = r + 2s + 3t From the material balances : in - out =
11、 accumulation,u = r/q,http:/ solved by integrating factor,There exists a factor by which the equation can be multiplied so that the one side becomes a complete differential equation. The factor is called “the integrating factor”.,where P and Q are functions of x only,Assuming the integrating factor
12、R is a function of x only, then,is the integrating factor,http:/ z = 1/y3,integral factor,http:/ of 1st O.D.E.,First order linear differential equations occasionally arise in chemical engineering problems in the field of heat transfer, momentum transfer and mass transfer.,http:/ O.D.E. in heat trans
13、fer,An elevated horizontal cylindrical tank 1 m diameter and 2 m long is insulated with asbestos lagging of thickness l = 4 cm, and is employed as a maturing vessel for a batch chemical process. Liquid at 95 C is charged into the tank and allowed to mature over 5 days. If the data below applies, cal
14、culated the final temperature of the liquid and give a plot of the liquid temperature as a function of time. Liquid film coefficient of heat transfer (h1) = 150 W/m2C Thermal conductivity of asbestos (k) = 0.2 W/m C Surface coefficient of heat transfer by convection and radiation (h2) = 10 W/m2C Den
15、sity of liquid () = 103 kg/m3 Heat capacity of liquid (s) = 2500 J/kgC Atmospheric temperature at time of charging = 20 C Atmospheric temperature (t) t = 10 + 10 cos (/12) Time in hours () Heat loss through supports is negligible. The thermal capacity of the lagging can be ignored.,http:/ of tank (A
16、) = ( x 1 x 2) + 2 ( 1 / 4 x 12 ) = 2.5 m2,Tw,Ts,Rate 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),t,At steady state, the three rates are equal:,Considering the thermal equilibrium
17、of the liquid,input rate - output rate = accumulation rate,B.C. = 0 , T = 95,http:/ O.D.E.,Purpose: reduce to 1st O.D.E. Likely to be reduced equations: Non-linear Equations where the dependent variable does not occur explicitly. Equations where the independent variable does not occur explicitly. Ho
18、mogeneous equations. Linear The coefficients in the equation are constant The coefficients are functions of the independent variable.,http:/ 2nd O.D.E. - Equations where the dependent variables does not occur explicitly,They are solved by differentiation followed by the p substitution. When the p su
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