An experimental method for determining local mass transfer coefficients at a solid surface, and its application to the case of

a circular cylinder in a transverse air stream

(Zleceived 6 October 1959)

INTRODUCTION

SEVERAL authors have described a technique for studying

interfacial mass transfer phenomena in which the rate of

sublimation or solution from the surface of a solid into a

stream of fluid is determined from measurements of the

dimensional changes of the solid ([l], [2], [3], [4], [5]).

This technique promises to be of particular value in

investigations of the fundamental mass and heat transfer

processes in two-phase two-component systems, for which

it offers important advantages over other methods. In

the first place, solid surfaces are free from the complicated

hydrodynamic instability effects associated with the

surfaces between fluids in relative motion, such as often

obscure the interpretation of experimental results for

gas-liquid systems. Secondly, the gauging of local changes

of dimension makes it possible to determine local mass

transfer rates at specified points in a solid surface ; whereas

at fluid interfaces, or at solid surfaces from which the loss

of material is estimated by weighing or by chemical

analysis of the fluid phase, only the average rate of mass

transfer over the whole area can be found.

In accurate work with this technique the dimensional

changes which develop in the course of an experiment

must be kept small enough to leave the flow pattern

round the solid essentially unchanged. It is therefore

necessary to use very delicate means of measuring local

configurational changes to obtain accurate mass transfer

data. No measuring device so far used in these investiga-

tions is sufficiently sensitive to yield data of high accuracy

compared with that attainable by other means. WINDING

and CHENEY [l] used slip gauges to determine the clearances

which developed between naphthalene cylinders exposed

to a transverse air stream and the moulds from which

they were originally cast. The absolute accuracy of this

method is limited by the gauges available to about

f 0.0025 cm, corresponding to an accuracy of f 10

per cent on a dimensional change of 0,025 cm - a change

likely to affect the fluid flow pattern in many cases. The

photographic techniques used by FROSSLIN~ [4] and

GARNER and GFLAFTON [5] are apparently similarly insensi-

tive. CHRISTIAN and KEZIOS [2] measured the sublimation

from the surfaces of sharp-edged naphthalene tubes,

exposed to an axial stream of air, by means of a dial

indicator capable of registering movements of the measur-

ing spindle of the order OOOOO5 cm. The accuracy of this

means of measurement is uncertain, however, since the

pressure of a measuring stylus fine enough to record

local changes of surface configuration is likely to indent

the soft surface of the volatile solid.

In the experiments to be described, the measuring

means was a fine jet of air impinging normally on the

surface under examination. The air, supplied from a

constant pressure reservoir via a restrictor or capillary

construction in the delivery pipe, issued from a nozzle

placed very close to the surface. Changes in the back

pressure of the air supply, measured at a point between

the nozzle and the restrictor, provide a sensitive and

nearly linear indication of changes in the distance between

the nozzle and the surface. This technique of pneumatic

gauging, which allows minute local recessions of the surface

to be accurately measured without mechanical contact,

was applied, for trial purposes, to the determination of

sublimation rates at points round the perimeter of a

circular cylinder placed in a turbulent transverse stream

of air.

EXPERIMENTAL

The cylinders used were of acenaphthene. approxl-

mately 1.4 cm in diameter and 7.5 cm long. Acenaphthene

was chosen as the volatile solid on the basis of data given by

MAIN-SMITH [0], because, (unlike naphthalene), it is not so

volatile as to sublime appreciably in the gauging jet or in

the atmosphere of the laboratory in the process of measure-

ment. Each cylinder was cast from a split wooden mould

around a steel spindle 1.25 cm in diameter and about 25 cm.

long. The spindle was then mounted between centres in an

ordinary lathe and the end faces of the acenaphthene

were turned until the cylindrical coating was made just

long enough to span the working section of the wind

tunnel used. The cylindrical surface of the acenaphthene

was also rough-turned in the lathe at this stage. It was

finally turned true and accurately concentric with the axis

of the gauging system in the apparatus shown in Fig. 1.

This apparatus comprised a rigid cast iron stand, having

uprights A, integral with the base, provided with vees in

which the steel spindle B could rest horizontally, as shown,

with one end bearing against the tip of the long horizontal

lead-screw, C. The turning tool D was mounted on a

screw-fed kinematically designed slide, E, having one

degree of freedom, viz. a sliding motion perpendicular to

142

An experimental method for determining local mass transfer coeficients at a solid surface

FIQ. 1. Turning and gauging apparatus.

A-Uprights with bearing vees. B-Ground steel spindle. C - Lead-screw. II - Turning tool.

E - Kinematic tool slide. F - Acenaphthene cylinder. G - Gauging nozzle. H - Parallel spring nozzle

mounting. J - Circular protractor. K - Fixed restrictor.

the axis of the work. The cutting edge of the tool was at the underside, the work being rotated against it in such a direction as to increase the reaction between the spindle and the supporting vees.

The requisite depth of cut having been set by means of the feed screw, the spindle and acenaphthene cylinder F were rotated by means of a handwheel secured to one end, while the lead-screw was slowly rotated by means of its knurled knob. In a few minutes’ operation the whole surface of the acenaphthene cylinder was made to pass under the fixed turning tool.

The same stand and vees were used for the turning and gauging, and the tool acted on the surface at a point near the gauging nozzle, G. Surfaces generated in this way, unlike those turned in a centre lathe, appear almost perfectly cylindrical from the standpoint of the measuring jet ; i.e. the nozzle-surface distance remains nearly constant at all positions of the spindle in the supporting vees, irrespective of small departures of the acenaphthene surface from the straight cylindrical form caused by irregularities in the form of the spindle. The spindles used were silver steel rods of stock size, which are readily available ground to close limits of circularity and diametral uniformity, and selected for straightness ; the acenaph- thene surfaces formed on them could therefore be taken to be perfect circular cylinders from the standpoint of their behaviour in the wind tunnel.

After turning, the cutting tool was withdrawn and the

finished cylindrical surface was freed from powdered material by wiping with cotton wool. The lead-screw was locked in such a position that the mid zone of the acenaphthene coating corresponded with the gauging jet. The gauging nozzle G was mounted in a flexure device, H, essentially a simple parallel spring mechanism furnished with a fine-pitched screw bearing against a reducing lever integral with one of the stiff parallel cross-members. Using this arrangement the nozzle could readily be set suitably close to the surface, to give a high back pressure of air. The mid zone was gauged with the jet at 5” intervals of rotation of the cylinder, a circular protractor J attached to the spindle allowing the angle corresponding to each gauge reading to be measured.

The spindle was then removed from the vees and mounted vertically in the wind tunnel in such a manner that the acenaphthene cylinder spanned the square horizontal working section. The turned ends of the volatile coating fitted close against the tunnel walls, and the spindle projected through close fitting holes drilled in them. The spindle was set so that the zero division of the attached protractor was aligned with the axis of the tunnel.

The air current in the tunnel was induced by a fan driven by belting and pulleys from a constant speed motor. The air intake section was provided with a paper honey- comb and two net gauzes to straighten the flow and promote uniform turbulence ; t,hc Qg& cootraction

143

N. MACIXOD

ratio was about 10 : 1, and the velocity profile in the 7 cm square, ‘70 cm long, perspex working section was found to be satisfactorily ilat, as determined by Pitot traverses 43 cm from the entrance, i.e. at the spindle mounting holes. A Pitot measurement of the wind speed at the axis of the working section and the air temperature were taken at the start and finish of each run, and the average of these was used in calculating the Reynolds number for the run.

After 12-24 hr exposure to the transverse air current, the acenaphthene cylinder and its central spindle were removed and replaced as before in the gauging apparatus. The acenaphthene surface was again gauged round the perimeter of the zone previously selected, in general at 5’ intervals, but at intervals of lo in regions where the gauge pressure was found to change rapidly with angle.

The jet nozzle itself was a thin-walled brass tube, approximately 0.05 cm in internal diameter, and was supplied by Messrs. Negretti and Zambra, together with the fixed restrictor, K ; these components are similar to those used in the flapper-nozzle assemblies of pneumatic control instruments made by this firm, and are designed to operate at an air supply pressure of 180 cm of merrury. Separate calibration experiments carried out with a bench micrometer calibrated in ten-thousandths of an inch indicated that the back pressure of air immediately upstream of the nozzle falls nearly linearly from 180 to approximately 107 cm of mercury as the distance between the nozzle and a cylindrical surface of 1.25 cm diameter is increased from zero to 0.0021Q cm. The constant pressure of the air supply (derived from a compressed air cylinder via a reducing valve) and the back pressure in the nozzle were measured on ordinary Bourdon gauges. As these were sensitive to changes of & 0.5 cm of mercury it appears possible (so far as the pneumatic system is concerned) to measure changes in dimension over the range 0 to OQO25 cm with an absolute accuracy rather better than 5 0*000020 cm, i.e. less than & 1 per cent of the largest measurable change.

In practice, this accuracy was somewhat reduced by frictional and hysteresis effects in the Bourdon gauges used. A further cause of scatter in the observations is thought to have been the presence of small pits and crevices in the surfaces ; this effect might be reduced by in creasing the wall thickness of the nozzle at the opening. The error involved in removing and replacing the spindles in the gauging apparatus appeared to be negligible compared with these effects. The reproducibility of determinations of relative local rates of mass transfer at different points round the perimeter of the cylinders was found to be of the order -J= 5 per cent.

RESULTS AND DISCUSSION

Local sublimation rates

Polar plots of rate of acenaphthene removal vs. angular distance from the forward stagnation point are shown in Fig. 2. The curves relate to air stream Reynolds numbers

(based on cylinder diameter) of 10,000, 15,500 and 21,500, corresponding to wind speeds in the range Q-21 m/set. The points on this figure were obtained by converting the pressure gauge readings into surface-nozzle distances by the use of the nearly linear calibration curve obtained with the bench micrometer previously referred to, and subtract- ing from each of these distances the corresponding distance obtained before exposure of the cylinder in the wind tunnel. The results, divided by the length of run (and the molecular volume of the solid), give relative measures of the sublimation rates. Smoothed curves were then constructed by averaging these relative rates of acenaph- thene removal for pairs of corresponding points symmetri- cally placed with respect to the wind direction. Curves I and II were obtained at the same Reynolds number (10,000). The maximum discrepancy between them is 75 per cent.

The curves are similar in general form to those obtained by other workers at similar Reynolds numbers. In the Reynolds number range of these experiments, from 10,000 to 21,500, it is seen from Fig. 2.

(1)

(2)

(3)

The maximum sublimation rate is at the forward stagnation point, and is approximately 18 times that at the rear. This compares with the ratio 1.5 of POWELL and GRIFFITHS [7] and 1.4 of Lonn~scn [8] for Reynolds numbers in the range used here.

The sublimation rate at the forward stagnation point is approximately I.8 times the average over the whole surface (obtained by graphical integra- tion).

The minimum sublimation rate is at the pair of points about 85’ from the forward stagnation point. It is about 30-40 per cent of the mean value, and there is some indication that it increases relative to the mean sublimation rate with, increase of Reynolds number.

For the flow of water past 1.25 cm soluble spheres in the Reynolds number range from 20 to 1,000, GARNER and GRAFTON [5] have shown that the point of minimum mass transfer corres- ponds to the point of flow separation, determined by visual observation of the flow pattern ; near the separation point the boundary layer within the wake appears almost stagnant. Similarly the minima on Fig. 2 occur at approximately the same position (85’ from the forward stagnation point) as the points of flow separation for circular cylinders in the same Reynolds number range determined by HIEMENZ, FACE and others [Q].

Rectilinear plots show indications of points of inflection, possibly of local maxima, in the sublimation rate curves at points between 1OW and 120’ from the forward stagna- tion point. The nature and location of these points is made uncertain by the exceptional scatter of the experimental data in their neighbourhood, They may have to do with the influence of the tunnel walls on the flow past the

144

An experimental method tar determining local mass transfer coefficients at a solid surface

FIG. 2. Local rates of mass transfer

from the surface of a circular cylinder

transverse to a turblent air stream.

Rate of acenrrphthene removal as a

function of angular distance from forward

stagnation point for 1.4 cm diameter

cylinder at room temperature:

Curves I and II for Re = 10.000

Curve III for Re = 15,500

Curve IV for I% = 21,500

(Air stream Reynolds numbers

based on cylinder diameter).

cylinder. It appears that two of WINDING and CHENEY’S curves for the same Reynolds number range and a similar ratio of tunnel width to cylinder diameter (4.2) show a similar inflection, though this is not discussed in [l].

Mass transfer coeficients. Vapour pressure of awnaphthene

For the calculation of mass transfer coefficients from measurements of the rate of dimensional change due to sublimation, data are required for the density and vapour pressure of the subliming solid at the temperature of the experiment. Here the density was taken to be that given for solid acenaphthene in International Critical Tahles. The density of the cast material used may have differed from this by a few per cent. The only published data for the vapour pressure of solid acenaphthene appear to he those given in a paper by LINDER [lo]. These values are evidently much too high, however, being of the same order as those for naphthalene, which rough pneumatic gauging experiments indicate to he far more volatile. To check this, the over-all mass transfer coefficient was calculated from the data from Run 3 (Curve III), by graphical in- tegration over the whole perimeter of the cylinder, assum- ing LINDER’S value for the vapour pressure of acenaph- thene ; j, was then calculated from this, assuming a Schmidt number of 2 for the system acenaphthene-air, and an equivalent heat transfer coefficient was found from the assumed equality of j, and j,. The calculated result, 2.0 kg cal/hr ms “C, is very much helow that given by well established correlations for heat transfer data from single cylinders. Thus MCADAMS’ equation 10-2 of [ll]

gives at the Reynolds number of Run 3 a heat transfer eo- efficient of approximately 88 kg cal/hr m2 “C. It is not likely that a large part of this discrepancy is due to an error in the assumed value of the Schmidt number, i.e. to an abnormally low value for the difhusivity of ace- naphthene ; it thus appears probable that LINDER’S value for the vapour pressure of acenaphthene is very much tot high.

CONCLUSIONS

The measurement of dimensional changes due to sublimation by means of pneumatic gauging has been shown to provide a delicate method for measuring local mass transfer rates from a volatile solid. The gauging apparatus can readily be made sensitive enough to measure configurational changes in the range 04.0025 cm with an accuracy of the order * 5 per cent ; i.e. it can be used to measure accurately changes small enough to leave the flow pattern round the solid essentially unchanged during an experiment.

Measurements of the relative rates of mass transfer at different points on the perimeter of an acenaphthene cylinder placed transverse to a turbulent stream of air have been determined by this means, and agree sub- stantially with those of other workers. Absolute values of local mass transfer coefficients for this system cannot be compared with data from other sources until the vapour pressure of solid acenaphthene is accurately known. The results of present experiments indicate that the literature value is several times too high.

N. MACLEOD

REFERENCES

[l] WINDING C. C. and CAENEY A. J. Induslr. En@@. Chcm. 1949 49 10.3’7.

[Z] CERISTIAN W. J. and Kmros S. P. J. Amm. Inst. Chem. En@% 1959 5 61.

[S] SHERWOOD T. K. and BRYANT H. S. Can. J. Chm. En&n& 1957 35 51.

[a] FR~~~LINC N. Be&. Geophys. 19.39 32 1’70.

[a] GARNER F. H. and GR~ON R. W. Pmt. Roy. Sot. 1954 A224 64.

[9] Mar~-Snrmz J. D. Chemical Solids as Diffusible Coating Films for Vi.w.01 Indicaliom of Houndmy-Layer Tranaitim in Air and W&r. Aeronautical Research Council Technical Report R. & M., No. 2755, 1954.

[7] POWELL R. W. and GRIFFITH~ E. Trans. I?&. Chem. En@.% 1935 13 175.

[a] LOARI~CH W. Mirt. Forsch. 1929 322 48.

[9] GOLDSTEIN S. (Editor) Modern DeuelopmenLP in Fluid Dynamics, Vol. I, p. 01. Clarendon Press, Oxford 1998.

[lo] LINDER E. G. J. Phys. Chm. 1981 35 591. [II] MCADAMS W. H. Heat !lbammineion McGraw-Hili, New York 1954.

Chemical Engineering Laboratories,

University of Edinburgh and

lieriot- Wait College

Edinburgh 1, Scotland.

N. i,lACLEOU

G. STEWART

146