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An Introduction toModern IndustrialMathematicsC. Sean Bohun
IntroductionWhen beginning this article one of the most dif-ficult questions I was faced with was to providea definition of industrial mathematics and todistinguish it from applied mathematics. Appliedmathematics is primarily concerned with usingmathematics as a tool for analysis. This is quitea broad characterization and encompasses notonly what one would typically consider the meatand potatoes of modelling physical phenomena,solving problems and interpreting the results, butalso finding connections across disciplines andutilizing the language of mathematics to unifydisparate fields of study. Theories and models de-veloped in applied mathematics lie at the interfaceof a deep understanding of mathematical insightand technique with a specialized knowledge ofthe particular process being studied. Historically,practical applications have led applied mathemati-cians to develop models which were then taken onby pure mathematicians, where the mathematicswas further developed for its own sake. The fieldhas been responsible in the last few decades foropening up completely new disciplines, includingeverything from string theory to mathematicalbiology.
There is a further distinction within the disci-pline of applied mathematics between those thatconcern themselves with the mathematical meth-ods and the application of mathematics withinother fields. This has led to terms such as “practi-cal mathematics” or “applied science” to describe
C. Sean Bohun is associate professor of mathematics atthe University of Ontario Institute of Technology. His emailaddress is [email protected].
DOI: http://dx.doi.org/10.1090/noti1096
the latter [7], [11]. In this same vein, a possibledefinition of industrial mathematics is to view it asa subset of applied mathematics where the focusis on the use and development of mathematics tosolve industrial problems. I contend that it is morethan this.
Industrial mathematics is concerned with real-world problems which focus on furthering theestablishment and dissemination of links betweenmathematics and the physical world. The termindustry is meant in a very broad sense consistingof any field with either a commercial or societalbenefit, whether it be the optimal design of anelectric motor, modelling financial options, or evenclassifying sociological interactions. The key pointto make here is that, despite the inherent com-plexity of being concerned with real-life problems,industrial mathematics is tasked with laying barethe essential mechanisms at work and effectivelydisseminating the logical consequences to thoseoutside the field of mathematics, significantlyincreasing understanding in the process.
This definition reflects the evolution of thecontact between the industrial community andthe mathematical community. What started asan indirect connection through industry hiringprofessors as consultants and supporting post-doctoral research positions has grown to a globalnetwork of modeling workshops where directcontact is made between mathematicians andmembers from industry [5]. Due in part to this effort,there has been a shift in the thinking of the appliedmathematics community and in government overthe last couple of decades. As evidenced by the2012 SIAM Report of Mathematics in Industry [19],the role of mathematics in industrial society is notonly relevant but required.
364 Notices of the AMS Volume 61, Number 4
1975 1980 1985 1990 1995 2000 2005 20100
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New doctoral recipients in the mathematical sciences
Applied mathematics
All categories
1975 1980 1985 1990 1995 2000 2005 20108
10
12
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New doctoral recipients in applied mathematics
Per
cen
tag
e
Figure 1. Trends in the number of new doctoral recipients in applied mathematics from 1973 to 2012.
• In 1996 manufacturing accounted for 15.4%of US GDP, while the combination of finance,insurance, and scientific and technicalservices jointly contributed 12.5%.
• By 2010 the order had reversed, with manu-facturing accounting for 11.7% and finance,insurance, and scientific and technicalservices accounting for 15.9%.
This shift identified by SIAM enhances the factthat the skills attained through the study of appliedmathematics are integral to the demands beingmade by this changing demographic. The SIAMCareers in Applied Mathematics and ComputationalSciences [18] lists a few of these observations thatare repeated below.
• Mathematics has burst the old boundariesthat limited what an engineer could design,a scientist could know, or an executivecould manage.
• Subtle interactions, masses of data, andcomplex systems are all within the scope ofthe tools and ideas of applied mathematics.
• The logical clarity and deductive skillsyou are being taught in your mathematicsclasses are exactly the tools required tostudy these complex real-world problems.
Figure 1 shows data for the last forty years for theU.S., extracted from the annual AMS survey [24].To the left are the number of new doctoralrecipients in applied mathematics compared tothose across all of the mathematical sciences, whileto the right is the corresponding percentage ofrecipients in applied mathematics. From 1973 to1985 the historical record shows a steady increasein the percentage of new graduates in appliedmathematics as the total number of graduates inthe mathematical sciences decreased, while thenumber of those in applied mathematics heldsteady. From 1985 to 1994 the total number ofgraduates increased, and this trend was matchedin the applied mathematics group. During this
time period there was a steady increase in thepercentage of unemployed, reaching 10.7% in1994 [8] and stabilizing in 1995 with a sharp dropin graduates in applied mathematics. This dropwas not reflected in the mathematical sciencescommunity as a whole. Since the year 2000 therehas been a steady increase in the number ofgraduates in the mathematical sciences, recentlysurpassing anything in the historical record, yetthe same trend has not been seen in appliedmathematics. What we as applied mathematiciansneed to ask ourselves is why. As I have describedabove, the current trend in funding for appliedmathematics is tied to industrial relationships [1],[9], [13], yet this is not a watershed for appliedmathematics. I believe this is primarily a problemof public relations and a lack of a unified messageacross the discipline to expose potential studentsto the wonder, excitement, personal sense ofaccomplishment, and basically fun of being apart of the worldwide effort to apply modernmathematics to real-life problems.
One of the main difficulties of modern appliedand industrial mathematics is not relevance or alack of interesting problems. Personally, I havefound that once students begin to have a sense ofthe applicability and relevancy of this discipline, theproblems themselves become the best recruitmenttool of all. In fact, the richness of the mathematicsfound with many of the problems brought tovarious study groups around the world haveformed the backbone of a number of texts ofadvanced modeling [5], [7], [10], [11].
In the remainder of this report I will consider twocase studies in industrial mathematics to give someinsight as to the variety of techniques availableto the industrial mathematician. One of theseproblems had its inception as a real-life problembrought to one of the global network of modelingworkshops, while the other began with a seeminglyinnocuous question from an interested student.
April 2014 Notices of the AMS 365
2 CHAPTER 1. OPTIMAL DESIGN OF GAS BURST GENE GUN
1.1 Introduction
A fundamental problem faced by geneticists is the ability to modify a given segment of DNA and to study theeffects of the induced genetic modification. There are several methods of introducing DNA into animal and plantcells. A gene gun is a device which can be used to accelerate DNA-coated micro-particles directly into livingtissue [8, 9, 12]. The introduced particles are incorporated into the target cells and can bring about a change in thecell’s genetic properties. In this work, we model the behaviour of such a device and search for ways in which tooptimize the design.
The gas burst gene gun consists of a source of high pressure gas, a shock tube, and a nozzle. Rather than usinga kapton disk to carry the micro-particles, they are instead placed on the interior wall of the shock tube near theentrance of the nozzle as shown in Figure 1.1. A short pressure pulse imparts a drag force on the micro-particleswhich accelerates them through the nozzle and subsequently into the target. Our specific modelling effort willsuppose that the accelerating gas is helium so that an ideal gas law can be used which significantly simplifies theanalysis. As for the micro-particles, we require a high density material that does not react with our biologicalsample. Many materials are available as a delivery particle including tungsten, gold, polystyrene, borosilicate andstainless steel [8]. Our model will assume the use of gold since this has the highest density and is available asessentially spherical particles.
Without the kapton disk, the underlying theory of operation of a gas burst gene gun is closely related tothe concept of a cold spray [1, 3, 4, 5]. This is a process whereby an ultra fine metal powder is accelerated tosupersonic velocities and deposited on the surface of the target material.
1.2 Modelling the Shock Tube
The underlying assumptions for the shock tube include:
• the diameter of the tube is much smaller than its length,
• viscosity of the the gas is ignored and no heat energy is exchanged with the walls of the tube,
• the tube is considered to be frictionless with respect to the flow of the gas.
The first assumption allows one to reduce the problem to one spatial dimension whereas the second and third as-sumptions allow us to assume that the gas flow is isentropic (adiabatic and frictionless). Under these assumptionsthe conservation of mass, momentum and energy can be expressed as:
ρt + (ρv)x = 0 (1.1)
(ρv)t +(ρv2 + P
)x
= 0 (1.2)
Et + (v (E + P ))x = 0 (1.3)
Gold particles
Nozzle
Gas
Valve
Figure 1.1: Gene gun geometry.Figure 2. A Helios gas burst gene gun and the underlying gene gun geometry.
The emphasis here is placed on the questionsand the process that one goes through when firstexposed to any given problem and not on thelevel of sophistication. In fact, overcomplicatingan already complicated problem seems to be askill that pervades most academic disciplines. Ifthere is a theme in what follows, it is that a deepunderstanding of simple models as well as theirlimitations should never be underestimated. Whatshould be clear after reading through the modelsis that each choice made when modelling hasboth benefits and disadvantages and the modelleris continually faced with balancing mathematicalwith practical issues. Effectively navigating throughthese waters becomes an art as opposed to ascience. The first example is taken from the fieldof biotechnology and is concerned with the designof a gene gun, and my introduction is the followingexchange I had with a previous student:
NJ: Dr. Bohun, why do they use helium ina gene gun? Why not just air?
CSB: What’s a gene gun?
Let’s find out!
Modelling a Gene GunA gene gun consists of a tube that is configured topropel DNA-coated microparticles, typically gold,into a wide range of biological samples. There area number of propulsion methods that have beenutilized in the past, including, but not limited to,electric discharge, gunpowder explosions, gas flowand gas burst methods [14], [15], [22]. Figure 2shows a Helios gas burst gene gun and a diagramof its functionality.
The details of how a gene gun works is acombination of biochemistry and physics. Undercertain chemical conditions DNA can become stickyand adhere itself to biologically inert particles,for example, gold. These gold microparticles canact as bullets if they are placed in the path of a
pressure pulse. The gas burst gene gun consistsof a high-pressure tank of helium, a shock tube,and a nozzle. By placing the microparticles onthe interior wall of the shock tube after the valve,the particles accelerate as the shock wave travelsdown the tube. They then pass through a nozzlemade up of a converging and a diverging section,which makes it possible to achieve high-speed gasmolecules without having a very high pressure.Different gases can be used to generate a shockwave inside the shock tube, including helium (He),carbon dioxide (CO2), and air—mainly nitrogen(N2). But why is helium used? The modeling mayprovide an answer to this. In this work, we modelthe behavior of such a device and search for waysin which to optimize the design as a first step indesigning a handheld device using a CO2 cartridge.
A short pressure pulse imparts a drag force onthe microparticles that accelerates them throughthe nozzle and subsequently into the target. Ourspecific modeling effort will focus on helium as theaccelerating gas but also briefly contrast with car-bon dioxide. As for the microparticles, we requirea high-density material that does not react withour biological sample. Many materials are availableas a delivery particle, including tungsten, gold,polystyrene, borosilicate, and stainless steel [14].Our model will assume the use of gold, sincethis has the highest density and is available asessentially spherical particles.
Much of the underlying theory of the operationof a gas-burst gene gun is closely related to theconcept of a cold spray whereby an ultrafine metalpowder is accelerated to supersonic velocities anddeposited on the surface of a target material. Theinterested reader is directed to the text by Papyrinet al. [16] to further explore this connection. Manyof the submodels that are cascaded together inthis problem contain material that is well trodden,and the reader will be referred to an appropriateclassical text when appropriate.
366 Notices of the AMS Volume 61, Number 4
−1000 −500 0 500 1000 1500
0
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Speed vs x/t
x/t (m/s)
v (
m/s
)
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/m3)
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x/t (m/s)
P (
atm
)
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1
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ρL
vL= 0
PL
ρ1
v12
P12
ρ2
v12
P12
ρR
vR
= 0
PR
rarefaction fan contact discontinuity shock
x (m)
t (m
s)
(x,t)−plane
Figure 3. A simulated shock flow of helium gas for an initial uniform temperature of T0 = 300T0 = 300T0 = 300 K. Aleading shock in all three variables is followed by a contact discontinuity in the density and finally atrailing rarefaction fan. The bottom right panel shows the structure of the solution in the (x, t)(x, t)(x, t)-planeand the piecewise constant values within each of the solution regions.
Shock Tube
The underlying assumptions for the shock tubeinclude:
• the diameter of the tube is much smaller thanits length,
• viscosity of the gas is ignored and no heatenergy is exchanged within the walls of thetube,
• the tube is considered to be frictionless withrespect to the flow of the gas.
The first assumption allows one to reduce the prob-lem to one spatial dimension, whereas the secondand third assumptions allow us to assume that thegas flow is isentropic (adiabatic and frictionless).Under these assumptions the conservation of mass,momentum, and energy can be expressed as:
∂ρ∂t+ ∂∂x(ρv) = 0, (mass)(1a)
∂ (ρv)∂t
+ ∂∂x
(ρv2 + P
)= 0, (momentum)(1b)
∂E∂t+ ∂∂x(v (E + P)) = 0, (energy)(1c)
where ρ, v , and P respectively denote the density,speed, and pressure of the gas. The variable Edenotes the total energy of the gas per unit volumeand can be written as the sum of the gas’s internal
energy and its kinetic energy. In the case of anisentropic gas,
(1d) E = 12ρv2 + 1
γ − 1P,
where γ is the ratio of the specific heats. Forhelium, γ = cp/cv = 5/3.
When the pulse of gas is initiated, we assumethat the shock tube is in thermal equilibrium witha room temperature of T0 = 300 K and that thegas is at rest, v0 = 0. Across the valve is a jump ofpressure:
(2a) P(x,0) =PL = 4 atm, x < x0;
PR = 1 atm, x > x0,
where x0 is the location of the valve in the tube.There is a corresponding initial jump in the gasdensity, with pressure and density related throughthe ideal gas equation (1 atm = 101325 Pa):(2b)
ρ(x,0) = P(x,0)RT0=ρL = 0.6505 kg m−3, x<x0,ρR = 0.1626 kg m−3, x>x0,
with the ideal gas constant for helium, R =8.314× 103 JK−1mol−1/Mw , Mw = 4.003 g mol−1,being the molecular weight of the gas. Finally, atthe edges of the spatial domain, we assign a zeroNeumann boundary condition that is consistent
April 2014 Notices of the AMS 367
provided no disturbance in any of the variableshas time to reach the boundary.
A conservation law together with piecewiseconstant initial data having discontinuities isknown as a Riemann problem, and to solve thesystem (1), it is written in the form
(3a) ut +Aux = 0
where u = (ρ, ρv, E)T and [12], [23](3b)
A = 0 1 0
(γ − 3) v2
2 (3− γ)v γ − 1
(γ − 1) v3
2 − (E + P)vρ (E + P) 1
ρ − (γ − 1)v2 γv
.The solution is usually presented in terms of thepressure P rather than the less intuitive (for thisproblem) yet mathematically more convenient totalenergy E with expression (1d) used to make thisexchange.
The described problem is a variant1 of Sod’s [20],and has a well-known exact solution that can befound either in any well-represented gas dynamicstext, for example [2], or more recently withinthe case studies by Danaila et al. [6]. Figure 3illustrates the structure of the solution, and Table 1contrasts He and CO2 where in all cases the initialtemperature, pressure differential, and velocity ofthe gas remain constant: T0 = 300 K, PL = 4 atm,PR = 1 atm, vL = vR = 0. A shock in all threevariables (ρ, v, P)T propagates at a speed vS, whichis then followed by a contact discontinuity in thedensity moving at vCD = v12 and finally a trailingrarefaction fan in all three variables, the edges ofwhich move at speeds vfan
L < vfanR .
The purpose of modeling the shock wave is toprovide initial conditions for the gas entering thenozzle. The values obtained are idealized, sinceboth the friction and viscosity have been ignored,yet they give some initial insight for why helium ispreferred. It is clear from the data in Table 1 that,for a fixed pressure differential, the initial shockspeed of helium is significantly larger than thatof air or carbon dioxide. However, it is not onlythe speed of the gas particles that determines thebehavior of the gold microparticles but the densityof the gas as well. Since the density of carbondioxide is nearly ten times that of helium, any clearadvantage of using helium remains elusive.
Nozzle Modelling
At the end of its travel down the shock tube thegas is passed through a nozzle, and this changesits speed, pressure, temperature, and density.The model for the shock tube, together with thenozzle, describes the behavior of the gas along thecomplete length of the gene gun, and within this
1Sod’s original problem has the initial conditions ρL = 1,PL = 1, vL = 0, ρR = 0.125, PR = 0.1, and vR = 0.
managed stream of gas, the gold microparticlesacquire the inertia to be propelled into a biologicalsample. We begin by illustrating how a nozzle isused to accelerate the gas particles.
An ideal isentropic gas satisfies the additionalrelationship Pρ−γ = const. and, as a result, thespeed of sound is given by
(4) c2 = dPdρ= γ P
ρ= γRT
using the ideal gas equation of state (2b). For asteady one-dimensional flow through the nozzle,the continuity equation reduces to the hydraulicapproximation ρvA = const., where A is the cross-sectional area at any point in the nozzle. Expressedin terms of differentials this gives
(5)dρρ+ dvv+ dAA= 0.
In the same steady one-dimensional limit, using (4),the conservation of momentum reduces to [21]
(6) v dv + 1ρ
dP = v dv + c2
ρdρ = 0.
Eliminating dρ/ρ from (5) and (6) and rearranginggive the expression
(7)dAdv= Av
(M2 − 1
),
where the Mach Number M = v/c.Expression (7) embodies the purpose of the
nozzle. Incident gas particles approaching thenozzle at subsonic speeds (v < c) are acceleratedwith the converging portion. If the particles reachthe sonic speed (v = c) at the throat of the nozzle,then they can be further accelerated to supersonicspeeds (v > c) with the diverging portion. Tosummarize, the maximum possible exit velocity ofthe gas particles is attained by having subsonicflow in the converging part of the nozzle andsupersonic flow in the diverging part, and this canonly be attained by having v = c at the throat.
Continuing the analysis yields a relationshipbetween the Mach number and the cross-sectionalarea at any point in the nozzle [2], [21]:
(8)AA∗= M∗M
1+ γ−12 M
2
1+ γ−12 M
2∗
γ+1
2(γ−1)
with A∗ and M∗ denoting the area and Machnumber at the throat of the nozzle. If M∗ < 1,then the gas speeds up as the nozzle narrowsand then simply slows down in the divergingpart, since A/A∗ can never be less than 1 fora converging/diverging nozzle. For M∗ > 1, thegas must be entering the nozzle faster than thespeed of sound and the converging nozzle actuallyslows the gas, contrary to one’s intuition. Only inthe case M∗ = 1, when the speed at the throatof the nozzle matches the speed of sound, can
368 Notices of the AMS Volume 61, Number 4
Gas γ Mw (g mol−1) P12 (atm) ρ1 (kg m−3) ρ2 (kg m−3)He 1.667 4.003 1.905 0.4169 0.2374CO2 1.289 44.01 1.941 4.081 2.969Gas v12 (m s−1) vfan
L (m s−1) vfanR (m s−1) vCD (m s−1) vS (m s−1)
He 421.4 −1019 −457.2 421.4 1338CO2 145.6 −270.3 −103.6 145.6 366.2
Table 1. Solution to the Riemann problem for He and CO2. In all cases T0 = 300T0 = 300T0 = 300 K, PL = 4PL = 4PL = 4 atm,PR = 1PR = 1PR = 1 atm, vL = vR = 0vL = vR = 0vL = vR = 0 and ρLρLρL, ρRρRρR follow from the ideal gas law (2b). The speeds of the initial shock,the contact discontinuity, and the left and right edges of the rarefaction fan are denoted by vSvSvS, vCDvCDvCD,vfan
LvfanLvfanL and vfan
RvfanRvfanR respectively.
0 0.2 0.4 0.6 0.8 1
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Nozzle geometry drawn to scale
x/L
r/L
0 0.2 0.4 0.6 0.8 150
100
150
200
250
x/L
T (
K)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
x/L
P (
atm
)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
x/L
ρ (
kg/m
3)
Figure 4. Pressure, temperature, and density for the indicated nozzle design with L = 2L = 2L = 2 cm andassuming the flow of helium gas from the shock tube.
the particles transfer from subsonic (M < 1) tosupersonic (M > 1) speeds. Subsonic particlesenter the nozzle, they speed up to Mach-1 at thethroat, and they continue to accelerate as thenozzle diverges.
This increase in speed of the gas is done at theexpense of a loss of internal energy. What thismeans is that the temperature, the pressure, andthe density of the gas all drop as it progressesthrough the nozzle. Assuming that M∗ = 1, theseare given by the expressions [21]
(9)
TT0=
1+ γ−12 M
20
1+ γ−12 M2
,
PP0=(TT0
) γγ−1
,
ρρ0=(TT0
) 1γ−1
,
where (M0, T0, P0, ρ0) are the values at the entranceof the nozzle. Connecting the behavior within thenozzle to the results of the shock tube analysisis now simply a matter of matching the nozzleinput parameters. That is, P0 = P12, ρ0 = ρ1,T0 = P12/ρ1R and setting M0 = v12/(γRT0)1/2,where the values are taken from Table 1 for theflow after the contact discontinuity has passed.For the simulations a simple converging/divergingnozzle is chosen with an entrance angle consistentwith the Mach number of the entering gas and anexit angle of four degrees. The resulting flow fora nozzle of length L = 2 cm assuming helium isshown in Figure 4.
One of the important design aspects with nozzledesign is to ensure that the pressure does notdrop too low, or the outside gas will rush into thenozzle, undoing any advantage of making the gassupersonic in the first place. We will not address
April 2014 Notices of the AMS 369
5
5
5
25
25
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50
75
75
75
100
100
125
125
150
M
v p (m/s
)
Level curves of f/1000
f = 0vp = vp max
Γ1
Γ2 Γ3
0 0.5 1 1.5 2 2.50
50
100
150
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5
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2 5
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M
v p (m/s
)
f = 0
Γ1
Γ2 Γ3
0 0.5 1 1.5 2 2.50
50
100
150
200
250
300
350
400
450
500
vp = vp max
He CO2
Figure 5. Behavior microparticles when utilizing the simple converging/diverging nozzle depicted inFigure 4. Note that vp/c(M) ≤ Mvp/c(M) ≤ Mvp/c(M) ≤ M for the assumed model of the drag. For this simulation themicroparticles are placed 101010 cm from the nozzle entrance and the nozzle is 222 cm in length.
this issue of what is referred to as a normal shockbut refer the reader elsewhere [21].
Drag Force
Having shown how the speed of the gas variesthroughout the gene gun apparatus, we can moveon to investigating how the gold microparticles areaccelerated. By treating the gold particles as tinyspheres that are dragged along by collisions withthe gas molecules moving past them and assumingthat the microparticles are always moving slowerthan the gas, we find that the equations of themotion of the gold particles are(10)
dxpdt
= vp, xp(0) = 0,
mpdvpdt
= 12CDApρ(v − vp)2, vp(0) = 0.
This model ignores gravity, since the gold particlesare quite small and it is easy to verify that thegravitational force is far outweighed by the forcedue to the pressure pulse. Quantities mp, vp, andAp are the mass, speed, and cross-sectional area ofthe gold particles, CD is the drag coefficient (CD ' 1if the Reynolds number ≥ 30) [21], and (v, ρ) arethe speed and density of the gas respectively. Theseare initially obtained from the shock tube modeland subsequently changing within the nozzle,as described above. Typical values for the gold
microparticles are [14]
rp = 3× 10−6 m, ρp = 16800 kg m−3(11a)
and the combination
(11b) CDAp
2mp= Γ0 = 3
2rpρp' 29.76 m2 kg−1
so that (10) can be written as
(12a)
dxpdt
= vp, xp(0) = 0,
dvpdt
= Γ0f (vp;M,ρ, c), vp(0) = 0,
for vp < Mc where
(12b) f (vp;M,ρ, c) = c2(M − vp
c
)2
.
Figure 5 displays the level curves of f × 10−3
for both He and CO2. Bounded above by 2c2, theadmissible domain contains an interior curve alongwhich f attains its maximum value for a fixed valueof vp. A straightforward calculation gives
(13a) ∂f∂M
=−γP0
(M− vp
c
)(M2− vp
cM−2
)(1+ γ − 1
2M2) 1−2γγ−1
with(13b)
c = (γRT)1/2 = (γRT0)1/2(
1+ γ − 12M2
)−1/2,
giving this curve as
(13c) vpmax(M) =(M2 − 2
) cM.
370 Notices of the AMS Volume 61, Number 4
surface
drainage radius (Rd )
end cap
EMIT
z = 0 L1 L2 z = L
η(0) = 0 P(L) = PP
pump
---
6 6 6
? ? ?
Figure 6. Overall geometry for the horizontal wellbore problem. A horizontal cylindrical well extendsfrom z = 0z = 0z = 0 to z = Lz = Lz = L with a pump located at z = Lz = Lz = L to pull the heated oil out of the rock formation.
In each case the trajectory of the microparticlesis computed assuming that they initially lie 10 cmfrom the nozzle entrance and the nozzle is 2 cmin length. Three distinct regions characterize thebehavior.
1. Approaching the nozzle, the gas flow isconstant and the gold particles acceleratetowards the nozzle entrance. In this regimethe speed of the microparticles is given by (12)with a fixed Mach number determined by thesolution of the shock tube equations. Thefinal speed in this region is determined by theinitial distance of the microparticles to thenozzle entrance. Within He the microparticlesonly achieve 50% (211 m s−1) of the gas flowspeed of (421 m s−1), whereas the higherdensity of CO2 allows the microparticles toreach 82% (120 m s−1) of the gas flow speed(146 m s−1) over the same distance.
2. In the converging section of the nozzle thegas particles rapidly approach M = 1, andin both cases of He and CO2 this occursover such a short timeframe (He : 32 µs,CO2 : 57 µs) that no significant change inmicroparticle speed occurs.
3. In the diverging section the shape of thenozzle can be modified to allow the micro-particles to track the curve of maximal dragcurve. Again this portion of the nozzle isquickly traversed (He : 55 µs, CO2 : 92 µs) sothat only a moderate increase is observed.
With the groundwork that has been laid, anozzle design that maximizes the exit velocity ofthe microparticles can be determined. This designwill clearly depend on the gas being used. Thereare other issues as well. First, the gold particlesare not uniform in size, and their diameterswill have some probability distribution. We maywish to investigate the effect of this fact in thecurrent model. Second, we have only begun the
investigation of contrasting the effect of the use ofother gases. And finally, we have not even startedto consider a model for stopping the particles andcontrolling the distribution of where they embedthemselves in a target.
Clearly, for validation of the model a setof data needs to be made available and thepredictions of the model compared to what isseen experimentally. Let’s return to the student’sinitial question: “Why do they use helium in a genegun?” The simple answer is the structure of thegas and its nonreactivity. Helium, being a noblegas, will not easily react with a biological sample,but this is not the only reason. The structureof the gas (essentially spheres) gives a value ofγ = 5/3 ' 1.67. Carbon dioxide molecules areslightly more complicated, with two oxygen atomsconnected to a single carbon atom. The result isa value of γ = 7/5 = 1.4. The larger value of γmakes helium more efficient in that less energygoes into the internal vibrations of the moleculesand consequently there is a larger Mach numberfor a given area ratio. So now you know!
Horizontal WellboreThis last case study was brought to the attentionof the author at a problem-solving workshopconcerned with primarily oil and gas problems.As oil wells age it becomes increasingly difficultto extract oil from them, and they are eventuallyabandoned when the cost of extraction becomesprohibitive. To increase the production rate forthese wells and ultimately increase their lifespan,one possible technique is to somehow heat thetrapped oil, allowing it to flow more easily. Theheating technique described to the working groupwas the use of electromagnetic heaters known asEMIT (electromagnetic induction tool) units. For ahorizontal well of up to a kilometre in length thepractice is to insert several EMIT units at intervals
April 2014 Notices of the AMS 371
of about one hundred meters and then power themall from a single cable protected by a steel housing.Figure 6 shows a cut-away section of a well with asingle EMIT region. Arrows indicate the flow of oilthat seeps into the well from the surrounding rocksubsequently removed from the rock formationwith a pump located at z = L.
So why do any modelling at all? In this case, theindustrialist has an expensive computational fluiddynamics (CFD) code that can be used to simulatethe process, but he wants a simplified modelthat still captures most of the physics. His keyrequirement is a formulation of the problem thatcan be solved rapidly so that one can quickly searchwide ranges of possible operating parameters.Attempting this with a large commercial CFDsoftware package would become prohibitivelyexpensive. Another application of the model wouldbe financial. Being able to quickly estimate theincrease in production by heating the well, one canuse the local cost of electrical power to determinethe expected profit in a systematic way. As inthe previous model, the problem naturally breaksinto several subtasks, but in this case they remainintimately coupled throughout the wellbore.
Axial Flow
We start with estimating the amount of oil that willflow into the well over a very short length of time.This is given by an expression originally developedby Peaceman [4], [17] that depends on the drainageradius of the formation; Rd , the outer radius ofthe well, Rc ; the axial length of the section underconsideration; and the pressure in the pipe P(z).The expression for a segment of the well with axiallength ∆z is
(14) ∆η(z) = 2πk(PR − P(z))µ0 ln (Rd/Rc)
∆z,where ∆η is measured in m3 s−1, and k, µ0 are thepermeability of the ground and viscosity of the oilrespectively. The subscript zero for the viscosityrefers to unheated oil, as we have assumed thatthe fluid seeping into the well is not effectivelyheated when compared to the fluid flowing alongthe well. PR is known as the reservoir pressure andis the minimum suction pressure required to pulloil out of the well. It can vary from one well to thenext and increases as the well ages.
Figure 7 illustrates a small section of the well ineither the EMIT region or a region with the cablehousing. Of course the regions with neither an EMITnor the housing will not have any blockage. Thevolumetric flux is given by η(z) = π(R2
w − R2z)v,
where v is the radially averaged axial velocity, Rw isthe inner pipe radius, and Rz depends on whetherthere is nothing (Rz = 0), an EMIT (Rz = Re), orthe cable housing (Rz = Rh) within the pipe. By
the conservation of mass, the amount of fluid thatemerges from this infinitesimal segment must bethe sum of the fluid that entered along the welland the fluid that seeped in through the sides. Inother words,
(15) η(z∗ +∆z) = η(z∗)+ 2πk(PR − P(z))µ0 ln (Rd/Rc)
∆zor, letting ∆z → 0,
dηdz= 2πk(PR − P(z))
µ0 ln (Rd/Rc), η(0) = 0.(16)
The condition η(0) = 0 simply reflects the fact thatat the end cap the only flux of oil is that drippingin from the sides.
Axial Pressure
Assuming a steady state solution, the Navier-Stokesequations provide a connection between the fluidflow and the pressure drop applied to the fluid.The axial velocity of the fluid depends on its radialposition in the well and for the EMIT and housingregions. By further assuming that the radial flowentering the well from the sides is much smallerthan the axial flow (strictly true only away fromthe sidewall), it then satisfies
µ1r∂∂r
(r∂v∂r
)= dP
dz, v(Rw ) = 0, v(Rz) = 0,
(17)
where Rz is either the EMIT or the housing radiuswith the solution(18a)
v(r) = 14µ
dPdz
[r2 − R2
w +R2w − R2
zln(Rw/Rz)
ln(rRw
)].
For regions where Rz = 0 we impose the conditionv(0) <∞ to find that
(18b) v(r) = 14µ
dPdz(r2 − R2
w ).
Computing the average radial velocity defined as
(19) v = 2
R2w − R2
z
∫ RwRzv(r)r dr ,
using η = π(R2w − R2
z)v, and solving for thepressure give(20)
dPdz= − 8µ
πR4w
[lnλ
(1− λ2)2 + (1− λ4) lnλ
]η(z),
P(L) = PP ,where λ = Rz/Rw is piecewise constant and PP isthe suction applied by the pump at the surfacez = L. Combining this with expression (16) weobtain a two-point boundary value problem
(21a)
d2ηdz2
− γ2(z)µµ0η = 0, η(0) = 0,
dηdz
∣∣∣∣∣L= 2πk(PR − PP )µ0 ln(Rd/Rc)
,
372 Notices of the AMS Volume 61, Number 4
η(Z*)oil
cable housing
EMIT
oilcasing
Qc
Z = Z*
Z = Z* + ∆Z
∆η(Z*)
η(Z* + ∆Z)
Qe
Figure 7. Axial flow within the well illustrating both the fluid flow and the power flux from the casingand the EMIT.
where(21b)
γ2 = 16kR4w ln(Rw/Rc)
lnλ(1− λ2)2 + (1− λ4) lnλ
is piecewise constant, reflecting the placement ofEMIT, housing, and free regions.
Heating the Wellbore
How do we deal with the heat that is beinggenerated in the wellbore? Since we know wherethe oil is flowing, we can determine the heat flux
(22a) Φ = −ko∇T + ρCpvT ,where T is the temperature, v is the fluid, and ko,ρ, Cp are respectively the thermal conductivity,density, and heat capacity of the oil. The heatequation can be written in terms of this flux as
(22b)∂∂t(ρCpT)+∇ · Φ = Q,
whereQ corresponds to an external heating source.To find an effective axial temperature the flux ineach of the four regions (reservoir, casing, well,EMIT) is determined and combined with a radialaveraging process. The resulting final expressionfor the average temperature is obtained in [3],where it is shown to satisfy(23)
ρCpd
dz(η(z)T(z))= π
(Qc(z)
(R2c−R2
w
)+Qe(z)R2
e
),
T (0) = 0.
The heat sources Qc and Qe have been writtenas functions of z; if one is not in an EMIT region,these functions are simply zero. As a result, the
Data Symbol ValueOuter casing radius Rc 69.85 mmInner casing radius Rw 63.50 mmEMIT radius Re 50.80 mmHousing radius Rh 31.1625 mmWellbore length L 1000 m
Permeability k 9.869× 10−12 m2
Ambient viscosity µ0 15 Pa sDrainage radius Rd 100 mReservoir pressure PR 5.0× 106 PaProducing pressure PP 5.0× 105Pa
Fluid heat capacity ρCp 2.8× 106 J m−3K−1
Casing power Qc 795.8 kW m−3
EMIT power Qe 79.58 kW m−3
Table 2. Wellbore, reservoir and thermalproperties for a single EMIT region extendingfrom L1 = 495L1 = 495L1 = 495 m to L2 = 505L2 = 505L2 = 505 m.
right-hand side of (23) is piecewise constant andnonzero only where an EMIT is located. In addition,the temperature scale is chosen so that T = 0corresponds to 25C. Integrating (23) for a singleEMIT region extending from L1 to L2 gives theresult that(24)
T(z) =
0, 0 ≤ z < L1,πΩη(z) (z − L1) , L1 ≤ z < L2,πΩη(z) (L2 − L1) , L2 ≤ z ≤ L,
Ω = Qc (R2c − R2
w)+QeR2
eρCf
,
April 2014 Notices of the AMS 373
0 100 200 300 400 500 600 700 800 900 1000500
1000
1500
2000
2500
3000
3500
4000
4500Pressure
z (m)
P (
kP
a)
0 100 200 300 400 500 600 700 800 900 100025
30
35
40
45
50
55Temperature
z (m)
T (
0C
)
0 100 200 300 400 500 600 700 800 900 10000
0.02
0.04
0.06
0.08
0.1
0.12
0.14Velocity
z (m)
v (
ms-1
)
outer casing
inner radius
0 100 200 300 400 500 600 700 800 900 10000
2
4
6
8
10
12
14
16Viscosity
z (m)
μ (
1000cP
)
Figure 8. The four figures are the pressure, temperature, velocity, and viscosity as a function ofdistance along the wellbore. Only the pressure and temperature for the CFD code were available.These are the dashed lines in the respective plots. More than one method was used to solve the
simplified model. Where they are distinguishable, the shooting method solution is a solid line,whereas the SOR method is indicated with a dashed dot. A longitudinal section of the wellbore is
indicated in the plot of the velocity.
and from this the viscosity µ satisfies the empiricalrelationship in units of thousands of centipoise2
through
(25) log10 µ(T) = −3.002+(
453.29303.5+ T
)3.5644
,
which closes the model.
Results
To compare the results of the model with the CFDcode, a single EMIT unit was placed in the well andthe resulting temperature and pressure profileswere calculated. Table 2 lists the various parametersfor the model, and Figure 8 shows the resultingpressure, temperature, velocity, and viscosity ofthe fluid in the well and compares it with the resultsof the CFD code (dashed lines) in the pressure
21 centipoise = 1× 10−3 kg m−1 s−1.
and temperature plots. The solid and dashed-dot curves (where discernible) correspond to twodifferent techniques for solving the boundary valueproblem: namely, a shooting method and an SORmethod. Both the pressure and the temperatureare reasonably well reproduced considering therelative simplicity of the new model. The simplifiedmodel underestimates the production rate atthe pump giving ηCFD(L) = 187.6 m3 day−1 whileη(L) = 115.2 m3 day−1.
The simplified model reproduces the overallcharacteristics of the full model described by theCFD code and does this with a comparatively lowcomputational cost. One of the primary benefitsof this reduction in the computational cost isthat it allows one to run a number of numericalexperiments cheaply and onsite in the vicinity ofthe well itself. The question of determining theoptimal placement and number of required EMIT
374 Notices of the AMS Volume 61, Number 4
regions for a given production rate is explored inBohun et al. [3].
Final WordsDespite the ever-increasing complexity and cross-disciplinary nature of modern real-life problemsand a redoubling of efforts by government andindustry to form meaningful collaborative bridgeswith mathematicians, the historical record showsthat such efforts are being achieved only modestly.Industrial mathematicians weave disparate threadsof science from many different fields to produceinnovative models which are typically utilizedoutside the discipline, enriching understanding.Leading the reader through two separate casestudies, I have attempted to expose some of theinner workings to encourage students and facultyto participate in the global network of modelingworkshops. What has not been emphasized isthe importance of building and maintaining work-ing relationships with both industry and otherdisciplines within academia. With collaboration,both academically and institutionally, we stand toincrease not only the range of real-world problemsthat are made available to us, but we also standto ensure our positive impact on commercial andsocietal benefits. Personally I take great satisfac-tion in seeing the technologies I know and lovebeing applied in productive and interesting waysto real-world problems.
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April 2014 Notices of the AMS 375
