THE CALCULATED EFFICIENCY OF MONOLAYERS IN RELATION TO INCREASED WATER TEMPERATURE

H . J . WOLBEER Associate Research Officer

Saskatchewan Research Council

RÉSUMÉ

L'échange d'énergie entre un réservoir et l'atmosphère est décrit comme un flux descendant, désigné «la puissance calorifique de l'atmosphère» et un flux montant, désigné «la puissance de refroidissement de la surface de l'eau». La première est une fonction des paramètres météorologiques, la deuxième est principalement une fonction de la température de la surface de l'eau. Leur différence est égale au changement de la teneur calorifique du réservoir, qui est associée à la température de la surface par l'entremise de «la profondeur effective».

En utilisant ces principes, il est possible de résoudre les équations pour évaluer la température de la surface et de déterminer le régime thermal et le taux d'évapori-sation. La méthode permet l'évaluation des effets d'une couche monomoléculaire sur la température de l'eau et l'évaporation. L'évaporation d'un réservoir couvert est calculé en divisant l'évaporation d'une surface libre par un facteur 1 + (1 + //) DR. L'intensité du mélange turbulent d'atmosphère est désignée par D, la résistance de la couche monomoléculaire par R, et n est un coefficient sans dimensions.

La réduction d'evaporation est accompagnée d'une augmentation de la tempé-rature, qui est proportionnelle à la réduction d'évaporation. Le rapport dépend principalement de la vitesse du vent. Les changements de la teneur calorifique au commencement et à la terminaison de l'application sont traités. Les effets sur l'éva-poration sont significatifs. Des couches monomoléculaires partielles sont évaluées. L'efficacité des applications intermittentes est relativement réduite.

SUMMARY

Energy exchange between a reservoir and the atmosphere is described as a total downward flux, called "Heating Power of the Atmosphere" and a total upward flux, called "Cooling Power of the Water Surface". The former is a function of meteoro-logical parameters only, the latter is primarily a function of the water surface temper-ature. Their difference equals the change in heat storage which is related to the water surface temperature by means of the "effective depth". Using these concepts it is possible to solve for surface temperature and to determine thermal regime and rate of evaporation from meteorological factors and reservoir characteristics.

The described method permits a simple evaluation of the effects of a monolayer on water temperature and evaporation. Evaporation from a covered reservoir is calculated by dividing the free water evaporation by a factor 1 + (1 — n) DR, where D is a turbulent mixing factor, R is the film resistance and n is a dimensionless coefficient. Evaporation reduction is accompanied by a temperature increase, which is propor-tional to the evaporation reduction. The ratio is mainly a function of wind velocity. The change in heat storage at the beginning and termination of monolayer application is discussed. Its effect on evaporation is shown to be significant. Partial monolayers are evaluated. Intermittent monolayers appear to be less efficient.

INTRODUCTION

It has been known for a long time that certain substances, when brought in contact with a water surface, will form a monomolecular layer. If sufficient material is available, the total system will have a minimum of potential energy when the mole-cules in the monolayer are under a certain compression. Such a monolayr will be cal-led a fully compressed monolayer. The effect of such a monolayer is to resist pene-tration by water molecules in either direction.

Application of monolayers to actual reservoirs, with the purpose of reducing

13

evaporation, has been studied intensively only during the last decade. Problems may be classified into 3 categories.

1. Problems related to the properties of a fully compressed monolayer as such. The effects of impurities and temperature, biological aspects and the magnitude of the resistance to penetration have been studied extensively for numerous materials. Other problems, such as emission and reflection of radiation, and the influence on wave forming and laminar air and water layers have received less attention.

2. Problems related to the application and maintenance of a fully compressed monolayer on actual reservoirs. These are engineering problems in which wind condi-tions play a major role.

3. Problems related to the effects of a given monolayer on thermal regime and evaporation. Obviously investigations of this nature require a fundamental approach to the interchange of mass and energy between the reservoir and its surroundings.

Evaluation of experiments is usually based on overall results and it is difficult to separate the various effects mentioned above. This paper is an attempt to describe the influence of a given monolayer on thermal regime and evaporation. Such influence is in general not independent of meteorological conditions and reservoir characteristics.

As a necessary prerequisite, thermal regime and evaporation as a function of meteorological conditions and reservoir characteristics, will be briefly reviewed for a non-covered reservoir. Certain concepts, developed by the author, will be introduced because they allow a simple visualization of the processes involved.

The changes in such a system of interactions, resulting from the presence of a monolayer, will then be investigated, utilizing the concepts previously developed.

Finally, the efficiency of partial monolayers will be evaluated.

THE HEAT BUDGET OF A RESERVOIR

The heat content of a reservoir varies due to energy exchange with the atmosphere and to a much lesser extent due to energy interchange with the bottom and in-and-out flow. Energy exchange with the atmosphere occurs in the form of radiative heat flow, sensible heat flow and vapour flow or latent heat flow. Each of these can be consi-dered (with much physical justification) as a net result of a downward and an up-ward flow. The downward flows can be combined and, for the sake of convenience, be measured with respect to a certain datum value. The datum values are those of a water surface at 0° C and the total downward flow is deno ted by H and called the "Heating Power of the Atmosphere". It is the amount of heat, transjerred per area unit per time unit from the atmosphere to a water surface at 0°C.

H = as Ras + ai (Ral — K 2734) + CTa + D{ea — 6.02)

Ra + CTa + D(.ea — 6.02)

where H is expressed in ly/hr.

as is the absorption coefficient of water for solar radiation. a\ is the absorption coefficient of water for longwave radiation = 0.97. K is the Stephan Boltzman constant = 4.75 X 10~9 fy/hr/°C4. Ras is the incoming solar and sky radiation in ly/hr. Ral is the incoming atmospheric radiation in ly/hr. Ta is the air temperature of the bulk of the air in °C. ea is the vapour pressure of the bulk of the air in millibar. C and D are turbulent mixing factors for sensible and latent heat respectively

in ly/hr/°C and ly/hr/millibar, respectively.

The upward flows can be combined and also measured with respect to a water surface at 0°C. The total upward flow is denoted by Q and called the "Cooling

14

Power of the Water Surface".

= cnK{(Ts = JR. + CTS

+• 273)4 — 273* } + CTS + D(es — 6.02)

D(es — 6.02)

where g is expressed in ly/hr. Ts is the water surface temperature in °C. es is the saturated vapour pressure at Ts in millibar.

Under conditions of neutral air stability and steady state the factors C and D are related by D = 1.6 C.

The energy exchange with the bottom and in-and-out flow are ignored in this paper, but can be included quite easily. We now introduce the concept of "Effective Depth d" which is,defined as the change in heat storage per cm2 surface area divided by the change in surface temperature (Tz — 7i). It appears that d is fairly constant for a certain reservoir, if the period is taken sufficiently long (from a day for very shallow reservoirs to 4 weeks for very deep reservoirs) and if the surface temperatures 7i and 72 are either smoothed or derived from average values over adjacent période. As a first approximation d may be taken as the actual average depth for non stratified reservoirs and as the depth of the epilimnion for stratified reservoirs.

If the increase in heat storage per cm2 per hour is denoted by S the heat budget of the reservoir is expressed by

JSdt = Hdt—jQdt

The Heating Power H is essentially a function of meteorological conditions only, except that the factors C and D must be adjusted for reservoir size and sheltered location.

The Cooling Power Q is primarily a function of the surface temperature for known values of C and D.

The Heat Storage change S is related to surface temperature through d. The heat budget equation can now be solved for the water surface temperature.

In practise average values of Q over a number of periods are calculated. The average

1 Or

C'c £J^

H i /

,pU) -'

/

/

s

..

10 15 20 25 Surface Temperature °C Fig. 1 — Cooling Power Q.

15

surface temperatures and the smoothed surface temperatures at the breaks between the periods are related by one of the well known interpolation formulae.

The value of Q is tabulated as a function of C and D and surface temperature Ts, and is graphically represented in figure 1.

The average rate of evaporation E over each of the periods can now be calculated from any of the familiar formulae,

EL = DOs — ea)

or EL = Ra — Rs + C(Ta — Ts) — S

or EL -Rs—S

C(TS — Ta) 1 +

D es ea which will all give the same result.

Many details of the described method, such as the corrections for the use of average values, must remain unmentioned, but the author wishes to introduce 2 coefficients which will prove to be of practical value.

These coefficients are defined as

~b(EL)

and

It follows that mn = Z)J, where 3 is the gradient of the saturated vapour pressure — temperature curve, as used by Penman. The coefficient m has the dimensions of ly/hr/°C and n is dimensionless. Both m and n are a function of Ts and C and D and are presented in figure 2 and figure 3.

At this stage it must be admitted that the relationships between C and D and the

m J5 zo 25 SL/rface Temper of lire aC

35

Fig. 2 — Coefficient m = ~ba%

16

wind velocity profile are left undetermined. Although at one time the problem of evaporation was almost identified with the problem of establishing those relationships, this is not the case in methods such as the procedure described above.

Surface 7~eûiperafure °C

Fig. 3 Coefficient n = - 5 i . OQ

An error in the estimation of C and D has a far smaller effect on calculated evaporation than a similar error on evaporation calculated from a mass transfer equation. Fortunately Nature is on our side and the effect of errors in the estimation of meteorological parameters is identical with the effect of real variations, if the described procedure is followed.

In the diagrams for n and m and also in further illustrations, values of D = 1.6 C are related to wind speeds at 2 m. above the water surface. This is done to indicate wind conditions, but it should not be inferred, that a fixed relation beween C, D and wind speed exists under various conditions.

The described method can be classified as a combined mass-tranfer heat-budget method, eliminating the need for surface temperature measurements. This principle was utilized for the first time by Penman C1) to derive his ingenious well-known formula. Kohler (2) and Kohler, Nordenson and Fox (3) used Penman's formula to calculate evaporation from a hypothetical pan and designed a set of nomographs based on an empirical turbulent mixing factor. It should be noted that the coefficient n is similar to the coefficient a used by Kohler, Nordenson and Fox to correct pan evapo-ration for advected heat. Penman's formula becomes less accurate for larger differences between air and water temperatures. A correction for this difference has been intro-duced by Baker and Linsley (4).

Trial and error methods to solve simultaneously the mass-transfer and the heat-budget equations have been utilized by Harbick (5) and Velz and Gannon (6). The change in heat storage was usually neglected or introduced as a measured quantity.

Mansfield (7), assuming a uniform water temperature and a sinusoidal annual variation of radiation and air temperature, calculated the effect of depth on the annual cycle of evaporation.

Raphael (8) assuming uniform temperature changes over periods of 3 hours and starting with some initial temperature proposes an accounting system, whereby the uniform temperature increase over each period is calculated.

17

THE INFLUENCE OF A MONOLAYER ON SURFACE TEMPERATURE AND EVAPORATION'

The resistance R of a monolayer is defined by EL — where e\ and e%

are vapour pressures at each side of the monolayer, expressed in millibars. E is the net flow of water molecules through the monolayer in grams/cm2/hr. L is the latent heat of vaporization in cal/gram.

It is assumed that a monolayer does not change the reflected or emitted radiation, or the sensible heat exchange. Conceivably wave forming under moderate wind conditions might be reduced and the thicknesses of a laminar air layer and a laminar water layer might be increased under calm conditions. This would also affect sensible heat exchange and heat flow into or from the body of water. Lacking sufficient evidence, such considerations are not taken into account.

Under steady state conditions a vapour pressure eo will be established just above the monolayer, such that

es — eo EL = = D(eo — ea).

D or EL = Os — ea)

1 + DR

The procedure, described before, may now be followed, simply by substituting

D D' = for D

1 + DR The film resistance R appears only in the dimensionless product DR.

It is possible to derive simple expressions for the reduced evaporation and the increased temperature, by utilizing the coefficients m and n.

First it will be assumed that the monolayer has been present for some time and does not affect the change in heat storage. Two identical reservoirs, one of which is fully covered by a homogeneous monolayer, are compared and the parameters referring to the covered reservoir are primed.

H = Ra + CTa + D (ea - 6.02)

H' = Ra + CTa + D'(ea — 6.02)

Q = £„ + CT„ + D(ee — 6.02)

Q' = R' + CT\ + D'(e - 6.02)

if AT =T'—T s s s

R'= R A SAT and e" = e + ÔA T s s dTx s s s s

H' - H =-.—(p~- D') (ea — 6.02)

Q' - Q = ^—S A T + CAT + DôA T - (D — D') (e' — 6.02) dTx s s s s

= AT - (D — D') (e' — 6.021 = mA T —(D — D') {e' — 6.02) ~à T« s s s s

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Since S = H— Q = H' — Q'

(Q'—Q) — (H' — H) = 0

or 0 = ma T -(D — D') (e' — ea) = mÂTs— (D — D') (e., — ea + ÔA Ts)

(D — D') (,es — ea) DRD (es — ta) ATS =

m — ô(D — D')

ELDR

m(\+DR)—ôDRD

EL DR

E'L = D' (e' s

D

m(l +DR) — mnDR 1 + (1 — ri) DR

D -e) =

1 + DR

1 + DR

EL { nDR

•ea) +

(e + ô ET — e ) s s a

ELDRô

i{\ +(!—«) DJî}

1 + Z>J? 1 + EL

1 + (1 — ri) DR)

EL

1 + DR

1 + DR [1 + () — n) DR l

1 + (1 - ri) DR

The free water evaporation must be divided by a factor 1 + (1 — ri) DR to obtain the reduced evaporation. As shown before the reduced evaporation is also equal to the free water evaporation at the same temperature, divided by 1 + DR.

If the water saving is denoted by Er

Er = E — E' or Er = (1 - ri) E

Ï/DR + (1 — ri) = (1 — ri) DR E'

The relative saving is larger for larger values of D and R and smaller values of n, that is better film resistance, higher wind speeds and lower water temperatures. The absolute saving, however, is larger at higher water temperatures and higher wind

•£

25 10 Z5" 20

Surface Temperature "C

Fig. 4 — Evaporation reduction

19

speeds. Unfortunately it is more difficult to maintain a certain film resistance at higher wind speeds.

Relative savings are shown in figure 4 as a function of DR and surface temperature. Since n is somewhat dependent on £>,the diagram is based on R = 1.25 millibar hr/ly. Other values of R will change the picture only slightly. It should be noted that generally R is a function of temperature, but this is a separate phenomena, not considered in this paper.

A few examples are shown below Wind Speed 3 m.p.h. 6 m.p.h. 9 m.p.h.

Ts = 10»C Ts = 30°C

ET = 26% 18%

38%

26%

46% 32%

The increase in surface temperature A Ts has been calculated before

EL DR x E'L-

DR

1 + (1 — n) DR m m

It is more convenient to express A Ts as a function of the evaporation reduction Er-

ErL /A J s —

m{\—n)

The factor m (I —n) is mainly a function of wind speed and the variation with surface temperature is very small.

Z 3 4 5 b 7 {Zfaporaf/on Redaction cm /month

Fig. 5 — Surface Temperature increase.

The ratio between temperature increase and evaporation reduction is shown in figure 5 for various wind speeds and surface temperatures varying bewteen 10° C and 30» C.

A method to evaluate the effect of a monolayer has been described by Harbeck and Koberg (9). Harbeck (10) and Koberg ( u ) (12). A basic feature of this method is that, in addition to the meteorological parameters, the surface temperature Ts and the change in heat storage S are measured. In other words the results Ts and S are measured in order to avoid the necessity of knowledge of the determining factors d and R. This

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has advantages in actual situations, because it is difficult to determine the correct values of film resistance and coverage. On the other hand, it is also difficult to measure the real average surface temperature, particularly under low wind conditions. The presence of a film might well increase the suspected steep temperature gradient in the top few millimeters. Naturally the described method does not apply in the planning stage and does not illustrate the influence of the basic factors.

Another method is described by Florey and co-workers (10) ( u ) (12). The relative evaporation reduction at a reservoir was obtained by calculating a weighted average of relative reductions during short periods (3 hours). The latter were assumed to be proportional to the coverage factor c, which assumption will be shown to be incorrect. The evaporation reduction factor/was measured at " class A " pans. This can be consid-ered as a combined measurement of the film resistance R and the influence of surface temperature as indicated in figure 4. However the influence of the wind velocity was ignored, although in the Lake Cachuma studies, this neglect was not as serious because pan and lake investigations were conducted concurrently.

The reduction factor / will be over-estimated in the method because of the heat flow through the pan wall and because the factor D is larger for a pan than for a lake.

THE EFFECT OF HEAT STORAGE IN THE RESERVOIR

It has been assumed before that a monolayer had been present for some time prior to the period of evaluation. The actual saving that results from an application is influenced by the change in heat storage.

The temperature rise ATS takes place during a certain period after application has been started. Assuming that the concept of effective depth is valid, this corresponds to an increase in heat storage equal to d x A Ts- Consequently the integrated value of Q' during this period of "warm u p " is decreased by d x A Ts. A fraction of this decrease is in the form of latent heat and this fraction is «', where n' is similar to », but with D' substituted for D.

It can be shown that

n' 1 E'

1 + (1 — n) DR E

After termination of monolayer application, the surface temperature will fall by A Ts and the heat storage will be decreased by d x A Ts. The situation is now rever-sed and the latent heat flow is increased by rid A Ts. Since the monolayer is no longer active, n has now the normal value. The overall effect is to increase evaporation by (n — n ') d A Ts

L

If the duration of application is t hours the average increase of EL is

(it — n') d A Ts Er nd  Er nd Er L Es L = = AT, = t E t E t m(l—r)

Es Er n d

Er E m (1 — n) t

For a surface temperature of 25° C and a wind speed of approximately 9 m.p.h p — = 0.18 dit or the saving is reduced by 18 dit %. Er The value of d/t might approach unity, e.g. if the depth is 24 ft. and the duration

of application is one month.

21

The situation is much worse however if an evaluation of a monolayer performance is based on the period of application. In such a case an evaporation increase equivalent

ltd A Ts to , which materializes after termination, is neglected. This represents a

Lt n

fraction of the apparent saving equal to djt, which in the above example m (1 — n)

amounts to 50 djt %. It is obvious that for short test periods monolayer performance can be greatly

over-estimated, if the increased evaporation after termination is no taken into account. In practice, meteorological conditions will not remain constant. The change in surface temperature A Ts is then superimposed upon changes not caused by the monolayer. This does not essentially change the above considerations. As an example the Lake Cachuma report may be referred to. The actual reduction was 170 acre ft and the increase of evaporation after termination of monolayer application was 87 acre ft. If the evaluation had been stopped with the monolayer application, the savings would have been over-estimated by 50%.

PARTIAL MONOLAYERS

In practice it is almost impossible to maintain a fully compressed monolayer over the entire reservoir surface for a prolonged period. A monolayer can be partial in three distinguishable ways.

1) The resistance of tl e monolayer is cR, where c < 1. 2) The monolayer covers a fraction c of the reservoir. 3) The monolayer is alternately destroyed and re-applied and is active during a

fraction c of the total period. A monolayer moving over a reservoir surface represents a combination of 2) and 3),

although small areas with reduced resistance exist. Case 1 Since R is replaced by cR, the free water evaporation must be divided by

1 + (1 — n) cDR. If the reduced saving is denoted by a Er, it can easily be derived that

a 1 + (1 — «) DR - = or a > c c 1 + (1 —- H) CDR

e.g. at a surface temperature of I5°C and a wind speed of approximately 9m.p.h, a 10% reduction in film resistance will result in a 6% reducing in saving. This is important when different materials with different resistances and different costs are compared. A material with twice the film resistance will increase the saving by only 40%.

Case 2 Due to consistent wind conditions it might be possible to maintain a mono-layer only on a portion of the reservoir at the leeward side. It appears that ge-nerally lateral heat exchange is limited and the covered and uncovered portions es-sentially maintain their own regime. The saving is obviously proportional to the fraction of the reservoir that is covered

a = c

Case 3 If a portion of a reservoir is intermittently covered by a compressed mono-layer and the periods during and between coverage are small (a day or less) the surface temperature will vary very little and will be established at a temperature between the temperatures of a covered and an uncovered reservoir. Again this picture must be superimposed upon the diurnal surface temperature variation,

22

TABLE 1

T'2 Q = 1.42 x 10~3 7*(273 + 1.5 T + ) + C {T + 1.6 (e — 6.02))

2 273 m = 1.42 x lu"3 (273 + 3T + —) + C (1 + 1.6 ô) n = hÊ.£A

91 m

T

°C

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

e

• mb

6.02 6.48 6.97 7.49 8.03 8.61 9.23 9.89

10.59 11.33 12.12 12.95 13.84 14.78 15.77 16.82 17.94 19.12 20.36 21.68 23.07 24.54 26.09 27.72 29.44 31.26 33.17 35.18 37.30 39.53 41.87 44.33 46.92 49.64 52.50 55.50 58.64 61.93

dT

mb/°C

0.43 0.46 0.49 0.52 0.55 0.59 0.63 0.67 0.71 0.75 0.80 0.85 0.90 0.95 1.01 1.07 1.13 1.20 1.27 1.34 1.42 1.50 1.58 1.67 1.76 1.86 1.96 2.06 2.17 2.28 2.40 2.53 2.66 2.80 2.94 3.09 3.24 3.40

Q = a + bC

ly/hr.

a

0 0.39 0.78 1.18 1.58 1.99 2.40 2.82 3.23 3.66 4.08 4.52 4.95 5.40 5.84 6.30 6.75 7.22 7.68 8.15 8.63 9.11 9.59

10.08 10.58 11.08 11.58 12.09 12.61 13.13 13.66 14.18 14.72 15.27 15.81 16.36 16.92 17.48

b

0 1.74 3.52 5.35 7.22 9.14

11.14 13.19 15.31 17.50 19.76 22.09 24.51 27.02 29.60 32.28 35.07 37.96 40.94 44.06 47.28 50.63 54.11 57.72 61.47 65.38 69.44 73.66 78.05 82.62 87.36 92.30 97.44 102.79 108.37 114.17 120.19 126.46

m = c

ly/hi

c

0.388 0.392 0.396 0.401 0.405 0.409 0.414 0.418 0.423 0.427 0.432 0.436 0.441 0.446 0.450 0.455 0.460 0.465 0.469 0.474 0.479 0.484 0.489 0.494 0.499 0.504 0.509 0.514 0.519 0.524 0.530 0.535 0.540 0.545 0.551 0.556 0.561 0.567

+ dC

/°C

d

1.688 1.736 1.784 1.832 1.880 1.944 2.008 2.072 2.136 2.200 2.280 2.360 2.440 2.520 2.616 2.712 2.808 2.920 3.032 3.144 3.272 3.400 3.528 3.672 3.816 3.976 4.136 4.296 4.472 4.648 4.840 5.048 5.256 5.480 5.704 5.944 6.184 6.440

C=0.25

0.212 .222 .232 .242 .253

0.264 .275 .286 .297 .308

0.319 .330 .342 .354 .366

0.378 .390 .403 .415 .427

0.439 .451 .463 .475 .486

0.497 .508 .519 .530

- .541 0.552

.563

.574

.585

.595 0.605

.615

.625

{d— 1) n ~ — ~

c/C + d

dimensionless

C=0.50

0.279 .291 .303 .316 .329

0.342 .355 .368 .381 .394

0.407 .420 .434 .447 .460

.0.473 .486 .499 .512 .525

0.537 .550 .562 .574 .586

0.598 .609 .620 .631 .641

0.651 .661 .671 .681 .691

0.701 .711 .721

C=0.75

0.312 .326 .339 .353 .366

0.380 .393 .407 .420 .434

0.447 .461 .475 .489 .503

0.517 .530 .543 .556 .569

0.582 .594 .606 .618 .629

0.640 .651 .662 .673 .683

0.693 .703 .713 .722 .731

0.740 .749 .758

C=1.00

0.332 .346 .360 .374 .388

0.402 .416 .430 .444 .458

0.472 .486 .500 .514 .528

0.542 .555 .568 .581 .594

0.607 .619 .631 .642 .653

0.664 .675 .685 .695 .705

0.715 .725 .734 .743 .752

0.761 .770 .779

due to the difference in the Heating Power H during the night and during the day (The equilibrium temperatures corresponding to day and night conditions might differ by as much as 20° C, whereas the actual temperature differences are only a few °C for depths of 4 ft or more). During the periods of coverage, heat is stored in the reservoir and this heat is released during the periods between coverage. It can be shown that the average rate of evaporation.

1 + (1 — c) DR E' = E , where

1 + (1 — en) DR E is the free water evaporation. In the limiting cases of c = o and c = 1, this reduces to

E' = E and E' = 1 + (1 — n) DR

If again the reduced saving is denoted by aEr

a 1 + (1 — n) DR - — or c 1 + (1 — en) DR

a < c

e.g. For a surface temperature of 15°C and a wind speed of approximately 9 m.p.h., coverage during 90% of the time will result in 86% of the saving, coverage during 50% of the time will result in 4 1 % of the saving and coverage during 10% of the time will result in 7% of the saving. It appears that gener-ally intermittent monolayers are not efficient. On deeper reservoirs the diurnal variation of H — S = Q is rather small and the above calculations apply with little correction. On very shallow reservoirs, however, the diurnal variation must be taken into account, particularly if the coverage pattern is strongly correlated to the diurnal surface temperature variation. Itisthen advantageous to apply a monolayer during the periods of high surface temperature, that is roughly from late morning to late evening.

AKNOWLEDGMENTS

The author is grateful to the Saskatchewan Research Council for the opportunity to prepare this paper, and particularly to Mr. W. H. W. Husband, head of the Engineer-ing Division, who reviewed the manuscript.

REFERENCES

C1) H.L. PENMAN, Natural Evaporation from Open Water, Bare Soil and Grass. Proc. Royal Soc. of London, Ser. A, Vol. 193, No. 1032, 1948.

(2) Water Loss Investigations, Lake Hefner Studies. U.S. G.S., Circular 229. (3) M.A. KOHLER, T.J. NORDENSON and W.E. Fox, Evaporation from Pans and

Lakes. U.S. Weather Bureau Research Paper No. 38, 1955. (4) D. BAKER and R.K. LINSLEY, The Felt Lake Evaporation Study. Stanford

University, Department of Civil Engineering, Technical Report No. 5, 1960. (6) G. E. HARBECK, Jr., The Use of Reservoirs and Lakes for the Dissipation of Heat.

U.S. G.S., Circular 282, 1953. (6) C.J. VELZ and J.J. GANNON, Forecasting Heat Loss in Ponds and Streams.

32nd Annual Meeting, Feder. tion of Sewage and Industrial Wastes, Dallas, Texas, 1959.

24

(7) W.W. MANSFIELD, The Influence of Monolayers on Evaporation from Water Storage. Austr. J. of Appl. Science, Vol. 10, No. 1, 1959.

(8) J. M. RAPHAEL, Prediction of Temperature in Rivers and Reservoirs. Proc. A.S.C.E., Vol. 88, No. PO 2, 1962.

(9) G. E. HARBECK and G. E. KOBERG, A Method of Evaluating the Effect of a Mono-molecular Film in Suppressing Reservoir Operation. / . of Geophysical Research, Vol. 64, No. 1, 1959.

(10) Water-Loss Investigations, Evaporation Reduction Investigations. Lake Hefner, 1958. U.S. Dept. of Int. Bureau of Reclamation.

( u ) Water-Loss Investigations, Evaporation Reduction Investigations. Sahuaro Lake, 1960. U.S. Dept. of Int. Bureau of Reclamation.

(12) Water-Loss Investigations, Evaporation Reduction Investigations. Lake Cachuma, U.S. Dept. of Int. Bureau of Reclamation.

25