Saville, D. A.

English

NASA Technical Reports Server

U.S. DEPARTMENT OF COMMERCE
 
National Technical Information Service
 
N81-74352
 
MATHEMATICAL MODELS OF CONTINUOUS FLOW 
ELECTROPHORE;SIS. 
Princeton University, 
NJ 
1981 
https://ntrs.nasa.gov/search.jsp?R=19810069929 2020-03-21T11:47:16+00:00Z
(NASA-CR-1617iUO 11) ATEAICAL MODETS OF 	 N81-743 5 2 
CONTINUOUS FLOW ELECTROPHORESIS ProgreSS 
Report, 1 May - 31 Dec. 1979 (Princeton 	 unclas 
3.) 24 pn~iv.-, N. 00/25 21586 
I. 	 Fundamental Studies 
(A) 	 Synopsis 
Since we are nearing the end of the contract it seems wrth­
while to summarize the task, the acccnplishments this far, and 
plans for the remainder of the contract period. 
Development of high resolution continuous flow electrophoresis 
devices ultimately requires ccmprehensive understanding of the ways 
various phenamena and processes facilitate or hinder separation. 
Seemingly small individual effects may, in cnbination, constitute 
serious limitations on a particular chanter design or operating 
configuration, especially when particles with small mobility dif­
ferences are to be separated. A carprehensive model of the actual 
three dimensional flow, temperature and electric fields is being 
developed to provide guidance in the design of electrophoresis 
chambers for specific tasks and means of interpreting test data on 
a given chamber. 
Part of theprocess of model development includes experimental 
and theoretical studies of hydrodynamic stability. This is neces­
sary to understand the origin of mixing flows observed with wide­
gap gravitational effects, the suppression of gravity may allow 
other processes to becm inortant. 
To insure that the model accurately reflects the flow field and 
particle motion requires extensive experimental work. Much of the 
experimental work can be done under terrestrial conditions if the 
roles of gravity are appreciated and taken into account properly. 
Even though the resolution of a terrestrial based machine nay be 
unsatisfactory, verification of the nodel will pro=vide thesupport
'EEIROD UIGE By 
- NATIONAL TECHNICAL 
INFORMATION SERVICE 
S&ODEPARtMENT"OF-COMMERCEVA 22161SPRINGFIFLfl. 
necessary for the interpretation of micro-gravity operations. 
Pecaendations will be made for the design and operation of the 
ground experiments. 
The research undertaken at Princeton is theoretical and ccMputa­
tional. Supporting work at MSFC under the direction of Dr. R.S. 
Synder aims at establishing the experimntal applicability of the 
theoretical work. 
The 	principal accomplishments thus far are: 
(1) 	 A one-dimensional model of the axial and electro-osmotic 
flow fields was constructed and couter programs written 
to implement the model. The nodel accurately reflects the 
effects of temperature on viscosity, buoyancy and electro­
osmotic mobility of the walls. 
(2) 	 Two-dixensional models of the temperature and flow fields 
were developed to evaluate edge effects. 
(3) 	 A two-dimensional model of the electro-osmotic cross-flow 
velocity field was constructed which includes effects of 
taperature on viscosity and the effect of convection on 
the tarnperature field. This model also enables one to 
predict the axial velocity resulting from the effect of 
buoyancy generated in the electro-osmotic flow. 
(4) 	 The axial developrent of the tenperature field in the re­
gion between the electrodes was modelled to disclose a 
significant entrance length, heretofore unexpected. 
ta 
(5) A first generation model of the separation process was
 
constructed to predict particle separation with various
 
chmber configurations.
 
(6) 	 Several modes of hydrodynamic instability were investigated 
theoretically and an extreme sensitivity to small temperature
 
gradients uncovered. 
(7) 	 Active interaction with the experimental program at MSFC 
was maintained. This program is a complementary effort 
designed to tstudy the flow fieid in electrophoresis chambers 
and establish their operating characteristics. It has been
 
confined that the aforementioned instability is the cause
 
of meandering in wide-gap chcabers at low power levels. 
(8) 	 A mathanatical model which employs the so-called Hele-Shaw 
assumptions was developed to predict the temperature and flow
 
fields after the meandering is established. There is good
 
qualitative agreement with experirental results. 
The 	following Publications have appeared: 
(1) 	 "FluidMechanics and Electrophoresis" in Physicochemical #ydro­
dynamics D. B. Spalding (Editor) Guernsey: Advance Publications, 
vol. 2 (1978) pp. 893-912. 
(2) 	 "Fluid Mechanics and Continuous Flow Electrophoresis" (COSPAR) 
Space Research, Proceedings of the 2XIst OSPAR Conference
 
(Innsbruck), vol, XIX, M.J. Rycroft (Editor) OXford: Pergamon 
Press (1979) pp. 583-597.
 
3
 
(3) 	 "Studies of Continuous Flow Electrophoresis" AIAA paper No. 
79-0032, New York: American Institute of Aeronautics and Astro­
nautics (1979) 4 pp. 
(4) 	 !'Flow and Thermal Effects in Cbntinous Flow Electrophoresis 
UAHINASA Workshop on the Fluids Ecperiment System (John Hendricks 
and Barbara Askins, Editors) Huntsville: Univ. of Alabama 
(1979) pp. 136-145 (with P.H. Rhodes and R.S. Snyder). 
Papers in Preparation 
(1) 	 An Overview of Continuous Flow Electrophoresis 
[Accepted for publication in J. Physiochemical Hydrodynamics]. 
(2) 	 Structure of Temperature Velocity Fields in Continuous Flow 
Electrophoresis. 
(3) 	 Development of the Tererature Field in a Fully Developed 
Laminar Flow. 
[Accepted for presentation at the Philadelphia Meeting of 
the 	AIChE]. 
(4) 	 Flow Structure and Temperature Fields in an Electro-osotically 
Driven 	Flow.
 
[To be suhmitted to J. Heat Transfer,, ASME].
 
(5) 	 Hydrodynamic Stability of the Flow in an Electrophoresis Chamber. 
(6) 	 A Mathematical Model of the Fractionation of Cell Populations in 
Continuous Flow Electrophoresis. 
A
 
During the remainder of the contract period we expect to: 
(A) 	 Continue our collaboration withMSFC to: 
(1) Complete the studies of meandering and stability. 
(ii) 	 Test the separations model. 
(iii) Develop a flight experiment and improve the chamber 
design.
 
(B) 	 Develop mathanatical models of the effects of concentration
 
polarization.
 
(C) 	 Investigate the effects of axial temiperature gradients on
 
fraction.
 
(D) 	 Investigate same of the effects of cell concentration, e.g.
 
agglceration in shear and electric fields.
 
(B) 	 Current Work 
(1) 	 Studies of the electro-osnotic cross-flow (Deiber) 
In our last progress report we described efforts to extend the 
results of a previous study of a weak electro-osmotic crossflow so 
as to include the effects of a stronger flow, including that due to 
leaky membranes. Since the last report we have completed an exten­
sive revamping of the caiputational technique so as to include a 
variety of modes of electrical heating. In the earlier studies it 
was assumed for simplicity that the electric field was uniform 
throughout the chamber cross-section. Now, it is possible to make 
computations based on a constant current or, more realistically, the 
conservation of charge. The latter coaputation requires the 
simultaneous solution of the equation for the conservation of charge 
5 
V - GE = 0, (1) 
the energy equation and the equations of motion. This has been 
inplinented and sane of the results are shown next. 
Figures 1, 2, 3 and 4 depict sane results from this study 
calculated using characteristics of the A-I buffer and the wide-gap 
chaber (SPAR). Figures 1 and 2 show the situation where manbrane 
leakage adds to the electro-osnotic flow to provide sharp teperature 
gradients at the right hand nembrane. Figures 3 and 4 depict matters 
when leakage opposes electro-osrmosis and indicates that a controlled 
leakage can diminish the sharp gradients produced by electro-osmotic 
convection. 
Of course the temperature field depicted in Figure 2 will 
produce severe buoyancy effects that can alter the sturcture of the 
axial velocity. To probe this a camputer program was devised to solve 
for the axial velocity due to buoyancy and pressuring forces. Figure 
5 shows contours of equal axial velocities arising from buoyancy pro­
duced by the tenperature field shown in Figure 2. In the absence of 
the asynetric temperature field and the attendant buoyancy the (up­
flow) axial velocity field (due to a pressure gradient) would be 
syirmetric with respect to the median planes (similar to that shown 
on the left of Figure 5). The added buoyancy due to warm fluid 
adjacent to the right hand imbrane causes channeling and enhances 
the upward flow here at the expense of flow elsewhere in the cross 
section.' 
G
 
The effects depicted here are of central importance in the design 
and operation of continuous flow devices in l-g and micro-g gravity 
fields. If the hydrodynamic instabilities due to small tanperature 
gradients can be suppressed by careful design then the next limits 
on l-g operation will be the structure of the fully developed 
temperature and axial velocity fields. Buoyancy effects in upflow 
or downflow must, at the very least, be taken into account in deciding 
on locations for sample stream injection and ranzval. Such effects 
will also alter same residence times and may engender other hydro­
dynamic instabilities due to the sharp gradients. 
Work on this aspect of the problem is complete and a manuscript 
has been prepared for submission to the Journal of Heat Transfer 
(ASME). That manuscript plus an appendix describing the ccoputer 
programs in being assenbled in the form of a project report to NASA. 
0.15 0.21 0.27 0.15 
0.09­
0.03 ' 
-0.09 
-0.15 -0.21 -0.27 -0.15 
Figure 1: Flow patterns (streamlines) in a rectangular chamber due to the combined effects of leakage
through the membranes and electro-osnosis. Note that the figure is not drawn to scale. 
The actual aspect ratio (width/depth) is 10. 
0.15 0.21 0.27 0.15 
0.03 
0.03 
-0.09 
_ 
e 
-0.15 
Figure 1: 
-0.21 -0.27 -0.15 
Flow patterns (streamlines) in a rectangular chamber due to the combined effects of leakage
through the mmbranes and ejectro-osmosis. Note that the figure is not drawn to scale. 
The actual aspect ratio (width/depth) is 10. 
0.10 0.20 0.30 0.41 0.51 0.61 0.72 0.81 0.92 0.0 
Isothem pattern in the chamber flow depicted in Figure 1. Note steep temperatureFigure 2: 
gradients at right. 
-0.01 -0.04 -0.06 -0.09 -0.11 -0.14 -0.15 
-0.04 0.06 0.09 0.11 0.14 
p.o 
-0.04 -0.06 -0.09 -0.11 -0.14 
Figure 3: 
0.01 P.04 0.06 0.09 
Flow patterns (streamlines) in a rectangular 
osmosis and leakage through the end walls. 
P.I 0.14 0.15 
chmber due to the opposing effects of electro­
0.0 0.08 0.17 0.25 0.33 0.42 0.50 0.58 
I
 
0.66 0.92 
Figure 4: Isotherm pattern corresponding to the flow depicted on Figure 3. 
Figure 5: Contours of constant axial velocity (into the paper) due to caobined effects of 
a pressure gradient and buoyancy for the temperature field whown on Figure 2. 
(2) Flow Stability and Structure (Saville) 
Recent experimental studies by P.H. Rhodes at MSFC [NASA 
TM-78178] disclosed, among other things, a notewrthy correspondence 
between the measured tenperature gradients associated with flow 
meandering and the hydrodynamic instability noted in earlier pro­
gress reports. According to the theory, a critical axial tanPerature 
gradient exists above which the flow is unstable. For perfectly 
conducting walls the critical value, expressed in tems of the 
Rayleigh number, is 
4Td 
6 (21 
and for insulating walls the value is 
33I 712' H >> . 
H is the aspect ratio (width/depth). Rhodes found strong evidence 
for the inception of meandering and (perhaps) recirculation at 
measured Rayleigh numbers in the range of 2-6. Furthernore, the 
structure of the flow appears to aqree qualitatively with the theory. 
Because the theory is constrained by several assumptions so as 
to yield analytical results we sought to construct an alternate 
model which would circumvent problems inherent in the linearized 
stability theory. The model developed employs the so-called Hele-
Shaw assumptions, viz. variations in the y-direction (the narrow 
dimension or gap thickness) are averaged so that one deals with a 
13
 
I, /I 
LLI
 
(a) 	 (b) 
Temperature Field 	 Flow 
Figure 7: (a) 	Assumed temperature field - maxitmn temperature difference 
between the hot spot and the rest of the chamber is I°C. 
(b) Flow field 	corresponding to the situation studied in Rhodes 
experiments. Note recirculation.
 
ilL 
(a) 
Temperature 
(b) 
Flow 
Figure 8: (a) 
(b) 
Assumed tenperature field - to 
Flow field showing two eddies. 
2G 
hot spots. 
two-dimensional problem. The equation governing the tc-dimensional 
stream function turns out to be
 
V2 = - Li 9/9Z (3) 
where N1 is a dimensional group involving, among other things, 
buoyancy and e is the temperature. In the computation a known 
temperature field is given as input and the equation solved for 
the flow pattern. Figure 6 shows the streamlines for downflow at 
a low heating rate where N1 = 5 with the square region 0.5 0 C hotter 
than the rest of the chamber. There is very little distortion. 
Figure 7 shows the state-of-affairs in a chamber with dimensions, 
flows etc. conforming to the SPAR chamber operating conditions used 
in Rhodes' experinents. The left hand side depicts the temperature 
field and shows a 'hot sport' with a maximum temperature approximately 
lC hotter than the surroundings. The streamlines on the right depict 
a recirculating eddy due to the effect of buoyancy in the warm region. 
Figure 8 shows the flow pattern which ensues when two hot spots are 
present, eahc with a maximum taperature 11C higher than the surround­
ings.
 
Another interesting aspect of the Hele-Shaw approach is that 
by adding an equation for the conservation of energy we can examine 
the stability of the chamber. Such a computation involves solving 
an eigenvalue probla and, for the case of an insulating lateral
 
boundary, the critical Rayleigh number is 
3-2 
42
 
-t y 
which is in exact agreement with the more ccnplicated theory for 
insulating boundaries. It is not possible to compare the Hele-Shaw 
model to the exact calculation for perfectly conducting boundaries 
since the require~ent of isothermal front and rear faces is incom­
patible iwth the Hele-Shaw assumptions. 
We believe the nodel has considerable utility both as a tool 
for design and for understanding the behavior of the chamber and 
therefore plan to pursue its development in collaboration with the 
experiments at MSFC. 
-10
 
(3) Inplenentation of the Separation Yodel (Duranceau) 
Our efforts to refine and verify the mathematical model have 
continued with xdbent work devoted to comparing separations achieved 
in the Beckman and Hannig CPE's by J.K. DcGuire of MSFC with results 
predicted using the model. At the same time we are preparing a 
documented 'users guide' so that the software can be turned over to 
Dr. R.S. Snyder's group at NSFC for their use. The results from the 
camputer program can be presented in several forms, e.g. 
(a) the mobility distribution in each collector strem 
(b) particle distribution at the outlet plane 
(c) outlines of regions containing preselected mobilities. 
Sane examples are given in Figures 9 and 10, which show predicted 
separation in the Beckman CPE. Figure 9 shows the number densities 
-(concentrations) of particles leaving the device as a function of 
position. Here the input stream (injected at Z - - .55) contained 
two cell populations with well separated average mobilities (cow: 
= 2.0 pm/s/v/cm and turkey, I = 2.9 vnVs/v/crn) and the separation 
is good. Figure 10 shows the outlines of the average mobilities of 
the two populations-again showing good resolution. 
19
 
t;mCWHAH C PG PEoior(CMt (Q4'uruuS)l
 
0.75 -0.50 -0.25 0.00 0.25 0.50 0.75
 
Z. 
Figure 9
 
20
 
'BCMAI4 ClPf " (Thibuppee 
F.amm &d43C&V0?I 
-0.70 -6.5 -0.40 -0.25 -0.10 05 0 0.35 
z 
"W1,141a,. McIBd4 1 2., 
Figure 10 
(4) Project Review 
On October 18, 1979 a project review was held at MSFC where 
progress, current/status and future plans were presented and 
reviewed by NASA management including Dr. John Carruthers. 
(5) The Fluids Fperiment System Workshop 
A paper on continuous flow electrophoresis coauthored with 
P.H. Rhodes and R.S. Snyder of MSFC was presented at the Workshop at 
the Universtiy of Alabama in July. 
II. 	 Plans for the next Reporting Period 
During the next reporting period we plan to focus efforts on 
(1) 	 preparation of the aforenantioned reports, 
(2) 	 collaborate with MSFC on studies of hydro­
dynamic stability; 
(3) 	 continuous testing of the separation model. 
23
 


Mathematical models of continuous flow electrophoresis

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