By Norman W. Loney

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 traditional DIFFERENTIAL EQUATIONS creation basic options of the Homogeneous Equation Homogeneous EquationsRead more...

summary: DIFFERENTIAL EQUATIONS creation traditional Differential Equations version improvement References FIRST-ORDER usual DIFFERENTIAL EQUATIONS Linear Equations more information on Linear Equations Nonlinear Equations challenge Setup difficulties References LINEAR SECOND-ORDER AND platforms OF FIRST-ORDER traditional DIFFERENTIAL EQUATIONS advent primary suggestions of the Homogeneous Equation Homogeneous Equations with consistent Coefficients Nonhomogeneous Equations Variable Coefficient difficulties replacement equipment precis functions of Second-Order Differential Equations

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**Extra resources for Applied Mathematical Methods for Chemical Engineers, Second Edition**

**Example text**

Henry’s Law gives the equilibrium solubility of CO2 in water CA ¼ pA =HA where CA (mol/L) is the CO2 concentration in solution, pA (atm) is the partial pressure of CO2 in the gas phase and HA is Henry’s Law constant. The rate of transfer of CO2 from the gas to the liquid per unit area of gas– liquid interface is given by ra (mol=cm2 s_ ) ¼ k(CÃA À CA ) where CÃA is the concentration of CO2 that would be in equilibrium with the CO2 in the gas phase (CÃA ¼ PA / HA). Suppose the gas-phase total pressure is P (atm) and contains yA mol fraction of CO2, and the liquid phase initially has V (cm3) of pure water with the agitation of the liquid phase sufficient to neglect spatial composition dependency, and if the amount of absorbed CO2 is low enough for P, V, and yA to be assumed constant, write a differential balance on CO2 in the liquid phase and solve the differential equation to show that CA (t) ¼ CÃA {1 À exp (À kSt=V)} 6.

The quantity r/t can now be treated as a new variable, and one of the solution techniques of the previous sections may now be applicable. So far, the examples have assumed that the differential equations are given. However, as chemical engineers, we know that more often than not, the main problem is in the derivation of the differential equation and the associated conditions. To address that aspect of mathematical methods in this chapter, a problem setup section follows. 4 PROBLEM SETUP The traditional approach of outlining the theory and presenting some supporting examples has been followed up to now.

Include a consistent set of units (same units in each term). Appropriately include given conditions (initial or boundary). Check the mode of operation (steady or unsteady process). Below are some examples for which the straightforward application of mass or energy balance is sufficient to set up the differential equation. 6 Transient Behavior of an Air-Cooling System [5] Consider an engine that generates heat at a rate of 8530 Btu/min. Suppose this engine is cooled with air, and the air in the engine housing is circulated rapidly enough so that the air temperature can be assumed uniform and is the same as that of the outlet air.