ON TWO-DHIMF NSTONAL TEMPERATURE MODELING

Joel i". Kvitky

Any views expressed in this paper are those of the author.They shoul' not be interpreted -ir reelecting the views of TheRand Corporation or the official opinion or policy of any of its

governmental or private research sponsors. Papers are reproducedby The Pand Corporation as a courtesy to members of its staf f.

S . = ===.

-iil-

PREFACE

This paper is Intended to provide the theoretical and practical

framework for a two-dimensional temperature model. The motivation

for this study is priovided by the existence, at Rand, of a general

two-dimensional flow-reaction simulation modei developed by J.

Leendertse and E. Gritton. The vertically integrated mass balance

equations of the simulation model are discussed and shown to apply

to temperature modeling. Terms contributing tu the total heat flux

across the water surface are described and approximated by empirical

relations which, in turn, are integrated into the temperature model

through an equilibrium temperature and heat exchange coefficient.

The sensitivity of these parameters to the meteorological variables

is discussed in the context of a preliminary calculation. Tables

are given for the heat exchange coefficient and equilibrium tem-

perature. Finally, expressions are derived for the relative iso-

baric slope as a function of chlorinity and temperature.

Although this paper is primarily a review, the approach of

Edinger and Geyer is modified for the case when data is averaged

over periods of less than a day. The following advantages of the

E-G approach appear to remain:

(i) Additional arrays to compute the surface heatexchange at each grid point are not required (althoughit may be necessary to divide the bay into large sub-regions with different T and K).

eq(2) Tho method is readily adaptable to long term cal-culatior;s.

Sections 3 and 4 of this report (on computation of absorbed

radiation) were in, luded to emphasize the need for further research.

The empirical formulations discussed are essentially useless in

the development of a temperature model, but improved verions of

these equations could be quite valuable in tapping the model's

predictive capacities.

CONTENTS

Preface ................. ............................. iii

SectionI. The Heat Transfer Equation ..... ............ .

II. Heat Exchange Coefficients and EquilibriumTemperature ................. .................... 3

III. Net Isolation ....... ....... ................... 8

IV. Long-Wave Radiation ......... ................ .. 12

V. Evaporation ............. .................... .. 14

VI. Convective Heat Exchange ....... ............. .. 16

VII. Preliminary Calculations ....... ............. .. 17

AppendixA. Temperature Dependence of Relative

Isobaric Slopes ............ .................. .. 45

B. Modified Dewpoint Temperature .... ........... ... 50

References .................. ........................ .. 52

I. THE HEAT TRANSFER EQUATION

It is desired to exploit as far as possible tne similarity between

the heat and mass transfer equations. The former may be written:

•f + (DOT) + (vT) + d (wT) .'(=)

=r (x -5•t D-X +y + z - X jx

6 y( )z z PC P

p

-3where p = density of water, assumed constant at 62.4 lb-ft 3

K ,K ,K = thermal eddy diffusivity, C = specific heat of water takenx y z " 1 -1 P

as 4.1 Joul-gI 1C-_ = .98 Btu--lb F (assumed constant although it

may vary as much a- 17 due to salinity and temperature distributions

in the bays under consideration, and far more in some estuaries), S QQlocal heat source and sink.

Next, introduce the following distributions and vertically aver-

aged variables:

T I Tdz 1 T) (1.2)H Ht

-h

T(z) = T [1 + ET (z)] (1.3)

u = U [i -, c (z)] , etc. (1.4)

(1)The exact vertically integrated continuity equation is

+ 7. 0 , where V = (u,v) and -

(1.5)

x •y

and the boundary conditions on the free surface and on the bottom are

w = - 7', , w_ - - . Th. (1.6)r t r," I h: t -

... Uum..---"•'-- . ,,m ... - - A-id

Vertically integrating (1.1) and substituting equations (1.5 -

1.6), we obtain

[P x(T)Ta Tm (['.-LT) +T .V -LThh.]

QT pp a K x D. y Ky @y + z K z

[D= () T-T 30 + 'V. T) - rNVN

+ T T -h _h tr + ' . T) (1.7)

Here, QT is the net heat flux into the vertical water column

through the surface and bottom. In terms of the quantities a uT, and

CvT first introduced by Leendertse, where (2)

T = - i (1 + E (z) CT (z)) , etc. (1.8)auT =H u T

the right hand side becomes

H FT) 3RHS = L ( + - i) T U) + 2 ( (H T V) (1.9)

a- 3x uT ay vT

If T is uniformly distributed over the ?rtical, then a = 1.

This assumption is adopted throughout this paper, but the general

equation may be the subject of future investigations. The vertically

integrated diffusion terms are discussed by Leendertse for the case

of the mass balance equation. The same analysis is assumed to apply

here.

11II

--3--

II. HEAT EXCHANGE COEFFICIENTS AND EQUILIBRIUM TEMPERATURE

The Leendertse-Gritton general mass-balance model for n reacting

constituents is described b, the matrix equation

(H) + (HL) + a(HVP) (liDP -)

(2.1)S(HD •P) ) KH H

- -Y + (K] H + = 0

where [K] reaction matrix, and S = source or sink vector.

For DO-BOD reactions, the K-matrix and S are written in the form

[K) = (2.2)K K21 K 221

-11 sat /H)S ( C(2.3)

This suggests an analogy with the linear heat exchange model of

Edinger and Geyer, where K/pCp replaces K and the equilibrium tem-

perature replaces C . Here K is the heat exchange coefficient. (4-7)satV bis no counterpart in the -G model but can be utilized to describe

the temperature dependence of pLocesses such as reaeration. For com-

putational and theoretical convenience, therefore, we shall employ the

following expression for heat transported across the surface:

QT = (Teq - T)K (2.4)

The remainder of this section will be devoted to deriving expressions

for K and T in terms of meteorological variables.eq

The energy budget equation is

Q'r = (Qnet + Q anet (Qbr + Q + Qd) (2.5)

-4-

where

Qnet = net solar radiation

Qanet atmospheric radiation

Qbr = long-wave back radiation

Qc convection heat exchange

Qe = rate of heat transport due to evaporation

The details of the energy budg-t terms will be discussed in later

sections. Our concern here is with the dependence on water surface

temperature. In particular, while (Qnet + Q anet) is independent of

T , the other terms may be written as follows:

Qbr 0D U0 + Ts)

Qe = (L - L T ) (es - ea )f (2.6)

Qc = p B(Lo - LT) (Ts - T a)f

where e , e are the saturation vapor pressure at T and the ambients a s

vapor pressure in inches 11g, respectively;

p = water density, D .1661 x 10.8 Btu hr-1 ft-2 OR14

To = 460°F

L0 = 1093.2 Btu/lb.

L, = .57 Btu/lb.-°F

and f, B are meteorological variables.

Whiereas (2.4) implies that, without advection, T instantaneously

decays toward TI" with decay time C pH/K, it is clear from (2.6) thateq p

considerable complexities are buried in K and Teq. Specifically, K

is not independent of T , and T is the solution of a transcendentals eq

equation.

-5-

Furthermore, T appears in (2.6) rather than the vertically aver-s

aged temperature, T. Even in supposedly well-mixed cases, it may not

be a good approximation to replace T with T. Measurements of thes

vertical temperature distribution may be needed to settle this ques-

tion (and the related matter of determining a uT and uvt)(6)

Substituting equations (2.6) into (2.4) we obtain

QT = (Qnet + Qanet D (T s + Ts)4

- p (L - L1 T) C(e - ea )f - p B (L0 - L T ) (Ts - Ta )f

K (T -T) (2.7)eq

We shall assume that T = T. Then T is defined by the equations eq

(Qnet + Q anet) -D (T + T )4ne aeto eq

-(L0 - L1 Teq I(e eq - ea) + B(Teq -Ta)] P f

- G(T ) = 0 (2.8)eq

which says that T is the surface temperature (= vertically averagedeqtemperature) at which the heat transported across the surface is zero.

Edinger and Geyer do not solve (2.8) directly. Rather, they

first linearize the back scattering term and then assume the relation

(e - ea) / (T - Td) = (e - e ) / (•T - T ). The latter relationeq a eq d s a s d

is not a bad approximation if temperatures are averaged over a day

or more, so that T an, T are nearly equal. For our purposes, iteq s

is a very poor approximation which introduces an unwanted T1 dependence

into T , and thereby necessitates computation of T at each grideq eq

point. Our approach is to find T1 numerically by Newton's method.eqThis is quite effective since G(T ) is a monotonically decreasing

function which can be expressed analvtically if the saturation vapor

pressure At T1 is evaluated with the Lamoreaux exponential approxi-eqmart ion. One may obtain the initial value of T for the iterationeqfrom the previous time step. The Lamoreaux approximation is given by(

x 6 7482.6e(Tj = 6.4129 x 106 exp (- 398.362 + T(2.9)

-6-

which, when combined with equation (2.8), yields the iteration formula:

T =r -G(T )/G'(T G) (2.10a)eq eq _ .

where G(Te) is given by equation (2.8) andeq

G'(Teq) -4D(T0 + Teq) 3 + L1 f I(eeq - ea + B(Teq Ta)

- fp(L0 - LTeq) B + e 7482.6 e)2 (2.b)

0-1e eIj (398.362 + Teeq1 q

The procedure outlined above is quite efficient. If the equili-

brium temperature is computed hourly so that it changes only a couple

of degrees at each time step, no more than two iterations are required

to update Teq.

Since the equilibrium temperature will characteristically pass

through the water temperature, T, twice daily on its roughly sinusoidal

path, it is evident that equation (2.7) is not well suited for direct

calculation of K (i.e. the coefficient of K might be zero).

Alternatively, one may subtract (2.8) from (2.7) setting L0 - LITs

constant 1050, to obtain:)4 _ 4 )

D [(T + Teq - (TO + T) 4 ] + 1050 pf f( -e) + B(Teq - T)J

= K(Teq -T) = K I(Teq + T) - (TO + T)l (2.11)

or,

eq [T +(T 0 + +T 0

I (T0 eq (To0 eq T)+ (T+ Teq

(T + T)2 + (T + T)31 + 1050 of iB + /(eeq eT

0T (T) T2.12)(Teq - T)j (2.12)

The term (e -eT) / (T - T) may cause troublE when (Teq - T)is near zero, but for ITeq - TJ

-7--

S/T+T\(e eq eT (T e - T) Le 2•

eq T eq ýT \ 2

173.004 T e (L;Lea) (2.13)

1796.724 + T + T eq

Since K depends on T, it would seem necessary to calculate it at

each grid point. In section VII we see that K is not very sensitive

to T, so that usually one value of K .will suffice for the whole bay.

The linear approximation to equation (2.11) devised by Edinger

3and Geyer amounts to replacing the first bracket by 4PT 0 , and the

factor (eeq - e T) / (Teq - T) by (es - ea ) / (Ts - T d). Neither

assumption is very good for T averaged over less than a day.

If our working hypothesis, that T=T , turns out to be untenables

we need merely replace T by T in (2.7), (2.11-2.12), and employ meas-s

ured values of T . It may also be possible to set T = T[I+cT(C)],s sT

where ET (z) is the vertical temperature distribution.

Before describing the general features of T and K, includingeqsensitivities to fluctuations in the various terms of the energy

budget, the next few sections will be devoted to calculating contri-

butions to the total surface heat transport.

We note in passing that an alternative scheme to that of Edinger

and Geyer would be to use equation (2.7) for the heat flux Q. and in-T

sert it directly into equation (1.7). This would require computation

of QT at each grid point. The evaluation of K and T is more complex

than that of Q., but the former does not require new spatial arrays. A

further advantage of the E-G scheme is Lhat average values of K and

T obtained from short term representative periods (e.g. tidal cycle)eq

can be used in long term calculations since these values characterize

the entire waterbodv, whereas QT is a local variable. In addition,

T possesses natural periodicitv which is amenable, to F,,urier re-eq C1)

presentation.

-8-

III. NET INSOLATION

The solar radiation on a horizontal surface at sea level is pri-

marily a function of solar altitude, cloud cover and cloud height.

Tile solar altitude can be obtained from the cate, the hour and the

latitude. In particular, we have

sin a = sin • sin 6 + cos 6 ccs ý cos B (3.1)

where P = latitude, 6 solar declination, and B = local hour angle

(LilA). The local hour angle is given in radians by

B - 180 (Greenwich hour angle - longitude)

-j 12(Local standard time + No. Time Zones to Greenwich

- longitude/15 - L - 12) (3.2)

Approximate values for A and 6 are tabulated in Table 1. (9,10)

The total incident solar radiation may be obtained from the fol-

lowing empirical formula based on Raphael's curves:

Qs 329 sini[l.05076 a - 5.049 + w (15 - a)

when sin u > 0;

Q = 0, when sin a < 0;

= 0, when ,in ac 0.26;

= .02244, when 0 < sin i - 0.26 . (3.3)

s as given by (3.3) gives sufficiently accurate values for clear

skies. For cloudy skies these values must be modified by an appro-

priate factor. The cloud factor given by Raphael and based on the

work of Fritz is (I - .0071 C 2), where C is expressed in tenths of

sk,, covered. Comparison with available data reveals th it this factor,

when cor)bined with 0 above, produces accurate results for mean

monthly solar radiation. The negativC bias observed by Anderson

for !osbv 's factor does not appear here. However, as pointed out

-9-

Table 1

Date A (min) 6 (deg)

Jan 1 + 3 -23.0611 + 8 -21.9021 +11 -20.04

Feb 1 +14 -17.2811 +14 -14.2321 +14 -10.78

Mar 1 F13 - 7.8311 +10 - 3.9621 + 8 - .01

Apr 1 + 4 + 4.3011 + 1 + 8.0821 - 1 +11.64

May 1 - 3 +14.8811 - 4 +17.7221 - 4 +20.06

Jun 1 - 2 +21.9711 - 1 +23.0421 + 1 +23.44

Jul 1 + 4 +23.1611 + 5 +22.2021 + 6 +20.60

Aug 1 + 6 +18.1911 + 5 +15.4721 + 3 +12.33

Sep 1 0 + 8.5211 + 4.8021 -7 + .96

Oct 1 -10 - 2.9411 -13 - 6.7821 -15 -10.47

Nov 1 -16 -14.2311 -16 -17.2521 -14 -19.78

Dec 1 -11 -21.7111 - 7 -22.9521 - 2 -23.29

-10-

by Anderson, large deviations may occur for daily solar radiation

unless this factor is further refined to include variations due to

cloud height. It is essential to accumulate an extensive record

comprising hourly pyranometer measurements and cloud cover and cloud

height readings. Although many workers would prefer to utilize

field data rather than to concern themselves with the intracacies

of cloud attenuation, the latter option may lead to greater flexi-

bility and predictive capability due to the existence of long term

meteorological records suitable for statistical analysis. The fol-

lowing empirical formula for insolation on a cloudy day should be

regarded as tentative:

Qi = (1 - 0.0071 C2 ) Qs (3.4)

The reflectivity of the water surface, R, is defined by

R = Qrefl./Qi (3.5a)

and

Qnet = Qi (I - R) (3.5b)

where Qnet is tne net solar radiation entering the water after re-

flection at the surface. R is principally a function of solar alti-

tude, type and amount of cloudiness, and atmospheric turbidity.

Turbidity has been shown to be relatively unimportant except in

notably extreme cases. Anderson has given the following general

empirical formula for reflectivity:

bR = au , (3.6)

where a is solar altitude, as before, and a and b are experimentally

determined parameteis related to the amount and height of clouds.

If the cloud dependency of a and b is ignored, errors of order 5%

or less are incurred in the total daily Q. To avoid accumulation

of error in computations for periods of a week or longer, the fol-

lowing empirical formula, based on Anderson's curves, may be used:

-h = 1.1073 Log 1 0 (N/D) (3.7a)

a = 1/2 (N/10b + D/109031b) (3.7b)

where

N = (.2004 F + .2341 F + .2006 Fb + .1172 F ) (3.7c)c s 0

D = (.04040 F + .03115 F + .04877 Fb + .04694 F ) (3.7d)c s bo

and Fc, Fs, F b F are the frequencies respectively for clear skies

(C=O), scattered clouds (C=1-5), broken clouds (C=6-9), and overcast

(C=10). To avoid infinite reflectivities, set R equal to its value

at 50 when 0 : a 5 5'.

In summary, the net solar radiation entering the water after

reflection at the surface is given by equation (3.4-3.6), i.e.

Qnet = (1 = 0.0071 C 2) (I - au ) Qs (3.8)

where a and b are given by equation (3.7) and Qs by equation (3.3)

S. - • i

-12-

IV. LONG-WAVE RADIATION

Two ways for obtaining the net atmospheric radiation will be

discussed. One is direct measurement. The other is through an

empirical formulation promulgated by Anderson. (12)

Atmospoeric radiation can be measured by subtracting pyranometer

(shortwave) readings from pyradiometer (long and short wave) readings.

As pointed out below, good measurements are prudent before acceptance

of any empirical scheme.

Not all factors determining the atmospheric radiation are known

or quantitatively understood. The primary ones, undoubtedly, are

the total moisture content of the atmosphere and the temperature of

the constituent water vapor. Any attempt to by-pass these factors

by using local vapor pressures and ambient temperatures is doomed to

partial failure. In particular, Anderson's study revealed that a 5%

to 10% error is inherent in such efforts.

Anderson assumes that the atmosphere acts like a "gray body" of

variable "grayness", obeying a modified Stefan-Boltzmann law:

Qa = 6 (ý (T0 + T ) (4.1)

where 3 = Stefan-Boltzinani. constant= .1712x19- 8Btu hr- ft 20R4

and = atmospheric moisture factor = a + b e . (4.2)a

The parameters a and b are given by the following formulas:

a = .740 + .025 C exp(-.0584h) (4.3a)

b = .164 - .018 C exp(-.060h) (4.3b)

where C is the cloud cover in tenths of sky covered, and h is the

cloud height in thousands of feet (1.6 - h- -).

The partial success of Anderson's anproach hinges on the appro-

ximate correlation between local vapor pressure and total moisture

i,, tent of the atmosphere. ULnhfr tunaate.lv, any parameter fit re-

ilecting mtiis relationship must be restricted to use in locations

wnich have similar air mass characteristics. Failure to oetermine

parameters locally may, tnierefore, introduce bias of unpredictable

-13- )magnitude, in addition to the random error of order 10% already

present.

In summary, Anderson's equations (4.1 - 4.2) may be used withreasonable accuracy provided they are calibrated by adjusting thenumerical constants in a and b for each new location.

The long-wave radiation reflected from the water surface isapproximately 3% of the impinging atmospheric radiation for thenormal range of water temperatures. Therefore we may write

Qanet = 0.97 Qa (4.4)

The water body itself radiates like a gray body of emissivity0.97 provided no film (e.g. oil slick) is present. This value isessentially independent of salinity. Therefore,

Qbr - 0.97 c (T0 + T ) 4(4.5)-8 -8 1 -2 o-4

so that D =(.97) (.1712x10) .1661x10 Btu hr ft R

-14-

V. EVAPORATION

The results of evaporation studies to date are not conclusive.(1,12-14)

Anderson's exhaustive Lake Hefner investigation produced an empirical

equation of the form

Qe = (L - L1Ts) (e -e) e f Btu ft-2 hr- (5.1)

where

-l -lf M W ft hr in-Hg (5.2)

and W = wind speed (m.p.h.) at a specified height, Hw, above the water

surface; M = empirical constant; e = saturation vapor pressure (in-5Hg) at the water surface temperature; and e a ambient vapor pressurea

of unmodified air (i.e. unmodified by passage over the waterbody).

Using equation (2.9), e is given by

e = e(T) (5.3)s s

while e is obtained either froma

e= e(Td) (5.4)

where Td is the dew point, or from

ea = e(Twb) - P(Ta - Twb) (3.59xi0 4 + 2.34x1O Twb) (5.5)

where Twb = wet bulb temperature, T. = air temperature or dry bulb

temperature, and P = atmospheric pressure in-Hg.

Anderson found that M = 2.08xi0 ft hr in-Hg . In addition,

the Lake Hefner work appeared to support the theoretical efforts(1 5 ' 1 6 )

of Sverdrup and Sutton. Later studies contradicted these theories,

however, and there presentIv exists no reliable formula derived from

either continuous or discontinuous mixing concepts. All that has

survived the series of investigations conducted bv the US( S is an

empirical correlation between A1 and the size of the waterbody for

the case when If = 2 meters: (13)W

-15-

.000397 -1 -IA = 0.05 ft hr in Hg (5.6)

[Note: to convert M in (in day-I b-I) to M in (ft hr- in-Hg-) mul-

tiply by .11759.] Here, A is the area of the waterbody in acres. The

standard error of (5.6) is given as 16%.

The reasons for the above correlation are clear in a qualitative

sense. First, there is a downwind decrease in evaporation rates.

Generally, this will decrease i for increasing area, and, in addition,

introduce a fetch dependence into 'I.

Secondly, the wind profiles depend on the size of the waterbody.

With increasing fetch, the wind shear resulting from surface drag

decreaaes. For example, if the wind speed at 8 meters were the same

for a large and small lake, then at 2 meters the wind speed over the

large lake would be greater. As a result, if one wishes to use wind

measurements made near the water surface (e.g. at 2 meters) one must

compensate by using a smaller 1 for larger water surface areas.

Since (5.6) gives only a fair approximation to these effects even

for regularly shaped lakes, this equation should be applied to geomet-

rically complex estuaries with some trepidation. It is quite likely

that for Tampa Bay evaporation rates depend in a complicated fashion

on location in the bay and on wind direction, as well as on wind speed.

Two final points should be made regarding the use of equation

(5.6). This equation applies only if the anemometer readings are

taken in mid-bay. Also, it applies only to measurements at 2 meters,

a height chosen to minimize the effects of atmospheric instability.

If some measurements are made at different heights, the wind speed

input must be adjmuted accordingly. For example, the wind speed at

4 meters is anpr(,ximately 67' greater than at 2 meters for a 50,000

acre bay. If a wide range of heights are used, detailed knowledge of

the wind prof ile may be required to obtain the appnropri ite correction

factors.

-16-

VI. CONVECTIVE HEAT EXCHANGE

There is no easy way to measure the convective heat transport nor

any reliable way to calculate it. Consequently, one must rely, how-

ever reluctantly, on a 45-year-old prescription due to Bowen:(12 ,1 7 ,18 )

Qc = B(Ta - T ) / (ea - e S) Qe (6.1)

where

B = 3.39xlO- 4P in. Hg.OF-l (6.2)

and P = atmospheric pressure in inches Hg.

The physical content of (6.1) is essentially that the evaporation

eddy diffusivity the eddy conductivity are equal.

Generally, one can ignore conduction in comparison with convection

in much the same fashion as one ignores the contribution of molecular

diffusion to evaporation (perhaps unless a strong thermal gradient

increases free convection to a significant degree). One further assumes

that the effects of spray on evaporation and the effects of radiative

transfer (in moist air above the water) on conduction are negligible.

The equality of eddy coefficients has been questioned for some

time and several researchers have modified the Bowen constant in in-

stances where unstable atmospheric conditions prevail. The magnitude

of deviation is thoroughly speculative, however, especially for large

lapse rates associated with high temperature outfalls. 5) Even the

careful work of Anderson did not reveal a patterned response for B to

unstable conditions. Consequently, we shall adopt (6.1) and (6.2)

without change.

-17-

VII. PRELIMINARY CALCULAT[ONS

In order to get a feeling for the equilibrium temperature and heat

exchange coefficient, we shall calculate T and K for a tYnical Augusteq

day in Tampa, Florida. A useful heuristic tool for discussing the(4)

response to T and K is the one-dimensicnal model o! Duttweiler.

eqHe considers the non-advective heat equation

h (T - T) (7.1)dt eq

with T expressed in a Fourier series and k a constant. Takinigeq

T (t) = T + T. sin(2rvt + #) (7.2)eq in i

where T = time average value of T , T. = amplitude of variation ofm eq 'i

T frequency, and € = phase angle, he obtainseqp )

T(t) = Tm + T. I1 + (27vh/k)21 -1/2 sin(27vt + • - a) (7.3)

+ F(O) exp(-kt/h)

-1where v = tan (21T\)h/k), O

-18-

Average Meteorological Input

Solar declination = 14*N

Maximum solar altitude = 760

Cloud cover = 6 tenths (C=6)

Cloud height = 7800 ft (h=7.8)

Cloud cover frequencies: Fc = 1.3%, Fs 23%, F = 51.1%,

F = 24.4%0

Wind speed, W = 9m.p.h.

Max (min) air temperature, T a 90/74*a

Vapor pressure = .819 in Hg

Dew point temperature, Td = 730F

Bowen constant, B = .0102 in Hg *F-I

Water temperature, T = 83°F

From equations (3.8) and (4.4) one obtains the maximum and minimum

radiation flux:

(Qnet + Q anet) = 365.5/123 Btu ft-2 hr- = max/min . (7.4)

Substituting the radiation flux and the meteorological variables into

equations (2.8) and (2.10), we obtain T (max) = 103.6°F after threeeq

iterations (beginning with a default value of 120*F) and T (min) =eq

71.4°F after two iterations (beginning with 70 0 F). Therefore, the max

and min equilibrium temperatures for the entire bay are 103.6 and 71.4

respectively. From equation (2.12), the corresponding max and mmn

values of the heat coefficient K are 9.23 and 6.68 Btu ft hr °F-4 -4

These yield values .419xi0 and .303xi0 ft/sec for K - K/&C11 p

Nearly all the difference in K between the maximum and minimum

temperatures is due to the increased evaporation associated with the

increased vapor pressure at the higher temperature. Verv little change

irn K is caused by augmented back radiation. For the typical meteoro-

logical conditions considered, the percentagc variation In K per degree

temperature viriation is approximately .K/(K/T) = I"/ 'F. This value

-19

declines to a minimum of 1/3% *F- when evaporation is zero, at which

point it is due to back radiation alone.

The previous remarks apply to variation of either T or T. Iteqappears therefore that high temperature outfalls wovld generate sub-

stantial deviation (10-20%) in K. Fortunately, most of this deviation(6)

is of no consequence. In particular, as stated by Edinger and Pritchard,

surface cooling is of relatively little importance in the initial and

intermediate mixing zones. Only a small fraction of the discharged

heat is dissipated into the atmosphere within the region where the

temperature excess is 20% or more of the initial temperature excess.

For example, if the water near the outfall is 20OF higher than the

average for the waterbody (the temperature difference is usually

smaller), most cooling occurs in water which has been diluted to less

than 40 above average. Thus, in the area of principal concern K is

less than 4% above its average for the bay. This discrepancy is

probably less than errors involved in evaluating evaporation and

conduction. The percentage variation in K per m.p.h. wind speed

variation is AK/KAW = 1O0/W for typical evaporation rates. Therefore,

if W is measured at 10 m.p.h. with 1/2 m.p.h. prccision, then K is

known with 5% accuracy. Values are given for the heat exchange co-

efficient and equilibrium temperature in tables 2 and 3 respectively.

Note that equilibrium temperatures above the boiling point of water

occur frequently for W=O. This may be taken to signify that molecular

diffusion becomes important in those instances. One should therefore

set W equal to some small value (e.g. take W = .4 m.p.h.) to

account effectively for free convection.

-20-

Table 2

Heat exchange coefficient in Btu per hour per square footper degree Fahrenheit as a function of water temperature=T, equilibrium temperature =E, and wind speed =W.Mass transfer coefficient is M = .000231 correspondingto wind measurements at 2 meters in waterbody of 50,000 acres.

WATER TEMPERATURE 55.0 DEGREES FAHRENHEIT

W=O W=2.5 W=5.0 W=7.5 W=10 W=15 W=20 W=25 W=30Lq Temp

40.3 .87 1.72 2.58 3.44 4.29 6.00 7.72 9.43 11.14

45.0 .88 1.78 2.67 3.57 4.47 6.26 8.05 9.85 11.64

50.0 .89 1.83 2.77 3.71 4.65 6.53 8.41 10.29 12.17

55.0 .91 1.89 2.88 3.87 4.85 6.82 8.80 10.77 12.7460.0 .92 1.96 2.99 4,03 5.06 7.14 9.21 11.28 13.35

65.0 .93 2.02 3.11 4.20 5.29 7.47 9.65 11.82 14.00

70.0 .95 2.09 3.24 4.38 5.53 7.82 10.11 12.40 14.70

75.0 .96 2.17 3.37 4.58 5.79 8.20 10.61 13.02 15.44

80.0 .98 2.26 3.55 4.84 6.13 8.71 11.29 13.86 16.44

85.0 .99 2.36 3.72 5.09 6.46 9.19 11.93 14.66 17.3990.0 1.00 2.46 3.91 5.36 6.81 9.72 12.62 15.53 18.43

95.0 1.02 2.56 4.11 5.65 7.20 10.29 13.38 16.47 19.56100.0 1.03 2.68 4.32 5.97 7.62 10.91 14.20 17.49 20.78

105.0 1.05 2.80 4.56 6.31 8.07 11.58 15.09 18.60 22.11110.0 1.06 2.94 4.81 6.68 8.56 12.31 16.05 19.80 23.55115.0 1.08 3.08 5008 7.09 9.09 13.09 17.10 21.10 25.11

120.0 1.09 3.24 5°38 7.52 9.66 13.94 18.23 22.51 26.79

125.0 1.11 3.40 5.69 7.99 10.28 14.86 19.44 24.03 28.61

130.0 1.13 3.58 6.03 8.49 10.94 15.85 20.76 25.67 30.58

-21-

Table 2 (continued)

WATER TEMPERATURE = 60.0 DEGREES FAHRENHEIT

W=0 W=2.5 W=5.0 W=7.5 W=10 W=15 W=20 W=25 W=30

Lq Temp

40.0 .88 1.78 2.67 3.57 4.47 6.26 8.05 9.85 11.6445.0 .89 1.83 2.77 3.71 4.65 6.53 8.41 10.29 12.1750.0 .91 1.89 2.88 3.87 4.85 6.82 8.80 10.77 12.74

55.0 .92 1.96 2.99 4.03 5.06 7.14 9.21 11.28 13.3560.0 .93 2.02 3.11 4.20 5.29 7.47 9.65 11.82 14.0065.0 .9S 2.09 3.24 4.38 5.53 7.82 10.11 12.40 14.7070.0 .96 2.17 3.37 4.58 5.79 8.20 10.61 13.02 15.4475.0 .98 2.25 3.52 4.79 6.06 8.60 11.14 13.69 16,23

80.0 .99 2.33 3.67 5.01 6.35 9.03 11.71 14.39 17.0785.0 1.00 2.44 3.87 5.30 6.74 9.60 12.47 15.33 18.20

90.0 1.02 2.54 4.06 5.58 7.10 10.15 13.19 16.23 19.28

95.0 1.03 2.65 4.27 5.89 7.50 10.74 13.97 17.21 20.44100.0 1.05 2.77 4.49 6.21 7.93 11.38 14.82 18.26 21.71

105.0 1.06 2.90 4.73 6.57 8.40 12.07 15.74 19.41 23.08110.0 1.08 3.04 4.99 6.95 8.91 12.82 16.74 20.65 24.57115.0 1.09 3.18 5.27 7.36 9.45 13.63 17.81 21.99 26.17

120.0 1.11 3.34 5.58 7.81 10.04 14.51 18,98 23.44 27.91

125.0 1.12 3.51 5.90 8.29 10.68 15.45 20,23 25.01 29.79130.0 1.14 3.70 6.25 8.81 11.36 16.47 21.59 26.70 31.81

WATER TEMPERATURE = 65.0 DEGREES FAHRENHEIT

W=O W=2.5 W=5.0 W=7.5 W=10 W=15 W=20 W=25 W=30Eq Temp

40.0 .89 1.85 2.80 3.75 4.71 6.C1 8.52 10.42 12.3345.0 .91 1.89 2.88 3.87 4.85 6.82 8.80 10.77 12.7450.0 .92 1.96 2.99 4.03 5.06 7.14 9.21 11.28 13.35

55.0 .93 2.02 3.11 4.20 5.29 7.47 9.65 11.82 14.00

60.0 .95 2.09 3.24 4.38 5.53 7.82 10.11 12.40 14.7065.0 .96 2.17 3.37 4.58 5.79 d.20 10.61 13.02 15.4470.0 .98 2.25 3.52 4.79 6.06 8.60 11.14 13.68 16.23

75.0 .99 2.33 3.67 5.01 6.39 9.03 11.71 14.39 17.0780.0 1.00 2.42 3.83 5.24 6.6b 9.48 12.31 15.14 17.9685.0 1.02 2.51 4.00 5.49 6.98 9.97 12.95 15.93 18.9290.0 1.03 2.63 4.23 5.82 7.42 10.61 13.80 17.00 20.1995.0 1.05 2.74 4.44 6.13 7.83 11.22 14.61 18.01 21.40

100.0 1.06 2.87 4.67 6.47 8.28 11.89 15.49 19.10 22.71

105.0 1.08 3.00 4.92 6.84 P.76 12.60 16.44 20.29 24.13110.0 1.09 3.14 5.19 7.23 9.28 13.38 17.47 21.57 25.66115.0 1.11 3.29 5.48 7.66 9.85 14.21 18.58 22.95 27.32

120.0 1.12 3.46 5.79 8.12 10.45 15.12 19.78 24.45 29.11125.0 1.14 3.63 6.12 8.61 11.11 16.09 21.07 26.06 31.04130.0 1.16 3.82 6.48 9.15 11.81 17.14 22.47 27.80 33.12

SIt I ll lil lililllii l IMIA

-22-

Table 2 (continued)

WATER TEMPLRATURE = 70.0 DEGREES FAHRENHEIT

W=0 W=2.5 W=5.0 W=7.5 W=10 W=15 W=20 W=25 W=30Eq Temp

40.0 .91 1.91 2.92 3.93 4.93 6.94 8.95 10.97 12.9845.0 .92 1.97 3.02 4.07 5.12 7.22 9.32 11,42 13.52

50.0 .93 2.02 3.11 4,20 5.29 7.,47 9.65 11.82 14.0055.0 .95 2.09 3.24 4.38 5.53 7.82 10.11 12.40 14.7060.0 .96 2.17 3,37 4.58 5.79 8.20 10.61 13.02 15.44

65.0 .98 2.25 3.52 4.79 6.06 8.60 11.14 13.68 16.2370.0 .99 2.33 3.67 5.01 6.35 9.03 11.71 14.39 17.07

75.0 1.00 2.42 3.83 5.24 6 66 9.48 12.31 15.14 17.9680.0 1.02 2.51 4.00 5.49 6.98 9.97 12.95 15.93 18.9285.0 1.03 2.61 4.18 5.76 7.33 10.48 13.63 16.78 19.93

90.0 1.05 2.71 4.37 6.04 7.70 11.02 14.35 17.68 21.00

95.0 1.06 2.84 4.62 6.40 8.18 11.75 15.31 18.87 22.43100.0 1.08 2.97 4.86 6.75 8.65 12.43 16.22 20.C0 23.79

105.0 1.09 3.11 5.12 7.13 9.15 13.17 17.20 21.23 25.25110.0 1.11 3.25 5.40 7.54 9.68 13.97 18.26 22.55 26.84115.0 1.12 3.41 5.69 7.98 10.27 14.84 19.41 23.98 28.55120.0 1.14 3.58 6.01 8.45 10.89 15.77 20.64 25.52 30.40

125.0 1.15 3.76 6.36 8.96 11.57 16.77 21.98 27.18 32.39

130.0 1.17 3.95 6.73 9.51 12.29 17.85 23.41 28.97 34.53

WATER TEMPERATURE = 75.0 DEGREES FAHRENHEIT

W=O W=2.5 W=5.0 W=7.5 W=10 W=15 W=20 W=25 W=30Eq Temp

40.0 .92 1.99 3.05 4.11 5.18 7.31 9.,43 11.56 13.6945.0 .93 2.05 3.16 4.27 5.38 7.60 9.82 12.04 14,.2750.0 .95 2.11 3.27 4.43 5.60 7.92 10.24 12.57 14.89

55,0 .96 2.17 3.37 4.58 5.79 8.20 10.61 13.02 15.44so.% .98 2.25 3.52 4.79 6.06 8.60 11.14 13.69 16.23

65.0 .q9 2.33 3.67 5.01 6.35 9.03 11.71 14.39 17.0770.0 1.00 2.42 3.83 5.24 6.66 9.48 12.31 15.14 17.96

75.0 1.02 2.51 4.00 5.49 6.98 9.97 12.95 15.93 18.9280.0 1.03 2.61 4.18 5.76 7.33 10.48 13.63 16.78 19.93

85.0 1.05 2.71 4.37 6.04 7.70 11,02 14.35 17.68 21.0090.0 1.06 2.82 4.57 6.33 8.09 11.60 15.12 18.63 22.14

95.0 1.08 2.93 u.79 6.65 8.50 12.21 15.93 19.64 23.35

100.0 1.09 3.O0 5.07 7.06 9.04 13.02 17.00 20.97 24.95105.0 1.11 3.22 5.33 7.45 9.56 13.79 18.01 22.24 26.47110.0 1.12 3.37 5.62 7.87 10.12 14.61 19.11 23.61 28.11

115.0 1.14 3.53 5.93 8.32 10.72 15.51 20.30 25.08 29.87120.0 1.15 3.71 6.26 8.81 11.36 16.47 21.57 26.67 31.78

125.0 1.17 3.89 6.61 9.33 12.06 17.50 2 2.c•4 28.39 33.83

130.0 1.19 4.09 6.99 9.90 12.80 18.61 24.42 30.23 36.04

-23-

Table 2 (continued)

WATER TEMPERATURE = 80.0 DEGREES FAHRENHEIT

W=O W=2.5 W=5.0 W=7.5 W=1O W=15 W:20 W=25 W=30Lq Temp

40,0 .94 2,06 3.19 4.32 b.45 7,70 9.9b 1. '? 14.4745,0 .95 2,13 3.30 4.48 5.66 8.01 10.37 12.72 15.0850.0 .96 2.19 3.42 4.65 5.88 8.35 10.81 13.27 15.7355.0 .98 2.26 3.55 4.84 6.13 8.71 11.29 13.86 16.4460.0 .99 2.33 3.67 5.01 6.35 9.03 11.71 14.39 17.0765.0 1.00 2.42 3.83 5.214 6.66 9.48 12.31 15.14 17.9670.0 1.02 2.51 4.00 5,49 6,98 9,97 12,95 15.93 18.9275.0 1.03 2.61 4,18 5.76 7,33 10.48 13.63 16.78 19.9380.0 1.05 2.71 4.37 6.04 7.70 11,02 14.35 17,68 21.0085.0 1,06 2.82 4.57 6.33 8.09 11.60 15.12 18,63 22.1490.0 1,08 2°93 4.79 6.64 8,50 12.21 15.93 19,64 23,3595.0 1.09 3.05 5.01 6.98 8.94 12.86 16.79 20.71 24.64

100,0 1.11 3.18 5.25 7.33 9,40 13,55 17.70 21.84 25.99105.0 1.12 3.34 5.56 7.78 10.01 14.45 18.89 23.33 27.77110.0 1.14 3.50 5,86 8,22 10.58 15.30 20.02 24.75 29.47115.0 1.15 3.66 6,18 8.69 11.20 16.22 21,25 26,27 31,29120.0 1.17 3.84 6.52 9.19 11.87 17.21 22.56 27,91 33,26125.0 1.18 4.03 6.88 9,73 12.58 18.28 23,98 29,68 35.37130.0 1.20 4.24 7.27 10.31 13.35 19.42 25,50 31.57 37,65

WATER TEMPERATURE = 85.0 DEGREES FAHRENHEIT

W=O W=2.5 W=5.0 W=7.5 W=10 W=15 W=20 W=25 W=30Eq Temp

40,0 .95 2.15 3,J5 4.54 5,74 8.14 10.53 12.93 15.3345.0 .96 2,21 3,46 4.71 5.96 8.46 10.96 13.46 15.9650.0 .98 2.28 3.59 4.89 6,20 8.81 11.42 14.04 16,6555.0 .99 2.36 3,72 5.09 6.46 9.14 11,93 14.66 17.3960.0 1.00 2.44 3,87 5.30 6.74 9.60 12.47 15.33 18,2065,0 1,02 2.51 4,00 5.49 6.98 9.97 12.95 15.93 18.9270.0 1.03 2.61 4.18 5.76 7.33 10.48 13.63 16.78 19.9375,0 1.05 2,71 4,37 6,04 7,70 11.02 14.35 17.68 21.0080.0 1.06 2.82 4,57 6.33 8.09 11.60 15,12 18.63 22.1485,0 1.08 2,93 4,79 6,64 8.50 12.21 15,93 19.64 23.3590.0 1,09 3.05 5,01 6.98 8.94 12.86 16.79 20.71 24.6395.0 1.11 3.18 5,25 7.33 9,40 13.55 17.70 21.84 25.99

100.0 1.12 3.31 5.51 7.70 9,89 14.27 18.66 23.04 27.43105.0 1.14 3.45 5.77 8.09 10,41 15.04 19.68 24.31 28.95110.0 1,15 3,63 6.11 8.60 11,08 16.04 21.00 25.97 30.93115.0 1.17 3.81 6,44 9,08 11,72 16.99 22.27 27.54 32.82120,0 1.18 3.99 6,79 9.60 12,40 18.01 23.62 29.24 34.85125,0 1.20 4.19 7,17 10,16 13,14 19,11 25.08 31.05 37.03130.0 1.22 4.40 7o57 10.75 13.93 20.29 26,65 33.01 39.37

-24-

Table 2 (continued)

WATER TEMPERATURE = 90.0 DEGREES FAHRENHEIT

W=O W=2.5 W=5.0 W=7.5 W=1O W=15 W=20 W=25 W=30Eq Temp

40.0 .96 2.24 3.51 4,79 6,06 8.61 11.16 13.71 16,2645.0 .98 2.31 3.64 4.96 6.29 8.95 11.61 14.27 16.9350.0 .99 2.38 3.77 E,15 6.54 9.32 12.10 14.87 17,6555.0 1.00 2.46 3.91 5.36 6.P1 9.72 12.62 15.53 18.4360.0 1.02 2.54 4.05 5.58 7.10 10.15 13.19 16.23 19.2865.0 1.03 2.63 4.23 5.82 7.42 10.61 13.80 17.00 20.1970.0 1.05 2.71 4.37 6.04 7.70 11.02 14.35 17.68 21.0075.0 1.06 2.82 4,57 6.33 8.09 11.60 15.12 18.63 22.1480.0 1.08 2.93 4.79 6.64 8.50 12.21 15.93 19.64 23.3585.0 1.09 3.05 5,01 6.98 8.94 12.86 16.79 20.71 24.6390.0 1.11 3.18 5.25 7,33 9.40 13.55 17,70 21,84 25.9995.0 1.12 3.31 5.51 7,70 9.89 14.27 18.66 23.04 27.43

100.0 1.14 3.45 5.77 8.09 10.41 15.04 19.68 24.31 28.95105.0 1.15 3.60 6.05 8.50 10.95 15.85 20,75 25,65 30.55110.0 1.17 3.76 6.35 8.94 11.53 16.71 21.89 27.07 32.25115.0 1.18 3.96 6.73 9.50 12.27 17.82 23,36 28.91 34.45120.0 1.20 4.14 7.09 10.04 12,98 18.87 24.76 30.65 36,55125.0 1.22 4.35 7.48 10,61 13.74 20.00 26.27 32.53 38.79130,0 1.23 4.56 7.89 11,23 14.56 21.22 27.88 34,54 41.21

S.. ... .... .... .. .. ... l ll l i / • im l li ill mm m B i i { .... ... ... ...

-25-

.nl 0nn: ! otIl I tn iI Iiin ý tt ii (tnci tn n ni ga0~ U ~ U~l~e V U4!

wl r-0 o

fl nckt .Ucl.N * 04 ~ U M0 *.N M it & a' a a.nininU .0 0~a0 v a .0 a- ewa . g' e0 Na N .."N a t-a r- 4",U a )W r b

.0~~ 3w or-'P Pna rlA~WQ . *0 3 N ' * * * ' 'Q0.0P-f

b0b hi i c l1n1tnn 11n

C4 *4 M .4 cm N4 Of.4,0 0 C4 M

'iC c t 1 t9 C!l iC i' iC NN44'! C!U,!,.0.0.0NtC!UC! 1.5 3 NI a UIli.0I

C'4. Oa O ft 0N 0 02 0C - Ia.l ' . U U. a 0 U

-! 0! 0! 3i 90.0-NQ W!.000..0-.f4. -M .. 4.2..4.4

m 0 C30 3NNL. N 9U .0 ft a a 40 U,

C64.,4...NNNN .4 4 4 4.cN N N N

000000000n0000n090iIt0i0o 000 O0 0 00It000 .

........ ........ ..... ....

34w a. U N .' UN0 m4N m m I 0'0 U

U. a.0 4, UN.4 U N . . . . .m NN E'4',,~ aa A n 2 0 am 2

4*k4. 0 ,M 1

C!4 0 N ý9C ýC ý9o!9c :c

06 s4N4),'4)~fUN.UUNo . U. 5N0aaUUUUU.0.

-26-

U.0ý go r- :CU. 8U~.. 4-4r '~.0: t-..4a MI-a ftsUI moil, :8o, 4)* ~ ~ ~ 4 10 w qil w. 0.0 .rW *qfF Uqu r- nOt)qI af0auqr ~

40

Cod a.0)* *s. . .. . i i*m": n. ON.Qtt 011Q80int)Q44 .D .91;C f t ý""ýr "f

.00 Qw ".s low riq. r-I -I8 t3 - - -- - - -- - - A-r -lilt qOiciý 8V i~ il .:C ' t- v

* ~ ~ ~ . Z80ý - ý1 ft..ni. C *U*)W0fW) C 1- 0ý t.) ~~~~~~ - T- a .ft C! nC ti0:iiI !0:Q 0w0:

31 as-r aa qieqE...., Inr--t 0 a Ca *qr ONNr r- I-,-.

% It. % :O I !C!1 :4tC

U. r 0q8 0)tN at-ft'' *0C4 0 ;.ow:mfIid A.nt " N- ff.4a~l'e.at8-Cfto)i..a0.4)UNa-f fti wo " a gQraf5 -. 40'Na0U

......................................... . .

1aa iýli1!I:C aCU,'' t-N W Q C! ai aUC!-N N t ,-.m 9 :e !1 l C8 ecofa a. .oQsoota.)rv-ooft. 2007020074N.08100020010*N

. . . . . . . . . . . . .. A..: N x tm4*Fr-t'e)ro0.a. at 3.m0 C4 Nt-4~.0q8 W 0

0)at D nt-wc s tU.Q--e~~.8.... o- : :'UU'NNQ a a880 m.C.

.4. fl4 4.4 04 1- w .n

o~ ~~ . .0 a f-.. . . . . .* U~r0aCi.a 8.Ntf ~ -. ,Uit.-o~.a 0 .fN .4 0 t

:t - ct -4 t t 4-t -t Wat O.-4F al -:q n 'tnhaftv 01 nT444444 ftttttf n on r- w a C4

31 8 8 8 88 8 0 0 8 o000880As*&0888008800

n.....t..1............ t......n.....-.....n.........................................

4~~~ ~ ~ ~ ~ 4 vI ý :: - 0 t- g t--1 i C *I;- - .- r a

V" so .. f N ffa 4"P .4 vs : Nft-t0o)M0 .

04-44ftC4C4f" - *

4-IiI !C C !ic iC 1!91 C 9C 9C C !l

AMS * , 0ý w

No ----

-27-

C r4,o In 444, V-0..4, a C-w .4c4* ft . 4.44 * o 0 C fwt n0 .a ft. i4444. sorrrrr a- 20S 0 0. 0 0---fCCCfC

3~~~~~ ~~ 000 00 00 IsCr---rC t. I0'0000---- -C-Cr r-r

.n n . 40(*C4ý

.0 3 l 0004004 orrrrrCrC0 30 004 oooor-r-rC-n - t C! C-nGee

n f 4, 0r-C 0 a a 9% -n f 0r-.0( " 0vC0c-40-4ecoco

'n"400 40" , 1 O40 a'* a

I f~4 O e e- 1% 1%r-r

a" a a W 0 C w 4* wýrn% i i i n % n t 0 aI- & a .4 " Co~or-r-x nn-ec~ w l~o0 9.r- me Ms cwccs

0 ~ ~ ~ ~ ~ ~ ~ ~ n0 nV 00 0 4-0C4,4O-c-4 "aa 2 -er .w4.oa.4rawa o*a

a rn a- onw o"CA40fsM0 1, 00,c4 . 0(4 WcO 4.1 ' 40 0 0-4 Ag 0 a;@.45 :gM

3 *000090 0:09000eec :0o00 a 000C.rrr-ce

4,w4,r-.4oc.4~~. .1040- . . .(0.04 4 4. .. ow . .

wo morrwwo..

P4Oc,. 44,0C0000 A0.4*21 I * S, a" am Z*0wWon 20ro- 2U1 wo cc-40 oco~r .4(444(.4001- a woe ar- 2 0l Pr0. 4% 44f~0-

0 n

-4 1

2 g in 4 0 or.r-t-r- ta" ar- 2 CR 2M-

.3 040013n ý0 .040. a~-0,444444,4 0b 0(4o-00000r04

*~~~ .44 9 O w0.4tot-0.C4 t nw oO4! 4, i ni n9 t

6080i00 80*r 00000(40020 03-0 4c404OO4oC1e0

ft 0.onfp

NI M z Wnr- r. a0.0.4Nft 3.M In wr-.S -0 04~~ 3nf~- 0 4

3B 0- a .F )F qNN. )8 In F) .0 W . 0D ca 00 do 0ý 0ý ID a Cfl an a ab & 0- N 5.

t- ww w w t 0, 09

9 N .4)I~ W: di9 4 i 4tt * 4nIt Nili t N 0004 F*II N i 11: C 0:

N N4o -C0 a' NN N N W. w W O O cc* ,00 0r 0 a3 N r O O U.8 0 0 m0 cnp

~ cc0 W! ia 8 :4C >n.NmrW99 I04FN C.N)) I C In-"MIGýn 0.)5~ *3

do aNoc " ,C .C

3~OW N- . 08 8 00 0 3 ; j.0; 0:00 00 0

n.- w'Io - . IaC-a

eg 4 ) * ) O n4-P @ I O)D.. ~ I I 44 0 *4 $ @ 5 .I Q4)n-

280200890:000020000 00000:000000000300c:

ON I.E"41 09*9NO*O. 9)nO.4CN ON

.44.

I- 19a t t9r 11 tIiI 0

NWON.4038000$5I000.4C (Pa45O N - )@ 4 I .fl

.n 0a0 mg a5010 C;NN4 r.I I 8I O l~O00.NN4C NI

.4~~4.4.4.4.4ft . . . . . .4 . . F N N N N

00

.00

n 1! i i i It 1: i 9 i i 0c n In O N N$ Wn i t i .,

1. C" ft 05 990NC4 14 el

99 iiSI9 99 9 % 9i.NUUi* 0 i908 00. 9N1 *in In ON N*0

-29-

1701ss 0.U ~ ~ ~ ~ ~ ~ ~ ý4 n Tre 4U a - 2eat. v4 7,~e : m. a r . a -. : .4f. t3 CIO 00~4 of Inn to3N n u~a-*eg u'

.n N . a * O . . N , 3 .0 U~m 0 4 0 , . , U N N l .i ii C t 0:11C4 OtItI IW!0.i i It1%4%t :

0K a V.00 f0Nf N4,W* 0 ft :.*Q.4neO a,,ft " in I

.0~W 3 NNU -nan aa aa O0e me-n 3ý NU,.,e ama**aa a "a. 1%

Qw 0 0

N~ ~ ~ N.f.mf .4 a & *: to I -U~~~C" no 0 toaI-aP* na MOacao-. 4. a C44 l @Q # NIP &~a an aaoee, aa aana a

r: J C. a UZ sao"" " *U 101 MNI V ~

In I. : cra . . t N Oalf C,4 " @ 0C NA 9-41 ON..NI"ft A A -0. 4N

U Ci* i ,-4e.*0 i , C! I.E Ci Ct li#~ 2 'il tl ! !l3 N0~0w :0 a a-- ew020a000e-Ni - *

W M . MNlaCQN*O @ 0004.4

0 M00000a00000000000000 0 04J 000000

0000000 M 0 QP 0000I 0000- 00 000 0 0 0 0 0 e oU *O * 0 O N4OC N& N 0 0

o:C

.U -:0C .

00002200000000 0 80000

0~~~~~~" NI*4 A 4A.4aQ:

-30-

0 m 0ý :UCOb4 S .4 c4e.ftfl@0.4e,= in4* ~w4 ain in 10400 0 a in flt n*4 1 I lb a~04.U

0.

" 0a "s t.: .4 0 vs'&*** an4ea won *E40 4f0 1- r- VCl 0 * 0(400 "AsW .4.40"S . .1 A in n * * * *. . .a 0 ",P n.n..0I It t tel in .ln tj 0 a

La U t r: m o000(.a(.4 0(.40 *0C44a~a in in in inr* 0000 0 t- r4

.44..4 (1 r.41n44.

@0 V 0 000 0*0-0 :.* p~ 2 s -v4 aeo. .0 J0 -

002

N -r- f- 0 4af f ý06 -

-S S i -n 0 -A on in in n 'A a 0 0~0 c 0 ~ ~ ~ o r .Ci C:00.. 00. C 0 0 0, 0 C-

C!mA.0 4 " 3 :,a a s OO O . 0 0 :0 3 .0a~~f~ l s o *a-r0 e,0

0~ 0 10l I 0I tC !ItIm-4 : 0(- .vr.4

mooa~~~~ *s0 v0 we--rm 0 va aa 100I,0-6-g1

(.-r-.0.4ft In C4 0 M 0 0 r-4 aa(. 0 r0.4 0at a .0.4 ft.W.4.I a

-. tr~ rý4.. sm- a04 m s* boooror oe6-rg-t -0..

'D a. o o.~ 0 4 -z ~ O 'O 00 ~ 0o 0w0 e .. 4 00t . e a o . v- 3 6 ,

C4 C. 4-.

. 00300000.40010 000V 00 4- .0 ... o.00.000-0CE J - -2 1*1024 A". On " 2C04 1 ID 0-C~l 0 2 C'

bl% t I t tI" IIt tI M In I -3 1- 1 :'tIi i !o ni tn

a nWsIi * m..6 4.meft..l-lIs; -l-ra-- s f a ft .- O cOOs0A 0cvmg

A an n ina 4o 0 .4 t a no- 16 t ti *c.4t.8Q-

.4 .; . ... ft * 4: *4 1" ni n ***ftgG**.4ai .. a c@f.4j44t08~0f

n n~llaQ tntn-1lC.1 t tril V: ~Q~ll

eftnelf . 4.4~c ". oc-aa-o44 .4e..4.4.4 .

i n *4 *4 4* ** e *4 ni i *4 t4 *4 iil t t4 I iCC******' * wnA f.48le~alt~ae o ... ,am a 4avs

mm CNN m A A 9 "NA2"

900403 10c :00:020200. a-s ctW W.4~~l0t...44uLI & SO ** C *4 4 * * C 4

we a r- tm mm ac a meP....V.sS

9 ti:c4Ifta.C.f~ tnninnni~*-4Oln V~n ttn.0 9 E.... 9 00 26 Co" 2 s n

C.. .4 ii*4 lwi t. to:~ 4 4 4 44 C~l It *4i i% t n'ttte

ftC,~ftWUIWIIWI88*8 ~ .44.4..4 ~ftCC4 .. 4. .4*fl

m -04- .4e41. 44"4 10w ti6:ni W

C-,-w

C".,- .4................. ................ ft.4 . *4 4

mel80u--4a.fi 0 a inIatrl.a~44m8~

0 8m.4810c4.0 mo44...ec0 eWa-.4 4aaaa 1328 .00.4....................................0-† † † † ††-

gm4 a o*.4e.. .ti 4 2 V) W " :100 C 0 4C.* " ft In.1% 0

r - 40 mo co me agoaco 4a cm 00 m Ie to a m ft aa~a. I -A0a, S4"a. 0,

C! C D C4 vVo :C .~O 0A.JN 0- .4 W .4 m s In 0 4. a 0 aI0*

.0 4 30 a-- e e m a- m 0 4, a. a .0 63 0mc emmged a a a, g. Wa, (P.o a. C' 0. 00 C.

a. 4" 0. *, 0 b00 e.4

ul na. - l~I I n ~ nIý i* ~ ~ r40 0 g z i 9 4a.0m s 4. 0 " s 0aa a 0 " m In r0(UrU.N4)* 41: .40

1 4 .4 . NN" V. .0I-a. .4 4

N J J A *fa- Q r; a.. a . I. , v 3 S

t' e a M 0- 4.Q4 4 .4.4. 44

14 .4 " .c4 04 c "

U,4 sA N. I.. . . . . . * OAP *. . .. * n *~ A A m* A A * At *j scu,.c.1r- E1 W4~a.0Ut

haA*--mj~e 04...NN -- m~a004444 CNCc444444444444............

044.eU0a.~~lU.t ,4.0WfIg.m4

.n I... . .. . n It t I 14 P; n *1 *4 n U1U~~~~~~~~~~~~~~~ e4C-.0O.04N.~IU M fC,0 00.CN.Oa

t3 01I.NUO@04O 1UaW4 tmWo.N1* V

00 0 00 0 00 00 00 0 00ge.0 0v000:0O0CA ar- . t m0N- t OC0 tO a a ~ t aC 0 *0C aE r-C NE r.4 a

C .44.44.4NNN(01W01001 #8*0' 4.4I.4NNNNO~fPIM1E:

QJp

.4.

U, 14r9.4. I-0 - wo .4A O E.* w .4 We mtf.4w).4 d1 a..WO4. 0'0 amt-a 0 0.4.n .erm. .M.4qU*UU~~ .4 aU .4 ý .44 4N01511

o~~. " M MN.0C U, f 0 *e1 UV E 0 1-4 1-4 ,

.4 4 ~ .t4UONU 04.t-. t4CfN,- #4 WI4 eC04 0 4'0N c.0 emi i Il a.. *. . . . . . . . . ....................... ! . .

0 U 0.4 U td r-c 0 01 ,* .C W 0 U . a m W 4 0 * e - . 0. V 0 1.4 -* aa.4 N--t-A m cc jl "s Zmcm *0 a q , .4 ~- - -mfmo4e44m Ame . r4

-33-

a a r Ca *.4 C04 It4* in tn *gab0% t4% t t V! n n **

40 x 4 3c s n amo. Cl ftg 5

a ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ... ".4.-4Omam4-4 l.ea-04qmn~aiim

0 4a0 ~lC49A0.44"Z4I : " A" 1

CP a, CP aol " a-C4p c 00 4wetmanws2r- go~e o a *mmraa-rea4 -.900a000 0000990~~~ %n in i Ci nInai It .% ii. % t ow- ia-

C4 a a- m. . £a..4aa Ci0 n %

L* * * * * * * * C * * * * * * * * LU - * * * **. . . .. . . .. . .. *

r- m m a- a-: 0C ft.,l * -ý:0v 1:C*a~e~ena-..enla-.nnC.aC . 051-C4" CVCCP a .40 .0 0aila-

3; 4,;aa. 0-1 ~ ee41 A a J rS.a 0.2..

a aaanaomena.n 1 an na.m~n enac eae

a am-e eae rCe. amM -IN mam4- ftCa rC.40CC:-m i n aaa m: i~ natiiiiiiiC iC ýin i i

-34-

a . . . . .. . . . . . am .% .t' 11 1 . %4 1 .t~ n101 in ****

a& sI t1 t a - t '!r 0 *0. @a 4r-11 n n 4% -. n

a ga dS 4 rs 0 P -- Pa - g2 dA sInSS ol: " .4 994.mi TOIfal-0. 4

if v *a id :I- gri .i -.0 .- .- .o .0 Up . . i 0 *U***U*

0 nInt t" 1~fi*4:1 t 4ificil.4i 4in 0t v: %nf t4f .i-tilli -i*..4I~fuIe In Ob.* *4gi---s. i

i a ni In 4 1It nW. 1 1 I I

lii if.. C:C iiC

n t :i 1. 9-0 11 1 ti t n l4*U .4 ifai*40ift0m-d.t..4 ~ ~~~~~~~~~~~~~~~~~ S U *

21 S 12,: S t,:M :n .n "I ,-: 2"s~~I.' 0bi Siif & in %ff.te-gt

lit. .n .i .~n~ . t ... * U*** 4 **** UA is T 50 *.e.4am.4aifA.*e,4UIq1f.U

.ft'A n es aei *, 4.Ea -ft A -b- if@.4U4.S4*

- m *- - ; A b 4 1 a0

.4.4.4.44.4.4 % if nif if if -i C: t- 444444i U fi fi f(

"-4-

C-,K

% iA 9r 4 *e U .4~f% . a0 .4a.0 44MUA Ut n Ve on219S' r- 2 0- On so.4 wa-0 .00

.0~~~ ~ n * n n n C! U n * 1 -t t * n U****UU *****0 £ 0l.Of~tr:4f£4~F-o

O ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ " .44. Sr0U .4ffff* g.a 0. C.U.4 U.4 ii.4i CPtAU, C! C! * i

*~~~~~~~~~800 E228C Mi~ne4~~f f, f f. .U*i ~fS4COOm aaa aaaa aa .wte.eo.44. aIeor

-35-

*~~~~~ -0 *i g"R .a *i gg ag 1 scir***a *4 ** * **4* **.

nl ntrg~- i4i%.n-4M ti% '00 a.

a I~oo Oinm. metn 1.00---. 4449630.4 t 1 f400.l4o

0 09

It 6i ~ id 4! % I " il V t4 1I a i1 d -.WO.. ... . .*.4 .60. . 4104 9 .4 .. t.0004.4-4f-4A No4 44 *4 4* 44 4 s- .4 44 44 44 44 *4 *4 x 444* ,0 .40"a

".4 ** ' "" . ".4.44.@09,4

0 4

".4 ".4 ft .. C4 .4 .4.4.44 544 .4 .44, C@4 AC. C. Co.

in-999 t t i 09inn n 0:1"0% 1414 in091-444 ai a* ro4 n141 a

CC

9, n4 .4 .44 4... . . n nn tt n n

A SO0 0 e 0-n ' VO- a 2:: ~ :~

9, *4. 9C, 44 4p4 4 *4"4 .44

m* n I4 tiQ44.4#109449a4.- Vt t049 -i -. 0-090149,a.40.

"" w, r- .0 4 414 *4 † † † † † † 4 14 *4 44 .4 * %. 4444.4444ii i%

04 4 , 4 4 0 1

aa1 -

104200 $8.4,42d1-9.Ta4-2 42 l5 003

3~~~~~Nw.~mu ---.40 -0-1 -----S a 4 4 00 0 1- 1

-36--

Wb~~~~fl Ca "a * a *** 4 * * * * *4

a en..

0~C,

*~***~** eq v(. I. .44C a .4q .04 V0

I-Mae O445

6l a0 00 ame coo. @00 00.. .4 -4.4

04.15.4 aqQ-04 t~0Cf

*f1ff 4* . PaoC( .92 4,0r 10001-441

000- 0 2 2 ea 0.04 .4.f44' 10- C21 I.4. t",(. C4C? 1

.4.4t.n.l 'n.d W, I t.4t n*in .44.0 C ~ .QI j4*~C .44re.e *gee .4qav. r ,Jn 4

" " g.4.qq O 0 T a .q4v-4.4e .4O. .ne~,.4a. 944 e404 .4 9 a O

0 *EI.*e-(.QQ4na(4,~**gV) 01 ft*e-Q4 .4.00A(4I',

9 0 00.

d * e9e9..i fli04# .Q n a .. 9aoeq...4 em.. .4omma:

0 ii n 5 .444e0440 .444.9 nil .A~qC4I.O

C C4

4 .. . . . . . .4 ~~~~IKI 4 p 4 . . . 4 * .

3~,nC C49 U U 4.4.4eq 42 *ee9 ft444eee

.4. l...... .44.44 i 4.4.4.4.4ici li

01.,.4t.C.-q-v4-eO.4 C * .4(

-37-

In in~~te ~ . f n v f ~ o f ~ t f E * - W4nf . NI s s I'. x s .0 "a . 4** *** 2 . 1 . 94

0 :4:I i~t il~il .4ft0n nit tnnOno4i I f- qtnOW1 0" *'a WS. :.4A ::s**e--: 0U4Ud " t4

ft 4* ** * .. Is 4* * 0- f- Vo "Moss &14::* 10 sea,

ft *4 *4 *4 444.-04-0444 4 . . ft 444 *- -S 4 s 4-am 4a 4.44.g 4.v4Pa.am*

1 ~ at f4MtfCt. ft"f

.4 .4 .4 .... 4...

Ina . m 4 .a t4a.4 S29 "4ftC-ft.4s*m-a

,200200002 128-2...eee 2392320900 " lit00

in inin i 1 "0 P4)4Sm4.Sfuta infi*a in .in m In40 40 to

,n t- ý4 ; r 0 sm : S"m n

' fi .42T ftlft.4E* 1C.ia~ o 0 4 .44a Oý44# (*ea nmaa ae . 40 aw woSSS aa

* ~ ~ ~ ~ ~ 0 on 0i4ateft4fffO~eaf IS cf~ -t O a: r", raa.gf! It I. It a:4 4 .ca' CpA P

to- 0r-a

0 f ma-..r.4.4...t~.. 0 ofa.,Sr.Ct.o 4r:.4ft44UMC 4pf~4e4-4a~4£~.4I aaa---.no

3 ~ ~ C .Cgaaa,,- 4~ C m,.4 C" e.* ft4 aeag.. A

- QE. - . m4 ~ ~444t* ~li l C! 04504 0Q ;0 0

3 r fta~-- C.C O .. 4 i 3 40* * u~ * ~ ~ e ....

S. .. .. ...

C * w,* O~a~ee~df~eeC e.a eee~dieeoo .i• 0 00f

-o 0 aoo oboe o o o o wo 0 aoo o om ~Ob:.. " .. . . . .. a..

O~~~ .4 ".C5 @.flC4*OCd . ~ V

*o o "Cd.4@i-C.4C0,C CO.9 .4 ". M a m .4r . 04 c•eo . d 4*

Cd * * * * * **..... ... ... ..... . Cd * 0"* .. . . . . .... a**

ft.. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ f -: It % .t*********** . 4 *S*S***** * n n * 4:**

++to.,o a v o'.o . . .. ... ....... ...... o ...... ..oI o -e... m..4Cd@Ma 3 "o ol 3 : 0 .4 ed - 31. 0 ama raa..4v.dmftA0,..

*.. ..... ........... . * * * .... . -- . t a * . ..v . .. ..m"--

0 c" 9- .4.w

an na . ft np* w) to *o ~ so & '* wt at a *~

4K 4

-0towZ0D 2 "2a n- , :s U

a . 4Cd no n- n.Cd i C i 11 6n40 mn we i 0. Ca .9 i-ftC a.

ad a f* A o 4 Aff** ft 4 4:*f:f w; *: af ... . t. . .

& ~ ~ ~ ~ .2fftAm ons

-39-

11 a a a 2 g,,4 A 9 2 a U

:4 0 "a . 0 a : :

ma ~ ~ 40 * aon*a C4@n n one 00a".4.4 U2 o4 0 Ia.moag.m..SA 00z 0s 1- m 000 a 0

a~a.44*..4aaa mecc "a0 ea - am O e a .4.4V.4a

A a414 P g * r* g r M a *0 s aV 9 .. A CE . . Pc a . a a. aa a . A . ..0 0 9---trm meA

.0 1- v r g. f a g W. -a s . a i. o *- 91.I~~~~~~~~~ .-,a am~ s~ 2~~ot0 0C 0n

anm

10

*9 * * * *p a * . *0 0............. a . . a*. a a 0 W a. .. 00u-me aaaa e * 0.4m .aaageS-

%* ------- IuIS.~m

0 ma.ne-~--.-ano -. . . a a . e m - a~M.Ma g Ot * N" a*.***.....ae,-eaae P a 0 e -m a am a. & @a aa "-

*ee "- I n- I-I --- * mm.. I--I u-I-u C" "4Aat 04e" " C%0

,I ig a o a-a aae~ at 'ta~ e 4a i a a % ma % %iiiCa *......... ac*.. ... cc ... .... .. aa

-40-

n . .%** * .* . . . . . . a . . .. . .t . . . . .** . * . .40 I wa A -%f 2: -0 -0 s :4 s : - - Sa IO S O" 0"3P9A

a 4 a a

A &rn f 0..@atoaldm PO a -0daO a- -0 1 .402 aw rca- a4d:

.4 a c.a e . a

At In* a Cdm dcmO.4. .d "mcac .4ce d 21 0 . . a a c . a .4 430 -

,4.41.4-dm.4 14 CI dC .4.4ualm a *a*4-dm. .4 fta A t*4

.4* &a0~~4c 0 aaa" a 0 F4 4 IEC24A saaaaot 1

OH ~ ~ ~ %t .4.4 a.a indO0C a a 0a 4am

cAaa.0 400 aeca40 . awm an in aCWcaa.

21 2 2 0 a a . Cd a 'sm .4P sd aZ sa on In-0 .4C4-0Qa O dIn. t41.n4..n .4%4.4c: do d t Idl 1 d .4441 to .44 4% l4 1 ol 4 It t

00000000000000000C-4 P)00 00 00an00 0-n 4- * n * 4- * - g g so 2 "Mf RA 0 egg** * 0 C:*t S * 9 .Mae) g AggAt on on a a o vý

a . * . * . -SSSSS*S* 9 *a ......... . . . . .o .4 .1 .i2 M

0 .C aac.4a No-m .Cda RtAcA C. 0 V Orda 9-ft0 . "4dft a cc

TICl.0" : tQ " f~t

a:& *% .l Ct 1 4 t 0i lot

a~~ 1 i e m 4 Q 1-. 4f;a a r " 2 n V! n It It I M.: * i* :.; .4 * * * 4 ... .'. .M. Z .1 ro .; 14"a& oo******0,a11 " a4040 t at a rý 1ýr ý10 a a a- a

0.

in~~ a nw

41 aa6 I 4t 0

*~~.. 0m .444 1S e1Q41'~ 0 1 . 44m 1 m o .. g . . . 4 e

31 ISO mr4,acp .. 0.4. n .a

.4 . "a**~ 4.. C24 C. * . l4~ ~ ". C 4 ...".a t Ct4 1nin4 t ai4%4 1inItC ~ %n4 nnnn4 t4

*aaa-g-min~ 0044.. in noota-Iman 0~064..1

a a 0 4a Vf a 4: aa a. af C4 J. m 6 a e~a a .

o ~~ ~~~ 41*4 4 n4 n 44. *4 *4t4C: ga m1eba 084.4' t4e-a .camo .a04 1144a~1 .4~~ 4D 1~~041~~~ o a-. a a a04l au a

me 41 0 aole re -a a-44*e &0 ino.1 3 m to-tvt4In;m:r: . 4.911114 41E4Il4 .444.. "4414

owa.I . -#" a

444444.444~~c~ 4.. . . 0l4 . . . . . .

2,.......... . . . . . . 4... . . . . . . . . . ..-

0 It nawlamie.N!m.041114 .401041: n n G:i It ilo! n* ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ C1 Sa"4.4 : 11mo4~ a -m*.11 la- .. 'Uaa- a

~~~~~t C :l c aaa aa ai lit 1--11 1--1 ci at a ac Itaa a ~11 1 1-1! na oos Rmaloe411 -.4am4aam.4a aUSO.TOOS09:320,ama0209

ft I

:: ft a r- ce4.O NMSn 09r-:%a

39 0 sn 1 .e a N 0 0 a* ~.0 * -10.0 A a.0 . ýI

101 .0 a C, C% "10 MO 'Do 110 C4 m 0~0

ei~v a4!4.

In- nUt-t ý % r n o C.t ý

24 -10m.e i **.'.l. 12 10.0.0PA.0g1 r

on n i s 4 s : P aa. 0t a NO 2, tmN oo0e i- I'M " Wl f.0 N.

.0Q.;.4I,0QeI0.m.0. : .4 c- 11 W c t 0.0.mfl.4a*Sa & nor# waof % l- o m

-In

0 t-t-t-1 % -a ~aca. .gw.a U4l.0.ggl.1.me~~e.a

0 a;.0

r- - r. Ma 0 P U . . . . .

0. o P -1.0 C c

It ~~a 9 ItO9 iI ... % t9, -' 10 0 0.0Iec4.0 a a*-c . . . . .. .Bi it t 1 *. . . . . . . . . . .

9 one N @NtO- a cI .a..,t. .. la ga cat0

-- --

it aa.. 0" C.4 a. a. ft m0 m eXIT

-43-.

O0 01VOOiei i.00)mft I! n Ct 000t 0)*t0i i in i t ii..0)**.C40*0))I~~e.e.0. or-= 0 " " 3 : I I, : : a)~t.

It2 t i l O)

4 0

00 t0 )

týeift t0nI io t!i ti ii00))0 n n0*~ ttn)t n*0 r-0*~~~a .0 0lt),.f

in Am :.0 "aas0 0~-.ft ) 0) *000)))0bli.b .ftm

ce . ft I .Ai d 0)* 0;.;-0- 0 t 40"ad-D 0 9, .A-. I ra- aG

0)0) P 992.42Ae.: 4

4 o)I-. -400 0) 0 ft0) A 0)0

O rooooooooooa 0oooo00oooo 0000004

...........................................................................................

CA

14O .

"C,ý!! 9 9ý C! 911!ý 991 1 t9C ! !l 1a

.2 02 98 13,f

It 4: li 0))00 t* 0 ItI! tI 0 !Itta I t 0 I!i i ri i i C: i

2pl :1223 a : so. ::0) 2 " "10)010 0)N0 C. )0Ap)

CIO .. .... ..... ... fSC0 rn 0)f0*0 20)0*100 to, 9 ; 11 P 0 U "0Il0f*)~0))0~ :.0 ~ ~ ~ ~ ~ ~ ~ ~ ~ An AA oa))))) 000I .- * 000000 0 0-tc - r- 1 I- 4

A cp o m ýe am ")

A ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 A0)aaC l .C waAAAa tA 0 " - Mq.) l-D* 0 0 0 S A O A V C % i1:% 0 Otn I i

U 0 ft0)0)ftflftf~e0)ftW*0*QI-00) 0. m0W OO0 CZ4 0) C ) aft. . . . . . . . . . . . .. . . . .ft 1 00 4 P-c O

OKU;Ut 1:P i n1 U iV I : - tC 9 v.t. t nnt t1

U ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - a *0@)0"ft* Rfo*'f )0@)0f* f0*4f

a 0)-ft- - - ft.*0C)ftt 0 0 -cmIOfO 9 aoS0m)aAo" 7

-44-

i tiM Cil C :1 - l!CItliýI !ýl ~r ticlno.JC-P aJp pa 00 r Cr -r 1r ~ ~ ~ ý ý ý w s

.0 r. r - % am . c1e~ CD .0 a~.. *. ~ - a a ~s ce .

od 0 ý C4 M 4ýK a M9 ý a Uff n r c o t c m o a Mr C. q - t * . 0 O p .$' a p. .

ID 1% r- e. r, rur a a2 o- o. C"(Po

.a,0 ~ -. 0 a: a a.a 0'. a. 0.ae aa ýa a ..

.4a a

r2 WOMO..('N .S. co o 0* - ft. ,-(W.4* S 0f .4

CP N , C r- 4 o c" '* a~a~r~.am~a.9 .NE~..4A~rd S @rIaia ~N.4at

9 9S~a Ict ftC Ct S! 9 -c 0C.(t *S n 19 9in-.r* n 1a.O-.r'.!9 P ýC!9 !9 !Ci ! C 99 !9 C

.4...r..r C.NN rlr.

-45-

Appendix A

TEMPERATURE DEPENDENCE OF RELATIVE ISOBARIC SLOPES

To include relative isobaric slopes in the vertically averaged

momentum equations, replace the terms A and A by the expandedax ' ayterms g(aW/ax + < i h) and g(a6/ay + < i y,p, > h) respectively.

Pritchard gives the following expression for the vertically averaged

isobaric slope: (19)

i 1/2 H H- = as (Al)II4 h p a-x 2ý a-s a-x

where o = vertically averaged density, and s = vertically averaged

salinity. In terms of the specific volume defined by a = i/p, this

becomes

i = (H/2) (as/ax) (-a1 " (A2)

1 ac)The quantity (- - ys is the coefficient of saline contraction andmay be obtained according to Eckart from the Tumlirz equation of

state given by( 2 0 )

(p + po) (a - a) =0 (A3)

where T is in *C and

= 1779.5 + 11.25T - .0745T - (3.80 + .01T)s

= .6980

0T2

P 5890 + 38T - .375 T+ 3s,

P total pressure (in atmospheres)

S= specific volume in ml/gm .

For H 100 m we can ignore p compared to po. For example, at 10

meters depth, p/p 0 is less than 10- Since the empirical formula

is accurate only to about 17, our approximation is well justified.

Evaluation of the saline contraction is now straightforward and

the result is:

-46-

3.8 + .01T + 3 (l/p0 ) (H/2) (as/ax) (A4)

x,h X + aoPo

A similar result applies for x replaced by y.

If data is given in terms of chlorinity, one must use the fol-

lowing equation relating salinity and chlorinity (both in parts per

thousand):(17)

s = .03 + 1.8050 Cl . (A5)

Chlorinity is defined as the chlorine equivalent in parts per thousand

of all precipitated halides. Equation (A5) does not apply to estuaries

whose constituents are altered by significant fresh water flow.

Let s' and Cl' denote the salinity and chlorinity, respectively,

esressed in g/liter sea water. Then

S =pS ,(A6)

Cl' = PCI

and

s' = .03p + 1.805 Cl'

.03 (A7)+ .03 + 1.805

Cl'ao0 + X/Po0

In terms of s', ix,h becomes

i (H/2) (Ds'/x -) (A8)1xh a as

We also obtain

1 Da -1 -I 1a as' = -(asv/aa) - as S 7 + (A9)

and

as'/ax = .03(Th/ax) + 1.805(¼C1'/,ix) (AlOa)

In (AlOa) the first term on the right hand side is roughly 105

times smaller than the second term, so we can write approximately

-47-

as'/ax = 1.805(aCl'/ax) (AlOb)

In terms of chlorinity, the isobaric slope is given by

ix,h = 1.805(H/2) {(aCl'/ax)/

1.805 C1,0T + . + (All)[185Xl + + 3.8 +A 01

0 o

where s appearing in the definitions of X and po is given by

s = .03 + 1.805 1.02V .03 + 1.770 Cl' (A12)

In (A12) we have used the value p = 1.02, which is correct within

1% for the range of temperature and salinity under consideration.

Errors of order 1% in s will produce errors of order only .001% in

X and P . Similar arguments lead us to drop the term .03(a0 + X/p)-

in (All).

Temperature gradients will also contribute to the relative iso-

baric slope. For example, (Al) is modified according to

i = (H/2) (lp as + p (A13)x,h (pas ax paT 3x) (A3

The coefficient of thermal expansion is not given as accurately by

our equdtion of state as is the saline contraction. However, this

is compensated by the smaller magnitude of the former effect.

From (A3) we find

1 ap _ 12 o_1 aT(•po)Jp ýT a ýT a

(/po~) (~' /aT) - (aW/aT)= X P0 P (A 14)

+ +ot Po

(A/p ) (38 - .750T) - (11.25 - .149T - .018 Cl')

A + aoPp

Finally, the relative isobaric slope as a function of chlorinity

'in g/liter), temperature (*C), the corresponding gradients, and

depth is given by

-48-

x = (H/2)[ 1ap aClI + 1 1P 2xT (A15)ix,h P ýCI' Dx P ýTax

= 1.805(H/2) 1 Cl/ax (A16)i. 805C1' + 38+.l

3.8 + .01T + (3X/Po0

+ (H/2) (aT/ax)

(A/p 0 )(38 - .750T) - (11.25 - .149T - .918 Cl')X + CeoPo

where p• p P and ao are defined in (A3) and the value of s used to

0evaluate A and Po is given by (A12)

Equation (A16) may be used for machine computation when large

salinity or temperature gradients make it desirable to employ local

values of the saline contraction or thermal expansion coefficient.

In most cases it will be better to use averaged values for the

coefficients over the entire field. Typical numerical results for

thermal and chlorinity coefficients are 2x1074 and 1.3xlO-3 respec-

tively.

-49-

Table A

Table of Z(T,C) where Z is the coefficient of chlorinitycontraction as a function of centigrade temperatureand chlorinity in grams per liter.

T=10 T=15 T=20 T=25 T=30Chlorinity in g/l

10.000 1.350-03 1.335-03 1.324-03 1.317-03 1.315-0310.250 1.349-03 1.334-03 1.323-03 1.317-03 1.314-0310.500 1.348-03 1.333-03 1.322-03 1.316-03 1.313-0310.750 1.347-03 1.332-03 1.321-03 1.315-03 1.312-0311.000 1.346-03 1.331-03 1.320-03 1.314-03 1.311-0311.250 1.345-03 1.330-03 1.319-03 1.313-03 1.310-0311.500 1.344-03 1.329-03 1.318-03 1.312-03 1.309-0311.750 1.343-03 1.328-03 1.317-03 1.311-03 1.308-0312.000 1.342-03 1.327-03 1.316-03 1.310-03 1.307-0312.250 1.341-03 1.326-03 1.315-03 1.309-03 1.306-0312.500 1.340-03 1.325-03 1.314-03 1.308-03 1.305-0312.750 1.339-03 1.324-03 1.313-03 1.307-03 1.305-0313.000 1.338-03 1.323-03 1.313-03 1.306-03 1.304-0313.250 1.337-03 1.322-03 1.312-03 1.305-03 1.303-0313.500 1.336-03 1.321-03 1.311-03 1.304-03 1.302-0313.750 1.335-03 1.320-03 1.310-03 1.303-03 1.301-0314.000 1.334-03 1.319-03 1.309-03 1.302-03 1.300-0314.250 1.333-03 1.318-03 1.308-03 1.301-03 1.299-0314.500 1.332-03 1.317-03 1.307-03 1.301-03 1.298-0314.750 1.331-03 1.316-03 1.306-03 1.300-03 1.297-0315.000 1.330-03 1.315-03 1.305-03 1.299-03 1.296-0315.250 1.329-03 1.314-03 1.304-03 1.298-03 1.295-0315.500 1.328-03 1.313-03 1.303-03 1.297-03 1.294-0315.750 1.327-03 1.312-03 1.302-03 1.296-03 1.294-0316.000 1.326-03 1.312-03 1.301-03 1.295-03 1.293-0316.250 1.325-03 1.311-03 1.300-03 1.2ý!4-03 1.292-0316.500 1.324-03 1.310-03 1.299-03 1.2S3-03 1.291-0316.750 4.323-03 1.309-03 1.298-03 1.292-03 1.290-0317.000 1.322-03 1.308-03 1.297-03 1.291-03 1.289-0317.250 1.321-03 1.307-03 1.297-03 1.290-C3 1.288-0317.500 1.320-03 1.306-03 1.296-03 1.289-03 1.287-0317.750 1.319-03 1.305-03 1.295-03 1.289-03 1.286-0318.000 1.318-03 1.304-03 1.294-03 1.288-03 1.285-0318.250 1.317-03 1.303-03 1.293-03 1.287-03 1.284-0318.500 1.316-03 1.302-03 1.292-03 1.286-03 1.284-0318.750 1.315-03 1.301-03 1.291-03 1.285-03 1.283-0319.000 1.315-03 1,300-03 1.290-03 1.284-03 1.282-0319.250 1.314-03 1.299-03 1.289-03 1.283-03 1.281-0319.500 1.313-03 1.298-03 1.288-C3 1.282-03 1.280-0319.750 1.312-03 1.297-03 1.287-03 1.281-03 1.279-0320.000 1.311-03 1.296-03 1.286-03 1.280-03 1.278-03

: .......... . P-- 'mmm n mmmmm m mm l l mmm llmmil~l l mmm mmmm A.

-50-

Appendix B

!IODIFIED DEWPOINT TEMPERATURE

Rather than regard T as a function of Q = absorbed radiation,eq

W, Ta and Td we can take advantage of the following procedure which

considerably simplifies interpolation in the tables. The air tem-

perature and dewpoint temperature enter the equation for Teq only in

the term (eeq - e + BT - BT a) where, as before, e = e(Td). Leta eq 1) a

us define a "modified dewpoint temperature, J, according to

e ea + B(T -Ta) = eq - e(J) + B(Teq - J) (BI)eeq a e qe

or,

e(J) = BJ = ea + BT (B2)a a

The modified dewpoint J may be found by Newton's method and tabulated

as a function of T and T The equilibrium temperature may thenad

be found as a function of 3, W and Q. In particular,

Teq (Ta' Td' W, Q) = Teq (J, J, W, Q) (B3)

so that we need only the first section (where V = 0 and T = R) of

Table 3.

J is tabulated in Table B as a function of T and V T - Tda a d

-51-

Table B

Table of modified dewpoint temperature J(TR) as a functionof air temperature T and V= (air temperature - dewpointtemperature], all in degrees Fahrenheit.

V=0 V=5 V=10 V=15 V=20 V=25 V=30 V=40 V=50 V=641

0. .0 -. 8 -1.4 -1.9 -2.3 -2.6 -2.8 -3.2 -3.4 -3.55. 5.1 4.2 3.4 2.8 2.3 1.9 1.6 1.2 .9 .810. 10.1 9.0 8.1 7.4 6.8 6.3 5.9 5.4 5.1 4.915. 15.1 13.9 12.8 11.9 11.2 10.7 10.2 9.5 9.1 8.920. 20.1 18.7 17.5 16.5 15.6 14.9 14.4 13.6 13.1 12.825. 25.1 23.5 22.1 20.9 20.0 19.2 18.5 17.5 16.9 16.6

30. 30.2 28.3 26.7 25.4 24.2 23,3 2?.6 21.4 20.7 20.235. 35.2 33.1 31.3 29.8 28.5 27.4 26.5 25.2 24.3 23.840. 40.2 37.9 35.9 34.2 32.7 31.5 30.5 28.9 27.9 27.245. 45.2 42.7 40.5 38.6 37.0 35.6 34.4 32.6 31.4 30.650. 50.0 47.5 45.1 43.0 41.2 39.6 38.3 36.2 34.8 33.855. 55.0 52.3 49.7 47.4 45.4 43.6 42.1 39.8 38.1 37.060. 60.0 56.9 54.3 51.8 49.6 47.7 46.0 u3.3 41.4 40.1b5. 65.0 61.7 58.8 56.3 53.9 51.7 49.9 46.9 44.7 43.270. 70.0 66.6 63.4 60.6 58.1 55.8 53.8 50.4 48.0 46.275. 75.0 71.4 68.1 65.0 62.3 59.9 57.7 54.0 51.3 49.380. 80.0 76.3 72.8 69.6 66.6 63.9 61.7 57.6 54.5 52.385. 85.0 81.1 77.5 74.1 71.0 68.1 65.S 61.3 57.9 55.390. 90.0 86.0 82.3 78,7 75.4 72.4 69.6 65.0 61.2 58.395. 95.0 90.9 87.0 83.4 79.9 76.7 73.7 68.8 64.6 61.4100. 100.0 95.8 91.8 88.0 84.4 81.1 77.9 72.4 68.1 64.5105. 105.0 100.7 96.6 92.7 89.0 85.5 82.2 76.3 71.6 67.7110. 110.0 105.7 101.5 97.4 93.6 89.9 86.5 80.3 75.1 71.0

-52-

REFERENCES

1. Leendertse, J. J., "Aspects of a Computational Model for Long-Period Water-Wave Propagation," The Rand Corporation, RM-5294-PR,

1967.

2. Leendertse, J. J., "A Water-Quality Simulation Model for Well-Mixed Estuaries and Coastal Seas: Volume I, Principles ofComputation," The Rand Corporation, RM-6230-RC, 1970.

3. Leendertse, J. J., and E. C. Gritton, "A Water-Quality SimulationModel for Well-Mixed Estuaries and Coastal Seas: Volume II,Computational Procedures," The Rand Corporation, R-708-NYC, 1971.

4. Edinger, J. E., J. C. Geyer, and D. W. Duttweiler, "The Responseof Water Temperatures to Meteorological Conditions," WaterResources Research, 4, No. 5 (1968).

5. Edinger, J. E., and G. E. Harbeck, Jr., "The Cooling of RiversideThermal-Power Plants, Discussion," Engineering Aspects of Thermal

Pollution, Vanderbilt University Press, 1969.

6. Edinger, J. E., "Estuarine Temperature Distributions," EstuarineModeling: An Assessment, Water Quality Office EnvironmentalProtection Agency, 1971.

7. Edinger, J. E., and J. C. Geyer, "Analyzing Steam Electric PowerPlant Discharges," Proc. ASCE, 94, SA4 (1968).

8. Callaway, J. R., K. V. Byram and G. R. Ditsworth, "MathematicalModel of the Columbia River from the Pacific Ocean to BonnevilleDam," U.S. Dept. of Interior, FWPCA, Pacific NorLhwest Water Lab.,1969.

9. Forsythe, W. E., Smithsonian Physical Tables, SmithsonianInstitute, Washington, D. C., 1954.

10. The Nautical Almanac, U.S. Government Printing Office, Washington,D. C., 1969.

11. Raphael, J. M., "Prediction of Tempcrature in Rivers andReservoirs," Proc. ASCE, 88, P04 (1962).

12. Anderson, E. R., et. al., Water-Loss Investigations: Lake HefnerStudies, Technical Report, USGS Professional Paper 269, U.S.Government Printing Office, Washington, D. C., 1954.

-53-

13. Harbeck, G. E., "Field Technique for Measuring Reservoir Evapo-ration, Hass Transfer Theory," USGS Professional Paper 272-E,U.S. Government Printing Office, Washington, D. C., 1962.

14. Parker, F. L., and P. A. Krenkel, "Thermal Pollution: Statusof the Art," Report No. 3, Department of Environmental andWater Resources Engineering, Vanderbilt University, 1969.

15. Sutton, 0. G., "The Application to Micrometeorology of theTheory of Turbulent Flow over Rough Surfaces," Royal Meteor-ological Society Quarterly Journal, 75, No. 326 (1949).

16. Sverdrup, H. U., "Evaporation from the Oceans," Compendium ofMeteorology.

17. Neumann, G., and W. Pierson, Jr., Principles of PhysicalOceanography, Prentice-Hall, 1966.

18. Jacobs, W. C., "Large Scale Aspects of Energy Transformationsover the Ocean," Compendium of Meteorology, American Meteoro-logical Society, 1951.

19. Pritchard, D. W., "Two-dimensional Models," Estuarine Modeling:An Assessment, Water Quality Office Environmental ProtectionAgency, 1971.

20. Fofonoff, N. P., "Physical Properties of Sea Water," The Sea,Vol. 1, ed. M. N. Hill, Interscience, 1962.

21. Brady, D. K., W. L. Graves, and J. C. Geyer, "Surface HeatExchange at Power Plant Cooling Lakes," Edison ElectricInstitute, 1969.

II