8/11/2019 Homework 4 2014 Draft_3

1/6

1

CHEE 3334Homework 4

This HW has only two problems - an elaborate problem for which you probably will choose to use MATLAB or a similartool, and a problem you can solve without MATLAB.

University of Houston MATLAB License:

The University of Houston now has a MATLAB site license. Students may download MATLAB R2014a for Students on theUH Software Downloads web page.

http://www.uh.edu/infotech/php/software/list.php?affiliation=student

Solving ODEs in MATLAB:

In practice, solving difficult numerical problems such as systems of coupled nonlinear ordinary differential equations(ODEs) is usually carried out using ODE solvers available in various programming languages and software packages suchas MATLAB. Many of these ODE solvers, such as the ode23 and ode45 functions in MATLAB, are based on Runge-Kuttamethods. Being able to solve ODEs in MATLAB is an invaluable skill in engineering, and understanding the numericalmethods considered thus far in CHEE 3334 is essential for helping you troubleshoot issues that arise in numericalsolutions to problems, choosing the right numerical method for solving a problem, and writing your own algorithms orfunctions for obtaining numerical solutions.

In this homework assignment, you will need to use MATLAB to solve various problems involving ODEs. There are variousresources on the internet with example problems on how to use MATLAB to solve differential equations. You may findsome of the following resources helpful.

MathWorks has extensive documentation on how to use different MATLAB functions.

https://www.mathworks.com/moler/odes.pdf

http://www.mathworks.com/help/matlab/ref/ode45.html

The MathWorks forums are a great resource for writing MATLAB code and solving problems in MATLAB.

http://www.mathworks.com/matlabcentral/

Various other resources on solving ODEs in MATLAB can be found on the internet.

http://www.math.tamu.edu/~phoward/m442/matode.pdf

http://www.uh.edu/infotech/php/software/list.php?affiliation=studenthttp://www.uh.edu/infotech/php/software/list.php?affiliation=studenthttps://www.mathworks.com/moler/odes.pdfhttps://www.mathworks.com/moler/odes.pdfhttp://www.mathworks.com/help/matlab/ref/ode45.htmlhttp://www.mathworks.com/help/matlab/ref/ode45.htmlhttp://www.mathworks.com/matlabcentral/http://www.mathworks.com/matlabcentral/http://www.math.tamu.edu/~phoward/m442/matode.pdfhttp://www.math.tamu.edu/~phoward/m442/matode.pdfhttp://www.math.tamu.edu/~phoward/m442/matode.pdfhttp://www.mathworks.com/matlabcentral/http://www.mathworks.com/help/matlab/ref/ode45.htmlhttps://www.mathworks.com/moler/odes.pdfhttp://www.uh.edu/infotech/php/software/list.php?affiliation=student

8/11/2019 Homework 4 2014 Draft_3

2/6

2

Problem 1: Silica Encapsulation of Nanoparticles

Nanoparticles play an increasingly important role in biomedical technology with applications in diagnostics, in vivoimaging, and targeted drug delivery. Silica, also known as silicon dioxide (SiO 2), is commonly used to preparenanoparticles for biomedical applications because it is highly biocompatible and has very low toxicity.

Suppose you are working for a company that has developed a new type of nanoparticle for biological applications. Thenanoparticles must be coated or encapsulated in a silica shell in order to make them biocompatible and usable in theirfinal application. You are tasked with designing a batch reactor to encapsulate the nanoparticles in silica (Fig. 1). Use theinformation and equations provided to help design the batch reactor and process for encapsulating the nanoparticles.

A. Create MATLAB files for solving the given set of differential equations modeling the silica shell thickness as afunction of time. (Hint: you will need one function file containing the ODEs and some of the given constants, anda separate file that uses the ode45 function to solve the set of equations).

B. Plot the concentration of tetraethoxysilane ( , -) as a function of time from t = 0 to t = 15 hours.C. Plot the concentrations of the tetraethoxysilane hydrolysis species ( , , and ) as a function of timefrom t = 0 to t = 15 hours.D. Plot the silica shell thickness as a function of time from t = 0 to t = 15 hours.E. How long does it take for the silica shell thickness to reach 20 nanometers? How long does it take for the silica

shell thickness to reach 50 nanometers?

Show all MATLAB code to receive full credit.

Fig. 1 Silica encapsulation of nanoparticles suspended in a liquid in a batch reactor.

8/11/2019 Homework 4 2014 Draft_3

3/6

3

Silica encapsulation is based on the Stber process and involves the reaction of tetraethoxysilane (TEOS) with water inthe presence of ethanol to form silica. The overall reaction is given below:

The reaction kinetics are complex and involve the sequential hydrolysis of TEOS to form 4 different hydrolysis products.The hydrolysis reactions are reversible. The hydrolysis reactions are presented below:

{1} {2} {3} {4}Assuming that only completely hydrolyzed TEOS (i.e. the 4 th hydrolysis product, Si(OH) 4) contributes to the growth of thesilica shell, we have an additional reaction to include in the kinetic model. This reaction is called condensation, and isassumed to be irreversible:

{5}Using Reactions {1} {5}, a set of coupled ordinary differential equations was derived with the concentrations of thedifferent species represented by the symbols in Table 1, and the rate constants for the reactions in Table 2.

Table 1: Chemical species and corresponding symbols in kinetic model

Species Symbol

, - , -

, - , -

, - Water , -

, - Table 2: Rate constants for kinetic model

Reaction Forward Rate Constant Reverse Rate Constant{1} {2} {3} {4} {5} N/A

8/11/2019 Homework 4 2014 Draft_3

4/6

4

, - , -, - , -, - , - , -, - , -, - , -, - , -, - , - , -, - , -, - , -, - , -, - , - , -,- , -, - , -, - , -, - , - , -,- , -, - , - , - , -*, - , - , - , -+, -*, - , - , - , -+ , - , -

A simple mass balance can be used to calculate the total volume of silica ( ) produced in the reactor as a function of

time:

, - Assuming that the nanoparticles are spherical with total radius , and the that all silica that is formed is distributeduniformly on all nanoparticles in the reactor, we can calculate the radius of a nanoparticle as a function of time due togrowth of the silica layer.

The silica shell thickness is equal to the difference between the total nanoparticle radius and the initial radius.

Table 3: Other constants in kinetic model

Parameter Value 60.08

2.65 10 L

200 nm

8/11/2019 Homework 4 2014 Draft_3

5/6

5

The 9 differential equations can be used to model the silica shell growth kinetics. Use these equations in MATLAB todetermine that transient values of the 9 different variables over a time scale from to hours. Use the initialconditions for the variables given in Table 4.

Table 4: Initial Conditions (at )Variable Value, - 0.02 M , - 0 M , - 0 M , - 0 M , - 0 M , - 13.6 M , - 11.1 M 0

Note : Carefully consider the units of the given parameters, constants, and the differential equations when typing yourMATLAB code. You may need to include unit conversions in your code.

8/11/2019 Homework 4 2014 Draft_3

6/6

6

Problem 2: Aerodynamic Drag and Atmospheric Density Effects on a Rocket

The position of a rocket accelerating vertically through the atmosphere can be approximated with the followingequation:

Where is the instantaneous mass of the rocket, is the thrust force exerted by the rocket, is the forceon the rocket due to gravity, and is the drag force on the rocket.

The drag force on the rocket depends on the instantaneous velocity of the rocket ( ), and the density of the air,

which changes as a function of altitude. can be approximated as follows:

, . /

{ ( )}()

Where the drag coefficient is , and the cross sectional area of the rocket is . The other constants arepresented below: , molar mass of dry air , temperature at

, acceleration due to Earths gravity , atmospheric pressure at , temperature lapse rate , gas constant

Assume that the rocket engine exerts a constant thrust at . If we neglect changes in the mass of therocket due to loss of propellant we can treat the rocket mass as constant at , and use the followingequation for :

The initial velocity of the rocket at takeoff is 400 m/s.

A. Use the numerical methods discussed in this class to solve this differential equation. Plot the position of therocket as a function of time from to seconds.

B. Plot the velocity of the rocket as a function of time from to seconds.C. How long does it take the rocket to reach 15 km (the edge of the troposphere)? What is the velocity of the

rocket when it reaches this altitude?D. What is the minimum velocity of the rocket (after 10 seconds) during the first 100 seconds of flight?