VISCOUS HEATING IN CAPILLARIES The Adiabatic Case J.
E. G E R R A R D , F. E. S T E I D L E R , A N D J . K. A P P E L D O O R N
Products Research Dmston, Esso Research and Engmeerzng Co.. Linden, >V. J .
In measuring viscosity or viscoelastic behavior with capillary viscometers, considerable heating can occur at
high shear rates.
An experimental program has shown that a temperature rise of as much as
350" F.
can
occur at the capillary wall when a 10-poise oil is forced through an insulated capillary at 2000 p.s.i. This heating can reduce the apparent viscosity by 7070, which completely vitiates viscosity measurements. A mathematical (computer) solution of the flow and temperature pattern in the capillary gives very close agreement with the experimental data over a surprisingly large range of conditions. Pressure drop, bulk temperature rise, and the axial and radial temperatures in the capillary are all computed to within 5% of the experimental values.
The data reported are all for an insulated capillary (adiabatic case); the case of the isothermal wall is to be the subject of a subsequent paper. However, experimental data show that the adiabatic case is far more prevalent than heretofore believed.
VISCOUS heating is always a possible source of error in viscosity and viscoelastic measurements at high shear stress, particularly with rotational viscometers where the entire sample is at the same shear stress and is sheared continuously throughout the entire measurement. Capillary viscometers are somewhat better. The sample is heated only for the length of time it takes to pass through the capillary, and only the liquid at the wall is subjected to the maximum heating effect. However. even with properly selected capillaries and good temperature control, viscous heating eventually will occur if the shear stress is high enough. Many attempts have been made to calculate the temperature rise in a capillary by mathematical techniques. The average temperature rise of the fluid can be very misleading. For adiabatic incompressible flow, the average AT is a function of the driving pressure only and is independent of the viscosity of the fluid or the dimensions of the capillary:
For a typical hydrocarbon ATb is about 7' F. for each 1000 p.s.i. However, heating is not uniform in the capillary. The total power input to each incremental element of fluid is: 9 =
T
. i. =
72/1.1
Inasmuch as the shear stress, T , is a function of the radial position, the heating is nonuniform across the radius. This results in a very complicated situation where the viscous drag causes a temperature rise, which causes a change in the viscosity, which in turn causes a change in the heating rate. The mathematics become so complicated that probably no analytical solution is possible. Several investigators have attempted mathematical analyses of this problem (2-4, 7-77). HoLvever, in most cases certain simplifying assumptions had to be made in order to reduce the problem to a mathematically tractable form. These included assuming that the viscosity does not change u i t h temperature or pressure ; assuming a parabolic flow profile ; assuming telescopic flow (flow parallel to capillary axis) ; assuming a constant pressure gradient down the capillary; or assuming a condition where the heat generation from viscous flow is 332
l&EC FUNDAMENTALS
balanced by the heat dissipation to the surroundings so that thermal equilibriuin is achieved. None of these assumptions is experimentally acceptable for typical viscosity measurements where heating is a problem. Large changes in viscosity \vi11 occur as a result of the temperature and pressure clianges; the flow profile will not be parabolic or telescopic; the pressure gradient will not be uniform; and the assumption of thermal equilibrium applies only at shear stresses so loiv that heating is negligible. Only Gee and Lyon ( 3 ) treated the case rigorously from an experimental viewpoint. Theirs was a computer solution and was concerned with the flow of polymer melts; they did nor include a correction term for the change of viscosity with pressure, Lvhich is important with most liquids. .Also they treated only the case of the isothermal Lvall and not the adiabatic case (insulated kvall). Experimentally very little \vork has been done to confirm the results of the mathematical analyses. Our own work on viscoelastic measurements (5) had shown that a considerable temperature rise \vould occur and that the heating was more adiabatic than isothermal. I t therefore seemed worthwhile to measure the temperature rise experimentally under a wide range of conditions, and at the same time devise a numerical solution of capillary heating using a digital computer. This has been done for both the adiabatic case and the case of the isothermal tvall. In this paper only the adiabatic case is considered. The isothermal wall will be covered in a subsequent paper. Experimental worK has shown? however, that a truly isothermal wall is rarely achieved and that in many cases the heating is much closer to the adiabatic case than heretofore believed. Apparatus
Oil was forced through the capillary by nitrogen pressure from a high-temperature bomb through a quick-opening valve. Temperature equilibrium was reached in about 30 seconds. The capillaries used in this experiment were made from stainless steel hypodermic needles. The tips were ground to a knife-edge to prevent any surface tension effects at this point. 'The capillaries have a negligible heat sink because'of their very thin walls and the temperature difference between the inside and outside bvall of the capillaries was negligible. Five capillaries were used with lengths varying from 1.4 to
19.5 c m . ; four of them had the same radius, 0.0425 cm. T h e dimensions are listed in Table I . Temperatures along the length of the capillaries were measured by thermocouples of 0.003-inch iron and constantan Lvire spot-ivelded to the outside wall a t various points. Temperatures across the radius of the capillary Lvere made using a thermocouple consisting of a n iron and a constantan wire butt-\velded to form a i.hermocouple junction of very small dimensions. This was mounted in a microscope feed so that it transversed the diameter of the stream as it emerged from the capillary. T h e temperature at the inlet of the capillary was meastired on a thermocouple located several centimeters upstream from the entrance to the capillary. As an independent check, Templestiks were used to measure the bra11 temperature a t the exit. .4greernent between the Templestiks and thermocouple measurements was within 2' F. T h e bulk or average temperature rise of the stream was measured by collecting a quantity of the effluent in a Dewar flask, stirring thoroughly. and reading the temperature lvith a calibrated thermometer. This temperature was the most difficult to obtain accurately and showed the greatest variance from prrdicted values. 'I'he oil used in all experiments \vas highly refined paraffinic bright stock having a viscosity of 10.25 poises at 77' F. Like most lic1"ids of this viscosity, it has a n appreciable temperature and pressure dependence of viscosity. T h e viscosity a t elevated temperatures and pressures can be closely calculated from the equation derived by Appeldoorn ( 7 ) .
rvhere '1' is in degrees Fahrenheit and P in pounds per square inch. 'l'he experimental oil had the following constants: p,> = 10.25 poises. a = -3.55, b = 8.2 X 10-5 p.s.i.-'> and c = -8.0 X 10-5 p.s.i.-' €or the computer program, the specific heat, C,, \vas assumed constant at 0.47 cal.,'gram C . ; the density. p , 0.8874 gram per cc.; and the thermal conductivity, cal. sec. . q.cm. . C./cm. k , 3.5 X
T h e boundary conditions applied to this equation are as follows: by virtue of symmetry at the capillary center
37- ( O j Z ) br
=o
At the wall of the capillary
if the wall is insulated. or
T(&)
=
(1b '1
f(2)
if the wall temperature is fixed. T h e third boundary condition must relate to the temperature distribution at the inlet of the capillary. Obviously, this distribution will vary M ith the history of the fluid before it is introduced in the capillary. In this development it is assumed the radial temperature gradient is zero at the inlet. Consequently \\hen z = 0,
T o develop the momentum balance it is assumed that:
7. Pressure variations in the radial direction are small compared to axial variations. Hence, for momentum considerations, forces in the radial direction can be neglected. (This does not mean, however, that radial flow is neglected. There is, in fact, an appreciable outward flow due to heating.) 8. Kinetic energy changes in the fluid are small compared to the energy of viscous dissipation. Under these assumptions a momentum balance in the axial direction reduces to the form
I
Mathematica I Development
l'he equations \\hich describe the flow of a fluid under nonisothermal conditions are statements of the conservation of mass. energy. and momentum. T h e exact form taken by each of these equations depends upon both the geometry of the flo\v and the simplifying assumptions which are made to facilitate solution. In this study it is assumed: 1.
1he flow is through a circular capillary and hence there
is symmetry about the capillary axis. 2. T h e fluid does not slip a t the wall. 3. Heat transfer occurs by both conduction and convection. Axial conduction, however, is assumed to be small when compared Lrith convective effects. 4. T h e fluid is Newtonian-that is! its viscosity is not a function of shear stress but only of temperature and pressure. 5. Density, thermal conductivity, and specific heat are independent of temperature and pressure. 6. T h r flow is laminar and in the steady state. Under these assump1,ions the energy equation in polar coordinates reduces to the form
pC,[Vb -bZ+ TVr-
C a p /lory S O
1 2 3 4
5
3br7 1 - k
[rr-
- + - -r1 3br7 1 = -J1 @
Table I. Capillary Dimensions Lenqth, Cm . Radius, Cm. 19 5 0 0425 10 4 0 0428 5 0 0 0425 1 4 0 0421 5 3 0 0208
(1)
T h e density of the fluid is assumed to be constant and the flow is symmetrical with respect to the capillary axis. Therefore, flow continuity at any point in the capillary can be expressed in cylindrical coordinates b y :
WVT)
+
dr
bJ bz
=
(3)
T h e assumption of laminar flow allows the use of Newton's viscosity law : TTZ
=
V, br
(4)
-/. l
Boundary conditions for Equations 2 to 4 follow from assumptions 1 and 2. Assumption 1 requires the axial velocity to pass through a maximum at r = C. 'Therefore, by virtue of Equation 4, T~~ is zero \Then r is zero. Assumption 2 required that V , be zero at the \\.all \\.here I;, is also zero. Thus, for Equations 2 to 4 T~,(O,Z)
=
VZ(R,z)= V,(R,z)
=
0
In addition to these boundary requirements, the mass balance expressed by Equation 3 requires the radial velocity to vanish at the capillary center, and its derivative Lvith respect to r to vanish at the wall-Le., LIR 459 242 118 33 260
Solution Technique
Equations 1 to 4 are solved simultaneously using numerical techniques on a digital computer. I t is assumed that a, the VOL. 4
NO. 3
AUGUST
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333
energy dissipation function for capillaries, can be closely approximated by the expression =
(5)
77r2//1
which has the units energy per unit volume per second. This expression can be written in terms of d P / d t by integrating Equation 2 at constant z to find
This result is substituted into Equation 1. The energy balance is then expressed in finite difference form using first-order differences. A nonuniform grid size was used. In this way it was possible to use a fine grid in regions where the temperature and pressures changed rapidly. Details of this procedure and other aspects of the numerical scheme are presented in the appendix. The system of linear algebraic equations which is generated when the enrrgy balance is expressed in finite difference form involves coefficients kvhich are functions of the velocity components. the viscosity, and the pressure gradient. An iterative process was developed to evaluate these coefficients. Initially, it is assumed that the capillary operates isothermally at T = Toand that p = po. Poiseuille’s equation is then used to arrive at first estimates of the pressure gradient and the velocity components. In this fashion first estimates are obtained for the coefficients of the energy balance equation. The energy balance is then solved numerically to find second estimates of the temperatures and viscosities in the capillary. A new approximation of the pressure gradient is obtained by combining Equations 2 and 4 and integrating the result with respect to r . The result is
pressures. The difference between computer and experiment was rarely more than 5% and usually within 2’%, for both flow measurements and temperature measurements (axial and radial). This agreement lends confidence to both the correctness of the computer program and the accuracy of the experimental data. The magnitude of the temperature rise and the resulting effect on apparent viscosity were surprisingly high. Temperatures at the wall as high as 425’ F. were recorded, with a resulting decrease in apparent viscosity of as much as 70%. Obviously, viscosity or viscoelastic measurements under these conditions are meaningless, and they point out the importance of maintaining good temperature control. Even with the best temperature control, however, conditions are rarely isothermal at high shear rates. In fact, the adiabatic case is much more prevalent in experiments than heretofore suspected. This will be covered in a subsequent paper. Temperature Measurements. MAXIMUM TEMPERATURE RISE. I n the adiabatic case considered here, the maximum temperature in the capillary is the exit temperature at the wall. Figure 1 is a plot of this temperature us. the inlet pressure for capillaries 1 to 4, which have the same radius but different lengths. The points are experimental; the lines are computed. The general agreement is excellent, but in the longer capillaries it can be seen that some heat loss at the wall did occur, as evidenced by lower experimental points than compu ted . The longer capillaries show less temperature rise at the wall, largely because the shear stress at the wall (which is a function
36C
(7) 320
This expression is combined with the macro-material balance over the capillary. which requires that 280
PR
Q
=
2s
J
rVzdr
0
From Equations 7 and 8 dP - (2) dz
240
-2
=
s
Q
W‘
(9)
(““7
Jo
k
1
Equation 9 yields a new estimate of the pressure gradient. Equation 7 is then used to find a new value for the axial velocity component. An improved value for the radial component is derived from the micro-material balance. Thus, integrating both sides of Equation 3 with respect to r
Y 200 0:
a
+
d W
160
+ -1 1
2 5 -a
120
X
a
I
80
Equations 7 , ‘9, and 10 together with the viscosity-pressuretemperature relationship form the basis of an iterative cycle \vhich successively approximates the coefficients appearing in the energy balance equation. Convergence is assumed when successive numerical solutions of the energy balance yield the same temperatures at each grid point in the space of the capil1arv .
40
0 INLET PRESSURE, PSI
Results and Discussion
Excellent agreement was obtained between the computer program and the experimental data for all capillaries and at all 334
l&EC
FUNDAMENTALS
Figure 1. Maximum temperature rise vs. inlet pressure for capillaries 1 to 4 Polntr experimental, lines computed
of R,'L) is correspondingly less. .Also the longer length allows a more uniform temperature to be reached inside the capillary because the residence time is longer, giving more opportunity for heat transfer. No compute1 points were obtained for capillary 4 above 1250 p.s.i. because of the excessive time for the computer program to converge. However, good agreement \vas obtained up to this point. The experimental points for capillary 4 begin to fall off above 1000 p.s.i. This is due to the kinetic energy of the liquid as it leaves the capillarv-that is, part of the driving pressure is not expended to viscous heating but remains in the fluid as kinetic energy (equal to V2,'68,950p.s.i.). For capillary 4 this correction is about l0YG at 1000 p.s.i. and 25Yc at 2000 p.s.i. Eventually. a point is reached where any further increase in pressure appears solely as an increase in kinetic energy rather than in temperamre. In the computer program, the kinetic energy correction is calculated separately and expressed as a pressure drop. \Yhen comparing the computer data with experimental data. this pressure drop \vas added to the pressure drop resulting from viscous flow. AXIAL TEMPERATURE: DISTRIBUTION. The temperature increase as the oil flows do\vn the capillary is marked by a rapid initial rise, followed by a more gradual rise as the oil approaches the exit. Figure 2 sho\\.j a typical axial temperature profile at the wall for capillary 3 at three different driving pressures. Again the good agreement benseen experiment and computer can be seen.
I
-
-1 5 0 I
i 2
I
/
I
75f4 50
7 4 0 psi
0.4 0.6 0.8 AXIAL D I S T A N C E , z/L
0.2
I
1.0
Figure 2. Typical profiles of temperature rise along capillary length a t three inlet pressures Capillary 3 Points experimental, lines computed
I
30
-
RADIAL TEMPERATURE DISTRIBUTION. The temperature profile across the diameter of the stream as it leaves the capillary is much mnre difficult to measure experimentally, partly because the size of the thermocouple, although very small, 0.003 inch, is still about 1070 of the diameter of the capillary. ,4s a result. the thermocouple measures the temperature not at a point but over a small Ar. .41so the capillary appears to be somewhat wider> because first contact is made when only the edge of the thermocouple touches the stream. A further apparent uidening of the capillary occurs because of the presence of the thermocouple in the stream; this causes some aberration of the floxv due to surface tension effects. Figure 3 compares the calculated and experimental values for one set of conditions. The apparent widening of the capillary is clearly seen. Although the agreement does not appear to be too good, if the maximum temperature rise observed experimentally is placed at the capillary wall, the agreement is within 5%. The maximum temperature rise observed experimentally is only 2' below that predicted by the computer, and is actually 2' belo\\. the temperature measured by the thermocouple welded to the wall. Figure 3 shows that the radial temperature profile closely resembles the shape of the theoretical adiabatic profile and does not at all resemble the profile for an isothermal wall. TEMPERATURE PATTERS INSIDE CAPILLARY.The computer program calculates the temperatures for each point inside the capillary; a typical temperature "contour map'' is shown in Figure 4. This is for capillary 2 at 1930 p.s.i., with an inlet temperature of 75' F. Over half of the capillary has a temperature rise of less than 5' but this increases rapidly toward the wall, reaching a maximum of 142' F. at the exit. The temperature at the center line is only 75.2' F.: so that the average temperature gradient across the radius a t the exit is 167' per mm. Flow Measurement. T h e computer program also calculates the hydrostatic pressure down the capillary. The pressure drop is not uniform but is, as shown in Figure 5, much steeper near the entrance. Viscosities are then calculated for each point within the capillary using the temperatures given in Figure 4 and the hydrostatic pressure from Figure 5. T h e resulting viscosit)- contours are given in Figure 6. The inlet pressure (1930 p.s.i.) is enough to raise the viscosity from 11.2 to 16.3 poises. Do\vn the center of the capillary, heating is negligible and the change in viscosity is almost entirely due to the change in pressure. ,4t the wall) however, the predominant effect on viscosity is that of temperature, and the outlet viscosity at the wall is 1.14 poises-only 107, that of the centerline viscosity. The over-all effects of the change in viscosity on flo\s rate are given in Figure 7, which is a plot of driving
f
1.0
1 1 2 0 - ~ ~ O
OBSERVED
0.8
1
w 3
W
0.6 U
a
VI
w
1.0
0 RADIAL. DISTANCE, r/R
1.0
Figure 3. Typical profile of temperatures across diameter of liquid stream emerging from capillary Circle a t right shows relative size of thermocouple Capillary 3, 800 p.s.i.
I 1.0
O O
0.2
0.4
0.8
0.6
A X I A L DISTANCE, z/L
Figure 4.
Computed temperature contours in r z plane Capillary 2, 1 9 3 0 p.s.i.
VOL. 4
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AUGUST
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335
20001
I
1
'
'
I
i
'
I
J
Table II. Bulk (Average) Temperature Rise ( 1 0 . 3 5 - c t i ~capillary. 1930 p,s,i,) Bulk 7'enip Rise. ' F.
Calculated from Af' Computer Esperi mt-n tal
e 0
,
1
0
.2
,
1
,
.4 .6 A X I A L DISTANCE, z/L
.8
1.0
Computed pressure drop along showing nonuniform pressure Capillary 2, 1 9 3 0 p d .
1.0
0.8
*
0.2
I
I-
\i 4
16.3
0
1.14
10
0.6
0.4
0
12
I,
Figure 6.
cs
1
0.2
0.4 0.6 A X I A L DISTANCE, z/L
\
0.8
1.0
Computed viscosity contours in rz plane Capillary 2, 1 9 3 0 p.s.i.
2000
1600
-
vr
n
ti
1200
rx VI 3 w
a +
900
2
z
400
0 FLOW R A T E , cc/sec
Figure 7.
11 2
1
~4
Figure 5. capillary, gradients
13.7 14 0
Flow curve for capillary 2
1 = 10.4 cm. R = 0.0428 cm.
'4 significant departure from the ideal case occurs at pressures as lo\\. as 500 p.s.i. At 2000 p.s.i. the flo\\- rate may be four times as high as if no heating had occurred. This discrepancy is so great as to make viscosity measurements or viscoelastic measurements under these conditions entirelv meaningless. l'his is not at all obvious from the bulk temperature rise, Lvhich in all cases is less than 14" F.and \vould cause a relatively minor effect. Table I1 compares the bulk temperature rise for the truly adiabatic case. ivith the computer solution and the experimental data (capillary 2, 1930 p s i , ) . The good check benveen computer and theory (and this is not a mathematical consequence in the computer program) is a further confirmation of the correctness of the computer solution. The lo\\-er value for the experimental A T b indicates that the capillary under these conditions is about 80yc adiabatic. Xfore probably. the experimental procedure allo\ved some cooling of the oil to occur before temperature measurement. and the measured A T o is in error. T h e effect of heating is shown more clearly in Figure 8. \There the apparent viscosity calculated by mcans of the Poiseuille equation is plotted against the shear stress for capillaries 1 to 4. All four capillaries have a radius of 0.0425 cm. I n each case there is an appreciable decrease in viscosit)at the higher shear stresses. T h e advantage of using a capillary Lvith a smaller L R ratio is also clearly she\\-n. For an equivalent viscosity decrease. the 1.4-cm. capillary allo\vs measurements up to five times higher shear stress than the 19,j-cm,capillary does. This is largelv because the maximum temperature rise as a function of shear stress is less for the shorter capillaries-that is. temperature rise is largely a function'of pressure only. so that all capillaries regardless of their length \vi11 have roughly the same heating at a given pressure. but the shear stress is not only proportional to the pressure but also inversely proportional to the length. Therefore. shorter capillaries \vi11 give a higher shear stress for the qame pressure. For the same reason. the radius of the capillary should be as large as possible. limited onlT by the requirement of having an L R ratio large enough to ensure a fully developed flo\\- profile bvithin the capillary. ('Ihis requirement is peculiar to the adiabatic case and \\-odd not be expected to hold if heat is lost through the capillar); xvall,) Figure 9 s h o w experimental
Points experimental, lines computed
pressure 1's. flo\\- rate for capillary 2. Similar plots \vere obtained for the other four capillaries. In these plots a straight line through the origin represents constant viscositv. l ' h e dashed line represents the so-called Poiseuille flo\v-that is. iiothermal. telescopic flo\\- neglecting any pressure-visco5ity effrct. *Ihe line marked "isothermal flo\v" takes into account the increase in viscosity due to pressure but assumes no heating anytvhere in the capillary. The curve marked '.adiabatic flo\v" is that calculated by the computer; the points are experimental. T h e effect of heating is the difference betiveen the adiabatic and the isothermal lines. 336
l&EC FUNDAMENTALS
M A X SHEAR S T R E S S , d y n e s / s q c n X l O - '
Figure 8. Effect of capillary length on apparent viscosity
12
L
t
4
:
21 4
0
L 0
1
i
.
2
3
A
4
4
5
MAXlMlJM SHEAR STRESS, dynes/cm2 x
K
0.2
I
Figure 9. Effect of length and radius of capillary on minimum viscosity error for adiabatic case
and computed data that bear this out. If the length of the capillary is decreased by half, the apparent viscosity loss is not as great; but if the radius is then decreased by half, the apparent viscosity loss again, increases. The optimum case is to use the shortest possible length and a radius as large as possible still giving a satisfactory L / R . Flow Pattern inside Capillary. The computer program also calculates the flo\v pattern inside the capillary. This cannot be confirmed experimentally, but it does allow some insight on the conditions of flow at the entrance and the effect of heating on the streamlines. In the standard computer program the input temperature is assumed constant. and s o the computer calculates a parabolic flow distribution at the entrance. There is then a general out\vard flo\\, (as sho\vn in Figure l o ) , which is the result of the decreasing viscosity at the wall compared to that at the center. This outxvard flow is coafined almost entirely to the first few per cent of the capillary, Xvhere the viscosity change is the grratest. For the remainder of the capillary, the flow is almost telescopic (parallel to the axis) and even tends to revert back slightly to the center as temperature equilibrium begins to have some effect. The dashed line in Figure 10 marks the point Fvhere the flow changes from an outward to an inward one. As flow proceeds down the capillary, the axial velocities near the \vall tend to speed up, whereas those near the center slow down. These changes in velocities result in a flattening of the velocity profile. The flow profile at the exit is considerably flatter and closely resembles the profile of a nonNewtonian fluid: as sho\vn in Figure 11. T h e computer program has also been used to test other velocity profiles a t the entrance and other flow patterns down the capillary. In each case pressure drop and maximum temperature rise at the \Val1 were calculated for capillary 2 at a flow rate of 3.64 cc. per second. Table I11 summarizes the results. Case 1 assumes a parabolic flow profile a t the inlet, but the profile changes as heating occurs. This case has been used in
Table 111.
Effect of Entrance Profile and Flow Pattern Max. Pressure Temp.,
Case
Entrance ProJle
1
Parabolic Final profile from case 1 Flat profile* Parabolic Experimental
2
3 4
Flatfrom r
=
-
J
10-5
Flow Pattern Nontelescopic Telescopic
Drop, P.S.I. 1936 1908
!we,
Nontelescopic Telescopic
1921 1512 1950
68.1 73.9 60.7
F. 68.2 68.1
I
Figure 1 1 . Change in flow profile from parabolic at inlet to flat at outlet Capillary 2, 1930 p.s.i.
5
1000
;5
-1
2
400 200 0
.2 .4 .b .8 1.0 RADIAL DISTANCE, r/R
all the calculations heretofore and agrees closely with the experimental data. Case 2 used an entrance profile the same as the exit profile from case 1, and assumed this did not change down the capillary-Le., telescopic flow. Gerrard (6) used this case in drawing an analogy between non-Newtonian isothermal flow and Newtonian nonisothermal flow. The legitimacy of this analogy is borne out by the close similarity between the results from cases 2 and 1. Case 3 assumed a flat input profile (modified somewhat a t the wall to obtain a satisfactory computer solution), and this also agreed with the experimental data. The only case to be eliminated was case 4, a parabolic entrance profile and telescopic flow; this gave a computed pressure drop far below the experimental value. In summary, at the relatively low Reynolds numbers being used, the inertial terms are negligible. the flow profile rearranges very rapidly, and the shape of the entrance profile is not important in determining the flow rate or the temperature rise. Conclusions
Viscous heating in capillary flow is potentially a serious experimentzil problem in viscometry, particularly in the adiabatic case. Good agreement was found between experimental data and a computed solution of viscous heating. Both give decreases in apparent viscosity as high as 70y0 and temperature rises up to 350’ F. T o minimize the problem, short capillaries and small L / R ratios are needed.
Appendix. Numerical Techniques for the Solution of the Energy Balance
The energy balance in dimensionless form is
0 t o r = 0.9 R.
VOL.
4 NO. 3
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337
where
L , 7 ’ = r/R> T’ = T / T O V,‘ j = V,/V, V,’ = (Rzl’p0VL)P. and p ’ = p l ’ p o , In these ratios To
t’ = 2,
V r / V )P’
=
is the inlet temperature, I‘ is the average axial velocity, and po is the viscosity when T = To and P = 0. Equation 1‘4 can be directly expressed in finite difference form. It was anticipated, ho\vever, that the temperatures and pressures which develop in the space of an insulated capillary change very rapidly near the entrance and near the wall of the capillary. It was decided, therefore. to use a finite difference grid \vhich is finer in these regions. Ordinarily the use of a nonuniform grid size complicates the calculation considerably. This difficulty can be avoided, ho\vever. by transforming the coordinates of the capillary in some suitable fashion, and expressing the differential equation in terms of the transformed coordinates. A uniform grid is then consrructed in terms of the new coordinates. Two advantages accrue from this device : More information is calculated in regions of rapid change, and the finite difference expressions used to approximate derivatives become more accurate. In this study. exponential transforms were used. Thus, in the radial direction let
and -
nz
The choice of a backward rather than a central difference formula in the axial direction assumes that the temperature of the fluid at any point in the capillary is not influenced by the temperatures at neighboring points bvhich are downstream. This is consistent with assumption 3 of the text, \
