By Norman W. Loney
DIFFERENTIAL EQUATIONS creation usual Differential Equations version improvement References FIRST-ORDER traditional DIFFERENTIAL EQUATIONS Linear Equations additional info on Linear Equations Nonlinear Equations challenge Setup difficulties References LINEAR SECOND-ORDER AND platforms OF FIRST-ORDER traditional DIFFERENTIAL EQUATIONS advent basic suggestions of the Homogeneous Equation Homogeneous Equations with consistent Coefficients Nonhomogeneous Equations Variable Coefficient difficulties substitute equipment precis functions of Second-Order Differential Equations. Read more...
summary: DIFFERENTIAL EQUATIONS advent traditional Differential Equations version improvement References FIRST-ORDER usual DIFFERENTIAL EQUATIONS Linear Equations additional info on Linear Equations Nonlinear Equations challenge Setup difficulties References LINEAR SECOND-ORDER AND structures OF FIRST-ORDER usual DIFFERENTIAL EQUATIONS creation basic options of the Homogeneous Equation Homogeneous Equations with consistent Coefficients Nonhomogeneous Equations Variable Coefficient difficulties substitute tools precis purposes of Second-Order Differential Equations
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Additional info for Applied Mathematical Methods for Chemical Engineers, Second Edition
Symbolizes the derivative of r with respect to t. 4 can be obtained by employing the following steps: 13 14 Applied Mathematical Methods for Chemical Engineers 1. 4 as r0 þ a1 (t) g(t) À a0 (t) r¼ , a2 (t) a2 (t) 2. Determine m(t) ¼ exp ð a2 (t) 6¼ 0 for all t a1 (t) dt a2 (t) (2:5) (2:6) where m(t) is called an integrating factor. 3. 5 by m(t) ! 7 can be recasted as d r exp dt ð ! ð a1 (t) g(t) À a0 (t) a1 (t) dt ¼ exp dt a2 (t) a2 (t) a2 (t) (2:8) 4. 8 with respect to the independent variable to get ð ð a1 (t) g(t) À a0 (t) a1 (t) r exp dt ¼ exp dt dt þ c a2 (t) a2 (t) a2 (t) or ð ð a1 (t) g(t) À a0 (t) a1 (t) r(t) ¼ exp À dt exp dt dt a2 (t) a2 (t) a2 (t) ð a1 (t) þ c exp À dt a2 (t) where c is the constant of integration.
1) are always in equilibrium with each other and can be expressed as y ¼ mx (2:56) where m is the distribution coefficient, x is the concentration of benzoic acid leaving the stage in the organic phase, and y is the aqueous phase benzoic mass concentration. 7. Assume that the composition of a stream leaving the stage is the same composition as that phase in the stage. 8. Assume that the stage initially contains V1 liter of toluene, V2 liter of water, and no benzoic acid. 10, that is Rate of accumulation ¼ Rate of input À Rate of output the quantities for any time t can be derived.
Assume that the size distribution of particles is uniform throughout the mill and equals that of the product, and let xif be the mass fraction of the feed that falls in the ith size range. Show that a mass balance on the jth size fraction in the tank yields X dxj ¼ ðQ=MÞ(xjf À xj ) À kj xj þ ki xi bij dt i¼1 jÀ1 Solution Accumulation ¼ Input þ Generation À Output À Consumption dxj d Accumulation ¼ (Mxj ) ¼ M dt dt Input ¼ Qxif Generation: the rate at which particles enter the jth size fraction from the ith size fraction by breakage is kimibif.
Applied Mathematical Methods for Chemical Engineers, Second Edition by Norman W. Loney