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Non-isothermal laminar pipe flow — II. Experimental
Chemicd~ngineering
Science, 1973. vol. 28, pp. 13 17-1330.
Pcrgamon PESS.
Print4
in Great Britain
Non-isothermal laminar pipe flow - II. Experimental P. B. KWANT,t R. H. E. FIERENSS and A. VAN DER LEE5 Laboratorium voor Fysische Technologie der Technische Hogeschool, Delft, The Netherlands (Received 10Augusr 1972) Abstract- Id a previous article [ 11theoretical predictions of velocity distributions, pressure drops and mean Nusselt-numbers are given for non-isothermal pipe flow of power-law fluids having a temperature dependent consistency-index. In the present investigation these solutions are checked for Newtonian fluids with extensive experimental data. A flow visualization technique was used for measuring the non-isothermal, laminar velocity profiles of glycerol in a round tube. Furthermore, accurate pressure drop measurements were carried out for non-isothermal tube flow of a viscous. Newtonian liauid. Finally logarithmic mean heat transfer coefficients were measured in laminar flow heat transfer. The exoerimental velocitv orotiles. nressure drops and heat transfer coefficients show a good agreement with those predicted iheoreti&lly. INTRODUCTION
IN A previous article[l] the hydrodynamics and the heat transfer rate have been calculated for laminar, non-isothermal pipe flow of power-law fluids of which the consistency-index varies exponentially with the temperature. The calculations are based on the following assumptions: 1. The flow is steady state. 2. The flow is rectilinear and axisymmetric. 3. The fluid thermal conductivity A, specific heat c,, and density p are independent of temperature and pressure. 4. The flow is fully developed at the entrance to the heat transfer section and has a uniform temperature at this point. 5. Axial diffusion of energy and momentum is negligible compared to axial convection of energy or pressure gradient. 6. Thermal energy generation in the fluid by viscous dissipation is negligible. 7. A constant wall temperature heats (or cools) the fluid. Due to these restrictions our results will usually not be valid for gases, but we think that they can be applied to most cases of laminar
flow heat transfer to liquids (see e.g. [2]). The assumption that the flow is rectilinear, which implies that we neglect radial velocities, is the most serious restriction. At the very beginning of the tube the distortion of the velocity profile is considerable and consequently the radial velocities may be important. In this region also the term &,/a, is not negligible, which, however, also has been neglected in the analysis. For greater distances in the tube the importance of the radial velocities will decrease gradually. In the present investigation we checked the validity of our results for Newtonian fluids with accurate measurements of non-isoviscousl[ velocity distributions, pressure drops and heat transfer coefficients. As experimental fluids viscous, Newtonian liquids were used of which the temperature dependence of the viscosity is very large. NON-ISOVISCOUS VELOCITY DISTRIBUTIONS
Experimental
set-up
A flow visualization technique was used for measuring the non-isothermal velocity profiles
tPresent address: Koninklijke/Shell Laboratorium, Amsterdam. $Present address: Badger, Den Haag. BPresent address: AKZQ, Arnhem. lIWe define isoviscous flow as the flow of fluids with a rheology independent of the temperature. Non-isoviscous flow is the flow of fluids having a temperature dependent rheology.
1317
P. B. KWANT,
R. H. E. FIERENS
of laminar, Newtonian tube flow. The technique used is essentially the same as that used by Atkinson et al. [3,4] who measured velocity profiles in developing liquid flows under isothermal conditions. Very small tracer particles are present in a transparant liquid. The particles are being illuminated from the side, viewed against a dark ground, and photographed. The particles used had an extremely high reflectance coefficient, a density very much alike the fluid and negligible settlement properties. For a scheme of the apparatus see Fig. 1. The fluid containing tracer particles is stored in a vessel (1). We used glycerol as the experimental fluid. As tracer particles we chose particles of Mearlin Luster Pigment Silkwhite M.S.W.-139 (Mearl Corporation, New York). The particles were added to the bulk of the glycerol from a suspension. Their average diameter was found to be about 10 pm. Vessel (1) is provided on the outside with an electrical heating wire (2kW), surrounded by thorough insulation. Due to the hygroscopic qualities of the glycerol it seemed useful to give vessel (1) an air-tight construction. This vessel is also provided with a ribbon-stirrer, which can be used to improve the heat transfer to the glycerol from the electrical wire and to minimize the temuerature differences in the glycerol. A gear pump (2), driven by a variable speed motor, pumps the liquid via a Multiflux-mixer (3) into the heating section (4) and via a by-pass back into the vessel (1). By va$ng the speed of the gear pump and by making use of this by-pass system the desired volume rate of flow can be adjusted. The Multiflux-mixer (3) is used to eliminate radial temperature differences in the fluid. It is a mixer without moving parts. The fluid flows through a series of similar elements in each of which the stream is divided, deformed and rearranged in such a way that inhomogeneities are evenly dispersed over the entire cross section. It is an excellent apparatus for mixing viscous fluids. For further information see e.g. Sluyters [5]. The Multiflux-mixer (3) has been constructed of copper.
and A. VAN DER LEE
After passing through the mixer the fluid enters into the heat transfer section (4) which consists of a copper tube of 12 mm i.d. and I.50 m length. The copper tube is constructed in a thermostat system in which water at constant temperature is circulating at very high speeds in order to make the heat transfer resistance at the outside of the tube negligible compared to the inner heat transfer resistance. Temperatures are measured by means of 1 mm thick chromel/alumel thermocouples (Pyrotenax). Immediately behind the mixer (3) a thermocouple (t,) measures the uniform temperature of the fluid entering the heat transfer section. Between the Multiflux-mixer and the copper tube a PVC section of about 3 cm length is fitted which has exactly the same inner diameter as the copper tube. This has been done in order to eliminate axial conduction of heat along the tube wall from the heat transfer section to the mixer or vice versa. Furthermore this intermediate piece of tube is a guarantee that in all our experiments the fluid entered the heat transfer section hydrodynamically fully developed. The Multifluxmixer (3) and PVC tube both are very thoroughly thermally insulated. In order to avoid errors in the measurement of the entry temperature due to conduction along thermocouple leads the thermocouple tl is inserted through the PVC tube and led for about 1 cm in the opposite direction to the flowing fluid and parallel to the tube axis. The wall temperature of the copper tube is measured with three thermocouples (t, ts and tJ which are inserted in sleeves in the tube wall along the axial direction. In all our experiments the temperature difference between tZ and t, was less than 0*2”C, usually less than O*l”C. At the end of the heat transfer section the copper tube is followed up by a perspex tube of exactly the same diameter and a length of 0.25 m. Everything feasible was done in order to obtain a very smooth transition of the copper tube into the perspex tube. The perspex tube is surrounded by a perspex box, together forming the optical system. After passing through the optical system (5) the glycerol is collected in a vessel (6) from which it can be pumped back to vessel (1) by means of the gear pump (2). The mass flow rate is determined by weighing the glycerol collected in a glass beaker during a known time-interval. The optical system is quite similar to that used by Atkinson et a1.[43.
Fig. 1. Scheme of the flow-visualization apparatus.
1318
Non-isothermal
laminar pipe flow- II
gtyc~!_--_____-~----_____~~~_~
glycerol in
1
mercury
2
collimating
3
adjustable
slit
4
focussing
Ions
vapour
lamp
lens
5
perspax
6
experimantal
7
camera
box section
Ians
Fig. 2. Optical arrangement. The refractive index of the perspex used is nearly equal to the refractive index of glycerol at room temperature. An area about 2 cm long and 0.5 mm deep on a plane along the tube axis is illuminated by the arrangement shown in Fig. 2. The illumination used was provided by a high-intensity mercury vapour discharge lamp (80 W) operated directly at mains frequency, providing illumination discontinuously at 100 flasheslsec. The path of each moving particle is thus seen as a discontinuous streak. As the paths of many particles are recorded simultaneously, a complete velocity profile may be derived from a single exposure. The discontinuous light-beam from the mercury lamp (1) was collimated through a lens (2) on to an adjustable slit (3), whose image was focused on to a second slit which was part of the perspex box (5) surrounding the experimental section of the tube. This perspex box has been applied for two reasons. In the first place it helps to minimize refraction effects due to the curvature at air-perspex-glycerol surfaces and in the second place as a means of minimizing further heat transfer to the perspex tube. Both difficulties were overcome to a major degree by filling the perspex box with glycerol of the same temperature as the wall temperature of the copper tube. By means of a gear pump glycerol is pumped at low speed from a thermostat through the perspex box. The normal surfaces of the box were carefully aligned and light passing through the tube was reflected away by a surface set at 45 degrees to the direction of the incident light. The illuminated plane was viewed by a camera at right angles to the optical axis of the main light beam. All surfaces of the box were painted black except for that facing the camera, and the slits where the main light beam entered and left the box. The camera used was a Hasselblad camera with Kodak Tri-X-PAN 27 DIN film. High-contrast enlarged prints of the flow field were made and from these the velocity profile could be measured. Each particle in the illuminated area appeared on the photograph as a series of dashes, each dash representing the path of the particle during a single flash of the mercury lamp (mains frequency being 50 Hz, this corresponds to a time interval of 0.01 set). The position of the
particle relative to the tube wall could easily be determined as could the distance of travel for a given number of discharges. The distance travelled by a particle in a single interval of 0.01 set could be determined. In order to check the scaling of the photographs due to refractive effects a test grid was made out of a thin (0.3 mm) piece of copper plate of length 1 cm. This copper plate was adjusted in small sleeves in the wall of the perspex tube on the horizontal plane along the tube axis and was provided with a calibrating system of horizontal and vertical lines at 05 mm intervals. We now made some photographs of this test grid, pumping glycerol without particles through the apparatus. We did these experiments under isothermal conditions and also with temperature differences (TU,-TO) from -40°C up to +4o”C. We could not detect any differences in the scaling systemat all on our photographs, so we concluded that refmction effects did not influence our measurements. After these preliminary experiments we removed the test grid and started our experiments with a new perspex tube.
Results
For the physical constants of the glycerol we refer to the Appendix. In all our experiments we compared the volume rate of flow calculated from the measured velocity profile with the volume rate of flow calculated from the directly measured mass flow rate at the exit of the experimental apparatus: R
I
27rrv, dr = QV= >.
0
The difference between these two quantities was nearly always smaller than 3 per cent, usually smaller than 2 per cent.
1319
P. B. KWANT, R. H. E. FIERENS
and A. VAN DER LEE
In order to check the optical part of the experimental apparatus we did some isothermal experiments at different temperature levels and with different volume rates of flow all of which agreed very well with the parabolic velocity distribution (see e.g. [2]). We performed about thirty non-isothermal experiments in which the dimensionless length varied from X+ = 3.6. 1O-3 up to X+ = 10-l. The corresponding Reynolds-numbers, calculated at an average temperature (To + ( T,) )/2, are in the range (Re) = 0.24-5.6 for heating and (Re) = 1.1-55.0 for cooling. Q-values varied from Q = -2.10 up to Q = +260. The quantity Q has been defined [ l] as follows:
Q = HT,-Tt,)t
(1)
in which b is the viscosity variation parameter: K = K’ exp {-b(T,-
T’)}.
(2)
From Eqs. (1) and (2) follows:
Q = In (&I&,).
I (3) Fig. 3. Non-isoviscous
Equation (3) reduces for Newtonian fluids to:
Q = ln (nhw).
velocity profile for Newtonian pipe flow (cooling).
(4)
In all experiments the agreement between the measured velocity profiles and the theoretically predicted profiles according to [I], calculated either numerically or with a first-order approximation, was very good. In the Figs. 3 and 4 we show two representative measurements of a cooling- and a heating run. We plotted the dimensionless velocity vs the dimensionless tuberadius. As a reference also the isoviscous, parabolic distribution has been given (Q = 0). Lines are according to theory[ 11, crosses and round points refer to measurements. Crosses refer to measuring points situated in the left part of the illuminated plane, round points refer to the right part of this plane, this distinction being useful as a confirmation of the absence of systematic maldistribution. As can be seen the 1320
flow (heating).
Non-isothermal
laminar pipe flow- II
Fig. 5. Experimental set-up for heat transfer- and pressure drop measurements.
agreement between theory and very good. It should be remarked experiments we could not detect radial component of the velocity photographs.
experiments is that in all the any significant on any of the
NON-ISOVISCOUS PRESSURE DROPHEAT TRANSFER MEASUREMENTS Experimental
AND
set-up
The experimental apparatus is shown in Fig. 5. As experimental fluid we used a very viscous Newtonian oil, Shell Nassa 89. The fluid is stored in a vessel (1) in which it can be thoroughly mixed by means of a ribbon-stirrer. The fluid can be heated in the vessel by means of a heating spiral (2kW), which is attached to the inside of the wall of the vessel. Two thermocouples were soldered on this heating spiral in order to check the temperature near this spiral, which never raised above 90°C in order to prevent deterioration of the oil. A gear pump (2), driven by a variable speed motor, transports the fluid either via a thermostat bath (3) and a multiflux mixer (4) into the heat transfer section or via a by-pass back into the vessel (1). By varying the speed of the gear pump and by making use of the by-pass system the desired volume rate of flow can be adjusted. In order to adjust the desired temperature of the fluid at the entrance to the heat transfer section, the oil is pumped through a spiral steel tube placed in a thermostat (3). Besides this, the Multiflux-mixer (4) is surrounded by a jacket through which water from the thermostat (3) is circulated at high speed by a pump (5). After passing through the mixer the fluid enters into the heat transfer section, which consists of one or two copper tubes of 1.50 m length and an i. d. of 12 and 6 mm.t This tube
tFor the heat transfer measurements we also used two spiral copper tubes with a length of 504 m and 1.05 m and a diameter of 12 mm each.
is constructed in a thermostat system (6) in which water from a thermostat is circulating at high speed. In order to make the heat transfer resistance at the outside of the tube negligible compared with the inner heat transfer resistance one has to take care of a great water velocity at the outside of the tube. We achieve very high heat transfer coefficients at the outside of the tube by employing a pump (100 I./mm) which via a shower system spouted water from a thermostat over the entire length of the tube. Temperatures are measured by means of 1 mm thick chromel/alumel thermocouples (Pyrotenax). Some of the important thermocouple readings are registered continuously on a recorder to make sure that all temperatures are stationary. If this steady situation has been reached all temperatures are being measured by a compensator. All thermocouples have one common reference in an ice-distilled water mixture. Immediately behind the first Multiflux-mixer (4) the entry temperature ZF,,is measured by a thermocouple (tl). The wall temperature T, of the copper tube is measured by three thermocouples (tz, t, and t4). Thermocouples tl, tS, t, and t4 have been constructed and inserted in the same way as described in the experimental apparatus for flow visualization. The difference in reading between t, and t, was always smaller than 0.2”C, atmost experiments smaller than O-1°C. The copper tube has been provided with eight pressure taps, which are connected via a central tube (7) with the pressure measuring device. The places of the pressure taps on the copper tube are 1.6; 18.3; 23.3; 38.3; 68.3; 103.3; 138.3 and 147.8 cm from the entry. The pressure measuring device is a Texas Instruments precision pressure gage model 156, of which the measuring gage is a quartz Bourdon tube, filled with silicone oil. The main advantage of this system is its quick response. After passing through the heat transfer section the meancup mixing temperature has to be measured. This is felt to be the most difficult part of the heat transfer measurement. In order to prevent any possible heat loss during the mixing process in the final Multiflux-mixer (8), we felt that it was necessary to operate with this mixer under adiabatic conditions. The procedure therefore was as follows. After passing
1321
P. B. KWANT,
R. H. E. FIERENS
through the heat transfer section the fluid enters via a PVCtube of 3 cm length and the same id. as the copper tube into the Multiflux-mixer (8). At the end of the mixer thermocouples tJ and ta are measuring the mean-cup mixing temperature of the fluid. In order to check the temperature uniformity achieved by the mixer the couples tJ and ts are inserted in the fluid leaving the mixer at the axis of the tube and near the tube wall. Both thermocouples again were led for about 1 cm into the opposite direction of the flow to eliminate conduction errors through the thermocouple leads. For all our experiments the difference in reading between t5 and ta was less than O*l”C. To ensure adiabatic operation of the mixer (8) we made a construction, which is schematically shown in Fig. 6. The Multiflux-mixer, the construction material of which is copper, the PVC-tube in between the mixer and heat transfer section and a piece of tube behind the mixer are wrapped around with glasswool insulation. Around this insulation a spiral heating wire was wrapped, supported by a copper gauze. The gauze is present to ensure that the heat developed by the wire is distributed as evenly as possible through the insulation. Around the heating wire again glasswool is present. At a distance of 1 cm from the outer wall of the Multifluxmixer three thermocouples t,, ts and ta are located. At every experiment the electrical current through the wire was adjusted so that the thermocouples t,, ts and ts indicated a temperature which was equal to the mean-cup mixing temperature measured by thermocouples t5 and tp. In all experiments the differences in reading between t,, ts and tg were less than 0.3”C once all the temperatures were stationary. It can therefore be concluded that we measured the average mean-cup mixing temperature correctly. The accuracy of this temperature measurement is estimated as O.l”C. After passing through the second Multiflux-mixer the oil is collected in another vessel (9), from which it can be pumped back into vessel (1) by means of the gear pump (2). The mass flow rate is determined by weighing the oil collected in a glass beaker over a known time-interval. It should be pointed out that the heat transfer and pressure drop measurements were usually carried out separately in different set-ups in order to concentrate better on each of them. We performed also a few heat transfer experiments with glycerol and pressure drop measurements with a pseudoplastic liquid (a mixture of 80 per cent wt glycerol/water to which 0.3 per cent wt Carpobol has been added).
Fig. 6. Detail of the second Multiflux-mixer.
and A. VAN DER LEE
The physical constants of all fluids are given in the appendix. We checked that for all experiments the fluid entered the heat transfer section fully developed and that viscous dissipation was absent.
Results of pressure drop measurements The quantity C#J’ has been defined [ 11as the ratio of the true and the isoviscous pressure gradient, the latter quantity taken at the tube wall temperature:
Inserting the known relation for (-dp/dx)O,,w for Newtonian fluids we find:
Analogously one can write for the total pressure drop ratio for Newtonian fluids: D4(-APlAx)
r ‘128.
(7)
We now operated as follows. For every x+ corresponding to the axial length in the middle of two pressure taps, the pressure gradient (-d&b) was approximated by the pressure drop (--AP/Ax) over these two pressure taps. In doing this we are introducing an error which however is very small because the distance of the taps deliberately has been made very small in the beginning of the heat transfer section where the changes in pressure are high. The lirst pressure tap at 1.6 cm distance from the beginning of the tube was not used in the determination of the pressure gradients. For some experiments we measured the real pressure gradient (-d&lx) from the graph of pressure against axial length. It appeared that the difference between both quantities was always smaller than 3 per cent. So, in one experiment we gather six @-data from seven readings. On every x+, corresponding to the axial length of the last of two pressure taps we determined the average pressure drop (-AflAx) from x = 0 to x = x. In order to determine the pressure drop in a correct way the first pressure tap from the entry of the heat transfer section has to be present near x = 0. We situated this tap at 1.6 cm from the beginning of the tube and therefore it is possible to extrapolate to x = 0 accurately to get the correct value of the pressure at x = 0. tFor Newtonian fluids the pressure gradient ratio C#J’ equals the ratio of the non-isoviscous and the isoviscous velocity gradient at the wall.
1322
Non-isothermal
Fig. 7. Experimental
laminar pipe flow- II
pressure gradient ratio for non-isoviscous,
The fourth pressure tap at x = 38.3 cm from the entry is the first one used for the calculation of (4’). We estimate the possible error in (-d&Lx) as 3 per cent and that of (-AP/Ax) as 2 per cent. The diameter D of the tube was determined by accurate isothermal measurements in which T, and T, differed by less than O.l”C from each other. This has been done at different temperature levels and for different volume rates of flow. Under all these conditions and for all axial parts of the straight tubes the spreading in D4 was less than 4 per cent. The possible error in D therefore has been estimated as I per cent. The accuracy of the determination of the viscosity is also estimated as 4 per cent (see Appendix). The volume rate of flow, calculated from the mass flow rate and the fluid density is probably accurate within 0.5 per cent. We estimate therefore that the maximum error in the dimensionless pressure gradient 4 is about 12 per cent and the maximum error in the dimensionless pressure drop is about I I per cent.
All experiments have been carried out with Shell Nassa 89 oil in either of two straight tubes of 1.50 m length and 12 mm and 6 mm dia. The Q-values were Q = +0*69; +1*12; + l-95; +2*60; -1.12; -1.95; -260 within 3 per cent. For the heating experiments (Q > 0) the inlet temperature was 20°C and the wall temperature either 30, 40 or 50°C. The heating runs with Q = +0*69 had an inlet temperature of 40°C and a wall temperature of 50°C. For the cooling experiments (Q < 0) the wall temperature was 20°C
Newtonian
pipe flow.
and the inlet temperatures used were 30, 40 and 50°C. Reynolds-numbers related to the entry temperature varied from Re, = 0.1 up to Re, = 3.5 for heating and from Re, = 0.25 up to Re,, = 53 for cooling. Some results of measurements of the pressure gradient ratio 4 are shown in Fig. 7, the results of the measurements of the total pressure drop ratio (4’) are given in the Figs. 8-10. The ratios 4 and (4’) have been presented as a function of the dimensionless length X+ with Q as parameter. All experimental results (crosses for the tube with D = 12 mm and points for the tube with D = 6 mm) are compared with the values predicted numerically (drawn line). It appears that from x+ = 5 . 10e4 up to x+ = 3.10-l the agreement between theory and measurements is very satisfactory. The greatest deviation occurs in the cooling experiments with the greatest temperature difference (Q = -2.60) in the region of X+ = lo+ up to x+ = 8. lo+. Except for these experiments the deviation of theory and experimental data is nearly always within 12 per cent. We performed also pressure drop measurements of non-isothermal flow of a pseudo-plastic
1323
P. B. KWANT,
‘6’1
R. H. E. FIERENS
and A. VAN DER LEE
I
I
w-2
UJJ Fig. 8. Experimental
pressure drop ratio for non-isoviscous,
fluid (0.3% wt Carbopol in an 80% wt glycerol/ water mixture), having a flow behaviour index n = 0.55. Measurements were done for Q = +l and Q = -1. The results of these experiments, which are not given here but can be found elsewhere [2] show the same good agreement between measurements and theory.
lo’-
I
I
w”
10-l
Newtonian pipe flow.
Results of the heat transfer measurements We can calculate the average Nusselt-number according to: In(&)
(Nu) =& We estimate
r1111,
=&ln(2<:“).
the accuracy I
I
I
I
of the temperature ,11,1-
-<‘PI> (:I
it
-=yi+k *t-%L13*+_ 8-w -yo. *“+sb
Fig. 9. Experimental
pressure drop ratio for non-isoviscous,
1324
(8)
Newtonian pipe flow.
Non-isothermal
Fig. 10. Experimental
pressure
laminar pipe flow - II
drop ratio for non-isoviscous,
measurements to be 0.1°C. The accuracy of the determination of the Nusselt-number is very dependent on the value of the average outlet temperature (T,) and consequently upon x+. At very small x+ the possible error will be great because (T,) is very near T,,, at high x+ the possible error is large again because (T,) now is near T,. One can calculate that for our experiments the accuracy is 2 lo-20 per cent for IT,-T,I= 15°C; 5-10 per cent for IT,-T,I= 30°Cand 3-6 per cent for IT,- ToI = 50°C. The results of the heat transfer experiments are shown in Figs. 1 l(a-h). In these graphs the measured Nusselt-numbers divided by the Nusselt-numbers predicted numerically are given as a function of x+ for different values of Q and different tube dimensions. This has been done in order to eliminate the spread in Q between the individual measurements of a series. Every experimental Nusselt-number now is compared directly with the Nusselt-number predicted numerically for the corresponding Q-value. In addition to this feature this way of presenting the data gives at once the deviation of theory and experiment.
Newtonian
pipe flow.
All heat transfer experiments have been performed in tubes with an i. d. of 12 mm. The experiments given in Figs. 1 l(a-e) have been carried out with Shell Nassa 89 in spiral tubes of length 504 m and l-05 m.t The temperature differences (T,- T,,) were: 15”C, 30°C 50°C -10°C and -15°C corresponding to Q-values of Q = 160, 2.70, 3.80, -140 and -1.50. In the heating experiments Reynolds-numbers referring to initial conditions varied from O-02 to O-30; in the cooling-runs Reynolds-numbers were between O-07 and O-90. The experiments given in Fig. 1 I(f) were performed with glycerol in a straight tube of 1.50 m tin these cases the ratio of the diameter of the spiral U&al ) and the inner tube diameter was about 25. It can be found in literature (see e.g. Richter[6] and Truesdell[7]) that such a curvature of the spiral tube does not influence the velocity distribution and hence does not influence the heat transfer if the Dean-number is smaller than one. De-Re
4-
D r
(9)
This implies that in our case the Reynolds-number should be smaller than five. In all our heat transfer experiments this condition was met.
1325
P. B. KWANT, R. H. E. FIERENS
and A. VAN DER LEE
Fig. 1 la.
I
I
I
III,,
I
I
II111,
I
~N-exp. I.*I.
0
Q
L=l.OzSm
c?=3.60
$ 0
_----P--,-o-_
--------
c-t
i
I
,
I,llll
I
I
I
l,lll,
I
QJ - 1.00
1.0
I
I
0 L=l.O5m
1.2_-axp. o 0
0
-_
~oO_O--_--
-
-
0 0
tt
I
Q8 lo-3
___t
#111#1
x1
I I IIII
I
I
I lc+
KY2
Fig. 1 Id. I
I
I
IIIII
I
I
I
I
IllI,
x
exp. I2
0
Q I 00 ID
---00 tt
0.8 lo-3
o
-_o__-
x-----_
x-
X
x’
L=5.04m L=l.O5m -1.50
I
I
Non-isothermal I
I
I
IllI,
I
I
I
I
!111l,
Glycerol
12-
D -._-o__-~-x~--x0 x
--
I
1.2 -
,
,
x Qz -1.30
~
l.0
1
I
I
I
I#(,
0 Qc -2.10
x x ---
Ic -
---
r
I
-
1
,111,
I
Straight tube x D=l?mm
_
Q =-2.60
K
xx 1.0
laminar pipe flow-II
_
-
X
L_
--x-
-------
_x_
--I
I
1.2 -
1.0
I
I
,
I
,111,
I,!,,
I
~ t
___---)E---
X X
I
I
Straight tube x Dd2mrr-1 0 = -3.70
ze ---
--
---
%
1-t
0s10-3
I
-
X I
I
I
II
I
-2 10
x* I
1 I
III
I
I
lc-’
Fig. 1 lh. Fig. 1 l(a-h). Experimental heat transfer results for Newtonian pipe flow.
length. Q-values were approximately Q = -1.30 and -2.10 corresponding to temperature potentials of about 18 and 40°C. Reynolds-numbers were in the range Re, = l-55-55. The experiments given in Figs. 1 l(g) and (h) have been carried out with Shell Nassa 89 in straight tubes of l-50 m length. Q-values were Q = -260 and -3.70 corresponding to temperature driving forces of 30 and 50°C. Reynoldsnumbers varied from O-75 up to 14. In the results of Fig. 1 l(h) (Q = -3.70) we applied a correction for the enhancement of the heat transfer rate due to free convection. This correction factor was proposed by Oliver[8].t- There may be some uncertainty in this correction term which amounts from -2.5 per cent up to - 16 per cent. For this reason we estimate the accuracy of the results in this cooling run somewhat lower than in the other series. As can be seen from the results given in Figs.
1 l(a-h) the agreement between theory and experiment is very good. With a few exceptions all measurements agree within 8 per cent with theory. It should be added that we also correlated our tTo allow for free convection has to be multiplied by a factor:
the mean Nusselt-number
, + 04083(GrmPr,)U’75 I’3 (b&L) 1 . [ The mean Grashof-number
is defined as follows:
(p), (p) and (n) are the coefficient of expansion of the fluid, its density and its viscosity, all taken at the average temperature (T,, + (T,))/2. In the mean Prandtl-number, Pr,, the physical quantities also have to be taken at this average temperature. With this correction term as a criterion we calculated that for all other heat transfer results, Figs. 1 l(a-g), the influence of free convection may be neglected.
1327
P. B. KWANT,
R. H. E. FIERENS
and A. VAN
LEE
NOTATION
results according to the wellknown, empirical Sieder and Tate correction method[91. This correction to allow for isoviscous heat transfer coefficients with a factor ( (~)/~~)“‘14 gives significantly larger errors for the measured heat transfer coefficients than those associated with the numerical prediction (see [2] for the details).
= hlpc,,
CP
D D spiral De g
CONCLUSIONS
Measurements were carried out of non-isothermal velocity profiles and pressure drops and of mean heat transfer coefficients in laminar flow of Newtonian liquids. The viscosity of these fluids is strongly dependent on the temperature. There is a good agreement between the experimental results and the theoretical values predicted by a numerical analysis in a previous article [ I]. We may conclude therefore that the neglect of radial velocities and changes in axial velocity from the transport equations is acceptable for viscous liquids under practical heat transfer circumstances (x+ > 10e3 and -3 < Q < +3). We expect that also for pseudo-plastic fluids the results from the analysis can be used with confidence, because there are no principal differences with Newtonian fluids concerning the deviation of the isoviscous behaviour. Pressure drop measurements with a pseudo-plastic fluid (n = 0.55) for Q = + 1 and Q = - 1 showed also a good agreement with theory. However, with increasing pseudo-plastic behaviour, especially for large negative values of Q, the distortion of the isothermal velocity profile increases considerably (see [ 11). The distortion being greater, the more important will be the radial velocities in the first part of the heat transfer tube. For small x+, therefore, more experimental work seems to be needed in the area of increasing pseudoplasticity (n < 0-S) and large temperature intervals in order to assess the region of validity of the theoretical results.
DER
Gr Gr, K Ko K’ L n (Nu) (NM), (Nu),xp C--dplW (-AplW Pr
temperature diffisivity, m2/s viscosity variation parameter defined by Eq. (2), “C-l specific heat, J/kg “C tube diameter, m diameter of spiral, m = Re (DIDspiral ) 112,Dean-number gravitational acceleration, m/s2 =
PATD3p2g
, Grashof-number r12 Grashof-number at an average temperature ( To + ( T,) ) /2 consistency-index of power-law fluids, Nsn/m2 consistency-index at entry temperature, Nsn/m2 consistency-index at a reference temperature T ’ , NsVm2 tube-length, m flow behaviour index of a power-law fluid logarithmic mean Nusselt-number logarithmic mean Nusselt-number for non-isoviscous flow logarithmic mean experimental, Nusselt-number pressure gradient in axial direction, N/m3 pressure drop in axial direction, N/m3 =- ‘, Prandtl-number a
Prandtl-number at an average temperature ( To + ( T,) ) /2 Q = b (T, - To), quantity giving the deviation from isoviscous behaviour r radial coordinate, m r+ = r/R, dimensionless radial coordinate R tube-radius, m
Pr,
Re
P(v,)D
= p,
Reynolds number
7 Acknowledgement-The authors thank Ir. P. H. F. van Voorst van Beest who performed most of the flow-visualization measurements.
1328
(Re) To
Reynolds-number at an average temperature (To + (T,) )/2 entry temperature, “C
Non-isothermal
laminar pipe flow- II
(Te) mean-cup mixing temperature T' AT (2 V+
X X+
at the end of the heat transfer section, “C reference temperature, “C temperature difference, “C axial velocity, m/s average axial velocity, m/set = v,.( v,) , dimensionless axial velocity axial coordinate, m dimensionless
=$j&
axial co-
Greek symbols
of expansion
dynamic viscosity at an average temperature (T,, + ( T,) )/2, Nsec/m2
(0)
=++4
dimensionless
mean-
cup” mix;ng temperature at the end of the heat transfer section A thermal conductivity, W/m”C Y kinematic viscosity, m%ec p density, kg/m3 (p) density at an average temperature (To+ (T,) )/2, kg/m3 4’ pressure gradient ratio, Eq. (5) (4’) pressure drop ratio, Eq. (7) 4Jm mass flow rate, kglsec 4” volumetric flow rate, m3/sec
ordinate
/3 coefficient
(q)
of the fluid
y-1 (p)
q q.
coefficient of expansion of the fluid at an average temperature (TO+ (T,) W, ‘C-l dynamic viscosity, Nsec/m2 dynamic viscosity at entry temperature, Nsec/m2
Indices
0 e w
referring to isoviscous conditions or referring to entry conditions referring to end conditions referring to wall conditions
REFERENCES
[II KWANT P. B., ZWANEVELD A. and DIJKSTRA, F. C., Chem. Engng Sci. 1973 28 [21 KWANT P. B., Ph.D. thesis. TH Delft 197 1.
M. P., CARD C. C. H. and SMITH J. M., A.1.Ch.E. .I1 1969 15 548. [31 ATKINSON B., BROCKLEBANK 1967 13 17. [41 ATKINSON B., KEMBLOWSKI Z. and SMITH J. M.,A.Z.Ch.E.JI [51 SLUYTERS R., De lngenieur 1965 77 Ch. 33. 161 RICHTER H., Rohrhydruulik Springer, Berlin 1958. 1970161010. [71 TRUESDELLL.C.JR.andADLERR.C.,A.I.Ch.E.Jf [81 OLIVER D. R., Chem. Engng Sci. 1962 17 335. 191 SIEDER E. N. and TATE G. E., 2nd. Engng Chem. 1936 28 1429.
APPENDIX Physical properties of the experimental&ids used The density of Shell Nassa 89 and of glycerol, measured by standard methods is given in Fig. 12 as a function of the temperature. The viscosity of Nassa 89 and glycerol was measured in a rotational viscometer (make Haake “Rotovisco”) as a function of the temperature. The results are given in Fig. 13. We see from this figure that for large temperature differences the deviation from the exponential viscosity-temperature rehttionship is considerable, especially for Nassa 89. One should
note, however, that in practice the value of Q is calculated by means of Eq. (4): Q = In (no/n,). In this way one calculates an average value of the viscosity variation parameter b (see Eqs. (1) and (2)) over the relevant temperature trajectory (T, - T,,). This is the reason that also fluids of which the viscosity does not follow exactly the exponential tempetature dependence will obey the predicted, non-isoviscous behaviour. The good agreement between theoretical predictions and the experiments with Nassa 89, even for the largest temperature differences, is another argument for this statement.
1329 Cl23 Vd. 26 No. 6-G
P. B. KWANT,
12
20
R. H. E. FIERENS
and A. VAN DER LEE
40
30
50
60
Fig. 12. The density of the experimental fluids as a function of the temperature.
“\
WE
I
“‘\.\ “\
“\
*\ -a.
%
ld“lc$
i,
*\ m
I M
I x)
I
1
40
50
xl
q-c]
I
I
63
70
m
Fig. 13. The viscosity of the experimental duids as a function of the temperature. The temperature diffusivity a (= A/PC,) of the experimental fluids is determined by means of a non-stationary heat penenation method, for the details of which we refer to [2]. Experiments at different temperature levels (20-50°C) did not show any difference in the values of the thermal diffisivity a
outside the experimental results are: Shell Nassa-89 Glycerol
1330
scatter (= 044. 10-7m2/sec). The a = 0.81 . 10-r mYsec a = 0.88 . IO+ m2/sec.
Science, 1973. vol. 28, pp. 13 17-1330.
Pcrgamon PESS.
Print4
in Great Britain
Non-isothermal laminar pipe flow - II. Experimental P. B. KWANT,t R. H. E. FIERENSS and A. VAN DER LEE5 Laboratorium voor Fysische Technologie der Technische Hogeschool, Delft, The Netherlands (Received 10Augusr 1972) Abstract- Id a previous article [ 11theoretical predictions of velocity distributions, pressure drops and mean Nusselt-numbers are given for non-isothermal pipe flow of power-law fluids having a temperature dependent consistency-index. In the present investigation these solutions are checked for Newtonian fluids with extensive experimental data. A flow visualization technique was used for measuring the non-isothermal, laminar velocity profiles of glycerol in a round tube. Furthermore, accurate pressure drop measurements were carried out for non-isothermal tube flow of a viscous. Newtonian liauid. Finally logarithmic mean heat transfer coefficients were measured in laminar flow heat transfer. The exoerimental velocitv orotiles. nressure drops and heat transfer coefficients show a good agreement with those predicted iheoreti&lly. INTRODUCTION
IN A previous article[l] the hydrodynamics and the heat transfer rate have been calculated for laminar, non-isothermal pipe flow of power-law fluids of which the consistency-index varies exponentially with the temperature. The calculations are based on the following assumptions: 1. The flow is steady state. 2. The flow is rectilinear and axisymmetric. 3. The fluid thermal conductivity A, specific heat c,, and density p are independent of temperature and pressure. 4. The flow is fully developed at the entrance to the heat transfer section and has a uniform temperature at this point. 5. Axial diffusion of energy and momentum is negligible compared to axial convection of energy or pressure gradient. 6. Thermal energy generation in the fluid by viscous dissipation is negligible. 7. A constant wall temperature heats (or cools) the fluid. Due to these restrictions our results will usually not be valid for gases, but we think that they can be applied to most cases of laminar
flow heat transfer to liquids (see e.g. [2]). The assumption that the flow is rectilinear, which implies that we neglect radial velocities, is the most serious restriction. At the very beginning of the tube the distortion of the velocity profile is considerable and consequently the radial velocities may be important. In this region also the term &,/a, is not negligible, which, however, also has been neglected in the analysis. For greater distances in the tube the importance of the radial velocities will decrease gradually. In the present investigation we checked the validity of our results for Newtonian fluids with accurate measurements of non-isoviscousl[ velocity distributions, pressure drops and heat transfer coefficients. As experimental fluids viscous, Newtonian liquids were used of which the temperature dependence of the viscosity is very large. NON-ISOVISCOUS VELOCITY DISTRIBUTIONS
Experimental
set-up
A flow visualization technique was used for measuring the non-isothermal velocity profiles
tPresent address: Koninklijke/Shell Laboratorium, Amsterdam. $Present address: Badger, Den Haag. BPresent address: AKZQ, Arnhem. lIWe define isoviscous flow as the flow of fluids with a rheology independent of the temperature. Non-isoviscous flow is the flow of fluids having a temperature dependent rheology.
1317
P. B. KWANT,
R. H. E. FIERENS
of laminar, Newtonian tube flow. The technique used is essentially the same as that used by Atkinson et al. [3,4] who measured velocity profiles in developing liquid flows under isothermal conditions. Very small tracer particles are present in a transparant liquid. The particles are being illuminated from the side, viewed against a dark ground, and photographed. The particles used had an extremely high reflectance coefficient, a density very much alike the fluid and negligible settlement properties. For a scheme of the apparatus see Fig. 1. The fluid containing tracer particles is stored in a vessel (1). We used glycerol as the experimental fluid. As tracer particles we chose particles of Mearlin Luster Pigment Silkwhite M.S.W.-139 (Mearl Corporation, New York). The particles were added to the bulk of the glycerol from a suspension. Their average diameter was found to be about 10 pm. Vessel (1) is provided on the outside with an electrical heating wire (2kW), surrounded by thorough insulation. Due to the hygroscopic qualities of the glycerol it seemed useful to give vessel (1) an air-tight construction. This vessel is also provided with a ribbon-stirrer, which can be used to improve the heat transfer to the glycerol from the electrical wire and to minimize the temuerature differences in the glycerol. A gear pump (2), driven by a variable speed motor, pumps the liquid via a Multiflux-mixer (3) into the heating section (4) and via a by-pass back into the vessel (1). By va$ng the speed of the gear pump and by making use of this by-pass system the desired volume rate of flow can be adjusted. The Multiflux-mixer (3) is used to eliminate radial temperature differences in the fluid. It is a mixer without moving parts. The fluid flows through a series of similar elements in each of which the stream is divided, deformed and rearranged in such a way that inhomogeneities are evenly dispersed over the entire cross section. It is an excellent apparatus for mixing viscous fluids. For further information see e.g. Sluyters [5]. The Multiflux-mixer (3) has been constructed of copper.
and A. VAN DER LEE
After passing through the mixer the fluid enters into the heat transfer section (4) which consists of a copper tube of 12 mm i.d. and I.50 m length. The copper tube is constructed in a thermostat system in which water at constant temperature is circulating at very high speeds in order to make the heat transfer resistance at the outside of the tube negligible compared to the inner heat transfer resistance. Temperatures are measured by means of 1 mm thick chromel/alumel thermocouples (Pyrotenax). Immediately behind the mixer (3) a thermocouple (t,) measures the uniform temperature of the fluid entering the heat transfer section. Between the Multiflux-mixer and the copper tube a PVC section of about 3 cm length is fitted which has exactly the same inner diameter as the copper tube. This has been done in order to eliminate axial conduction of heat along the tube wall from the heat transfer section to the mixer or vice versa. Furthermore this intermediate piece of tube is a guarantee that in all our experiments the fluid entered the heat transfer section hydrodynamically fully developed. The Multifluxmixer (3) and PVC tube both are very thoroughly thermally insulated. In order to avoid errors in the measurement of the entry temperature due to conduction along thermocouple leads the thermocouple tl is inserted through the PVC tube and led for about 1 cm in the opposite direction to the flowing fluid and parallel to the tube axis. The wall temperature of the copper tube is measured with three thermocouples (t, ts and tJ which are inserted in sleeves in the tube wall along the axial direction. In all our experiments the temperature difference between tZ and t, was less than 0*2”C, usually less than O*l”C. At the end of the heat transfer section the copper tube is followed up by a perspex tube of exactly the same diameter and a length of 0.25 m. Everything feasible was done in order to obtain a very smooth transition of the copper tube into the perspex tube. The perspex tube is surrounded by a perspex box, together forming the optical system. After passing through the optical system (5) the glycerol is collected in a vessel (6) from which it can be pumped back to vessel (1) by means of the gear pump (2). The mass flow rate is determined by weighing the glycerol collected in a glass beaker during a known time-interval. The optical system is quite similar to that used by Atkinson et a1.[43.
Fig. 1. Scheme of the flow-visualization apparatus.
1318
Non-isothermal
laminar pipe flow- II
gtyc~!_--_____-~----_____~~~_~
glycerol in
1
mercury
2
collimating
3
adjustable
slit
4
focussing
Ions
vapour
lamp
lens
5
perspax
6
experimantal
7
camera
box section
Ians
Fig. 2. Optical arrangement. The refractive index of the perspex used is nearly equal to the refractive index of glycerol at room temperature. An area about 2 cm long and 0.5 mm deep on a plane along the tube axis is illuminated by the arrangement shown in Fig. 2. The illumination used was provided by a high-intensity mercury vapour discharge lamp (80 W) operated directly at mains frequency, providing illumination discontinuously at 100 flasheslsec. The path of each moving particle is thus seen as a discontinuous streak. As the paths of many particles are recorded simultaneously, a complete velocity profile may be derived from a single exposure. The discontinuous light-beam from the mercury lamp (1) was collimated through a lens (2) on to an adjustable slit (3), whose image was focused on to a second slit which was part of the perspex box (5) surrounding the experimental section of the tube. This perspex box has been applied for two reasons. In the first place it helps to minimize refraction effects due to the curvature at air-perspex-glycerol surfaces and in the second place as a means of minimizing further heat transfer to the perspex tube. Both difficulties were overcome to a major degree by filling the perspex box with glycerol of the same temperature as the wall temperature of the copper tube. By means of a gear pump glycerol is pumped at low speed from a thermostat through the perspex box. The normal surfaces of the box were carefully aligned and light passing through the tube was reflected away by a surface set at 45 degrees to the direction of the incident light. The illuminated plane was viewed by a camera at right angles to the optical axis of the main light beam. All surfaces of the box were painted black except for that facing the camera, and the slits where the main light beam entered and left the box. The camera used was a Hasselblad camera with Kodak Tri-X-PAN 27 DIN film. High-contrast enlarged prints of the flow field were made and from these the velocity profile could be measured. Each particle in the illuminated area appeared on the photograph as a series of dashes, each dash representing the path of the particle during a single flash of the mercury lamp (mains frequency being 50 Hz, this corresponds to a time interval of 0.01 set). The position of the
particle relative to the tube wall could easily be determined as could the distance of travel for a given number of discharges. The distance travelled by a particle in a single interval of 0.01 set could be determined. In order to check the scaling of the photographs due to refractive effects a test grid was made out of a thin (0.3 mm) piece of copper plate of length 1 cm. This copper plate was adjusted in small sleeves in the wall of the perspex tube on the horizontal plane along the tube axis and was provided with a calibrating system of horizontal and vertical lines at 05 mm intervals. We now made some photographs of this test grid, pumping glycerol without particles through the apparatus. We did these experiments under isothermal conditions and also with temperature differences (TU,-TO) from -40°C up to +4o”C. We could not detect any differences in the scaling systemat all on our photographs, so we concluded that refmction effects did not influence our measurements. After these preliminary experiments we removed the test grid and started our experiments with a new perspex tube.
Results
For the physical constants of the glycerol we refer to the Appendix. In all our experiments we compared the volume rate of flow calculated from the measured velocity profile with the volume rate of flow calculated from the directly measured mass flow rate at the exit of the experimental apparatus: R
I
27rrv, dr = QV= >.
0
The difference between these two quantities was nearly always smaller than 3 per cent, usually smaller than 2 per cent.
1319
P. B. KWANT, R. H. E. FIERENS
and A. VAN DER LEE
In order to check the optical part of the experimental apparatus we did some isothermal experiments at different temperature levels and with different volume rates of flow all of which agreed very well with the parabolic velocity distribution (see e.g. [2]). We performed about thirty non-isothermal experiments in which the dimensionless length varied from X+ = 3.6. 1O-3 up to X+ = 10-l. The corresponding Reynolds-numbers, calculated at an average temperature (To + ( T,) )/2, are in the range (Re) = 0.24-5.6 for heating and (Re) = 1.1-55.0 for cooling. Q-values varied from Q = -2.10 up to Q = +260. The quantity Q has been defined [ l] as follows:
Q = HT,-Tt,)t
(1)
in which b is the viscosity variation parameter: K = K’ exp {-b(T,-
T’)}.
(2)
From Eqs. (1) and (2) follows:
Q = In (&I&,).
I (3) Fig. 3. Non-isoviscous
Equation (3) reduces for Newtonian fluids to:
Q = ln (nhw).
velocity profile for Newtonian pipe flow (cooling).
(4)
In all experiments the agreement between the measured velocity profiles and the theoretically predicted profiles according to [I], calculated either numerically or with a first-order approximation, was very good. In the Figs. 3 and 4 we show two representative measurements of a cooling- and a heating run. We plotted the dimensionless velocity vs the dimensionless tuberadius. As a reference also the isoviscous, parabolic distribution has been given (Q = 0). Lines are according to theory[ 11, crosses and round points refer to measurements. Crosses refer to measuring points situated in the left part of the illuminated plane, round points refer to the right part of this plane, this distinction being useful as a confirmation of the absence of systematic maldistribution. As can be seen the 1320
flow (heating).
Non-isothermal
laminar pipe flow- II
Fig. 5. Experimental set-up for heat transfer- and pressure drop measurements.
agreement between theory and very good. It should be remarked experiments we could not detect radial component of the velocity photographs.
experiments is that in all the any significant on any of the
NON-ISOVISCOUS PRESSURE DROPHEAT TRANSFER MEASUREMENTS Experimental
AND
set-up
The experimental apparatus is shown in Fig. 5. As experimental fluid we used a very viscous Newtonian oil, Shell Nassa 89. The fluid is stored in a vessel (1) in which it can be thoroughly mixed by means of a ribbon-stirrer. The fluid can be heated in the vessel by means of a heating spiral (2kW), which is attached to the inside of the wall of the vessel. Two thermocouples were soldered on this heating spiral in order to check the temperature near this spiral, which never raised above 90°C in order to prevent deterioration of the oil. A gear pump (2), driven by a variable speed motor, transports the fluid either via a thermostat bath (3) and a multiflux mixer (4) into the heat transfer section or via a by-pass back into the vessel (1). By varying the speed of the gear pump and by making use of the by-pass system the desired volume rate of flow can be adjusted. In order to adjust the desired temperature of the fluid at the entrance to the heat transfer section, the oil is pumped through a spiral steel tube placed in a thermostat (3). Besides this, the Multiflux-mixer (4) is surrounded by a jacket through which water from the thermostat (3) is circulated at high speed by a pump (5). After passing through the mixer the fluid enters into the heat transfer section, which consists of one or two copper tubes of 1.50 m length and an i. d. of 12 and 6 mm.t This tube
tFor the heat transfer measurements we also used two spiral copper tubes with a length of 504 m and 1.05 m and a diameter of 12 mm each.
is constructed in a thermostat system (6) in which water from a thermostat is circulating at high speed. In order to make the heat transfer resistance at the outside of the tube negligible compared with the inner heat transfer resistance one has to take care of a great water velocity at the outside of the tube. We achieve very high heat transfer coefficients at the outside of the tube by employing a pump (100 I./mm) which via a shower system spouted water from a thermostat over the entire length of the tube. Temperatures are measured by means of 1 mm thick chromel/alumel thermocouples (Pyrotenax). Some of the important thermocouple readings are registered continuously on a recorder to make sure that all temperatures are stationary. If this steady situation has been reached all temperatures are being measured by a compensator. All thermocouples have one common reference in an ice-distilled water mixture. Immediately behind the first Multiflux-mixer (4) the entry temperature ZF,,is measured by a thermocouple (tl). The wall temperature T, of the copper tube is measured by three thermocouples (tz, t, and t4). Thermocouples tl, tS, t, and t4 have been constructed and inserted in the same way as described in the experimental apparatus for flow visualization. The difference in reading between t, and t, was always smaller than 0.2”C, atmost experiments smaller than O-1°C. The copper tube has been provided with eight pressure taps, which are connected via a central tube (7) with the pressure measuring device. The places of the pressure taps on the copper tube are 1.6; 18.3; 23.3; 38.3; 68.3; 103.3; 138.3 and 147.8 cm from the entry. The pressure measuring device is a Texas Instruments precision pressure gage model 156, of which the measuring gage is a quartz Bourdon tube, filled with silicone oil. The main advantage of this system is its quick response. After passing through the heat transfer section the meancup mixing temperature has to be measured. This is felt to be the most difficult part of the heat transfer measurement. In order to prevent any possible heat loss during the mixing process in the final Multiflux-mixer (8), we felt that it was necessary to operate with this mixer under adiabatic conditions. The procedure therefore was as follows. After passing
1321
P. B. KWANT,
R. H. E. FIERENS
through the heat transfer section the fluid enters via a PVCtube of 3 cm length and the same id. as the copper tube into the Multiflux-mixer (8). At the end of the mixer thermocouples tJ and ta are measuring the mean-cup mixing temperature of the fluid. In order to check the temperature uniformity achieved by the mixer the couples tJ and ts are inserted in the fluid leaving the mixer at the axis of the tube and near the tube wall. Both thermocouples again were led for about 1 cm into the opposite direction of the flow to eliminate conduction errors through the thermocouple leads. For all our experiments the difference in reading between t5 and ta was less than O*l”C. To ensure adiabatic operation of the mixer (8) we made a construction, which is schematically shown in Fig. 6. The Multiflux-mixer, the construction material of which is copper, the PVC-tube in between the mixer and heat transfer section and a piece of tube behind the mixer are wrapped around with glasswool insulation. Around this insulation a spiral heating wire was wrapped, supported by a copper gauze. The gauze is present to ensure that the heat developed by the wire is distributed as evenly as possible through the insulation. Around the heating wire again glasswool is present. At a distance of 1 cm from the outer wall of the Multifluxmixer three thermocouples t,, ts and ta are located. At every experiment the electrical current through the wire was adjusted so that the thermocouples t,, ts and ts indicated a temperature which was equal to the mean-cup mixing temperature measured by thermocouples t5 and tp. In all experiments the differences in reading between t,, ts and tg were less than 0.3”C once all the temperatures were stationary. It can therefore be concluded that we measured the average mean-cup mixing temperature correctly. The accuracy of this temperature measurement is estimated as O.l”C. After passing through the second Multiflux-mixer the oil is collected in another vessel (9), from which it can be pumped back into vessel (1) by means of the gear pump (2). The mass flow rate is determined by weighing the oil collected in a glass beaker over a known time-interval. It should be pointed out that the heat transfer and pressure drop measurements were usually carried out separately in different set-ups in order to concentrate better on each of them. We performed also a few heat transfer experiments with glycerol and pressure drop measurements with a pseudoplastic liquid (a mixture of 80 per cent wt glycerol/water to which 0.3 per cent wt Carpobol has been added).
Fig. 6. Detail of the second Multiflux-mixer.
and A. VAN DER LEE
The physical constants of all fluids are given in the appendix. We checked that for all experiments the fluid entered the heat transfer section fully developed and that viscous dissipation was absent.
Results of pressure drop measurements The quantity C#J’ has been defined [ 11as the ratio of the true and the isoviscous pressure gradient, the latter quantity taken at the tube wall temperature:
Inserting the known relation for (-dp/dx)O,,w for Newtonian fluids we find:
Analogously one can write for the total pressure drop ratio for Newtonian fluids: D4(-APlAx)
r ‘128.
(7)
We now operated as follows. For every x+ corresponding to the axial length in the middle of two pressure taps, the pressure gradient (-d&b) was approximated by the pressure drop (--AP/Ax) over these two pressure taps. In doing this we are introducing an error which however is very small because the distance of the taps deliberately has been made very small in the beginning of the heat transfer section where the changes in pressure are high. The lirst pressure tap at 1.6 cm distance from the beginning of the tube was not used in the determination of the pressure gradients. For some experiments we measured the real pressure gradient (-d&lx) from the graph of pressure against axial length. It appeared that the difference between both quantities was always smaller than 3 per cent. So, in one experiment we gather six @-data from seven readings. On every x+, corresponding to the axial length of the last of two pressure taps we determined the average pressure drop (-AflAx) from x = 0 to x = x. In order to determine the pressure drop in a correct way the first pressure tap from the entry of the heat transfer section has to be present near x = 0. We situated this tap at 1.6 cm from the beginning of the tube and therefore it is possible to extrapolate to x = 0 accurately to get the correct value of the pressure at x = 0. tFor Newtonian fluids the pressure gradient ratio C#J’ equals the ratio of the non-isoviscous and the isoviscous velocity gradient at the wall.
1322
Non-isothermal
Fig. 7. Experimental
laminar pipe flow- II
pressure gradient ratio for non-isoviscous,
The fourth pressure tap at x = 38.3 cm from the entry is the first one used for the calculation of (4’). We estimate the possible error in (-d&Lx) as 3 per cent and that of (-AP/Ax) as 2 per cent. The diameter D of the tube was determined by accurate isothermal measurements in which T, and T, differed by less than O.l”C from each other. This has been done at different temperature levels and for different volume rates of flow. Under all these conditions and for all axial parts of the straight tubes the spreading in D4 was less than 4 per cent. The possible error in D therefore has been estimated as I per cent. The accuracy of the determination of the viscosity is also estimated as 4 per cent (see Appendix). The volume rate of flow, calculated from the mass flow rate and the fluid density is probably accurate within 0.5 per cent. We estimate therefore that the maximum error in the dimensionless pressure gradient 4 is about 12 per cent and the maximum error in the dimensionless pressure drop is about I I per cent.
All experiments have been carried out with Shell Nassa 89 oil in either of two straight tubes of 1.50 m length and 12 mm and 6 mm dia. The Q-values were Q = +0*69; +1*12; + l-95; +2*60; -1.12; -1.95; -260 within 3 per cent. For the heating experiments (Q > 0) the inlet temperature was 20°C and the wall temperature either 30, 40 or 50°C. The heating runs with Q = +0*69 had an inlet temperature of 40°C and a wall temperature of 50°C. For the cooling experiments (Q < 0) the wall temperature was 20°C
Newtonian
pipe flow.
and the inlet temperatures used were 30, 40 and 50°C. Reynolds-numbers related to the entry temperature varied from Re, = 0.1 up to Re, = 3.5 for heating and from Re, = 0.25 up to Re,, = 53 for cooling. Some results of measurements of the pressure gradient ratio 4 are shown in Fig. 7, the results of the measurements of the total pressure drop ratio (4’) are given in the Figs. 8-10. The ratios 4 and (4’) have been presented as a function of the dimensionless length X+ with Q as parameter. All experimental results (crosses for the tube with D = 12 mm and points for the tube with D = 6 mm) are compared with the values predicted numerically (drawn line). It appears that from x+ = 5 . 10e4 up to x+ = 3.10-l the agreement between theory and measurements is very satisfactory. The greatest deviation occurs in the cooling experiments with the greatest temperature difference (Q = -2.60) in the region of X+ = lo+ up to x+ = 8. lo+. Except for these experiments the deviation of theory and experimental data is nearly always within 12 per cent. We performed also pressure drop measurements of non-isothermal flow of a pseudo-plastic
1323
P. B. KWANT,
‘6’1
R. H. E. FIERENS
and A. VAN DER LEE
I
I
w-2
UJJ Fig. 8. Experimental
pressure drop ratio for non-isoviscous,
fluid (0.3% wt Carbopol in an 80% wt glycerol/ water mixture), having a flow behaviour index n = 0.55. Measurements were done for Q = +l and Q = -1. The results of these experiments, which are not given here but can be found elsewhere [2] show the same good agreement between measurements and theory.
lo’-
I
I
w”
10-l
Newtonian pipe flow.
Results of the heat transfer measurements We can calculate the average Nusselt-number according to: In(&)
(Nu) =& We estimate
r1111,
=&ln(2<:“).
the accuracy I
I
I
I
of the temperature ,11,1-
-<‘PI> (:I
it
-=yi+k *t-%L13*+_ 8-w -yo. *“+sb
Fig. 9. Experimental
pressure drop ratio for non-isoviscous,
1324
(8)
Newtonian pipe flow.
Non-isothermal
Fig. 10. Experimental
pressure
laminar pipe flow - II
drop ratio for non-isoviscous,
measurements to be 0.1°C. The accuracy of the determination of the Nusselt-number is very dependent on the value of the average outlet temperature (T,) and consequently upon x+. At very small x+ the possible error will be great because (T,) is very near T,,, at high x+ the possible error is large again because (T,) now is near T,. One can calculate that for our experiments the accuracy is 2 lo-20 per cent for IT,-T,I= 15°C; 5-10 per cent for IT,-T,I= 30°Cand 3-6 per cent for IT,- ToI = 50°C. The results of the heat transfer experiments are shown in Figs. 1 l(a-h). In these graphs the measured Nusselt-numbers divided by the Nusselt-numbers predicted numerically are given as a function of x+ for different values of Q and different tube dimensions. This has been done in order to eliminate the spread in Q between the individual measurements of a series. Every experimental Nusselt-number now is compared directly with the Nusselt-number predicted numerically for the corresponding Q-value. In addition to this feature this way of presenting the data gives at once the deviation of theory and experiment.
Newtonian
pipe flow.
All heat transfer experiments have been performed in tubes with an i. d. of 12 mm. The experiments given in Figs. 1 l(a-e) have been carried out with Shell Nassa 89 in spiral tubes of length 504 m and l-05 m.t The temperature differences (T,- T,,) were: 15”C, 30°C 50°C -10°C and -15°C corresponding to Q-values of Q = 160, 2.70, 3.80, -140 and -1.50. In the heating experiments Reynolds-numbers referring to initial conditions varied from O-02 to O-30; in the cooling-runs Reynolds-numbers were between O-07 and O-90. The experiments given in Fig. 1 I(f) were performed with glycerol in a straight tube of 1.50 m tin these cases the ratio of the diameter of the spiral U&al ) and the inner tube diameter was about 25. It can be found in literature (see e.g. Richter[6] and Truesdell[7]) that such a curvature of the spiral tube does not influence the velocity distribution and hence does not influence the heat transfer if the Dean-number is smaller than one. De-Re
4-
D r
(9)
This implies that in our case the Reynolds-number should be smaller than five. In all our heat transfer experiments this condition was met.
1325
P. B. KWANT, R. H. E. FIERENS
and A. VAN DER LEE
Fig. 1 la.
I
I
I
III,,
I
I
II111,
I
~N-exp. I.*I.
0
L=l.OzSm
c?=3.60
$ 0
_----P--,-o-_
--------
c-t
i
I
,
I,llll
I
I
I
l,lll,
I
QJ - 1.00
1.0
I
I
0 L=l.O5m
1.2_-
0
-_
~oO_O--_--
-
-
0 0
tt
I
Q8 lo-3
___t
#111#1
x1
I I IIII
I
I
I lc+
KY2
Fig. 1 Id. I
I
I
IIIII
I
I
I
I
IllI,
x
0
Q I 00 ID
---00 tt
0.8 lo-3
o
-_o__-
x-----_
x-
X
x’
L=5.04m L=l.O5m -1.50
I
I
Non-isothermal I
I
I
IllI,
I
I
I
I
!111l,
Glycerol
12-
D -._-o__-~-x~--x0 x
--
I
1.2 -
,
,
x Qz -1.30
l.0
1
I
I
I
I#(,
0 Qc -2.10
x x ---
Ic -
---
r
I
-
1
,111,
I
Straight tube x D=l?mm
_
Q =-2.60
K
xx 1.0
laminar pipe flow-II
_
-
X
L_
--x-
-------
_x_
--I
I
1.2 -
1.0
I
I
,
I
,111,
I,!,,
I
___---)E---
X X
I
I
Straight tube x Dd2mrr-1 0 = -3.70
ze ---
--
---
%
1-t
0s10-3
I
-
X I
I
I
II
I
-2 10
x* I
1 I
III
I
I
lc-’
Fig. 1 lh. Fig. 1 l(a-h). Experimental heat transfer results for Newtonian pipe flow.
length. Q-values were approximately Q = -1.30 and -2.10 corresponding to temperature potentials of about 18 and 40°C. Reynolds-numbers were in the range Re, = l-55-55. The experiments given in Figs. 1 l(g) and (h) have been carried out with Shell Nassa 89 in straight tubes of l-50 m length. Q-values were Q = -260 and -3.70 corresponding to temperature driving forces of 30 and 50°C. Reynoldsnumbers varied from O-75 up to 14. In the results of Fig. 1 l(h) (Q = -3.70) we applied a correction for the enhancement of the heat transfer rate due to free convection. This correction factor was proposed by Oliver[8].t- There may be some uncertainty in this correction term which amounts from -2.5 per cent up to - 16 per cent. For this reason we estimate the accuracy of the results in this cooling run somewhat lower than in the other series. As can be seen from the results given in Figs.
1 l(a-h) the agreement between theory and experiment is very good. With a few exceptions all measurements agree within 8 per cent with theory. It should be added that we also correlated our tTo allow for free convection has to be multiplied by a factor:
the mean Nusselt-number
, + 04083(GrmPr,)U’75 I’3 (b&L) 1 . [ The mean Grashof-number
is defined as follows:
(p), (p) and (n) are the coefficient of expansion of the fluid, its density and its viscosity, all taken at the average temperature (T,, + (T,))/2. In the mean Prandtl-number, Pr,, the physical quantities also have to be taken at this average temperature. With this correction term as a criterion we calculated that for all other heat transfer results, Figs. 1 l(a-g), the influence of free convection may be neglected.
1327
P. B. KWANT,
R. H. E. FIERENS
and A. VAN
LEE
NOTATION
results according to the wellknown, empirical Sieder and Tate correction method[91. This correction to allow for isoviscous heat transfer coefficients with a factor ( (~)/~~)“‘14 gives significantly larger errors for the measured heat transfer coefficients than those associated with the numerical prediction (see [2] for the details).
= hlpc,,
CP
D D spiral De g
CONCLUSIONS
Measurements were carried out of non-isothermal velocity profiles and pressure drops and of mean heat transfer coefficients in laminar flow of Newtonian liquids. The viscosity of these fluids is strongly dependent on the temperature. There is a good agreement between the experimental results and the theoretical values predicted by a numerical analysis in a previous article [ I]. We may conclude therefore that the neglect of radial velocities and changes in axial velocity from the transport equations is acceptable for viscous liquids under practical heat transfer circumstances (x+ > 10e3 and -3 < Q < +3). We expect that also for pseudo-plastic fluids the results from the analysis can be used with confidence, because there are no principal differences with Newtonian fluids concerning the deviation of the isoviscous behaviour. Pressure drop measurements with a pseudo-plastic fluid (n = 0.55) for Q = + 1 and Q = - 1 showed also a good agreement with theory. However, with increasing pseudo-plastic behaviour, especially for large negative values of Q, the distortion of the isothermal velocity profile increases considerably (see [ 11). The distortion being greater, the more important will be the radial velocities in the first part of the heat transfer tube. For small x+, therefore, more experimental work seems to be needed in the area of increasing pseudoplasticity (n < 0-S) and large temperature intervals in order to assess the region of validity of the theoretical results.
DER
Gr Gr, K Ko K’ L n (Nu) (NM), (Nu),xp C--dplW (-AplW Pr
temperature diffisivity, m2/s viscosity variation parameter defined by Eq. (2), “C-l specific heat, J/kg “C tube diameter, m diameter of spiral, m = Re (DIDspiral ) 112,Dean-number gravitational acceleration, m/s2 =
PATD3p2g
, Grashof-number r12 Grashof-number at an average temperature ( To + ( T,) ) /2 consistency-index of power-law fluids, Nsn/m2 consistency-index at entry temperature, Nsn/m2 consistency-index at a reference temperature T ’ , NsVm2 tube-length, m flow behaviour index of a power-law fluid logarithmic mean Nusselt-number logarithmic mean Nusselt-number for non-isoviscous flow logarithmic mean experimental, Nusselt-number pressure gradient in axial direction, N/m3 pressure drop in axial direction, N/m3 =- ‘, Prandtl-number a
Prandtl-number at an average temperature ( To + ( T,) ) /2 Q = b (T, - To), quantity giving the deviation from isoviscous behaviour r radial coordinate, m r+ = r/R, dimensionless radial coordinate R tube-radius, m
Pr,
Re
P(v,)D
= p,
Reynolds number
7 Acknowledgement-The authors thank Ir. P. H. F. van Voorst van Beest who performed most of the flow-visualization measurements.
1328
(Re) To
Reynolds-number at an average temperature (To + (T,) )/2 entry temperature, “C
Non-isothermal
laminar pipe flow- II
(Te) mean-cup mixing temperature T' AT (2 V+
X X+
at the end of the heat transfer section, “C reference temperature, “C temperature difference, “C axial velocity, m/s average axial velocity, m/set = v,.( v,) , dimensionless axial velocity axial coordinate, m dimensionless
=$j&
axial co-
Greek symbols
of expansion
dynamic viscosity at an average temperature (T,, + ( T,) )/2, Nsec/m2
(0)
=++4
dimensionless
mean-
cup” mix;ng temperature at the end of the heat transfer section A thermal conductivity, W/m”C Y kinematic viscosity, m%ec p density, kg/m3 (p) density at an average temperature (To+ (T,) )/2, kg/m3 4’ pressure gradient ratio, Eq. (5) (4’) pressure drop ratio, Eq. (7) 4Jm mass flow rate, kglsec 4” volumetric flow rate, m3/sec
ordinate
/3 coefficient
(q)
of the fluid
y-1 (p)
q q.
coefficient of expansion of the fluid at an average temperature (TO+ (T,) W, ‘C-l dynamic viscosity, Nsec/m2 dynamic viscosity at entry temperature, Nsec/m2
Indices
0 e w
referring to isoviscous conditions or referring to entry conditions referring to end conditions referring to wall conditions
REFERENCES
[II KWANT P. B., ZWANEVELD A. and DIJKSTRA, F. C., Chem. Engng Sci. 1973 28 [21 KWANT P. B., Ph.D. thesis. TH Delft 197 1.
M. P., CARD C. C. H. and SMITH J. M., A.1.Ch.E. .I1 1969 15 548. [31 ATKINSON B., BROCKLEBANK 1967 13 17. [41 ATKINSON B., KEMBLOWSKI Z. and SMITH J. M.,A.Z.Ch.E.JI [51 SLUYTERS R., De lngenieur 1965 77 Ch. 33. 161 RICHTER H., Rohrhydruulik Springer, Berlin 1958. 1970161010. [71 TRUESDELLL.C.JR.andADLERR.C.,A.I.Ch.E.Jf [81 OLIVER D. R., Chem. Engng Sci. 1962 17 335. 191 SIEDER E. N. and TATE G. E., 2nd. Engng Chem. 1936 28 1429.
APPENDIX Physical properties of the experimental&ids used The density of Shell Nassa 89 and of glycerol, measured by standard methods is given in Fig. 12 as a function of the temperature. The viscosity of Nassa 89 and glycerol was measured in a rotational viscometer (make Haake “Rotovisco”) as a function of the temperature. The results are given in Fig. 13. We see from this figure that for large temperature differences the deviation from the exponential viscosity-temperature rehttionship is considerable, especially for Nassa 89. One should
note, however, that in practice the value of Q is calculated by means of Eq. (4): Q = In (no/n,). In this way one calculates an average value of the viscosity variation parameter b (see Eqs. (1) and (2)) over the relevant temperature trajectory (T, - T,,). This is the reason that also fluids of which the viscosity does not follow exactly the exponential tempetature dependence will obey the predicted, non-isoviscous behaviour. The good agreement between theoretical predictions and the experiments with Nassa 89, even for the largest temperature differences, is another argument for this statement.
1329 Cl23 Vd. 26 No. 6-G
P. B. KWANT,
12
20
R. H. E. FIERENS
and A. VAN DER LEE
40
30
50
60
Fig. 12. The density of the experimental fluids as a function of the temperature.
“\
WE
I
“‘\.\ “\
“\
*\ -a.
%
ld“lc$
i,
*\ m
I M
I x)
I
1
40
50
xl
q-c]
I
I
63
70
m
Fig. 13. The viscosity of the experimental duids as a function of the temperature. The temperature diffusivity a (= A/PC,) of the experimental fluids is determined by means of a non-stationary heat penenation method, for the details of which we refer to [2]. Experiments at different temperature levels (20-50°C) did not show any difference in the values of the thermal diffisivity a
outside the experimental results are: Shell Nassa-89 Glycerol
1330
scatter (= 044. 10-7m2/sec). The a = 0.81 . 10-r mYsec a = 0.88 . IO+ m2/sec.