Deposition of Aerosols from Turbulent Pipe Flow BY I. WILLIAMS AND A. B. HEDLEY Dept. of Chemical Engineering University of Sheffield Mappin Street Sheffield Yorkshire Received 1 lth January 1973 Several approaches to the calculation of the rate of deposition of particles from a turbulent fluid stream on to the boundary walls are discussed. In a turbulent flow system containing suspended particles the need to consider the relationship between the particle eddy diffusivity and the fluid eddy diffusivity in calculating the deposition rate has been indicated. A quantitative estimation is made of the effect of temperature gradients between the fluid and the wall surface in inducing a radially- directed thermophoretic velocity on the particles. A flow system is described in which a turbulent air stream passes through a cylindrical duct.The flow was assumed to be two-dimensional and was characterized by measuring the mean velocities and the fluctuating turbulent velocities of the fluid in the axial and radial directions and also the shear stress profile in the radial direction. From the latter measurement the eddy diffusivity of the fluid was determined. The measurements were carried out at Reynolds numbers of 1.27 x lo5and 2.67 x lo4 and at several duct wall temperatures between 279 and 317 K. Droplets of approximately 1 pm diam. were injected into the turbulent fluid and the results indicate the effect of the flow conditions and wall temperatures on the particle deposition rates. The prediction of the rate of deposition of particles from a turbulent fluid on to boundary surfaces has many important applications.Examples include deposition in atomic reactors spray dryers particle sampling lines and in any flow system where suspended particles are transported from a generating source to the place of applica- tion. A straight smooth walled cylindrical duct offers the most convenient means for studying the rate of particle deposition in turbulent flow systems. The processes by which particles deposit on a pipe wall include,l eddy diffusion gravity settling thermophoresis diffusiophoresis electrostatic effects and inertial effects such as impaction and interception. The parameter K which describes the deposition rate has been defined as K=-amount of particulate deposited per cm2 of surface s-I ~-.the airborne particulate concentration above the surface The approach to the problem has usually been to evolve methods of predicting K for different systems and to correlate the predictions experimentally. In previous investigations the following assumptions were often made. The structure of the turbulent fluid in pipe flow consisted of a laminated boundary layer and a turbulent core. The boundary layer was characterized by three regions (i) laminar sub layer y + <5 ; (ii) buffer layer 5 <y + <30 ; (iii) main boundary layer y + >30 where y+ is the dimensionless variable yu /v y is the distance normal to the surface measured outwards and u is the fluid friction velocity defined as 1.WILLIAMS AND A. B. HEDLEY zo is the tangential shearing stress on the surface over which the fluid flows p is the fluid density and v is the fluid kinematic viscosity. The equation used to describe the rate R of transport of particles from the turbulent core to the wall is R = (D+c,)dC/dy (2) where D is the molecular diffusivity and E is the particle eddy diffusion coefficient due to the turbulence ;C is the concentration of the diffusing substance at a distance y from the surface. The particles were usually assumed to diffuse by eddy diffusion from a constant particle concentration in the turbulent core of the pipe up to and in some theories into the boundary layer. In the diffusion process the eddy diffusivities of the particle and fluid were assumed equal.At the point where the eddy diffusion process was assumed to end the particle was associated with a free flight velocity v and a stop distance ds where ds = vi+z and z is the particle relaxation time. For particles obeying Stokes law of resistance where r is the particle radius rn the particle mass ppits density and q is the viscosity of the fluid. The value of v was usually equated to a function of the root-mean- square radial resolute of the fluid fluctuation velocity vi+. Friedlander and Johnson derived deposition velocities on the basis of the above postulate. They assumed that 2’ = 0.9 u,. This figure seemed unreasonably high and according to the fundamental turbulence measurements made by Laufer this velocity existed at a distance y-t = 80 which was within the turbulent core and not within the boundary layer.Even using such a high initial velocity the particle stopping distance was often less than the thickness of the laminar sublayer. This led Friedlander and Johnson to use the hypothesis of Lin et aL6 who determined the following empirical expression for Ef the fluid eddy diffusivity within the laminar sublayer cf/v = (~‘/14.5)~. (4) According to this model eddies from the turbulent core at a distance y+ = 80 penetrated the boundary layer and retained their momentum until they were within a distance S+,from the wall where S+ = 0.9z+ and z+ the dimensionless particle relaxation time was equal to zu,2/v. A finite eddy diffusivity within the laminar layer was assumed.When S+ was calculated using the actual values of u’ at y+ = S+ transport coefficients were obtained which were four orders of magnitude lower than those found experimentally by Friedlander and Johns~n.~ Davies ’derived a deposition scheme in which he considered both inertial depo- sition and deposition by Brownian diffusion. The particle radius was taken as the distance of closest approach to the deposition surface. The main difference between this theory and that mentioned previously for inertial deposition was that Davies calculated his free-flight particle velocity from an analytical expression derived from the measured turbulent velocity data in fully-developed turbulent pipe flow derived by La~fer.~ He determined the free-flight velocity at a distance from the wall where he considered free flight began not as previ~usly,~ in the turbulent core.Lawrence and Huang * adapted this theory and obtained solutions valid for a cylindrical coordinate system rather than the rectangular coordinate system used by Da~ies.~In all the above work re-entrainment of particles from the boundary walls was assumed to be absent. Reviews of these theories and of others AEROSOL DEPOSITION FROM TURBULENT FLOW differing little from the above have been given by Montgomery and Corn l3 and Sehmel.14 In a recent theory Lawrence and Huang considered that the size of the particles relative to the scale of the turbulence was of importance and they defined a relative entrainment factor as x = ds/l (5) where I is the fluid mixing length I5 at a point within the fluid.If this ratio was greater than unity the concept of a particle stop distance was used; however if the quantity a was less than unity then the mixing length was used as a measure of the particle free flight distance. On the basis of work by Tchen l6 and So0 and Tien the authors assumed equality of particle and fluid diffusivities but included a specifi-cation of the particle root-mean-square turbulent fluctuation velocity uL+ with respect to the r.m.s. fluid fluctuation velocity u;+ in the form of a non-linear differential equation relating the latter two quantities and the particle relaxation time in the following manner (6) The authors calculated the discrete particle deposition flux for fully-developed turbulent pipe flow.The results deviated widely as did those in all the previous work reviewed from the small mount of experimentally obtained aerosol deposition data available from other sources. Rouhiainen and Stachiewicz used the concept of frequency response developed by Hjelmfelt and Mockros l9 to obtain a quantitative evaluation of ep!eF. They showed that for 30pm diam. particles of lycopodium spore a fourfold increase in Reynolds number Re of the suspending fluid which caused a more than fourfold increase in gF only resulted in a twofold increase in E,. A more important result of their work for small particles was their quantitative evaluation of the shear flow induced transverse lift force on a particle in the laminar sublayer. They considered that for a vertical flow system if the particle radial velocity was sufficient to carry the particle to such a distance from the wall that the particle velocity in the x coordinate direction was higher than the local stream velocity in this direction then the lift force was directed towards the wall.For lycopodeum spheres of 2 pm diam. they calculated that for Re> 1 x lo4 the particle velocity at the edge of the sublayer such that deposition on the wall took place was three orders of magnitude lower when considering the lift force effect than when a purely inertial mechanism was considered. Further work was needed to apply this mechanism to horizontal pipe flow to determine the distance from the wall at which the lift reversal takes place and to clarify the mechanisms which propel the particles to within the latter distance from the wall.Sehmel examined the effect of removing the assumptions of regarding an equality of diffusivity of the particle and fluid and an equality of particle and fluid root- mean-square turbulent fluctuation velocities. He determined what dependence these variables had upon other parameters of the problem in order that theoretical calculations agreed with the experimental data i.e. he described the combined effect of the two parameters as an " effective eddy diffusion coefficient " and gave empirical correlations for predicting this quantity for various flow conditions. He also made deposition measurements on all surfaces of a duct and introduced a gravitational factor into the correlations.Finally an effect was investigated by Byers and Calvert *O which had been subject to few previous investigations. They determined the particle deposition from turbulent streams by means of a thermal force. Experi- I. WILLIAMS AND A. B. HEDLEY mental work carried out measured the deposition rate of 0.3-1.3 pm diam. particles from pipe flows at Re = 1.376 x lo4 when the gas temperature was several hundred degrees above the pipe wall temperature. High particle collection efficiencies were measured and compared with negligible particle collection efficiencies under similar experimental conditions with the temperature gradient removed. Unfortunately there seems to be few experimental data relating to the thermal deposition of micron size aerosols from fully-developed turbulent pipe flow incorporating small tempera- ture gradients.It was apparent from the current state of aerosol deposition studies in turbulent flow that certain aspects of the problem warranted further investigation ;these were (1) more experimental results of particle deposition rates from turbulent pipe flow under closely controlled conditions were needed. (2) The relationship between the particle diffusivity and the fluid diffusivity needed clarification. (3) The relative importance of thermal electrostatic and diffusive forces should be investigated. (4) The reverse lift force l8 warranted further theoretical investigation along the previously suggested lines. VT/lO-* K m-* FIG.1 .-The dependence of the particle therrnophoretic velocity upon the temperature gradient.It was decided to construct a variable flow system in which fully-developed turbulent pipe flow was achieved. An initial investigation was designed to characterize the flow in terms of the mean and fluctuating velocities U,u’ V,v’ in the axial and radial directions respectively and to allow the determination of the shear stress -% as a function of y and hence the eddy diffusivity of the fluid from the relationship where AEROSOL DEPOSITION FROM TURBULENT FLOW and a is the pipe radius Uis the mean axial velocity at a point and U is the maximum mainstream velocity at the centre line. The measurement of the above quantities was carried out using hot-wire anemo- metry.Non-volatiledroplets were chosen as the disperse phase since in the deposition measurements particle evapouration would be minimized. By carrying out concen- tration traverses of the aerosol injected into the turbulent flow the diffusivity of the particles in the fluid was deter~ined.~~ The measurement of the aerosol concentration was carried out by sampling the aerosol isokinetically and using a multi-channel light-scattering counter which was developed for the purpose. 24 For the purpose of this experiment the effect of charge was minimized by generat- ing a condensation aerosol examining it for charge using a charge analyzer and if necessary neutralizing the aerosol using a charge generator designed to produce equal numbers of +ve and -ve ions. Fig.1 indicates the magnitude of the thermo- phoretic velocity V,, induced in a particle by a temperature gradient VT. The quantities were calculated from the equation of Brock l2 derived from the slip-flow region corresponding to Knudsen numbers Kn in the range 0.1-0.01 where Kn = A./r, and 1 was the mean free path of the gas molecules. The particles used in the present work varied from 0.8 to 4 pm diam. and correspond to Kn = 0.11 to 0.05. The equation is where the constants A Q and b are dependent upon the gas-particle system. T is the absolute gas temperature CT and C are constants related to the thermal and momentum coefficients respectively and k and k are the thermal conductivities of air and the particle respectively. In order to induce thermophoretic particle velocities within the duct provision was made for heating or cooling the duct walls within the temperature range 0-50°C while maintaining the fluid temperature constant.Fig. 2 shows temperature profiles obtained from the duct centre to the wall measured using a thermistor probe within the test section and thermistors embedded in the duct wall at that point. The temperature gradients were appreciable and are shown in table 1. Also shown in table 1 are the times taken for a particle to traverse the laminar sub- layer in turbulent pipe flow under the action of a temperature gradient. Since the thermophoretic velocity of a particle acts towards the cooler region the results obtained when the duct wall temperature was raised above the fluid temperatures are prefixed with a negative sign.In this case any particles within a distance r’laG0.05 or 0.25 cm from the wall were subjected to a thermophoretic velocity moving away from the wall. Appreciable thermophoretic velocities were induced when the duct wall and the fluid flow were ostensibly at room temperature. The velocity referred to acted on a particle from some distance into the flow although the maximum value occurred over a distance r’la-0.05 as shown where r’ is equal to a-r and r is the coordinate in the radial direction r = 0 is the pipe centre. This velocity was maintained through the laminar sublayer so no question of a stop distance arose. It was considered that the magnitude of the approximate velocities calculated were sufficient to warrant an experimental investigation of this additional driving force acting on the particles.The experimental unit is shown diagrammatically in fig. 3 and consisted of three units the flow system the aerodynamic analysis system and the aerosol generation 1. WILLIAMS AND A. B. HEDLEY I67 and analysis unit. A general view of the duct assembly is shown in fig. 4. The duct assembly consisted of a valve regulated blower which passed up to 0.4 m3 s-' of cooled air through an absolute filter unit a 0.6cm mesh screen and a 35-cm-long section of paper honeycomb into the first of six interlocking sections of 10.16 cm i.d. 154.2 cm long stainless steel tubes each of which was mirror polished internally. L, 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0 r'la FIG.2.-The distribution of the fluid temperature within a circular duct.For curve 1 wall temper- ature was 317.0 K and Re = 2.67x lo4; curve 2,313.0 K and Re = 1.27 x lo5; curve 3,295.2 K and Re = 1.27x lo5 ; curve 4 296.0 K and Re = 2.67 x lo4; curve 5 284.0 K and Re = 1.27x lo5; curve 6,279.8 K and Re = 2.67~ lo4. TABLE VELOCITY OF PARTICLES IN TURBULENTFLOW UNDER THE INFLU- 1.-THERMOPHORETIC ENCE OF A TEMPERATURE GRADIENT timejs taken to Reynolds number laminar sublayer thickness at y+ = 5.Ocm duct walltemp./K fluid temp./K at r' = 0.25 cm VT/K cm-1 Vmi/cm s-1 traverse laminar sublayer 2.67x lo4 4.18 x 296.0 296.6 2.6 4.5~10-3 9.29 2.67~lo4 4.18~ 279.8 292.6 51.2 9.2~lo-' 0.45 2.67 x lo4 4.18~ 317.0 304.2 51.2 -9.2~ 0.45 1.27x lo5 1.23x 295.2 297.3 8.4 1.6~ 0.79 1.27x lo5 1.23x 284.0 293.2 36.8 6.8~ 0.18 1.27~lo5 1.23~lo-' 313.0 302.2 43.2 -8.0~ 0.15 The turbulent boundary layer was instigated by an annular protuberance of 1.5 mm depth in the first section.Each tube was fitted with the facility to accommodate an aerosol injection point in the form of an airfoil wedge section across the duct. The last two sections of the duct acted as test sections and were fitted with 20 equispaced I68 AEROSOL DEPOStTtON FROM TURBULENT FLOW pressure tappings and each section had facilities for fitting a probe scanning unit shown in fig. 5. Around the periphery of the duct at the corresponding axial position to the internal probe tip eight thermistors recorded the internal wall temperature and eight adjacent removable plugs were fitted flush with the inside tube wall to act as droplet sample holders which were subsequently examined microscopically.The wall temperature of the last section of the duct was controlled by passing ethylene glycol through eight 1.27 cm i.d. copper tubes fastened to the outer tube wall. The flow Reynolds number range available with the unit was between 2.67 x lo4 and 1.27 x lo5 with wall temperatures between 279 and 317 K. AEROSOL HOT WRE FENLRATSR AkEhtOMETER AVERSING SAWLING PROBE STOR TFlRERATURE ESOLUTE AIR FILTER CIPCULT PROBE FIc;. 3.-A diagrammatic view of the experimental system. The second unit in the experiment a DISA hot-wire anemometer type 55D01 was used to determine the aerodynamic characteristics of the fluid flow.These quantities were the fluid shear stress -uV where u and v were the instantaneous values of velocity fluctuations in the x or y directions respectively the mean axial velocities U and U, and the root mean square fluctuating velocity components in the axial and radial directions u’ and v’. In order to determine v’ it was necessary to determine the double correlation coefficient uU/u‘v’. The probes used in the experiment were Disa gold-plated miniature probes types 55F14,55F12 and 55Fl1 to measure the average velocity profiles the shear stress and the fluctuating root-mean-square velocities respectively. The correlation coefficient was measured at several points during a traverse in the test section using a DISA cross-wire probe.In the aero- dynamic measurements two dimensional duct flow was assumed. The effect of the fluid temperature variation on the hot-wire results was taken into account by calibrating the probes at several temperatures within the range of interest. A plot of the calibration constants against the temperature was then made. For calibration purposes the DISA calibration wind tunnel was used with a modification. The air was heated or cooled by passing through an automobile radiator at the inlet to the tunnel. Ethylene glycol acted as heatant or coolant and the temperature of the air in the wind tunnel was monitored’using a thermistor. The third part of the experimental apparatus was the aerosol generation sampling and analysis unit. The aerosol was generated from two materials dioctyl phthalate and di-2-ethyl hexyl sebacate respectively.The generator is shown in fig. 6. This was a condensation generator a description of which had been given previou~ly.~~ Some modifications to the generator described 25 have been made. The two most important were the provision of additional flow controllers at the outlet of each gas supply and a more sophisticated temperature control system. The temperature controller incorporated an electronic proportional control circuit with a fine differ- ential control applied. The sensor units were negative temperature coefficient FIG.4.-A view of the duct showing the diffuser/sampling unit. FIG.5.-The end of the duct test section showing the scanning unit and thermistor and sampling plug positions.[Toface page 168 FIG.6.-The aerosol generator. I. WILLIAMS AND A. B. HEDLEY thermistors which were fitted into the boiler and reheater flasks respectively. This unit enabled temperature control within +_O.S"Cto be maintained. Two different nucleii sources were used in the condensation generator. Incoming air was passed over a heated wire coated with Apiezon W wax and an alternative method of introduc- ing anthracene into the boiler flask was also used. The use of a condensation generator precluded the formation of charged droplets. That this condition was satisfied was tested by passing the aerosol in a laminar air stream between two plates with a potential of -5 kV between them.The plates were examined for deposited droplets using photo-micrography. The aerosol could be neutralized by passing through a charge apparatus designed to generate equal numbers of +ve and -ve ions both the charge analyzer and charger were designed after Linger and Radnik.26 Particles were pumped through the sampling probe and then through a conical diffuser the included angle of which was 5". The diffuser reduced the velocity to a level acceptable to the sensing system. The particles were classified into ten size- ranges and total counts in each range were indicated digitally. The anemometer hot-wire data was processed using a statistical approach to the signal analysis developed by Dvorak and S~red.~' The spacial resolution of the velocity vectors acting on a hot-wire probe in each of three 45" mutually differing positions provided a set of three non-linear equations whose analytical solutions represented the three velocity components as functions of three random variables.The random functions were processed to obtain the mean velocities and the various turbulent components. This method was applied to the two-dimensional system under discussion and necessitated measurements from a straight wire probe in two 45" mutually differing positions. The equations were solved for the mean and fluctuating velocities in the axial and radial direction at a number of points on a traverse across the duct test-section. The correlation coefficient was measured at several points across the duct using a cross-wire probe.The shear stress component was evaluated from measurements I f I 11 r'la FIG.7.-The effect of wall temperature on the fluid velocity profile in turbulent pipe flow at Re = 2.67 x lo4. Curve 1 wall temperature = 279.8 K; curve 2 296.0 K ; curve 3 317.0 K. 170 AEROSOL DEPOSITION FROM TURBULENT FLOW made with a 45O-slant wire probe 28 rotated through 180° assuming that the heat transfer from the wire depended only upon the flow velocity normal to the wire. In a fully-developed pipe flow the velocity distribution across a pipe is independent of the stream wise position. Under these conditions the pressure drop along a pipe is balanced by the shear stress ZO = ~dP/2d~ (9) zo = laminar stress -pZ (10) + where -pZ is the apparent turbulent stress.Except very close to the pipe wall zo is composed entirely of the turbulent stress. In this experiment the static pressure tapping along the last two pipe sections enabled a measurement of dP/dx the pressure drop to be made and so a direct determination of the shear stress was possible. -. --. ‘ .r .-..I . ,. ,. I I I I I I I -0 0.2 0.4 0.6 0.8 I.o //a FIG.8.-The effect of wall temperature on the fluid velocity profile in turbulent pipe flow at Re = 1.27 x lo5. Curve 1 wall temperature = 284.2 K; curve 2 295.2 K; curve 3 313.0 K. This compared well with the values obtained from hot-wire anemometry. Fig. 7 shows the variation with temperature of the velocity profile across the test section at Re = 2.67 x lo4 and fig. 8 shows similar data for Re = 1.27 x lo5.The duct wall temperatures correspond to those shown in fig. 1. A plot of the shear stress non-dimensionalized with the friction velocity as a function of r’/ais shown in fig. 9. Four sets of data points and two curves are shown. The curves correspond to Reynolds numbers of 2.67 x lo4 (upper curve) and 1.27 x lo5 (lower curve). The data points shown as triangles and squares both correspond to a Reynolds number of 2.67 x lo4 but at wall temperatures of approximately 317 and 280 K respectively. The effect of an alteration in pipe wall temperature on the shear stress profile at a Reynolds number of 1.27 x lo5 was very small. The distribution of the axial root-mean-square fluctuating turbulent velocity component u’ non-dimensionalized with U,,is shown in fig.10. The upper curve shows the distribution of a Reynolds number of 1.27 x lo5 and the lower curve was determined for a Reynolds number of 2.67 x lo4. I. WILLIAMS AND A. B. HEDLEY The last of the turbulent quantities the distribution of the radial root-mean- square turbulent fluctuating velocity component u‘ is shown in fig. 1 1. This quantity is again non-dimensionalized with the friction velocity. The upper curve corre- sponds to a Reynolds number of 1.27 x lo5and the lower curve to one of 2.67 x lo4. 0.a 0.6 NCI ;s is 0.4 0.2 I I I I I I I OO 0.2 0.4 0.6 0.0 I.o t I I I I 1 I I 72 AEROSOL DEPOSlTION FROM TURBULENT FLOW I .6 $ 0.8 a 0 0 0.2 0.4 0.6 0.0 1.0 r'/a FIG.11.-The distribution of u' within turbulent pipe flow at Re = 1.27~lo5 and 2.67~lo4 res-pectively. Curve 1 represents c' at Re = 1.27~ lo5 curve 2 represents 13' at Re = 2.67~lo4. FIG.12.-The fluid eddy diffusivity distribution in turbulent pipe flow. 0 results obtained at Re = 1.27~10'; 0,results obtained at Re = 2.67~104; A are theoretically derived from a correlation in ref. (7). I x~64 1x162 1.0 1x102 rlv FIG. 13.-The fluid eddy diffusivity distribution in turbuient pipe flow. 0,Results obtained at Re = 1.27~lo5; + results obtained at Re = 7.03x lo4; 0,results obtained at Re = 2.67~lo4; A theoretically derived results from ref. (7). I. WILLIAMS AND A. 3. HEDLEY ref. (7). Fig. 12 shows the results at the elevated pipe wall temperatures shown in fig.1 ; fig. 13 gives the results with the pipe wall at nominal room temperature again shown in fig. 1 and fig. 14 shows the results obtained at the low pipe wall temperature and again the numerical wall temperature can be obtained from fig. 1. Due to the finite residence time of the fluid in the test section of the duct it was necessary to derive an approximate particle trajectory in order to evaluate the distance a particle travelled down the duct before deposition on the wall occurred due to its radial thermophoretic velocity. The trajectory of the particle has been considered in the following case to be a function of the axial velocity at the edge of the boundary layer denoted by the directional coordinate x and the thermophoretic velocity in the radial direction denoted by the directional coordinate r ; other diffusive forces were neglected.3 1x10 ' x lo2 + ?I I x 10 1.0 I 1x10 1x1,c2 1.0 1x102 EIV FIG.14.-The fluid eddy diffusivity distribution in turbulent pipe flow. 0,Results obtained at Re = 1.27 x lo5; 0,resuIts obtained at Re = 2.67 x lo4 ; A theoretically derived results from ref. (7). The axial velocity of the air equalized VJr) = dx/dt =f(r) (1 1) wheref(r) described the fluid velocity profile across the duct. It was assumed that the particle was completely entrained by the fluid and that the axial velocity of the particle was equal to the axial velocity of the fluid. The radial velocity of the particle drldt equals the thermophoretic velocity YTH hence On integration we obtain f(r)dr VTHdx, = rZL x=o where a is the distance from the duct centre line to the wall and r is the distance from the duct centre line to the edge of the laminar sublayer.L is the distance a particle moving under the axial velocity would travel from the time it was subjected to the thermophoretic velocity to time of deposition on the tube wall. The expression AEROSOL DEPOSITION FROM TURBULENT FLOW chosen forf(r) to represent the velocity profile across the duct was the empirically obtained expres~ion,~~ U/U = [(a-r)/a]l/". (14) According to Schlichting 29 the exponent has values of 7.0 and 6.6 at Reynolds numbers of 1.1 x lo5 and 2.3 x lo4 respectively. Substituting for Jlr) in eqn (13) for a Reynolds number value of 1.1 x lo5 which on integration gives 7 uo 7 uo --&a -r)817 =-T7(a-ra)8i7 = vTHL 8a 8a from which the length L was obtained.The results from eqn (1 6) are shown in table 2. Although these calculations were approximate they indicate that quite small temperature gradients cause particles to deposit within the duct system at points depending upon the position of the aerosol injection point. The establishment of known temperature gradients in the present work should help to determine experimentally the influence of thermophoresis on particle deposition. TABLE 2.-AXI[AL DISTANCE TRAVELLED BY DROPLETS SUBJECTED TO A RADIAL THERMOPHORETIC VELOCITY Reynoldsnumber duct wall temp./K distance L before impactionlcm 1.27~105 295.2 1931.3 1.27~105 284.0 443.4 2.67~104 296.0 6970.1 2.67~104 279.8 342.0 With regard to the determination of the fluid dynamic characteristics of the flow in the duct it was necessary to determine how closely the present system approached fully-developed turbulent flow and also the effect of the boundary wall temperature on the fluid turbulent characteristics in particular the fluid eddy diffusivity.That the fluid flow in this experiment did closely approach fully developed turbulent flow was indicated by several features. For both values of Reynolds number the velocity profiles were typical flat turbulent profiles as opposed to the parabolic profile expected from laminar flow. The effect of a decrease in the wall temperature in each case resulted in a "flatter " profile.The shear stress profiles shown in fig. 9. showed little dependence on Reynolds number and varied linearly across the duct cross-section. The dependence of the profiles on the wall temperature was only significant for the lower Reynolds number when an increase in the dimensionless shear stress corresponded to an increase in wall temperature. The dimensionless fluid eddy diffusivity profile was correspondingly affected by temperature E/V increasing with decrease in temperature. Fig. 10 indicated a significant decrease in the axial root-mean-square turbulent component with a decrease in Reynolds number although the radial component profiles were of a similar magnitude for both values of Reynolds number.The turbulent intensities u'/Uowere calculated for r'/a = 0.1 and 1.O for Reynolds number of 1.27 x lo5 and 2.67 x lo4. The intensities were compared with those calculated by Laufer for Reynolds numbers of 5 x lo5 and 5x lo4 and the compari- I. WILLIAMS AND A. B. HEDLEY son is shown in table 3. The similarity of the magnitude of the turbulent intensities in the present system and those measured by Laufer at higher Reynolds numbers indicated that fully-developed turbulent flow was achieved in our system. TABLE OF RELATIVE TURBULENT INTENSITIES 3.-COMPARISON Reynolds number r'ln 1ilUo 1.27~105 0.1 0.079 1.27x 105 1.o 0.029 2.67~104 0.1 0.069 2.67~104 1.o 0.021 5x lo5 0.1 0.070 5~ 105 1.o 5~ 104 0.1 5~ 104 1.o 0.027 The work so far has established reasons for and provided a system within which the relationship between the diffusivity of the fluid and of the particles can be deter- mined.Furthermore the dependence of particle deposition on a thermophoretic force due to temperature gradients existing between the duct wall and the fluid is clarified. The authors wish to acknowledge the financial assistance of Shell Research Ltd. and in particular the help of Prof. T. M. Sugden F.R.S. which enabled this work to be carried out. G. A. Sehmel Meeting SOC.Eng. Sci. (Tel Aviv June 1972). A. C. Chamberlain Proc. Roy. SOC.A 1966 290 236. C. N. Davies Aerosol Sci. 1966 1 418. S. K. Friedlander and H. F. Johnstone Znd. Eng. Chem. 1957 49 1151. J. Laufer The Structure of Turbulence in Fully Developed P@e Flow N.A.C.A.Report 1147 1954. ti C. S. Lin R. W. Moulton and G. L. Putnam Ind Eng. Chem. 1954 45 636. C. N. Davies Aerosol Sci. 1966 1 393. W. R. Lawrence and A. B. Huang A.Z.A.A. 10th Aerospace Sci. Meeting (San Diego Cali- fornia January 1972) A.I.A.A. paper no. 72-81. S. K. Beal Nucl. Sci. Eng. 1970 40 lo V. E. Levich Physiochemical Hydrodynamics (Prentice Hall New Jersey 1962) p. 155. ' A. C. Wells and A. C.Chamberlain Brit. J. Appl. Phys. 1967 18 1793. l2 P. R. Owen Int. J. Air- Water Pollution 1960 3 8 50. l3 T. L. Montgomery and M. Corn Aerosol. Sci. 1970 1 185. l4 G. A. Sehmel J. Geophys. Res. 1970,75 1766. l5 L. Prandtl 2.angew. Math. Mach 1925 5 136. l6 C. M. Tchen Ph.D. Thesis (Delft 1947).l7 S. L. So0 and C. L. Tien J. Appl. Mech. 1960 27 5. lS P. 0.Rouhiainen and J. W. Stachiewicz J. Heat Transfer 1970 29 C 169. l9 A. T. Hjelmfelt and L. F. Mockros Appl. Sci. Res. 1900 16 149. 2o R. L. Byers and S. Calvert Znd. Eng. Chem. Fund. 1969 8 646. 21 N. A. Fuchs The Mechanics 0fAerosol.s (Pergamon London 1964) p. 56. 22 G. M. Hidy and J. R. Brock The Dynamics of Aerocolloidal Systems (Pergamon Oxford 1970). 23 W. L. Towle and T. K. Sherwood Znd. Eng. Chem. 1939,31,457. 24 I. Williams and A. B. Hedley Aerosol Sci. 1972 3 363. 25 I. Williams M.Sc. Thesis (Sheffield 1970). 26 G. Langer and J. L. Radnik J. Appl. Phys. 1961 32 955. "K. Dvorak and N. Syred DZSA Conference (Leicester 1972); also Internal Report (Dept. of Chem. Eng. University of Sheffield).28 J. 0.Him Turbulence (McGraw Hill London 1959) chap. 2. 29 H. Schlichting Boundary Layer Theory (McGraw Hill London 1 1968) p. 563.