ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Dye Displacement Technique for Velocity Distribution Measurements J. K. FERRELL, FRANCES M. RICHARDSON, AND K. 0.BEATTY, JR. Department of Chemical Engineering, North Carolina Stofe College, Raleigh, N. C . HEN a fluid flows past a solid surface, velocity gradients are
'
set u p in the fluid by the drag of the surface that result in a variation of point velocity with distance from the solid surface. An accurate knowledge of the relationship between this velocity distribution and the conditions affecting it is of well established importance in both practice and theory, and the measurement of the velocity distribution profile in a flowing fluid has been the object of study for many years. For the case of a fluid flowing inside a circular duct, the velocity distribution in the central portion of the stream has been well established. Recently, much effort has been attached t o the wall-adjacent region, since the velocity distribution here controls friction losses and transfer coefficients. Experimental measurements suffer because of the limitations of the available techniques. The most common procedures involve the use of sensitive elements inserted directly into the fluid stream and such procedures all exhibit to some degree several difficulties. The presence of the sensing element disturbs the stream a t the very point at which a measurement is being attempted; measurements cannot be made in very close proximity to a solid wall because of the finite size of the element; and the precise positioning of the sensing element in the stream, especially in the vicinity of a solid wall, is difficult. Much careful design and ingenuity have been expended to reduce these limitations. Very tiny hot wire anemometers and Pitot tubes have been ured with good result and the best of these devices can be made to give data of high probable accuracy in all regions except those within a few hundredths of an inch of a solid surface. Measurements made recently by Seneca1 and Rothfus (3), using a very small impact tube, provide data to within 0.025 inch of the wall. The dye-displacement method described in this paper is free from all these difficulties and is capable of measurements extremely close to the wall. All measuring or sensing elements are external to the tube and exact positioning of the sensing clement is not a n important factor. The experimental results presented in this paper are for the laminar flow of a fluid in a circular tube. The experimental method is not, however, limited t o full laminar flow but may be used for measurements within the laminar sublayer of a fluid in turbulent flow. Preliminary work indicates that the method can be used for measuring molecular diffusion coefficients when the fluid is in laminar flow and eddy diffusion coefficients in a fluid in turbulent flow. The displacement technique may be used with tracers other than dyes, and work is now under way in the authors' laboratories using radioactive materials. The use of a radioactive tracer allows measurements to be made in opaque tubes and the use of both radioactive and dye tracers provides a wide selection of tracer materials. Variation of Dye Thickness with Time Establishes Velocity Profile
The principle of the technique is illustrated in Figure 1. A vertical tube is filled with a dye solution to a level indicated by January 1955
dashed line 0-0. Above this level, the tube and storage system are filled with dye-free solvent. A spectrophotometer is mounted around the tube so that its light beam passes through the tube a t
B--B. After the tube has been filled as indicated, flow is started. The effect of the flow is to move the liquid junction, originally a t 0-0, downstream. Because of the variation of velocity across the tube the liquid junction is continuously stretched and distorted. To an observer located a t the position B-B, this appears as a displacement (from the center of the tube outward toward the wall) of the fluid already in the tube a t this position by the fluid entering from above. When a dye is used, the rate of such displacement may be followed with the spectrophotometer by measuring the change in the absorption of a narrow beam of light passing across a diameter of the lower section of the tube. This rate of displacement may be used directly t o calculate the velocity profile.
0--
e-t ( to
Figure 1.
t >to
t >>to
Streamline Flow Patterns Showing Positions of Wall-Adjacent Dye Layer
Before flow is started, the beam of light must pass through a layer of dye essentially equal in thickness to the diameter, d, of the tube. With true laminar flow (and ignoring any radial molecular diffusion of the dye), the dye will be gradually displaced from the center of the tube outward, thus leaving a ring or annulus of dye. If the instantaneous thickness of this annulus measured along a diameter of the tube is y, the light beam must pass through a layer of dye equal in thickness to 2y and through a layer of clear solvent equal to d - 2y. The relationship between light absorption and time is thus a measure of the relationship between dye thickness y and time. If the incident beam of light is located a t a distance x below the original dye-solvent junction and the velocity of the fluid in the tube a t any position y from the tube wall is v, for symmetrical flow in a tube of circular cross section, v is a function of y alone. It is apparent from the sketches in Figure 1 t h a t if the thickness
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT of the dye layer is observed to be y a t time t, measured from the instant of the start of flow, the mean velocity, V , a t this thickness over the period represented by t is given by
For conditions such that the accelerat,ion period from the start of flow to steady velocity is small compared t,o the total time, I, the mean velocity is equal to the velocity at, time t or
(la)
It is only necessary to measure y as a function of t in order to establish the relationship between v and y. According to Beer's law, the extent of absorption of a beam of light by a solution of a dye is an exponential function of the number of moleculee of dye in the pat,h of the beam. Spectrophotometric observat,ions give measurements of the constant,ly decreasing layer of dye solution which is imniediately adjacent to the tube, as depicted in Figure 1. When the incident light beam is confined to a narrow slit of moderate length, the light passes through two dye layers of essentially uniform thickness which may be regarded as rectangular liquid layers between two flat plates. For solutions in which Beer's law is valid, the transmittance of a light beam through two such dye layers surrounding a core of clear solvent is equivalent, to t,hat through the same total volume of liquid containing the same number of dye molecules dispersed across the whole diameter of the t.ube. Consequent,ly, readings of t,ransmittancy through a column filled with stationary dye solution of known concentration can be relatedin the following manner to readings obtained under flow conditions. Let Corepresent the concentration of dye per unit volume when the tube is full. Xft,er f l o ~begins and the wall-adjacent dye layer has reached thickness y, the light transmittance, T , of the light beam vi11 be an exponent,ial function of the product of the length of path, 2y, through the dye multiplied by the concentrat,ion, Co. If Beer's law holds, the transmit,tance will be the same as that for any other equal path-concentration product,. I n particular, the transmittance will be the same as that for a path length d and concentration C given by the equation Cd = 2yCo
(2)
or
If the tube is filled successively with a series of dye solutions of different concentrations and the transmittance is measured for each one, a curve of T versus C is obtained. Ideally this curve is a straight line on semilogarithmic paper and obeys the equation
T , = Toe+c
(3)
where T , is the transmittance reading, b is a constant including the absorption coefficient and geometry factors, and To is a constant numerically equal to the transmittance reading when C is zero. From Equation 3, In
T To
r-
=
T, To
2.303 log -
=
-bC
(4) (5)
Combining Equations 4 and 2,
L ! = -2.303 log T d
2bCo
where T = T,/To is the relative transmittance from which the dye layer thickness a t any time may be calculated from the value
30
of the relative transmittance, the known initial concentration of the dye solution, and the value of b from Equation 5. Dupont Pontamine Bluo 6BX and potassium permanganate were selected for test runs after eolutions of a number of dyes were carefully checked for conformity to Beer's law. Since the maximum dye concentration used for the runs reported was only 0.2 gram per liter, these solutions had essentially the same density as the pure solvent. Viscosities for these eolutions were determined experimentally and found to be the aame as the pure solvent. Changing Dye Concentrations Are Measured by Spectrophotometer
The velocity distribution measurements reported in this paper TTere made in a ljja-inch (inside diameter) vertical glass tube with gravity flow. Feed to the observation section of the column is from a st'orage tank and constant head tank located at the top of the column approximately 30 feet above ground level. Liquid from the storage tank flows by gravity to the feed tank, vihere a constant head is maintained by a n overflow pipe. A sniall pump returns the overflow t,o the storage tank and is also used for mixing solutions prior tBoa run. Liquid flows from t,he bottom of the feed tank through a 6-foot length of I/*-inch brass pipe and then into the uppermost section of the three sections of glass tubing that' make up the central part of the column. The central portion of the column consists of three sections of 1/2-inch (inside diameter) gage glase tubing joined together through devices for forming t8he junctions between the dye solutions and the clear solvent. This central port,ion is connected to t,he brass pipe from the feed t,anlr and to the brass pipe leading to the drain by glass-to-inetal adapters. Although the inside diameter of the glass tubing and brass pipe are not precisely matched, the upper and lower lengths of glass tubing act as calming sections. A schematic flow diagram is shown in Figufe 2. A view of the central portion of t'he glass column is shown in Figure 3. The changing dye concentrat.ions are measured by a spectrophotometer specially modified so t h a t it is mounted around the lower glass section. A liquid junction-to-spectrophotometer distance, z,of 2.19 or 5.67 feet can be obtained by using t,he lower or upper liquid junction device, respectively. The velocity for each run was controlled by a fixed orifice located at. the foot of the column. Five orifices were used to obtain the series of flow rates reported. The flow rate was spot checked during each run hy direct collect'ion of fluid. The success of the experimental method depends on forming, prior to the initiation of floiv, a horizontal plane interface betvieen the solvent' and the dye solution. Preliminary test3 shoived that a satisfactory interface could be formed by separately filling tx-0 sections of tubing and slowly sliding t,hem together. The interface-forming devices were construct,ed on this principle of operation. The only complicating features of the sliding block arrangement were those necessary for separate filling of the various sect,ions of the column and for holding the blocks in proper alignment. The blocks are made from a sheet of 1-inch-thick Plexiglas machined flat to give a good bearing surface. After machining, the bearing surfaces were lapped together and the outside surfaces were polished. The glass tubing of the column is joined flush to the outer surface of the top and bottom blocks. The holes in the blocks werc machined to 0.500 k 0.001 inch and great care mas taken to align properly the holes, both in the filling position (position A of Figure 2 ) and in the flow poEition (position B of Figure 2). The top and bottom block are held in permanent alignment by 3/a-inch pins through the four corners of the blocks. The sliding center block is held in alignment by guide rollers and in the flow position by a stop. These alignments wcre made by driving a carefully machined steel pin through the holes. To fasten the sections of glass tubing to the blocks a Plexiglas flange was shrunk onto the end of the tube with its bearing face flush with the end of the tube. Both tube and flange were then ground square on the end. The glass tube is aligned wit8h the hole in the block by means of hhe steel pin described above. Since the inside diameter of the glass is not precisely '/z inch, there is a possibility of a small disturbance at the wall of the tube.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. I
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT pressure when the interface is formed and will produce a distorted interface. To prevent this, the pressure must be equalized on both sides of the interface before it is formed. This is done by returning the plug cock to the flow position, closing valve 2, and opening valve 1, after the line from the constant head tank to valve 1 has been previously filled. The sliding block of the interface device is then carefully moved into the flow position. Flow is started by opening the solenoid valve 8 immediately after the interface is formed. A signal from the solenoid valve sn-itch fed through the amplifier to the current recorder gives a small pen deflection which marks the time of the start of flow. The current recorder gives a continuous record of per cent transmittance of the light beam passing through the column a t the location of the spectrophotometer. Two interface forming devices were installed on the column, one a t a distance of 2.19 feet, and one a t a distance of 5.67 feet above the spectrophotometer; however, all of the results reported in this paper were taken using the lower interface device.
S‘CRAGE
k
Wall-Adjacent Velocity Measurements Are Limited by Molecular Diffusion of Dye
Data obtained during a run consist of a continuous strip-chart record of the amplified output of the spectrophotometer phototube. To convert this record to velocity distribution, points are
A Y5
, I
TO
T
DRAIN
7
1 6
Figwre 2.
TO
DRAIN
Flow System
This irregularity would not exceed 0.004 inch where the glass joins the plastic block. The center block is moved into position by means of a feed screw device operated by a hand wheel. By using a dye solution as a visual indicator, a large number of interfaces were made by operating these devices. It was found that reasonable care ensures the formation of sharp interfaces between the two parts of the liquid. In order t o obtain a continuous record of the light transmittance, spectrophotometer-photocell output is sent to a current amplifier and the output of this recorded on a commercial strip chart potentiometer. Details of this instrumentation will be described in another paper. Procedure. Initially the entire system is filled with clear, dye-free solvent. The instruments are allowed to warm up until a continuous record of current versus time is constant over a period of several minutes. For the spectrophotometric analysis, the dye-free solvent is taken as the reference solution. At the end of the warm-up period the current is set a t lOOyotransmission by adjustment of the output of the amplifier. The interface device is moved to the filling position (position A of Figure 2 ) and plug cock 4 is opened to the side outlet. Valve 1 is closed, valves 2 and 5 are opened, allowing the liquid to drain from the section between the interface blocks and valve 4. This section of the column is filled a i t h dye solution by forcing the solution up through the plug cock until it runs out a t valve 2 . The section is rinsed several times with dye solution of the concentration to be used and finally filled t o valve 2 . Small amounts of air which are sometimes trapped in the lower parts of the column will be compressed because of a n increase in January 1955
Figure 3. Central Section of Column with Liquid Junction Forming Devices and Spectrophotometer taken from the recorder chart and from these the corresponding values of ?J and v are calculated. The “data points” shown in Figures 4,5, 6, and 7 , correspond to those selected from the chart. For a given run any number of points may be selected, but it is only necessary to plot asufficient numbertodefine the shape of the experimental curves. The experimental technique is particularly useful in measuring velocities very near the wall of the tube. The extremely rapid displacement in the center of the tube reduces the accuracy of velocity measurements in this region. The range of data ob-
INDUSTRIAL AND ENGINEERING CHEMISTRY
31
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
y
Equation 7 is n straight line with a slope of unity passing through the point 1,l. For convenience the abscissa scale has been marked t o read values of y d i r e d y . The normalizing factors in the scaies on Figure 6 cause both sets of data to lie on a single straight line for values of y greater than 0.007 inch. This line coincides with the theoretical equation. For smaller values of y the data deviate from this straight line because of the error associated with molecular diffusion. This deviation is greater for the lower flow rate because of the longer time required to reach a given value of y. To evaluate the limitations impoped on the method by molec. ular diffusion of the dye, runs were made using dyes of different diffusion constants. The four runs shown in Figure 7 are for potassium permanganate and water, Pontamine Blue and water, Pontamine Blue and a 67% glycerol-miter solution, and Pontamine Blue and a 7670 glycerol-water solution. For the range of concentrations used, the diffusion constant for potassium permanganate in water is approximately 17.4 X 10-6 square em. per second ( 1 ) . For the Pontamine Blue dye the diffusion constant in water is approximat,ely 5.0 X eqmre ern. per second ( 2 ) . The Stokes-Einstein equation for diffusion constants in liquids states that the diffusion constant is ini,ersely proportional t o the viscosity of the solvent. Baaed on t h k relation the diffusion constant of Pontamine Blue dye in the 67y0 glyccrollvater solut,ions (viscosity 11.53 cp.) is 0.34 X 1 0 F square cni. per second, and in the 'i6% solution (viscosity 24.15 cp.) is 0.16 X square cm. per second. The curves of Figure 7 show graphically the effect of the variation of diffusion constant. The experimental data for the solu-
in i n c h e s
Figure 4. Velocity Profiles in %-Inch Glass Tube 024
020
ci I C 0'
r
E' 0 12 L >
008
0 04
000 GOO
001
032
033 Y
8N
004
005
306
007
002
INCHES
Figure 5. Velocity Distribution n e a r Wall of %-Inch Glass Tube tained for five different flow rates using water and Pontamine Blue dye is shown in Figure 4. These same data in the walladjacent region are shown on an expanded scale in Figure 5. The solid curves are plots of the parabolic velocity distribution for each of the measured flow rates. This equation in terms of y and v is
v = 4t10[2//d- ( y / d ) ' ]
35
07
1
Y
2 IN
3 4 5 6 B 10 THOUSANCThS OF
20
AN
30 40
60
CO
250
ZNCd
Figure 6 . Effect of Flaw Rate on Measured Velocity Disfribution
(7)
where vo is the velocity a t the axis of the tube. In the derivation of Equation 6, molecular diffusion of the dye was ignored. However, diffusion is only negligible for reasonably short values of the time, after which the error increases as the time increases and the value of y decreases. A qualitative picture of the effect of molecular diffusing may be formed as follows. Ket diffusion of the dye molecules is from the wall toward the center of the tube, Since the velocity of flow is greater in the center, molecules diffusing in this direction are carried dormstream more rapidly. Consequently, the number of dye molecules in the path of light beam a t any given time is less than the number in the nondiffusing annular ring assumed in Equation 6. The use of this equation with observed transmission gives calculated values of y smaller than the true values. The data are compared with parabolic distribution in Figures 6 and 7, in which v/vo has been plotted against 4[y/d - ( ~ / d ) ~ ] using logarithmic coordinates. By use of these coordinates,
32
05
06 0 4
-0 PCNTAWNE BLUE IN 02
0
G7 %
GLYCEROL SOLUTION, \!0=033 PONTAFriiNi BLUE IN 76 % GLYCEROL SOLUTION, V e O 3 4
01
38
p
,
06 C4
Y
Figure 7.
IN
THOUSANDTHS
OF
Ah
IhCH
Effect of Diffusion Constant of Dye on Measured Velocity Distribution
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 1
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT tion in which the diffusion constant is 0.16 X 10” square cm. per second show excellent agreement with parabolic velocity distribution down to values of y of 0.002 inch. In addition to the deviation caused by diffusion, it has been suggested that the observed deviations may be inherent in the flow system or in the instrumentation; however, changes in both of these have had no effect on the data. This observation, together with the data shown in Figure 7, has led to the conclusion that molecular diffusion is the only significant factor.
greatly expedited the work, and the assistance is gratefully acknowledged. The authors also wish to acknowledge the support of the Beckman Instrument Co., which made a spectrophotometer available in the later stages of the work. Nomenclature
b
= constant in Beer’s law
C = concentration of dye Co = initial concentration of dye d
Conclusion
The dye displacement technique has been successfully applied to the measurement of the velocity distribution of a fluid in laminar flow in a smooth circular tube. Velocity measurements have been made by this technique to within 0.002 inch of the wall of a ljAnch-diameter tube. These measurements are in a,greement with the parabolic velocity distribution predicted by the theory of laminar flow. In this technique, wall-adjacent velocity measurements are limited by molecular diffusion of the dye. Measurements extremely close to the tube wall may be made if molecular diffusion is sufficiently reduced. Acknowledgmen!
The experimental work reported in this paper is closely related to a project being carried out by Wright Air Development Center a t Iiorth Carolina State College. Cooperation with this project
= = = T =
t to
T, To
= =
= uo = z = y =
v
diameter of tube time time for a point moving with velocity Z J to ~ travel distance z relative transmittance transmittance a t concentration C transmittance a t C = 0 velocity of any point in tube velocity at center of tube distance along axis of tube radial distance measured from wall of tube
References
(1) Furth, R., and Ullman, E., ,RoZZoid-Z., 41,307, (1927). ( 2 ) Robinson,C., and Hartley. G. S.,Proc. Roy. Soc. 134 A, 20 (1931). (3) Seneca],V. E., and Rothfus, R. R., Chem. Eng. P r o g ~ . ,49, 533, (1953). RIPCEIVED for review hIsy 26, 195-1. ACCEPTEDAugust 20, 1954. Presented before the Division of Industrial and Engineering Chemistry at the 125th kIeeting of the h 1 E R I C A N CHEMICAL SOCIETY, Kansas City, hl0.
Direct Construetion Nomonranhs V
.
W. H. BURROWS Engineering Experiment Station, Georgia lnstifufe o f Technology, Atlanfa, Ga.
A
LTHOUGH a formula may be available for use as a defining
equation for the construction of a nomograph, it happens occasionally that the defining equation is not readily manipulated to produce the form of nomograph desired. It is then desirable to prepare from the formula a table of values which may in turn be used for the construction of the nomograph. Some freedom of choice may be exercised in the selection of one of the scales, or in the spacing of the scales. The consequences of making a particular selection are best illustrated in symmetrical nomographq of which the logarithmic mean temperature difference nomograph is a typical example. I n the case of unsymmetrical tables, the selection of a suitable scaIe function for the dependent variable is critical, since it vastly affects the ultimate shape and arrangement of the scales of the nomograph. Consequently, it is necessary to have not only a general method for the construction of nomographs from data tables, but also a means of selecting the critical scale function. logarithmic Mean Temperature Difference I s Example of Development for Symmetrical Case
An excellent example of a formula which yields an undesirable form of defining equation and for which it is desirahle to construct a nomograph by other means is that for the logarithmic mean temperature difference January 1955
M.T.D. =
AT2
- Ti
In ATg - h
where AT1 and AT2 are the temperature changes experienced by two fluids as they pass simultaneously through a heat exchanger, and M.T.D. is the logarithmic mean temperature difference. This formula takes the determinant form ( 1 )
The nomograph constructed on the basis of this defining equation is shown in Figure 1. A modified form of this nomograph is given by Helves and Seward ( 4 ) . The ATl and A T z scales are identical, making readings of M.T.D. difficult and uncertain for cases in which AT1 and AT, have nearly the same value. Aside from this, however, the unsymmetrical form of this nomograph makes adjustment of scale moduli and positions especially unwieldy Consequently, the construction of an accurate nomograph by the use of this defining equation is very difficult. A much more desirable form would be one in which the AT, and AT2 scales were separate, with the M.T.D. scale between them. However, conversion of Equation 1 into a defining equation for such a form involves determining the nature of defining scale functions, such as f1 in Equation 2.
INDUSTRIAL AND ENGINEERING CHEMISTRY
33
