Fall 2002 10 34 Numerical Methods Applied to Chemical Engineering Homework 2 Nonlinear algebraic equations Assigned Friday 9 13 02 Due Wednesday 9 25 02 Problem 1 Tank draining problem Let us consider the system below in which a cylindrical tank of diameter D tan k is drained by a cylindrical pipe of length L and diameter Dpipe The fluid in the tank is water and the tank is exposed to standard atmospheric conditions FIGURE 1 Geometry for tank draining problem To calculate the resistance to flow of the pipe use the relation from fluid mechanics that the change in dynamic pressure P p gz across a pipe of length L and diameter D for flow with a mean velocity V is L 1 2 P f D V D 2 The Darcy friction factor fD EQ 1 is related to the Reynolds number in laminar flow by 64 f D Re Re 2100 September 24 2002 EQ 2 1 and in turbulent flow is related to the Reynolds number and the surface roughness the Colebrook equation 1 e D 2 51 2 log 10 3 7 fD Re f D Re 2100 e by EQ 3 Our pipe is assumed to be commercial steel for which the effective surface roughness 0 045 mm e is You may neglect viscous effects within the bulk of the tank but include the minor loss due to entrance flow into the pipe 1 2 p entrance 0 5 V 2 EQ 4 Question 1 A Plot the volumetric flow rate out of the tank as a function of h The diameter of the tank is 2 5 m the diameter of the drain pipe is 5 cm and the length of the drain pipe is 2 m Question 1 B From the results of 1 A starting with an initial depth of water in the tank of 2 m plot as a function of time the height of water in the tank until it empties Hint Try rearranging the equation h t t h t t September 24 2002 dh dt t h t 2 Problem 2 Modeling steady state behavior of a Nylon reaction system Background This question considers the application of numerical methods for solving nonlinear algebraic equations to the study of a continuous process for the polymerization of Nylon The problem statement gives a sufficient description of polycondensation kinetics to answer the questions however a more general formulation of a polycondensation kinetic model may be found in Beers and Ray J Appl Polym Sci v 79 246 265 2001 The most common form of Nylon Nylon 6 6 is made by polycondensation of the two monomers hexamethylene diamine HMD and adipic acid ADA The general reaction for the formation of polymer from the diamine and diacid monomers is n HOOC CH 2 4 COOH n H 2 N CH 2 6 NH 2 HO OC CH 2 4 CONH CH 2 6 NH n H 2n 1 H 2 O EQ 5 The symbol denotes that this reaction is reversible and is driven to the right by the removal through evaporation of the water coproduct Here we show n moles of diacid monomer and n moles of diamine producing a polymer that contains on average n number of repeat units the quantity within the brackets that is comprised of one monomer unit each of diacid and diamine Such a polymer contains 2n monomer units and is said to have a chain length of 2n or to be a 2n mer Because water is produced as a condensate i e a volatile species that evaporates and is recovered through condensation the method used to synthesize Nylon 6 6 is known as condensation polymerization although a name that better describes the nature of the process is stepwise polymerization In this approach we build up high molecular weight polymer step by step where the first step is the combination of two monomer units to form a dimer according to the reaction HOOC CH2 4 COOH H 2 N CH 2 6 NH 2 HOOC CH 2 4 CONH CH 2 6 NH 2 H 2 O EQ 6 The double sided arrow denotes that this reaction is reversible with an equilibrium constant on the order of 100 The mechanism of this reaction under acid catalysis is shown in the figure below The acid catalyst may be either supplied externally e g HCl or the reaction may be self catalyzed by the carboxylic acid end groups The equilibrium constant of this reaction is on the order of 100 because in the amide linkage CONH the less electronegative nitrogen is able to donate some electron density to the electron deficient carbon of the carbonyl group The dimer produced by this reaction still has functional groups on both ends and so may react further to produce even larger molecules Condensation among larger molecules occurs through the same mechanism as that shown below for monomers and the reactivity of an end group COOH or NH2 is essentially independent of the size of the molecule to which it is attached September 24 2002 3 FIGURE 2 Acid catalyzed condensation among diamine and diacid monomers To simplify our description we use the following short hand notation to represent the reaction among monomers P 1 P1 P 2 W EQ 7 refers to a polymer chain comprised of one monomer unit i e a monomer and P 2 refers to a chain comprised of two monomer units a dimer W represents a water condensate molecule Because the dimer itself has acid and base groups on both ends it may react with other monomers dimers trimers etc to produce larger molecules The hierarchy of additional reactions that occur during polymerization is P1 P2 P2 P4 W P1 P3 P4 W P2 P3 P5 W P3 P3 P6 W and so on or in general for m 1 2 3 n 1 2 3 Pm Pn Pm n W EQ 8 In the forward direction the condensation reaction takes the form P m P n Pm n W September 24 2002 EQ 9 4 Because the reaction may be catalyzed by either an added external acid or be self catalyzed by the carboxylic acid end groups the rate constant is written as ext self k fc k fc k fc A EQ 10 A is the total concentration of acid end groups Since each chain contains on average one acid group and one base group at a given conversion the total concentration of chains of any length is equal to the concentration of unreacted acid base end groups Px A B EQ 11 x 1 Here we have assumed that the concentrations of acid and base end groups are equal throughout the course of the polymerization If their initial concentrations are equal and there is no significant loss of either monomer to side reactions or evaporation the stoichiometry of the reaction will retain the equimolar balance The net rate of change of the m mer concentration due to the forward direction condensation reaction among all species is r P m fc 2k fc P m m 1 Pn k fc Pn Pm …