Museum Management and Curatorship (1991), 10,373-383

Wall Temperature and the Soiling of Murals

DARIO CAMUFFO

Overview. A balance is made here of the intensities of the main deposition mechanisms (i.e. Brownian diffusion, thermophoresis, inertial impaction, gravitational settling and Stefan flow) in the case of vertical wall surfaces, and thus frescoes or other murals. When the wall is colder than the air, the temperature gradient forces negative thermophoresis and develops a downward free-convection layer. This generates inertial impaction and increases gravitational settling. A further contribution due to the Stefan flow may be generated. When the wall is warmer, thermophoresis partially counteracts other deposition mechanisms for fine particles, but the temperature gradient generates an uprising flow, and inertial impaction of the largest particles occurs. The deposition rate is reduced for fine particles, but increased for the largest ones. As the mass of the suspended particulate matter is concentrated in the coarsest particles, the net effect is a heavy soiling. Up-rising flows transport upwards the majority of particles, causing soiling also of the ‘downward facing’ parts of the surface irregularities. Warm floor or cold ceiling may cause atmospheric instability and convective motions, with the above mentioned negative effects. Cold floor and warm ceiling form stable temperature profiles and attenuation of turbulence. Under the warm ceiling a cushion of stable air forms without motion (and inertial deposition) and with thermophoresis (partially) counteracting Brownian diffusion. The best situation appears to be offered by cold floor, warm ceiling, walls in thermal equilibrium with ambient air, and air with stable temperature gradient.

Introduction

In general, in historic buildings, the wall thickness is much greater than the thermal damping depth, so that the temperature of the inner surfaces tends to be constant. The diffusivity of the thick walls acts as a low pass filter which damps the external temperature variations: very well for the diurnal wave, much less so in the case of the seasonal one. This causes a phase lag between air and wall temperature. In average, the walls tend to be warmer during autumn and winter and cooler during spring and summer, causing important fluxes of heat and gradients of temperature. On the shorter time-scale, the diurnal cycle of temperature forced by solar radiation, the interior lighting of works of art, the heating system and the presence of visitors are responsible for exchanges of heat between air and walls which affect the deposition of airborne pollutants and other particles, causing mechanical stresses and many other adverse effects. Soiling is one of the main problems for murals, and several field tests have been carried out to study the microclimatic factors that may reduce deposition on the most outstanding frescoes and paintings (Camuffo, 1983), such as the Giotto frescoes in the Scrovegni Chapel, Padua (Camuffo and Schenal, 1982), the Michelangelo frescoes in the Sistine Chapel, Rome (Camuffo and Bernardi, 1986; 1988; 1991a, b), Leonardo’s The

0260.4779/91/04 0373-11 0 1991 Butterworth-Heinemann Ltd

374 Wall Temperature and the Soiling of Murals

Last Supper, Milan (Camuffo and Bernardi, 1991~). A general conclusion was that the inner surface of the walls and the ambient air should be in thermal equilibrium, whereas the ceiling should be slightly warmer than the underlying air. These results have a wider application within the museum environment.

It has been known since 1884, from the observations by Aitken, Lodge and Clark, that air velocity tends to carry particles to a hot plate, but thermophoresis tends to repel them, and in the case of very great temperature gradients a dust-free space near the surface is formed. This was theoretically analyzed and experimentally measured, by several observers, in the case of hot bodies (e.g. Watson, 1936; Zernik, 1957; Slinn and Hales, 1971; Goren, 1977; Talbot et al., 1980). The free-convection layer of hot bodies is described in several books (e.g. Isachenko et al., 1977; Luikov, 1980; White, 1984; Welty et al., 1984; Incropera and De Witt, 1985; Tritton, 1988; Holman, 1989). On the basis of laboratory experiments performed on heated flat vertical plates, Nazaroff and Cass (1987) suggested that this result has an application in preventing the soiling of indoor surfaces, including works of art. However, in their experimental apparatus (i.e. a vertical, flat plate) the balance was between thermophoresis and Brownian diffusion, excluding other important deposition processes. However, in the real case of murals, given the small temperature gradients, this suggestion provoked an exchange of views (Camuffo, 1989; Nazaroff and Cass, 1989a) aimed at pointing out that a more complete balance inclusive of other deposition processes may lead to different conclusions, but the Editor of that Journal decided to stop the discussion, and no resolution to this very important

problem was then reached. The aim of the present paper is to state some general ideas about the relative

importance of the deposition processes and the unsuccessfulness of using thermophoresis as a means to counteract deposition in the case of conservation of murals, etc.

Microphysical Phenomena Induced by Differences in Temperature

(a) Balance between Thermophoresis and Brownian Diffusion

Nazaroff and Cass (1987) made observations as to when the thermophoresis acting on a vertical, flat plate dominates Brownian diffusion. The experimental data of the equilibrium between thermophoresis and diffusion are plotted on a logarithmic paper where the differences in temperature AT (scaled: O.Ol-100°C) are plotted against the particle diameter d (scaled: 0.001-5 ym). In the Log-Log paper, the graph is substantially diagonal, its extremes connecting the corner small AT and large d (i.e. AT z 0.015”C; d 2 4 pm) with the corner high AT and small d (i.e. AT g 45 “C; d z 0.002 pm). The physical meaning is that particles suspended in a fluid undergo irregular motion due to bombardment by surrounding fluid molecules. The terminal velocity depends on the size of the particle and viscosity of the fluid (air). For very fine particles, where Brownian diffusion dominates, very high temperature gradients are needed to counteract diffusion; coarser particles mediate the random molecular bombardment and are more affected by the drag that opposes to the motion, therefore they are much less sensitive to Brownian diffusion, and may be transported by low thermophoretic forces, although thermophore- sis is not the dominant deposition mechanism for large particles.

In the more common range of temperature differences, the equilibrium between the two mechanisms is characterized by the pairs of values: AT = 1 “C; d g 0.035 pm and AT = 0.1%; d ^- 0.3~ m. From the graph, it can be noted that at these values of AT the equilibrium is reached in the transition between free molecule range (i.e. when d<<h,

DARIO CAMUFFO 375

where h is the mean free path of the air; h = 0.065 ym at 25 “C) and the continuum range (d>>h), as well as in the continuum range itself when the particles become to be, or are, too large (i.e. from d - h to d>>h) to have an appreciable Brownian diffusivity, and other deposition mechanisms dominate. In fact, it is well known that for dC0.01 pm diffusion dominates; for 0.01 pm<&1 pm thermophoresis dominates; for d>I ym inertia forces and gravitational settling dominate (e.g. Slinn and Hales, 1970, Prodi and Tampieri, 1982).

(b) The Free Convection Layer

When there is a difference in temperature between walls and ambient air, the heat transfer generates temperature gradients and density differences Ap in the ambient air. The gravity g causes an Archimedean force and a free boundary flow along the wall. When the vertical surface is warmer, the buoyancy forces induce a free convection layer starting from the edge of the floor; but when colder, a descending, cold flow is generated in a symmetrical way, starting from the ceiling. The latter situation should be assiduously avoided, as thermophoresis adds to Brownian deposition and the downward flow is responsible for increasing aerodynamic deposition and gravitational settling, as we see later.

For both up-rising or downward flows, the initial boundary layer development is laminar, but at some distance z from the leading edge the free stream becomes turbulent, depending on the conditions of the ambient air and temperature difference. Surface roughness or atmospheric disturbances may favour instability and the onset of turbulence. In the case of a smooth surface, transition to turbulence occurs when the Rayleigh number Ra is Ra - 109. As this number is represented by the product of the Prandtl (PY) and Grashof (Gr) numbers, and for air Pr E 0.7, turbulence occurs when the local Grashof number Gr, = g /3 AT z3/v2 (where l3 is the expansion coefficient of the air, and Y the kinematic viscosity) becomes Gr,>lO 9; for instance, when AT = 0.1 “C at a distance z E 4 m; when AT = 0.5 “C, z g 2.5 m; when AT = 1 “C, z E 2 m. However, in the case of a real room, with a rough surface to the walls and with air subject to several disturbances, the free-convection boundary layer will undergo transition to turbulence with much smaller values of z (Schlichting, 1979).

The thickness 6 and the maximum velocity wZe of the free laminar flow for a vertical plate can be computed by means of the formulae (see Isachenko et al., 1977: 241-249):

and

6 = 423 (nkzlc,J3p2g AT)-“4 [cm]

w:$ = 3.95pgl%‘A T/p [cm/s]

where : p the dynamic viscosity, k the thermal conductivity, cP the specific heat at constant pressure.

The maximum velocity is reached at the distance 6’; = 0.386 from the surface and, of course, is not tangent to it. On logarithmic paper the thickness 6 versus z is represented by a straight line for each value of AT (Figure l), and the same for w’” versus z (Figure 2). When AT = 0.1 “C, w’+ = 4.8 cm/s when z = I m and w’: = 15 cm/s when z = IOm; when AT = 1 “C, w’:‘ = 15 cm/s when z = 1 m and w’~ = 48 cm/s when z = 10 m. By considering the distance x normal to the surface of the wall, and integrating the vertical

376

10

:

5

Lo

2

1

Wall Temperature and the Soiling of Murals

1. Thickness 6 [cm] of the free convection boundary layer versus the distance z [m] after the leading edge.

2. Maximum velocity w” [ /I’ hf cm s m t e ree con- vection boundary layer versus the distance z [m] after the leading edge.

speed w(x) across the thickness 6 of the boundary layer, one obtains the integral mean velocity <w> of the convective flow, which results in <w> = 0.633 w’:‘. Field

measurements may be likely to give a similar value when insufficient care is taken in putting the anemometric sensor at the exact distance from the surface or when the sensor causes some turbulence around it.

The free-convection layer causes heavy aerodynamic deposition on the wall surface. When the flow is slow and laminar, it causes interception of the airborne particles on the rough parts of the wall; at low-impact velocities the kinetic energy of the particles is not sufficient to escape attractive forces at the surface, and nearly all the particles striking the surface adhere. When the flow becomes turbulent, w’; reduces but the mean velocity profile rises steeply at the wall; fluctuating velocity components normal to the surface

DARIO CAMUFFO 377

0

8

3 16

24

32

60 set I

I

3. Turbulence in the free-convection boundary layer measured at 3 mm from the Giotto fresco in the Scrovegni Chapel (w is in cm/s; after Camuffo and Schenal, 1982).

occur, and when the inertia forces of the particles dominate the viscous force of the air, particles impact on the surface. This deposition mechanism is particularly active for large particles, i.e. with diameter d> 1 pm. A typical velocity profile is reported in Figure 3.

Waldmann and Schmitt (1966) and Prodi et al. (1979, 1982) demonstrated that

thermophoresis drops rapidly at the transition range when the Knudsen number Kn becomes Kn<O. 1 (KYZ = 2hld): for large particles, thermophoretic contribution becomes smaller and smaller whereas impaction gets greater and greater. By looking at the spectrum of the distribution of the particle sizes, this method may appear advantageous, as in general the number of fine particles dominates. However, the right balance should not be made by comparing the number of particles per each spectral width, but the total mass associated with them. The normalized number of particles is maximized at the smaller diameters; the volume and mass are maximized at the larger diameters. In usual conditions, the total volume and mass of the few large particles generally exceeds by several orders of magnitude the total mass of the many fine particles (Figure 4). Consequently, the less abundant coarse particles are responsible for severe soiling. This is especially true when a fast convection layer is generated along a rough wall. It should be noted that the visual effect of soiling is a very complex phenomenon (Eastaugh, 1990) although, for simplicity, particular attention has been generally paid to the total deposited mass of particles (e.g. Nazaroff and Cass, 1989b, 1991). When the total mass (or volume) is considered, then it is the same as assuming that all the particles are of the same nature, semi-transparent, and the soiling is due to the thickness of the layer of particles deposited onto a given surface.

(c) Gravitational Settling

A free convection layer affects also the gravitational settling on surface irregularities or slant parts of frescoes or murals. The balance between the gravitational force and the drag exerted by air on the falling particles causes particles to move through the air at constant

378

1oc

8C

60

2

40

Wall Temperature and the Soiling of Murals

1 .oo

d

80

60

b

40

4. Spectral distribution of the number of particles N [number of particles/cm31 and total volume of them V [um31cm3] per unit volume of air (4 August 1988, 12.00 LT at Palazzo Chiericati,

Vicenza); d = particle diameter [urn].

velocity. For large particles (d>>h) the falling velocity can be computed with Stokes’ law, but for the smaller ones a slip correction factor (the Cunningham correction factor) must be used. However, the falling velocity ZIP becomes important only for micronic particles, e.g. at d = 1 urn, Vf = 0.035 mm/s; d = 2um, wf 0.13 mm/s; d = 5 urn, of = 0.8mm/s;d= 10~,~~=3mm/s;d=20~m,v~= 12mm/s;‘d=50um,vf=76mm/s

1 o.ooooc

1 .ooooc

0.1 oooc

2 0.01000

0.00100

0.00010

0.00001 I I I I I 0.1 0.2 0.5 1.0 2.0 5.0 10.0

d

5. Spectral distribution of the volume of airborne particulated matter SV settled per unit time and unit surface [pm3/cm2s]; d [pm] (4 August 1988, 12.00 LT at Palazzo Chiericati, Vicenza; d =

particle diameter).

DARIO CAMUFFO 379

(Friedlander, 1977; Seinfeld, 1986). Given this distribution of the settling velocities, although the small-size particles are much more numerous, the mass of airborne particulate matter that is settled per unit surface and unit time increases with the particle size (Figure 5).

When the wall is colder than the air, the convection is downward and ‘aiding’, because the motion of the convective airflow develops in the same direction as the gravity force: the two velocities of the medium (ZU) and falling particles (vf) add, increasing the bulk deposition rate. When the wall is warmer, the buoyant convection is upward and forms an ‘opposing’ flow, acting against the gravitation; the particles may fall or rise, and their final velocity is YO - vf As ZJ~ is a function of d, i.e. vf(d), in general: for fine particles wf(d)<<w, and they are freely transported by the uplifting airflow; for coarse particles in general wf(d)<w and their rising velocity is attenuated. For very small temperature gradients and very large particles it may also be vf(d)>w and in this case the settling velocity is attenuated; some other particles in the intermediate interval of moderately large diameters for which vf(d) -w can stay in equilibrium, floating as if the gravity were absent. The overall result is that, in the presence of uprising flows, soiling may occur both in the upward- and downward-facing parts of mural irregularities.

(d) Forced Evaporation

Heating a wall means lowering the relative humidity (RH) inside the micropores, favouring evaporation according to the Kelvin law (Camuffo, 1984). Some evaporation occurs in the micropores, until the new molecules of water vapour that have evaporated re-establish the equilibrium. If the inner pore and capillary system is open, the external airflow removes the moisture that diffuses out of the surface, and this process may continue, possibly with formation of efflorescences or subflorescences (Camuffo, 1984, 1988). If, on the contrary, the inner system is closed, an overpressure is generated in the inner cavities that may damage the work of art (Camuffo, 1991). Heating a wall with open pores means forced evaporation from the inner pores, thus causing diffusiophoresis and the associated Stefan flow. Their net contribution initially works against deposition, but when heating is stopped, the situation may reverse with daily cycles. For this reason it is inadvisable to force temperature cycles. The opposite occurs when cooling a surface. The deposition induced by the Stefan flow is irrespective of the particle size.

(e) A Field Observation

An example of the relative importance of the different deposition mechanisms may clarify this discussion. In order to show how important the dynamic regime of the air near the wall is, a photograph of the soiling above a heater is reproduced (Figure 6a). Note that the soiling is not uniform, as could be expected when considering only AT but is more enhanced in V-shaped regions whose vertices begin from the wall hooks which support the heater. The greater soiling is caused by greater inertial impaction due to greater turbulence in the uprising flow originated by the hooks. The analogous hydraulic situation can be observed in the overflow of a dam, with the obstructions in the overflow that cause the onset of turbulence (Figure 7).

Soiling is much greater in the V-shaped turbulent region, but it is difficult to separate the contribution due to impaction from that due to thermophoresis. However, it can be seen that very slight unevenness in the wall surface (Figure 6b) causes a micro-departure in the airstream, which then comes back into contact with the wall surface some 4 mm

380 Wall Temperature and the Soiling of Murals

later. In this cleaner region only thermophoresis (of course, in addition to the Brownian deposition) acts. A comparison of the three regions, i.e. the relatively clean AA, the grey background and the black V, shows how important the contribution of turbulence is in comparison with the difference in temperature between the wall surface and free air, which is the same in the three cases under consideration.

(b)

6. (a): Soiling over a heater. The deposition is enhanced in the V-shaped regions whose vertexes begin from the wall hooks which support the heater. (b): A slight unevenness of the wall surface, i.e. a crack in the paint, indicated by the letters AA, causes a micro-departure on the airstream, which then comes back into contact with the wall surface some 3-4mm after. In this cleaner region thermophoresis acts without con- tribution of inertial impaction.

DARIO CAMUFFO 381

7. Hydraulic situation showing a V-shaped re- gion analogous to that in Figure 6a, but reversed: the overfall of a drowned dam with disturbances in the overflow that cause the onset of turbulence (drop: 1.5m).

Role of Horizontal Surfaces: Floor and Ceiling

Having analysed the effects on a vertical surface, the horizontal surface should also be considered. When the floor is warmer than the air, an unstable, convective circulation is generated which causes some mixing in the room, with one or more rising plumes in the centre of the room and descending along the walls, with all the negative effects that have been discussed. The same effect may be generated by a colder ceiling, with a symmetrical circulation. When the ceiling is frescoed, the situation is very negative: the convective motion causes a flow of warmer air moving tangentially to the ceiling, causing aerodynamic impaction, thermophoresis and, possibly, diffusiophoresis with the associated Stefan flow. When the floor is colder, the overlying air is stable, the fluid is quiescent, and turbulent motions are suppressed. A warm ceiling forms below it a cushion of air less warm than the surface but warmer than the environmental air, so that no convection is possible. This layer stays quiescent, with no impaction; thermophoresis and gravitational settling contribute to keeping away the particles below.

Conclusions

Brownian diffusion occurs in all directions, is slightly dependent on air temperature and is important for fine particles, reducing when the particle sizes increase. To this mechanism may be added (or subtracted) the other deposition processes, and a balance can be made only by considering all the possible induced mechanisms, not just some of them. The situation may be summarized as follows. In the absence of charged particles, on a vertical surface, in thermal equilibrium with the environmental air, only Brownian diffusion occurs, and possibly sedimentation onto the upward-facing side of protruding irregularities. Up-rising flows transport upwards the majority of particles, causing soiling also in the downward-facing parts of the surface irregularities, i.e. the parts usually seen by visitors.

When a vertical fresco is colder than the air, the temperature gradient forces negative thermophoresis and develops a downward free-convection layer, This generates inertial

382 Wall Temperature and the Soiling of Murals

impaction and increases gravitational settling. A Stefan flow may be generated, thus increasing deposition. In brief, when the surface is colder, all the deposition mechanisms are active, leading to the greatest deposition rate. However, when the fresco is warmer, the thermophoretic forces partially counteract other deposition mechanisms for fine particles, but the temperature gradient generates an up-rising flow, and inertial impaction of the largest particles occurs. The deposition rate is reduced for fine particles, but increased for the large. Although the normalized number of particles is maximized at the smaller diameters, the mass is maximum at the larger diameters and the soiling provoked by the deposition of relatively few coarse particles is much more severe than that of the many fine particles.

Thus, a wall colder than air offers the worst condition for conservation; when the wall is warmer thermophoresis partially counteracts other deposition mechanisms, but the final balance that includes the contribution of the largest and more massive particles suggests one should avoid this situation too. Thermal equilibrium seems to be the best way to preserve murals and all wall surfaces. The vertical surfaces are important but so too are the temperatures of the floor and the ceiling. Warm floor or cold ceiling may cause atmospheric instability and convective flows, with the above mentioned negative effects, while the combination of cold floor and warm ceiling forms stable temperature profiles and attenuation of turbulence. Under the warm ceiling a cushion of stable air forms without motion (and inertial deposition) and with thermophoresis (partially) counteracting Brownian diffusion. This best situation, i.e. cold floor, walls in thermal equilibrium with ambient air, which is characterized by a stable temperature gradient, and warm ceiling has been suggested for the conservation of murals, especially in the case of totally frescoed rooms, such as the Scrovegni or the Sistine Chapels.

Acknowledgements

The field studies for the conservation of frescoes and murals were supported by the commission of the European Communities, DG XII (STEP contracts No. EV4V-0051-I and CT90-0107-SSH); the present study of dry deposition has been supported by the CNR-ENEL Project, Interactions of energy systems with human health and environment, Rome, Italy. Special thanks are due to Dr Peter Brimblecombe for very useful discussions.

References

Aitken, J. (1884) Journal of the Transactions of the Royal Society Edinburgh, Vol. 32, p. 239, in J. Aitken (ed.), Collected Scientific Papers. Cambridge University Press: Cambridge, 1923.

Camuffo, D. (1983). ‘Indoor Dynamic Climatology: Investigations on the Interactions Between Walls and Indoor Environment’, Atmospheric Environment, 17: 1803-1809.

Camuffo, D. (1984). ‘Condensation-Evaporation Cycles in Pore and Capillary Systems According to the Kelvin Model’, Water Air Soil Pollution, 21: 151-159.

Camuffo, D. (1988). ‘Surface Moisture and Conservation’, European Cultural Heritage Newsletter on Research, 2 (5): 6-10.

Camuffo, D. (1989). ‘Discussion: Particle Deposition from a Natural Convection Flow onto a Vertical Isothermal Flat Plate’, Journal of Aerosol Science, 20: 137-138.

Camuffo, D. (1991). ‘Environment and Microclimate’, in N. Baer, C. Sabbioni and A. Sors (eds), Science, Technology and European Cultural Heritage, pp. 37-50. London: Butterworths.

Camuffo, D. and Bernardi, A. (1986). ‘D inamica de1 micro&ma e scambi termoigrometrici tra pareti e atmosfera interna della Cappella Sistina’, Bollettino Monumenti Musei Galleie Pontificie, 2: 211-257.

Camuffo, D. and Bernardi, A. (1988). ‘Microclimate-Induced Weathering on the Sistine Chapel’, European Cultural Heritage Newsletter on Research, 2 (1): 36-42.

DARIO CAMUFFO 383

Camuffo, D. and Bernardi, A. (199la). ‘Indoor and Outdoor Microclimate: the Trajan Column and Sistine Chapel’, in N. Baer, C. Sabbioni and A. Sors (eds), Science, Technology and European Cultural Heritage, pp. 295-305. London: Butterworths.

Camuffo, D. and Bernardi, A. (I99Ib). ‘11 microclima della Cappella Sistina’, in Michelangelo e la Cappella Sistina. Morgano: Giacometti and Cappeller (in press).

Camuffo, D. and Bernardi, A. (1991~). ‘The Microclimate of Leonardo’s “Last Supper”‘, European Cultural Heritage Newsletter on Research and Bollettino Geofisico (joint edition, in press).

Camuffo, D. and Schenal, I’. (1982). ‘M icroclima all’interno della Cappella degli Scrovegni: scambi termodinamici tra gli affreschi e I’ambiente’, in Minister0 Beni Culturali (ed.), Giotto a Padova, special issue of the Bollettino d’Arte, pp. 107-209. Rome: Poligrafico dello Stato.

Camuffo, D. and Valcher, S. (1986). ‘A Dew-Point Signaller for Conservation of Works of Art’, Environmental Monitoring Assessment, 6: 165-170.

Eastaugh, N. (1990). ‘The Visual Effects of Dirt on Painting’, in V. Todd (ed.) Dirt and Pictures Separated, pp. 19-23. Gainsborough, UK: Belton.

Friedlander, S.K. (1977). Smoke, Dust and Haze. New York: Wiley. Goren, S.L. (1977). ‘Thermophoresis of Aerosol Particles in the Laminar Boundary Layer on a Flat

Plate’, Journal of Colloid Interface Science, 61: 77-85. Holman, J.P. (1989). Heat Transfer. Singapore: McGraw-Hill. Incropera, F.P. and De Witt, D.P. (1985). F un d amentals of Heat and Mass Transfer. New York: Wiley. Isachenko, V.P., Osipova, V.A. and Sukomel, A.S. (1977). Heat Transfer. Moscow: Mir. Lodge, O.J. and Clark, J.W. (1884). Philosophical Magazine., Vol. 17, p. 214, see W. Zernik, ‘The

Dust-Free Space Surrounding Hot Bodies’, British Journal of Applied Physics, 8 (1957): 117-120. Luikov, A. (1980). Heat and Mass Transfer. Moscow: Mir. Nazaroff, W.W. and Cass, G.R. (1987). ‘Particle Deposition from a Natural Convection Flow onto a

Vertical Isothermal Flat Plate’, Journal of Aerosol Science, 18: 445-455. Nazaroff, W.W. and Cass, G.R. (1989a). ‘Reply: Particle Deposition from a Natural Convection Flow

onto a Vertical Isothermal Flat Plate’, Journal of Aerosol Science, 20: 138-139. Nazaroff, W. W. and Cass, G. R. (1989b). ‘Mathematical Modeling of Indoor Aerosol Dynamics’,

Environmental Science and Technology, 23 : 157-166. Nazaroff, W. W. and Cass, G. R. (1991). ‘Protecting Museum Collections from Soiling Due to the

Deposition of Airborne Particles’, Atmospheric Environment, 2jA : 841-852. Prodi, F., Santachiara, G. and Prodi, V. (1979). ‘M easurements of Thermophoretic Velocities of Aerosol

Particles in the Transition Region’, Journal of Aerosol Science, 10: 421-425. Prodi, F. and Tampieri, F. (1982). ‘The Removal of Particulate Matter from the Atmosphere: the

Physical Mechanisms’, Pageoph, 120: 286-325. Schlichting, H. (1979). Boundary-Layer Theory. New York: McGraw-Hill. Seinfeld, J.H. (1986). Atmospheric Chemistry and Physics of Air Pollution. New York: Wiley. Slinn, W.G. and Hales, J.M. (1970). ‘Phoretic Processes in Scavenging’, AECSymposium Series No. 22,

US Atomic Energy Committee, pp. 41 l-422. Slinn, W.G. and Hales, J.M. (1971). ‘A Re-evaluation of the Role of Thermophoresis as a Mechanism of

In- and Below-Cloud Scavenging’, Journal of Atmospheric Sciences, 28: 1465-1471. Talbot, L., Cheng, R.K., Schefer, R.W. and Willis, D.R. (1980). Th ermophoresis of Particles in a

Heated Boundary Layer’, Journal of Fluid Mechanics, 101: 737-758. Tritton, D.J. (1988). Physical Fluid Dynamics. Oxford: Clarendon Press. Waldmann, L. and Schmitt, K.H. (1966). ‘Th ermophoresis and Diffusioforesis of Aerosols’, in C.N.

Davies, (ed.) Aerosol Science, Ch. 6. New York: Academic Press. Watson, H.H. (1936). ‘The Dust-Free Space Surrounding Hot Bodies’, Transactions of the Faraday

Society, 32: 1073-1084. Welty, J.R., Wicks, C.E. and Wilson, R.E. (1984). Fundamentals of Momentum, Heat and Mass

Transfer. New York: Wiley. White, F.M. (1984). Heat Transfer. Reading, MA: Addison-Wesley. Zernik, W. (1957). ‘The Dust-Free Space Surrounding Hot Bodies’, British Journal of Applied Physics,

8: 117-120.