Heat transfer coefficients for natural water surfaces

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International Association of Scientific Hydrology, 62, pp. 203-212, 1963-11-01

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Heat transfer coefficients for natural water surfaces

Williams, G. P.

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S

er

TH1

N21r2

no. 202

c.

2 BLDG

ANALYZED

NATIONAL

RESEARCH

COUNCIL

C A N A D A

D I V I S I O N O F B U I L D I N G R E S E A R C H

HEAT TRANSFER COEFFICIENTS FOR

NATURAL WATER SURFACES

G.

P. WILLIAMS

R E P R I N T E D FROM

INTERNATIONAL ASSOCIATION O F SCIENTIFIC HYDROLOGY PUBLICATION N O . 62, P. 203 - 212 R E S E A R C H P A P E R N O . 202 O F T H E

D I V I S I O N O F B U I L D I N G R E S E A R C H

" v " i J c

-

'-,

-L v

~2

- - 5 . P R I C E 2 5 C E N T S

O T T A W A

N O V E M B E R 1963

N R C 7 4 9 5

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Extract of pnblicatiott tio. 62 of the I.A.S.H. Comtnittee for Euaporatiotl, pp. 203-212

HEAT TRANSFER COEFFICIENTS FOR

NATURAL WATER SURFACES

G. P. WILLIAMS

(*) National Research Council Canada

Division of Building Research

It is shown in this paper that heat losses from open water surfaces will be approximately proportional to the difference in temperature between the water surface and the air, provided the energy gained by short-wave radiation equals that lost by evaporation. The heat transfer coefficient for a small, sheltered lake at Ottawa was measured during 4 autumn cooling periods and found to be 23.4 cal/sq cm/24 hr/oC. Calculated values from other studies for latitudes 40 to 60°N indicate that the coefficient varies from 40 to 60 cal/sq cm/24 hr/OC for average exposure conditions. Lower values occur when short-wave radiation exceeds evaporation loss; the higher values during mid-winter conditions result when evaporation loss exceeds short-waves radiation.

R f s u ~ i

On indique dans cette etude que les pertes de chaleur, qui se produisent a la surface des Ctendues d'eau, sont approximativement proportionnelles a la difference de tempkrature entre la surface de l'eau et de I'air, a condition que I'energie obtenue p a r radiations (ondes courtes) soit Cquivalente 2 celle perdue par evaporation. Le coefficient de transfert de chaleur pour un petit lac abrite a Ottawa a CtC mesure durant quatre pCriodes de refroidissement en automne et il s'es! trouvC Ctre de 23.4 cal/cm2/24 hr/oC. Les valeurs calculCes provenant d'autres Ctudes, pour des latitudes allant de 40 i 600N, indiquent que le coefficient varie de 40 h 60 cal/cm"24 hrl0C dans des conditions moyennes d'exposition. Des valeurs plus faibles sont obtenues lorsque les radiations (ondes courtes) sont supCrieures aux pertes par evaporation; au cceur de l'hiver des valeurs plus tlevCes sont obtcnues lorsque les pertes par Cvaporation sont supkrieures aux radiations (ondes courtesl.

A method of estimating the rate of heat loss from water surfaces is needed for such problems as predicting frazil and surface ice formation o r determining the heat required to keep ice from forming under given weather conditions. Newton's Law of cooling, which states that the rate of cooling of a body is proportional to the difference in temperature between the body and its su~roundings, has been the basis of many empirical formulae used in practice. Because of the difficulties associated with measuring the rate of heat loss from natural water surfaces, it is usually not feasible to check the accuracy and limitations of these formulae. It is the purpose of this paper t o present some measurements of heat losses from a lake and to use these measurements, along with other observations reported in the literature, to evaluate the validity of Newton's Law of cooling for predicting the rate of heat loss under given weather conditions.

The reason for the relationship between rate of heat loss and the difference in temperature of a body and its surroundings has been explained for a dry surface under laboratory conditions. Under these conditions heat loss is by convection and long-

(*) Snow and Ice Section, Division of Building Research, National Research Councll, Ottawa, Canada.

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wave radiation. T h e heat loss by convection is proportional to 11 (To - T,), where To and T, are t h e temperatures of the body and the air surrounding the body, a n d 11 is a coefficient which will vary with t h e shape and dimensions of the surface and the characteristics of t h e air stream. T h e heat loss by long-wave radiation is proportional t o a (Ti-I - T?"), where T2 and T i arc t h e a b s o l ~ ~ t c temperatures of t h e body and its surroundings, and cc is a constant that depends o n the eniissivity of t h e surface. If the temperature difference ( T i - T2) is small in comparison t o TI or T?, the expression for long-wave cooling will be approximately equal to 4 T $ ( T l - T?) (I). TIILIS both convection and long-wave radiation components of heat transfer a r c approximately proportional to teniperature difference.

Although there is justification for t h e use of this relation for a dry surface under laboratory conditions it is not s o apparent why there s h o ~ ~ l c l be a correlation between rate of heat loss and temperature difference for a wet surface i~ncler atmospheric conditions. Under these conditions t h e surface can receive heat by short-wave solar radiation and condensation and lose it by evaporation. Solar radiation varies with c l o ~ ~ d amounts, latitude, time of year, condition of surface - factors not dependent o n the difference between surface and air temperature. T h e rate of evaporation depends o n vapour pressure gradient, wind velocity,surface roughness and surface temperature, t h e latter through t h e dependence of saturated vapour pressilre of air o n temperature. As solar radiation and evaporation a r e essentially independent of air-surface teniperature differenec, it is necessary to consider the heat balance at a water surface in detail in order t o determine under what conditions Newton's Law of cooling is valid.

T h e net loss of heat from a water body for a given period must equal t h e heat lost by long-wave radiation, convection and evaporation less the heat gained by short- wave radiation. This requirement can be expressed by t h e following energy balance equation :

where

Ql = total heat loss

Q,, = net short-wave radiation

Ql,, = net long-wave radiation

Qc = heat associated with evaporation

Qc = heat associated with convection

Using available forniulae, simplified expressions for QSlo, Q I ~ , ~ , QL' and Q , are a s follows :

(a)

Qsu, = ( 1 - a) R,,, cal/sq cni/24 h r where

n = albedo of surface

R,, = incoming short-wave radiation (cal/sq cni/24 hr) (b)

Qz, = 12.0

-1

T cal/sq cni/24 hr

./I

T = TLo - T ,

where

T, = teniperature of water surface (OC)

T, = temperature of air P C )

(This formilla was obtained from Johnsson's approximation for long-wave radiation; it assumes the surrounding objects and tcrrain t o be a t the same temperature as t h e air, a n d water to have an emissivity of 1.0 (").)

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(c)

Qe = 23 A e cal/sq cm/24 lir (wind speed of 160 kmlday) at 2-metre level

= 38 i l e (wind speed of 320 kmlday) where

/le = (e", - es)

e,, == saturation vapour pressure of air a t temperature of surface (oib) e, = vapour pressurc of air (nib)

(This simplified formula for evaporation from Pennia~i (3)) (dl

Qe = 14 ilT cal/sq cm/24 h r (wind speed 160 km/day) = 24 A ~ c a l l s q cm/24 h r (wind speed 320 k~ii/day) where

A T = difference in temperature between air a n d water surface ( o C ) (obtained from Q, by assuming that Bowen's ratio (4) is valid) Substituting these expressions for the various conlponents of the heat balance into equation (1) gives :

(cal/sq cm/24 lir) (wind spced 160 km/day)

and Qt = (1 - n) R,, - 38 d e - 36

fl

T(wind speed 320 km/day) From these form~llae it is apparent t h a t if the energy gained by short-wave radiation balanced that lost by evaporation equation (2) w o ~ ~ l d simplify to :

Qt = ello

+

Qc = 26 ilT(average wind speed 160 km/day) ₍₃₎ = 36 d T(average wind speed 320 km/day) (4) Undcr this condition, t h c heat loss from water surfaces would be approximately proportional to t h e diflercnce in temperature tctween the air and the surface, in agree- ment with Newton's Law of cooling.

From thc preceding discussion it may be seen that a n over-all heat transrer coeficient /I", rclating nct heat loss to air-water tcmpcrature difference (Qt

-

110 LIT), can be expected to vary not only because of wind velocity but also because of thc size of t h e evaporation loss in relation to incorning short-wave radiation. If incoming short-wavc radiation exceeds evaporation, l r , s h o ~ ~ l t l be smaller for a given temperature dificrcnce, (i.c. Qt reduced); if evaporation cxceeds incoming short-wave radiation, then 11, should be largcr for the samc t e m p e r a t ~ ~ r c difference (i.e. Qt increased).

T H E IrlEASUREMENT OF NEAT TRANSFER COEFFICIENTS FOR A SMALL L A K E

During t h e past four years obser\,ations have bcen made o n water temperature

a t McKay Lake, a small lake located within t h e city of Ottawa (latitude45oN longitude 75oE). These observations were used to chcck t h e validity of Newton's cooling equation and to obtain information o n t h e possible variations in t h e coefficient /lo.

This small lakc was chosen for heat loss observations beca~lse of several advan- tages. I t is within 4 kilometres of n standard meteorological station and the weather records from this station could bc ~ ~ s c d in t h e analysis. 11 is about 450 mctres long a n d 200 metrcs wide, with a maximum depth oT 10 metres, ancl did notrequire a n excessive number of man hours to obtain the necessary water temperaturc records. T h e lake is fed by ground water with n o marked inflow o r o ~ ~ t f l o w of surface water except during periods of heavy rain o r mclting snow. It is s u r r o ~ ~ n d e d by trees and houses and thus

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Water temperature profiles were obtained at several representative locations at weekly intervals during the autumn cooling periods for the years 1959, 1960 and 1961, and at bi-weekly intervals during all of 1962.

The surface temperature at a given time was assumed equal to the average of several water temperature readings measured in the upper 10 to 20 cm at several locations in the lake. The average water surface temperature for a given one-week period was assumed equal to the mean of the average water surface temperature at the beginning and end of the period. Average air temperature was obtained by averaging the daily maximum and minimum air temperatures recorded at a standard weather station located on National Research Council grounds, Ottawa.

The depth of the lake was measured at a number of sites and a bottom contour map drawn. The volume of water contained in consecutive 5-ft layers was then calculated and the heat loss in each layer was estimated from the change in water temperature that took place over a given period. T h e rate of heat loss from the lake surface for a given period was obtained by adding the contributions from all the layers and dividing by the surface area. In these calculations it was assumed that the amount of heat brought into or out of the lake by ground water c o ~ ~ l d be neglected. T h e heat loss from snow that fell into the open water was allowed for in the calculations.

I SEPT 61 AT' O C 2 - WEEK AVERAGE S 200 U

bOOL

I I I I

,

1

b- a W L

/2;

0 W

"

0 a _K 0 2 4 6 8 10 12 14 16 18 2 0 w AT, " C 2 3

-

0 0 a 200 I 0 0 I I I I I I I I ~ I-MONTH AVERAGE 1959-62 - _,'/',U=215

,q.'

_{CAL/SO chl/24HR} _- *'/,'

,/%/

,'//' - ,i'd/' O S E P T 62

-

d 40,' OSEPT 61

/'/,.'

+

h : 2 = 2 3 4 ca~/so C M / Z ~ HR /'c / AT O o

Q

l b 1; 1; Ik 1; 2 0 AT, '=C

Fig. 1 - Relation between heat loss and the difference between air and water temperatures for various periods of time.

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T h e values obtained for t h e rate of heat loss from McKay Lake were compared t o t h e differences between t h e water surface a n d the air temperature for wcekly, bi-weekly and monthly periods (Figure 1). It is apparent that t h e correlation between measured heat loss and temperature difference i~nproves with length of observation period. O n a monthly basis t h e variability of factors that affect heat loss, such a s wind velocity, incoming short-wave radiation, and air vapour prcssure, tend to average o u t and a reasonably good relationship between heat loss and temperature difference c a n be established.

T h e slope of t h e line in each of t h e graphs is a measure of ha. O n a monthly basis t h e average heat transfer coefficient 11, equals 23.4 cal/sq cm/24 hr/oC. T h e standard error was calculated to be 21.5 cal/sq cm/24 h r . Individual values of 11, tend to be unreliable when t h e temperature differences a r e small.

T h e deviations shown for weekly periods, September 1961, can be explained In part by above-average incoming short-wave radiation and below-average evaporation during this month. Part of t h e explanation may also be that during this month there were periods of unusually high day-time air temperatures and above-average wind speeds, s o that convective heat gains during daylight hours possibly equalled o r even exceeded t h e heat losses at night. Under these circun~stances it is possible to have a net gain of heat by a water body even though average weekly air temperature is less t h a n the water surface temperature. As appropriate weather records were not available over t h e lake surface, these September anomalies could not be analysed in greater d e t a ~ l . T h e deviations during September 1961, and to a lesser degree those in Sep- tember 1962, suggest that t h e relation between heat loss and temperature difference will not be the same during early autumn (and presumably summer months) as during t h e late cooling periods.

T h e previous section showed that Q L = 11,

AT

is reasonably valid on a monthly basis for McKay Lake. This suggests that during the autumn cooling period evaporation losses are approximately equal to net short-wave radiation and that total heat loss equals t h c s u m of convection plus long-wave radiation. In this scction the validity of these relationships is further investigated by analysing published observations a n d comparing them with the McKay Lake observations.

In t h e McKay Lake calculations, values for incoming short-wave radiation were obtained from the standard meteorological records for Ottawa (j). T h c short-wave radiation absorbed was estimated assuming values of albedo given by Budyko ( 6 ) for water surfaces a t latitude 4 j o N , months September to July. Evaporation was calcul- atcd with Penman's evaporation formula and appropriate weather parameters.

Combined long-wave radiation and convection losses were then obtained for each observation period by subtracting the sum of net short-wave radiation and evaporation from the total heat loss.

T h e first comparison of the size of t h e heat exchange components is shown o n Figure 2, where monthly short-wave radiation is plotted against monthly cvaporation. Values for short-wave radiation evaporation, long-wave radiation, and convection ("a8) for latitudes 40° t o 60°N have been used in this analysis directly as reported Figure 2 shows that, o n the average, monthly values of incoming short-wave radiation d o balance calculated evaporation during the autumn cooling period, b u t not during the mid-winter months. During winter months incoming short-wave radiation is relatively small and evaporation from open water is relatively large because o f large air-water vapour-pressure differences.

If evaporation loss is approximately equal to short-wave radiation then Qt =

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OPEN WATER IN MID WINTER

0 2 0 0 400 600 8 0 0 1 0 0 0 a,= CALCULATED EVAPORATION (CAL/SQ ~ ~ / 2 4 H R I

LEGEND: R E F NO. MCKAY LAKE 0 L A K E K L A M M I N G E N 2 L A K E ONTARIO 7 0 ST. L A W R E H C E R I V E R 6 A L A K E M E N D O T A 8

Fig. 2 - Comparison of nlonthly evaporation and short-wave radiation

MID - WINTER

0 2 0 0 4 0 0 6 0 0 800 TOTAL HEAT LOSS Q t ICAL/SQ CM/24 HRI

LEGENO: REF NO.

H C K A I L A K E

0 L A K E K L i U U I H G E N 2 L A K E ,OXTAR10

0 S I . L A W R E N C E R I V E R 6

A L A K E U E H D O T A 8

Fig. 3 - Total heat loss versus sum of net long-wave radiation and convective heat loss.

and convectio~l loss, with the total heat loss Qt. This ill~~strates that during the autumn months the sum of

01,

+

(3, tends to exceed the net heat loss, whereas during the mid-winter months Qt excecds the sum of Ql,

-t

Q,. Monthly average sums of Qllv

+

Q,, obtained from the same sources, are plotted against monthly average

(10)

air-water temperatlire difference in Figure 4. Equations (3) and (4) are also plotted on Figure 4 as straight lines :

--

""

'

'5 = 26 and 36 cal/sq cm/24 hr/oC.

4

T A T 6 0 0

-

4 0 0 -

/

-

/

- 2 0 0

-

- I 1 0 5 10 15 2 0 AT, OC TEMPERATURE DIFFERENCE

LEGEND: REF NO.

M C K A Y L A K E

0 L A K E K L A M M I N C E N 2

L A K E O N T A R I O 7

0 S T . L A W R E N C E R I V E R 6 A L A K E M E N D O T A B

Fig. 4 - Sum of net long-wave and convective heat loss plotted against temperature* difference.

QLLO

+

ec

T h e majority of the values o f - - obtained from t h e l i t e r a t ~ ~ r e are some-

il T d? ,

what higher than 36, the value calculated for a n average wind speed of 320 km&; t h e v a l ~ ~ e s obtained for McKay Lake are close t o 26, the value calculated for average wind speed of 160 km/day. For small differences of temperature (less than 50C) there is considerable scatter t o the values obtained for t h e sum of Qzlu f Qe.

From the foregoing analysis it may be c o n c l ~ ~ d e d that d ~ l r i n g the cooling period t h e sum of long-wave and convective heat is approximately proportional to t h e difference between the average air and water surface temperature. T h e net short-wave radiation, for latitudes 40° to 60°N, is approximately equal t o t h e evaporation loss; it exceeds it during t h e autumn cooling period and is less than it for open water surfaces during mid-winter months. As a consequcnce, t h e sum o f the convective a n d long-wave radiative heat loss is approximately equal to t h e total heat loss Qt. T h i s indicates that Newton's law of cooling can be a useful approximation of the rate o f heat loss from a n open water s~lrface during autumn and winter if proper account i s taken of site and average weather conditions.

LIMITS TO HEAT TRANSFER COEFFICIEN7

It is evident that if Newton's law of cooling is used t o estimate the rate of cooling of water bodies, it is necessary to know t h e approximate variation f /lo u n d e r

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different conditions of exposure. In Table I are given valucs for 11, obtained from observations reported in the literature (q7 to I"). Values a r e listed as measured if the heat loss was obtained by taking temperature profiles a t regular intervals; and listed as calculated i f t h e heat loss was obtained by another method, usually t h e energy balance method. All values are for latitudes 40° to 60° N, except those for the ice surface (10) which were apparently calculated for Inore northerly locations.

Some of t h e information contained in Table I is presented in graphical form on Figure 5. Reported monthly values of heat loss a r e plotted against air-water temperature differences t o show the range in over-all heat transfer coefficients for different degrees of exposure and time of year.

TABLE I

approx. 7 47 nlwsured (I1)

River weekly 1

I

i

-P _PI

I

!

10-ft tank

1

i 5 t o 10 h r evening

1

20

1

1 36.5

1

I measured

1

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Slrrnninry o f cnlcrrlnted nrrd r~rensirrecl ~nlrres of 11, ocer-011 lrent irnrrsfer coefficierrr /or vnriorrs ,rPn/er bodies

I

River Lea

1

1.0 daily 3 52

i

mcasured (13)

1 Water Body McKay Lake L. Klam- Maximum Depth No. (metres)

i

. 10 monthly I 0 23.4

1

nieasi~red

T h e average value for the over-all heat transfer coefficient h, varies from 40 to 60 cal/sq cm/24 hr/OC for most situations. For a sheltered lake o r ice surface under relatively still air conditions the average value is from 20 to 30 cal/sq cm/24 hr/O C. River Meuse

Ice Surface

mingen 40

1

monthly 7 1 43.5

1

calculated

,

_{( 9}

L. Mendota 25 3

1

47 nieasured

1

(9) L. Ontario 225 49.5

/

calculated 4.0

1

a v e n g e

'

40 measured

1

conditions

1

i -- (8) (13

St. Lawrence 10 nionthly 8

1

57 calculated (7)

River

_i

avcrage 2 4 (still a i r ) calculated

,

_(I0) conditions 6 0 (15 mph

1

(12)

For an extremely large lake under mid-winter conditions the calculated average value of 11, is about 75 cal/sq cm/24 hrloC. As it is not known how representative the reported values for this lake will be for other water bodies, the upper limit for 110 of 75 must be used with caution.

1 I I

/

-

ho=75 /nuA

-

LARGE LAKE

/---

ho= 5 0 - AVERAGE EXPOSURE - -

-

/

Po

a/ - X O O / 4 h ~ = 2 5

-

&/=

* SMALL SHELTERED L A K E

-

/s:**

-

P

I I I

0 5 10 15 AT. "C

LEGEND: REF NO.

M C K A Y L A K E 0 L A K E K L A M M I N G E N 2 L A K E O N T A R I O 7 0 ST. L A W R E N C E RIVER 6 A L A K E M E N D O T A 8 A ST. LAWRENCE RIVER 10

Fig. 5 - Relation between total heat loss and air-water temperature difference.

Heat losses from open water surfaces are approximately proportional to the difference in temperature between the water surface and the air, provided the energy gained by short-wave radiation equals that lost by evaporation. In this case heat losses would be equal to the sum of convection and long-wave radiation and be approximately proportional to 26 hTcal/sq cm/24 hr for average wind speed of 160 kilometres per day, and 36 h T for average wind speed of 320 kilometres per day.

Average values of 110, the over-all heat transfer coeflicient relating total heat losses t o temperature difference ( Q t = 110 AT) were measured for four autumn cooling periods for a small sheltered lake. It was found that relatively large errors can be expected if calculations were made on a weekly basis. The average monthly value of 11" obtained was 23.4 cal/sq cm/24 hrtoC.

V a l ~ ~ e s of /lo obtained from other studies, latitude 400 to 600N, indicate that for average conditions of exposure 110 is probably between 40 to 60 cal/sq cm/24 hr/oC, with lower values when short-wave radiation exceeds evaporation loss and higher values during mid-winter conditions when evaporation loss exceeds short-wave radiation.

In conclusion it can be staled that if site conditions and air-water temperature differences are known, Newton's law of cooling can be used as an approximate method

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of estimating rate of heat loss from natural water surfaces on a monthly basis during the autumn cooling period.

T h e writer is indebted to several members of t h e Division of Building Research, particularly Mr. P . Montford, for assistance in obtaining water temperature measurements and analysing weather records. H e is also indebtcd to Mr. D . Boyd act1 Mr. L. Gold for most helpful discussions during the preparation of this paper. This is a contribution from the Division of Building Research, National Research Council, Canada, and is presented with the approval of t h e Director of the Division.

R E F E R E N C E S

(1) ALLEN, H.S. and R.S. MAXWELL. A textbook of heat. Part 1. MacMillan, 1939.

(2) JOHNSSON, H . Termisk-hydrologiska studier i sjijn K l l m ~ n i n g c n (Thermic- hydrological studies of Lake Klammingen). Geogrnjisltn Atrtrnler, Stockholm, Vol. 28, 1946, p. 149-154.

(3) PENMAN, H.L. Evaporation: a n introductory survey. Netlr~rlnrrcls Jo~trricrl of

Agricultrirnl Scietrce, Vol. 4, No. 1, Feb. 1956, p. 9-29.

(9

BOWEN, I.S. T h e ratio of heat losses by conduction and by evaporation from any water surface. Plrysicnl Review, Ser. 2, Vol. 27, J u n e 1926, p. 779-787.

(5) Canada, Department of Transport, Meteorological Branch, Mo~rtlrlj~ Rn(1intiotz Slittztnary, Sept.-Nov., 1959-1962.

(6) BUDYKO, M.I. T h e heat balance of the earth's surface. Washington, Dept. of

Commerce, Weather Bureau, 1958, 259p. (Translated from Russian by NINA A. STEPANOVA, from Teplovoi balans zemnoi poverkhnosti, Gidrometcorolo- gischeskoe izdatel'stvo, Leningrad, 1956)

\ (7) WARDLAW, R . L. A study of heat losses from a water surface as related to winter navigation. Proceedings, Enstertz Strow Cotrferetrce, Vol. 2, 1953 and 1954. (8) BRUCE, J.P. and G.K. RODGERS. T h e water balance of the Grcat Lakes systcm.

Symposiun~ o n Great Lakes. Attr. Assoc. for tire Aclucrtrcetnetrt of Scietrce, Chicago, Dec. 1959, 22p.

(9

DUTTON, J . A . and R . A . BRYSON. Heat flux in L a k e Mcndota. University of Wisconsin. Dept. of Meteorology, Teclr. Report No. 2, June 1959, 59p.

(10) ADAMS, C.M., D . N . FRENCH, and W.D. KINGERY. Solidification of sea ice.

Jorirtrnl of Glnciology, Vol. 3, No. 28, 1960, p. 745-761

(11) St. Lawrence Waterway Project, Report of Joint Board of Engineers, Ice formation on t h e St. Lawrence and other rivers, Appendix E, p. 406-453, Nov. 1924. (12) WILLIAMS, G.P. Two notes relating to frazil ice formation. Proceedings, Eishtlr

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