ChE626 Mathematical Methods in Chemical Engineering

Prof. Ecevit Bilgili

Term Project (Due: Dec. 07, 2020)

Mathematical Modeling of Transport Phenomena Problems via Numerical

Method of Lines (NUMOL)

The purpose of this project is to derive/construct mathematical models of two transport

phenomena problems, and then solve them via a numerical method of lines technique along with

finite difference discretization using Matlab.

Problem 1 Start up of Laminar Flow in a Circular Tube

Please read 4D.2 with Part (a) in BSL and

1. Derive 4D.2.-1 by writing down all assumptions, postulates, and boundary/initial conditions

following the standard approach as discussed in the class (assumptions and postulates must be

indicated). Convert the equation into dimensionless form as indicated in the problem statement.

2. Develop the NUMOL equations for each node including the boundaries.

3. Write down a Matlab code (main+subroutine).

4. Generate the curves in Fig. 4D.2 using NUMOL and the code by choosing N = 21, record results

from τ = 0 to 2 with intervals of 0.05 and ξ [0–1].

5. Is the above solution grid-independent? Find grid-independent solution to at least 4 digit accuracy

after decimal point.

Problem 2 Viscous Heating in Laminar Tube Flow

Please read 11B.2. with Part (a) in BSL and

1. Derive 11B.2.-1 using the step-by-step assumptions, postulates and write down boundary/initial

conditions following the standard approach as discussed in the class (assumptions and postulates

must be indicated). Assume that the fluid has a constant, uniform inlet temperature of T0 and the

tube wall has a constant, uniform temperature of Tw. Convert the equation into dimensionless

form via:

( ) ( )

Pe

z R

k

C V R

z R

z

r R

r

T T

T T

w p z

= = =

– –

=

0 ,max

0 , * , *

The final equation must have Brinkman (Br) number and r*, z*, and only.

2. Develop the NUMOL equations for each node.

3. Write down a Matlab code (main+subroutine).

4. Generate dimensionless temperature profiles for z* values of 0 to 2 with 0.1 intervals and r*[0,1]

using NUMOL and the code by choosing N = 21. Perform this for Brinkman # of 1 and find gridindependent solution to at least 4 digit accuracy after decimal point.

5. Repeat step 4 for Brinkman # of 0, 0.1, and 10 and comment on the impact of Br #. Here use the

same N that has yielded grid-independent solution in step 4 for all other Br #.

SUBMISSION: submit the report via Canvas and email the codes to bilgece@njit.edu by Dec. 07th.

Note 1: You are required to submit a report which has all equations/derivations and results/discussion

(figures) following the steps mentioned above and including what each step asks. Typing equations in

2

Word may allow you to get bonus points (up to 25 points), but be careful not to skip any

derivations (show all work). I prefer handwritten complete derivations to typed pages that miss several

steps vecause students usually do not want to type all equations. If not done properly, writing in Word

may result in loss of points, rather than gaining 25 bonus points.

The report must be submitted as a PDF in Canvas. Matlab codes (two files: Main + subroutine

(function m file) for ODEs in each problem must be submitted via email (bilgece@njit.edu). There

will be 4 files in the email. Do NOT submit codes in Canvas. Make sure that the Matlab codes were

clearly identified for each problem and they are in running conditions, capable of replicating the results in

the report. Do not artificially rename files as they are interconnected. The instructor will test each code. If

your codes do not run or yield different results than those in your report, you will lose 40% of your

project score and an investigation will be opened to elucidate what happened.

Note 2: You are required to clearly identify axes, different curves by different markers and labels in your

figures adding a caption to each figure. Your report should be professional and include all relevant graphs

as well as equations/derivations.

Note 3: Late submissions may be accepted only under extenuating circumstances, at the full discretion of

the instructor, with a penalty of 40%/day (maximum score 60%). So, do not wait for the deadline.

Additional Specific Information on the Project:

150 Chapter 4 Velocity Distributions with More Than One Independent Variable

(a) Find the pressure distribution, radial flow velocity, and mass rate of flow for an incompressible fluid.

(b) Rework (a)for a compressible liquid and for an ideal gas.

9 – 9, – ln (r/RJ – g2- 9, In (R,/R,) |
K 9 2 – 91 vOr=– Pr In (R2/Rl) |
zm~h(p2- p1)p | Answers:(a) | w = |

p In (R2/R,) |

4D.1 Flow near an oscillating wa1L8 Show, by using Laplace transforms, that the complete solution to the problem stated in Eqs. 4.1-44 to 47 is

4D.2 Start-up of laminar flow in a circular tube (Fig.4D.2). A fluid of constant density and viscosity is contained in a very long pipe of length L and radius R. Initially the fluid is at rest. At

time t = 0, a pressure gradient (Yo- YL)/Lis imposed on the system. Determine how the velocity profiles change with time.

Tube center = 00

Tube wall

Fig. 4D.2. Velocity distribution for the unsteady flow resulting from a suddenly impressed pressure gradient in a

circular tube [P.Szymanski,J. Math. Pures Appl., Series 9,

11,67-107 (1932)l.

(a) Show that the relevant equation of motion can be put into dimensionless form as follows:

in which 5 = ?/A, r = pt/pR2, and 4 = [(Yo- 9L)R2/4pLl-‘v,.

(b) Show that the asymptotic solution for large time is 4, = 1- t2.Then define 4,by +((, r) =

+m(t) – +r(e,r), and solve the partial differential equation for 4,by the method of separation

of variables.

(c) Show that the final solution is

in which J,@ is the nth order Bessel function of t, and the a, are the roots of the equation

Jo(an)= 0. The result is plotted in Fig. 4D.2.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 2nd edition

(1959),p. 319, Eq. (a), with E = $.rr and G = KU’.

Problems 363

for methane are:a = 2.322 cal/g-mole.K, b = 38.04 X lop6cal/g-mole K3. |
cal/g-mole K2,and c = -10.97 X |

Answers: (a) | exp[-(b/R)T – (c/2R)~~l = constant; |

(b)270 atm

llB.2. Viscous heating in laminar tube flow (asymptoticsolutions).

(a) Show that for fully developed laminar Newtonian flow in a circular tube of radius R, the

energy equationbecomes

if the viscous dissipation terms are not neglected. Here v,,,, is the maximum velocity in the

tube. What restrictions have to be placed on any solutionsof Eq. 118.2-I?

(b) For the isothermal wall problem (T = Toat r = R for z > 0 and at z = 0 for all r), find the asymptotic expression for T(r)at large z. Do this by recognizing that dT/dz will be zero at large

z. SolveEq. 118.2-1and obtain

(c) For the adiabatic wall problem (9, = 0 at r = R for all z) an asymptoticexpression for large z

may be found as follows: Multiply by rdr and then integrate from r = 0 to r = R. Then integrate the resulting equation over z to get

in which T, is the inlet temperature at z = 0. Postulate now that an asymptotic temperature

profile at large z is of the form

Substitute this into Eq. llB.2-1 and integrate the resulting equation for f(r) to obtain

after determining the integration constant by an energy balance over the tube from 0 to z.

Keep in mind that Eqs. llB.2-2 and 5 are valid solutions only for large z. The complete solutions for small z are discussed in Problem llD.2.

11B.3. Velocity distribution in a nonisothermal film. Show that Eq. 11.4-20 meets the following

requirements:

(a) At x = 6, v, = 0.

(b) ~t x = 0, av,/ax = 0.

llB.4. Heat conduction in a spherical shell (Fig. llB.4). A spherical shell has inner and outer radii

R, and R,. A hole is made in the shell at the north pole by cutting out the conical segment in

the region 0 5 8 5 81. A similar hole is made at the south pole by removing the portion (.rr –

8,) 5 8 5 T.The surface 6 = is kept at temperature T = T I ,and the surface at 8 = T – 81 is

held at T = T2. Find the steady-state temperature distribution, using the heat conduction

equation.