Optimum conditions for evaporation control by monolayers

Journal of Hydrology, 145 (1993) 165-173

165

Elsevier Science Publishers B.V., Amsterdam

[1]

Optimum conditions for evaporation control by monolayers G.T. Barnes Department of Chemistry, University of Queensland, St. Lucia, Qld. 4072, Australia (Received 11 March 1992; accepted 5 October 1992)

ABSTRACT Barnes, G.T., 1993. Optimum conditions for evaporation control by monolayers. J. Hydrol., 145:165-173. The ability of a monolayer to retard the evaporation of the water on which it is spread is often reported as the fractional reduction in evaporation rate. This fraction varies with the experimental conditions. Its relationship with evaporation resistance, an absolute measure of the effect, and with various environmental factors is described.

INTRODUCTION

Interest in using insoluble monolayers to reduce evaporative losses of water in open storages has diminished since the period of activity that followed the demonstrations of Mansfield (1955) and Vines (1960, 1962) that it could be an effective and practical procedure. To a large extent this reduced interest is attributable to the failure of the monolayers (usually cetyl or stearyl alcohol or a mixture of these) to withstand the conditions in which they were required to work. However another, more subtle, factor could have been an expectation of performance that was often unrealistic because the conditions for optimum performance had not been appreciated and because of the custom of reporting test results as percentage reductions in evaporation loss. It is the purpose of this paper to provide a rigorous theoretical framework linking the fundamental property of a monolayer, its permeation resistance, to the conditions of operation. THEORY

The rate of evaporation of water is governed by the driving force for Correspondence to: G.T. Barnes, Department of Chemistry, University of Queensland, St. Lucia, Qld. 4072, Australia.

0022-1694/93/$06.00

© 1993 - - Elsevier Science Publishers B.V. All rights reserved

166

G.T. BARNES

evaporation and by the total permeation resistance of the transport pathway in an equation analogous to Ohm's law for electrical conduction (Archer and La Mer, 1955)

J = Ac/Zr

(1)

where J is evaporative flux, Ac ( = ceq -- cV) is the difference in water vapour concentrations driving the evaporation (ceq being the equilibrium vapour concentration for the surface layer of water and c" being the actual vapour concentration in the atmosphere some distance above the surface), and Zr is the total permeation or evaporation resistance, being the sum of the resistances that arise from sections of the transport pathway (Barnes, 1986). With SI units in eqn. (1), J is in moles (or kilograms) per second per metre squared, c is in moles (or kilograms) per metre cubed, so resistance is in seconds per metre. The spreading of a suitable monolayer on the water surface will increase the total evaporation resistance by an amount that depends on the nature of the monolayer substance, the surface pressure (defined as the lowering of surface tension due to the monolayer (Gaines, 1966)), and the temperature. It is not the present purpose to explore these factors, but a summary has been published elsewhere (Barnes, 1986). This paper concentrates on the environmental factors that influence the performance of a monolayer: the total permeation resistance of the transport pathway, the magnitude of the driving force, and the heat balance of the system.

Effect of total permeation resistance With an appropriate choice of monolayer the total resistance to evaporation can usually be increased by perhaps 100-300 s m-~ (Archer and La Mer, 1955; Rosano and La Mer, 1956), a quantity called the monolayer resistance, r m. This is an intrinsic property of the monolayer: it depends on the surface pressure, the temperature, and the composition of the monolayer and is independent of conditions of measurement such as the driving force. Whether the monolayer resistance has a significant effect on the evaporation rate depends on the magnitude of the total resistance. The total resistance is usually considered to be a set of component resistances in series, with one resistance for each segment of the transport pathway, namely the bulk liquid phase (zero resistance for the evaporation of a pure liquid), the surface, the surface film if present, and the air (Barnes, 1986). For a monolayer-free surface, the total resistance is denoted by Erw (w for water), while for a monolayer-covered surface the total resistance is Err (f for film), where Err =

Zrw + rm

(2)

167

EVAPORATION CONTROL BY MONOLAYERS

The performance of a monolayer is often reported as the ratio, 49, of evaportion rate with the monolayer present to the rate for a monolayer-free surface 49 =

(3)

Jr/Jw

or as the relative decrease in evaporation rate, defined by 1 - 49 =

(Jw - Jr)/Jw

(4)

where the fluxes Jw and Jr refer to evaporation through a monolayer-free water surface and a monolayer-covered surface, respectively. Provided that the conditions for evaporation remain unchanged when the monolayer is spread, and specifically that the driving force, Ac, and the other resistances, Zrw, are unaltered, combination of eqns. (1) and (3) gives 49 =

Zrw/Zr f =

(5)

Z r w / ( Z r w + rm)

while (1) and (4) yield 1 -

49 =- r m / Z r r =

r m / ( Z r w + r m)

(6)

It is apparent that the values of 49 and (1 - 49) depend not only on the resistance of the monolayer (rm), but also on the resistances of the other sections of the transport pathway, collected together as Zrw: the larger the value of Zrw the closer the relative evaporation rate to unity (no effect of the monolayer) and the smaller the relative reduction. Thus, the monolayer is effective, as measured by 49 or (1 - 49), when the value of Zrw is small and, consequently, the evaporative flux is large (eqn. (1)). That is, it is most effective when it is most needed. The value of Zrw can be altered, for example, by turbulence in the air above the water surface, altering the effective diffusion path length and hence this c o m p o n e n t of the total resistance. An estimate of the value of Zrw can be obtained from the observation (Vines, 1962) that monolayers of cetyl alcohol (hexadecanol) can reduce evaporation losses from reservoirs by up to 50%. Substituting this value and a conservative value for r m (100 S m - I (Barnes and La Mer, 1962)) into eqn. (5) yields Zr w = 100 s m - ' Effect o f changes in the driving f o r c e

In economic terms, the a m o u n t of water saved may be more important than the relative saving. This can be measured by the reduction in evaporation rates Jw - Jr =

(1 - 49)Jw -

using eqns. (4), (6) and (1).

Fm

mc

Zrw + rm Zrw

(7)

168

G.T. BARNES

TABLE 1 Water saved by the addition of a monolayer with r m Zrw = 100sm l (in all cases l -- 0 = 0.5)

=

100sm-J to a water surface where

Property

Value at temperature and relative humidity shown

T(K)

283 0.9 0.94 4.7

c/c eq

Ac (gm 3) (Jw - J 0 ( m g s - l m

2)

283 0.6 3.76 18.8

283 0.3 6.58 32.9

303 0.9 3.04 15.2

303 0.6 12.15 60.8

303 0.3 21.3 106.5

Equation (7) shows that the reduction in evaporation rate depends not only on the resistances in the transport pathway, but also on the driving force for evaporation. If the evaporation rate is low because of a low value for the driving force, the relative effect of the monolayer will still be determined by eqns. (5) and (6), which show dependence on the resistances only, but the absolute magnitude of the reduction in water loss will be small (eqn. (7)). Thus, for example, Ac depends on the humidity of the air above the water surface, and for fixed values of Y~rwand r m, this governs the evaporation rates, Jw and Jr, in accordance with eqn. (1). In conditions of high humidity Ac is small, and although the amount of water saved will, therefore, be small the rate of evaporation will be low. Conversely, in dry conditions there will be a high rate of water loss from an untreated surface, but a large reduction when a monolayer is spread. Both 4) and (1 - q~) should, however, be independent of the humidity. Thus, in the example discussed above where (1 - ~b) = 0.5. Eqn. (7) shows that the actual saving varies with the humidity. Some values are shown in Table 1, where the calculations have been based on the resistance values given above.

Changes in heat transport A reduction in the net evaporation rate is associated with a reduction in the rate of transport of latent heat from the liquid to the vapour. Consequently the cooling of the liquid surface will decrease when a monolayer is present, but the effect will be partially offset by increases in the heat loss from the surface by radiation and conduction. The rise in surface temperature will also lead to a rise in the equilibrium vapour pressure and to an increased evaporation rate. Thus, the overall effect on the evaporation rate is that the reduction achieved by the addition of a monolayer is partly negated by the consequent rise in surface temperature (Mansfield, 1955, 1958). A detailed energy budget for this situation is complex and depends on

169

EVAPORATION CONTROL BY MONOLAYERS

factors relating to the specific body of water and to conditions at the time: depth, fetch, wind velocity, cloudiness, time of year, time of day, and so forth. The contributions of such factors are expressed in the so-called combination equation (Brutsaert, 1982; Monteith and Unsworth 1990), but in the present context only those terms that are altered by the addition of a monolayer are relevant. This simplifies the analysis. Two principal assumptions are made. To compare the monolayer-covered and the monolayer-free surfaces the environmental conditions must be the same, and so it is assumed that certain rates of heat transfer to or from the water surface (principally from solar radiation and by convection) are unaffected by the spreading of the film. For the most commonly used monolayers (long-chain alcohols) there is no absorption of solar radiation (Pretsch et al., 1983) and hence this assumption is reasonable. Also, although convection is an efficient mechanism of heat flow, the surface is insulated from it by effective boundary layers in both the liquid and gas phases (Sherwood et al., 1975), so it is unlikely that the addition of a monolayer will significantly alter the rate of heat loss by this means (Barnes and Hunter, 1982). It is also assumed that measurements are made only when there has been sufficient time for steady-state conditions to be established. Experimentally this can be ascertained by repeating the measurements at time intervals selected by experience. In the laboratory, steady-state conditions are established after about 30 s (Barnes et al., 1980). For a steady-state situation all of the heat fluxes to or from the interface must sum to zero, both before and after the addition of the monolayer. Thus, for those heat fluxes that do change when the monolayer is added, the sum of all the changes must be zero AOV + A4c + Aq r =

0

(8)

where A0 = qf - Ow is the difference between the rate of heat transfer with film and the rate without film; and superscripts v, c and r refer to vaporization, conduction, and radiation, respectively. The customary sign convention is adopted where heat flow away from the surface is positive. For the vaporization process aq v =

A H V ( J r - Jw)

(9)

for conduction Aqc =

(h w + h a ) ( T f - Tw) =

h(rf- Tw)

(10)

and for radiation a¢ r =

¢-T

4)

(11)

where AH ~ is the enthalpy of vaporization, T is the surface temperature of the

170

G.T. BARNES

water, e is the thermal emissivity, a is the Stefan-Boltzmann radiation constant, h w and h" are the heat transfer coefficients for water and air, and h=hW+h" Substitution of eqns. (9), (10) and (11) into (8) gives A H V ( J r - Jw) + h ( T r -

rw) + e,a(T4 -

T4) =

0

(12)

The following approximations can be introduced 7,4 -

7-;

4rw

(rf- L )

which is valid if(Tr - Tw) < 1. Also, from the Clausius-Clapeyron equation Aceq AT

dceq dT

A H v c eq

RT 2

Further, referring to eqn. (1) A(Ac)

=

(c~ q - -

=

C~q -

c~) Ceq

=

(c?

--

c'w)

A C eq

as the assumption of the same environmental conditions implies that cv is the same above the monolayer-covered surface and above the monolayer-free surface, i.e. that c[ = c~. Now, from eqns. (1) and (2), ACeq

=

A(Ac) =

=

(Jr-

J f Z r r - JwEr.

Jw)Erw + .]frm

Grouping terms into a function, B, given by B =

B(T)

(AHV): ceq , = (h + 4eaT~)RT ~

(13)

eqn. (12) becomes (Jr - Jw)(B + Zrw) + Jfrm

=

0

(14)

Rearrangement and use of eqns. (2) and (3) yields ~

m

Er w + B Y,rr + B

(15)

and I-4~

rm

Err + B

(16)

Equations (5) and (6) were derived without making allowance for a change in surface cooling, with the assumption, now shown to be incorrect, that

171

E V A P O R A T I O N C O N T R O L BY M O N O L A Y E R S

TABLE 2 Data used in the calculation of B, and values of B Property

Value at temperature shown

T (K) AH v (jg 1) ccq (g m-3 ) h ( W K im 2) s a ( W K 4m 2) 4saT3 (W K i m 2) B ( s m 1)

283 2471 9.40 230 0.96 5.67 x 10 ~ 4.93 6.61

303 2425 30.38 230 0.96 5.67 x 10 s 6.06 17.84

Reference

I.C.T. (1933) Weast (1978) Barnes and Hunter (1982) Siegel and Howell (1981) Weast (1978)

C~ q = C eq w • Comparison of eqn. (15) with eqn. (5) shows that the effect of diminished surface cooling on the relative evaporation rate, qS, is to introduce a virtual resistance, B, to be added to both Zr w and Zrf. However, it is important to note that, despite its formal placement in eqn. (14), B is not one of the c o m p o n e n t resistances of Zrw as it only appears when the monolayer is present. In order to estimate the significance of B, its value is calculated for two sets of conditions: water surface temperatures of 283 K and 303 K. The data used are shown in Table 2, together with the calculated values of 4eo-T,~ and B. First, it is useful to point out that the term arising from radiative heat transfer (4eaT 3) is small relative to the term for conduction (h), and can generally be ignored. The significance of B can be assessed by comparing its values with values of Erw. For the Langmuir-Schaefer method used in the laboratory (Archer and La Mer, 1955; Barnes, 1986) Zr w lies between 200 and 300 s m -1 . For open water storages there are few data available (see Table 1), but Er,v will depend strongly on the environmental conditions. For example, r for diffusion through stagnant air is about 40 s m - t for each 1 m m thickness (Barnes, 1986), so variations in the thickness of the boundary layer caused by changes in wind speed will have a major effect on Zrw. It can be seen from Table 2 that B increases with temperature, thus reducing the effect of the monolayer on q5 and (1 - qS). This effect is in addition to the normal decrease of rm with temperature (Barnes, 1986).

CONCLUDING REMARKS

The spreading of a monolayer on a water storage may significantly reduce the a m o u n t of water lost by evaporation when the driving force for evaporation is high and the total resistance to evaporation is low (eqn. (7)). In other

172

G.T. BARNES

words monolayers are potentially most effective in conditions where the rate of evaporation is high. Such conditions include hot dry weather, where the driving force is large because of a low humidity, and windy weather where the permeation resistance of the air is reduced by turbulence. Whether monolayers function effectively in such conditions depends on monolayer properties other than their permeation resistances. In hot conditions monolayer material can be lost from the surface by evaporation, but usually the rate is low so this is unlikely to be a problem. In windy conditions the monolayer can be blown across the surface and collapsed on the lee shore. Experimental evidence shows that this occurs with monolayers of cetyl and stearyl alcohols at quite moderate wind speeds. Thus, one of the most important properties in the design or selection of improved monolayer materials for evaporation control is the ability to withstand wind stress. The introduction of polymeric materials into the monolayers may provide an answer to this problem (Fukuda et al., 1979; Drummond et al., 1992). ACKNOWLEDGEMENT

Financial support from the Australian Research Council is gratefully acknowledged. REFERENCES Archer, R.J. and La Mer, V.K., 1955. The rate of evaporation of water through fatty acid monolayers. J. Phys. Chem., 59: 200-208. Barnes, G.T., 1986. The effects of monolayers on the evaporation of liquids. Adv. Colloid Interface Sci., 25: 89-200. Barnes, G.T., Costin, I.S., Hunter, D.S. and Saylor, J.E., 1980. On the measurement of the evaporation resistance of monolayers. J. Colloid Interface Sci., 78: 271-273. Barnes, G.T. and Hunter, D.S., 1982. Heat conduction during the measurement of the evaporation resistances of monolayers. J. Colloid Interface Sci., 88: 437-443. Barnes, G.T. and La Mer, V.K., 1962. The laboratory investigation and evaluation of monolayers for retarding the evaporation of water. In: V.K. La Mer (Editor), Retardation of Evaporation by Monolayers: Transport Processes. Academic, New York, pp. 35-39. Brutsaert, W., 1982. Evaporation into the Atmosphere. Reidel, Dordrecht. Drummond, C.J., Elliott, P., Furlong, D.N. and Barnes, G.T., 1992. Water permeation through two-component monolayers of polymerised surfactants and octadecanol. J. Colloid Interface Sci., 151: 189-194. Fukuda, K., Kato, T., Machida, S. and Shimizu, Y., 1979. Binary mixed monolayers of polyvinyl stearate and simple long-chain compounds at the air water interface. J. Colloid Interface Sci., 68: 82-95. Gaines, G.L., 1966. Insoluble Monolayers at Liquid-Gas Interfaces. Interscience, New York, p. 44. I.C.T., 1933. International Critical Tables, Vol. 5. McGraw-Hill, New York, p. 138. Mansfield, W.W., 1955. The influence of monolayers on the natural rate of evaporation of water. Nature, 175: 247-249.

EVAPORATION CONTROL BY MONOLAYERS

173

Mansfield, W.W., 1958. The influence of monolayers on evaporation from water storages I. The potential performance of monolayers of cetyl alcohol. Aust. J. Appl. Sci., 9: 245-254. Monteith, J.L. and Unsworth, M.H., 1990. Principles of Environmental Physics, 2nd edn. Edward Arnold, London. Pretsch. E., Clerc, T., Seibl, J. and Simon, W., 1983. Tables of Spectral Data for Structure Determination of Organic Compounds. Springer, Berlin, p. B140 (translated by K. Biemann). Rosano, H.L. and La Mer, V.K., 1956. The rate of evaporation of water through monolayers of esters, acids and alcohols. J. Phys. Chem., 60: 348-353. Sherwood, T.K., Pigford, R,L. and Wilke, C.R., 1975. Mass Transfer. McGraw-Hill, Kogakusha, Tokyo, Chapter 5. Siegel, R. and Howell, J.R., 1981. Thermal Radiation Heat Transfer, 2nd edn. Hemisphere, Washington, p. 835. Vines, R.G., 1960. Reducing evaporation with cetyl alcohol films: a new method of treating large water storages. Aust. J. Appl. Sci., 1I: 200-204. Vines, R.G., 1962. Evaporation control: a method of treating large water storages. In: V.K. La Met (Editor), Retardation of Evaporation by Monolayers: Transport Processes. Academic, New York, pp. 137-160. Weast, R.C. (Editor), 1978, Handbook of Chemistry and Physics, 58th edn. CRC, West Palm Beach, pp. E-41, F-243.