56000
#000
#000
“30600

20000

~ 20000

34000
~ 34000

10.000

~ BeGOO-

_. 99000 |

5-000
“100000
44000
22000
94000
5 e000
26000
46000
10-000
10+00u
10-000
_ 100000
5.000
8000
74000

#1000

54000

96000.

690000 810000
649000 93000
612000 974000

639000 100.000

684000 100+000°

690000 1002000

839000 67-000
880000 616000
740000 820000
740000 820000
710000 90+000
602000 674000
560000 75 +000
660000 756000

600000 62000
57

40000
_ 36000
3000

~ &C00

774000 742000
856000 644000
870000 57000

_ 814000 796000
764000 824000
720000 934000
738000 874000.
75000 854000
844000 674000

908G0C 564000
910000 50-000

7396000 694000

74000 824000.
700000 934000

8.000

11.000
90000

5000

5-000
32000 >

42000

D°0CO
94000
9°000

10+C00

8eCO00

4.000 |

36000
46000
56000.
5sC00
#000

12+6co00
3*000
7000
74000
66000
76000
94000

769500
76000
75°700
75*500

759500

76+000
77+000
77000
77+000
76+500

766200.

76°000
76#200

77000 _

77*500
77°700

779200.

76*700
77000
77+000
77*500
77°500
78+500
79*000
79*000
710200
709700
70200
70+500



2b

12572000
255+090
2334000
2694000
2514000
242900
263000
254.000
265-0900
2562000
267+900
268+000
2539+090
272+030
2714900
2726900
2734990
274000

2754900
275+900
277.000
278-000
279-000
2802000
2314900
2320000
2334000
234.000

205

35

80
210

C1909) sec

TA(1090) eTAIR
PH1(1900) sRH
w(1000)<wSP
TN(1G90)sTWA

Da 205 ICAS=4sNCAS
PRUICAS) 306
C3NT]NUF

DS 219 Tel NTA
DCECTA(T) CC
OTARTATA(I)=TAIR
DPHT=PHITA( I) «RY
DYEWTA (Ty =WSP
DTWeTNTACT) =TWA
STT3L1(1)=TK(1000)
STTSL1(2)5Tx(1000)
XTBLI(1)506
XTBL4(2)225009%
NTEMP1=?.

INVS=1090

CALL COMP (ENUMs EN)
D3 40 ICAS=1,NCAS
Tf (ABS(Y4 (42) eDFLT(ICAS)) 85285480
CANTINUE

PR(TCAS) sPR(TCAS) $40
C3NTINUE

CONTINUE

FNTAaNTA


tA

Table 2. PROBABILITIES OF MEETING GIVEN ERROR REQUIREMENTS BY AN OBSERVER AT
THE VERNON SITE AVERAGING TWO DAYS AND PREDICTING ONE DAY AHEAD AT A
DISTANCE OF 25,000 FEET (7620 METERS) DOWNSTREAM

Error Requirement Or (°c)

19F (0.56°C) 2°F (1,119) 38°F (1.67%C) OF (2.22°C)
June 21, 1973 0.469 0.694 0.898 0.980
September 27, 1973 0.533 0.867 0.933 0.967
October 30, 1973 0.5 0.833 0.833 0.917
November 15, 1973 0.522 0.957 1.0 1.0

Table 3. PROBABILITIES OF MEETING GIVEN ERROR REQUIREMENTS BY AN OBSERVER AT
STATION 7 ONE DAY AHEAD AT THE VERNON SITE(SAME POSITION AS STATED
FOR TABLE 2)

Error Requirement °F (°c)

19°F (0.56°C) 22°F (1.11%C) 39°F (1.679C) 4° (2222°C)

June 21, 1973 0.520 0.800 0.880 0.960

October 30, 1973 0.846 0.923 0.923 0.923



x4
In = [i dz = y(xth) - y(x-h),
x*h

2n3 2hn5 ,(IV) __
2 ne(x) + Be ty + Be (x) t---10.

(D-10)

From Taylor's theorem and equation (D-6)
h2 +,
y(xth) = y(x) + h£(x) + 37 £(%) (D-11)

and the error is proportional to the maximum value of the
second derivative. With these fundamentals set down the
error for the equation

ar _ of _
ax wApe (D-12)

can be readily found. From the general analysis the error
is bounded by the expression

. xk_ dQ
ITapproximate Texact! < wApe lax! 4x .
max (D-13)

An expression similar to (D-13) is easily developed for
.baunding the station length

|STa110wablel
Ax a a D-14
Xnax * 192 ‘ )
uApc ax
max

From experience, a convergent Ax can easily be obtained
for this differential equation since the heat fluxes are
generally very small and have small derivaties.


8hL

1542.050
15490009

15370990
1553-0900
15596050
15624900
1561-900
15624300
1563-6000
5 000
1543-990
5560300

1568-000
1563-000
1579+990
1571.50
15724000
4573900
15742009
15754000
15754900

40
30

PdIS=PHIS + PHICINT)
3eNS + wCINT)

CANTINGE

CONTINUE

C3= CS#FNDA

TAS= TAS*FNDA

P41S=PHIS#FNDA

WSF WS*¥EXDA

TAS=eT7 (120)

INT? = F9ILOWING DAY

INT2= 1?0+NDA

C1Ss06

Talgede

pPdliSs0-

wiSsOe

Tulssde

D3 30 12198

INTRs THE ACTUAL PASITI8N IN THF DAY
INT32INTA #5 <R41

C1S=C,S+c CINT3)

TAlSeTALS+TACINTS)

PHI1S#P411S+PHI(INT3)

WiS=WiS+wCINT3)

Tw1S2TW1S+TWCINT3)

TALSSTAIS#01?5
PHI 1S=PH11S#6125


RESEARCH REPORTING SERIES

Research reports of the Office of Research and Development,

U.S. Environmental Protection Agency, have been grouped into
five series. These five broad categories were established to
facilitate further development and application of environmental
technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in
related fields. The five series are:

Environmental Health Effects Research
Environmental Protection Technology
Ecological Research

Environmental Monitoring
Socioeconomic Environmental Studies

APWN—

This report has been assigned to the ECOLOGICAL RESEARCH STUDIES
series. This series describes research on the effects of pollution

on humans, plant and animal:species, and materials. Problems are
assessed for their long- and short-term influences. Investigations
include formation, transport, and pathway studies to determine the
fate of pollutants and their effects. This work provides the technical
basis for setting standards to minimize undesirable changes in living
organisms in the aquatic, terrestrial and atmospheric environments.

EPA REVIEW NOTICE

This report has been reviewed by the Office of Research and
Development, EPA, and approved for publication. Approval does
not signify that the contents necessarily reflect the views and
policies of the Environmental Protection Agency, nor does mention
of trade names or commercial products constitute endorsement or
recommendation for use.


BLL

14346500
1446990
145-3ou
146+9509
1474900
1434000
1494590
159-000
431-900
+990
#9090
+900
155+090
1560900
1576340
060
159-900
169+990
1614990
1526000
1634900
1644900
165-000
166+000
1672090
1686900
169+000
1794000

40
390

cccce
ccccc
ccccec
ecccc
ccecc

03 40 ICAS=1sNC4S
PRP( IDAs TCAS) 20+
CANTINUE

CaNTINUE

RETURN

END

SUBROUTINE PR3B

C3MM8N YP1(5)2Y1(5)2KPR1.XEND1,OXPRLAXLeDX1sNELS INTL sy INTCAs INI 4 184
1sEL1sARF AL Ut

CAMMY ERRAR1DXMAX1, TEMPLyHFGTS1(10)sC9NS11

CAMMON RHGtsewteWMDis dL TATRLaCLO 1s RHIZALBART A TNS1aTALs THSALZEWLe
1€A1,H9S1,HS1

COMMBN HSR1,HIBLyERR1ZEVAP Le HET HOt» HLYMNIsPTELL(10)2TTBLI(10)s
1w4S1,C8NST1

CSMMON Cl IF1sHFG1,H3C1sCLO114Cl 021, CLO312DALP120ALPR1,DCLO1sDRH1,
LOTAIRISDTALZ OWL »DAMS1 2» OWMD 1) DEA1s0EN120LYMN120BTUSIsNTEMP Lo
2STTALLC1N)sxT3L1 (10)

CSMMBN C¢1000)sTAC1009)sPHI (1000) »W(1000)sTW( 1000)

C3MM4EN NCASs NDas DELT(5O)» 1349150)

CAMMBN NI Na INe T9UsDTs NTA, IDA

CAMMEN. CTA(50)s TATA(50)sPHITA(50) sWTA(50) 4 THTA(50)

COMMBN UAsELsCCeTAIRs TWAs RH) WSP

COMMBN INT2s TNT32DC20TAsDPHI sDWsDTWs INVSs IFLAG


7.000
e000
0000

+000
“600
10,600
10,000
10-c00
10.000
10000
~~~"104000
10-600
42000
“24000
5.000
8-000

~ 8.000
102600
94000
10.000
10.000
10.000
8000
5000
8000
100000
10-000

10-000

50e000
610000
620006
52¢000

43,000 —

54.000

47,000

" 45000

440000
442000
440000
400000

38¢000

"310000

30*000
272000
39600
420000
430000
370000
410000
420000
400000
362000
45000
496000
439000

470000

96000
674000
67000
934000

‘100,000

96,000
36,000

832000

802000
74000

76.000
674000
~ 922000
96+000
96000
~ 79.000
632000
634000
826000
734000
714000

626000 ©

794000
682000
594000
596000

74000
66

710000

36000

"60000

5°C00
3-000

9.000

“00060 —

11.000
13-000

72000
3-000

~"""34000°

30000
5000
42000 —
8000
52000
3*CCGO
66C00
92900
126000
3-000
11.000
126000
13-000
11-000

"3,000
_..10.000

~9+000

512000
"510500
529000
52°000
~"$24000
524000
514700
"540500
519700
524000

$1+700
51500

51000
509700

“524000

“519000 7

“"§00500 —

500700
50+700
50*500
50+200
50*000
“50#000
50*000
50*000
50+000
50000

50°000



DA - =
Z (T(x + Ax) = TO] = [Ong - Spee ~ Spack ~ evap

* Sconv + Wise] 4x '

= at
Tix + dx) = TOx) + Gy [naa ~ Sree ~ Back ~ Snvap

* Gconv + Misc] 4x + » (D-3)

It is possible to then incorporate heat fluxes not covered
by those at the air/water interface in the Qyise term,
simplifying the coding and the model considerably. ‘Two
aspects of this concept should be emphasized: 1) all models
are similar to this model and are within a small error that
is a function of the mesh size, and 2) the convergence of
the procedure can be shown from previous literature.

Suppose that an alternate model is used in which the fluxes
are redistributed, i.e., the total flux may be the same but
the model assumptions causes a difference in the flux at
any position x. Then this difference can be represented
as a quantity proportional to the station length, Ax.

For example:

Qpag(* + Ax) = Qpag(x) + O(Ax) . (D-4)

But the introduction of this expression into equation (D-3)
will introduce a term of order Ax? (0(Ax2)), which can be
made appropriately small by a suitable choice of Ax. Thus,
all correctly formulated models will be a measurable error
away from the general model described by equation (1).

The question of numerical convergence of the solution is
discussed in general by Salvadori and Baron [1961], p. 116.
Much of the information provided by them can be directly
applied to this problem.

Suppose a general problemitakes the: form


APPENDIX C
PROGRAM LISTINGS

This appendix presents complete program listings for the
STREAM, SENSIT, and MONT codes.

77


(=; |-—I— T
Compare Dally Average eS WS airport Data and

With Vernon Average Stream Temperatures

Average Vernon
Data

Average Over A Day

Compare With Vernon Average!

J

Compute 6T Downstream

Numerator of Probability

COMPUTE AND
PRINT PROBABILITIES

Figure 4, Model using site and station 7 temperature data


Leb

473-090
4742990
4756900
4764900
477-500
473-900
473+900
4394990
4310009
4326900
4834200
4842900
485900
4354990
4876990
4R8<000
4896900
499-000
4916020
4924900
4931000
4246900
4950059
4964000
437+900
498+900
4994000
529900
5914000.

ceccc
eccce
cceccc

SUBROUTINE HUNDIL

STREAM THERMAL MBCEL Loaded SENSITIVITY ANALYSIS Slated
caMMeN YP(5)4¥(5.)4XPR» XENDs OXPRaXsDXaNEy INTs INTCe INs 18s EL s AREAS U
COMMON ERRGRsOXMAXs TEMPS HFGTB(10),CANSL

COMMBNR HAs Ws WMD» HDs Ta IRs CLD# RHs aLARd Tigy Tas THSASEWPE AD HOSSHS
CaMM6N ASRsHABsFBRsEVAPs HES 4CoHLYMN» PTBL (10)4 TTBL (10) s MSs CONST
CaMMAN CI If + HFG» HECsCLO1sCL92s6L0 9. DALP, DALPR,DCLD,DRHsDTAIRe
ADTAs DAs DWMSs WMO DE Ay DEW, OLYMN, DBTUSs NTEMPs STTSL (10) sXTBL (10)
Cule = 1466-6

CaLL INPUT

TFCINTC - 4) 49105+105

XPR 2 OXPR

CONTINUE

03 100 UF t,INT

CALL DIFFO

G3 TE. (1010.20). INTC

CALL EULFRIYPs Ys OXsNEs JE aX)

G3 Te 100

CALL RK4(YPsyeDXaNEs JES xX)

C3NTINUE

IF(X = XPR+ DX/4e3 54250250

C3NTINGE

XPR = XPR 4 DXPR

TF(X © XFND) 523002300

CONTINUE

JUMP = 94


eel

6446000
6450000
649000
6472000
643-000
6432000
6599900

6366500
6574000
6386500,
6526900
6539990

80
990

ccecce
ceccc
ccecce
cccce

6514900 C

662-990
6630590

6564000
6§7+000
658400)

6694009
672-690
6714000
6724000

WRITE (18430) DLYMNs DBTUS

FORMAT(' OLYMN #t€12e5" = DBTUS w1E22653/)
WRITE( 18,90)

FARMAT(1XsEOCTHNDA//)

RETURN

END

SUBRSUT INE: DIFEA

COMMON NDT+YP C4) OTs DUM (4) 2 XPR» XENDs DXPRas Xs OXsNEo
LINTSINTCs Is 18¢EL AREA, Us ERROR» OXMAXe TEMP, HFGT8(10) sCBNS1
C3MMBNRHG 2 Wa WMD 9 HDs TATRs Co R49 Al BARD TWSs TAs THSAs EWS EAS HBSS HS
COMMEN SR) H8Bs ESRs EVAP SHE SNCs HLYMNs PTEL(10)s TTBL (10) » WMSs CONST
COMMBN Cl IFsHFGsHAC + CLO1sCLO2sCLDA» DALPs DALPRs DCLDsDRHs DTAIRy
LDTAs OW, DWMS s DWMDs CEAs DE Ws OLYMN, DBTUS+NTEMPsSTTBL (10) sXTBL(1C)
THERMAL EXCHANGE WLTH ENVIRSNMENT.

TEMP © Sr (XTRLsSTTALs XaNTEMP)

TAS (TEMP O2P 0) 450/90

TWSA 2 TwS + 2730

DTWS = DU#5+/9e

FTA = SICTT&L+PTBLsTEMP2 40)

TIN = TEMP. + OT

FTB = SI(TTBUsPTSLsTTWs 10)

EW 2 (1D1Z6/1he 7) ¥FTA

DEW 2 (101g*/1y97)*#(FTB = FTA)

EFFECTIVE BACK 2ADTATIONe

H9Bs (14638 009*TWS=»046#RH) 69472


8e600 450000 684000 11-000 50*000

y 40.000 490000 596000. +12@c00 + S50e000.
bo 106600 490000 $96000 130000. _  500U0
10.090 470000 746000 11.000 50+000
7 40,000 ~ 47,000 ~ 93,000 8,000 50.0000 ~~
. 10600 394000 934000 30000 ... 499000
10.000 480600 80+000 5.000 49+000
“=== “doyco0 #60000 B38G00 BB e0G0. 8090
10-000 47000 932000 66000 49000
10.000 490000 934000 9-000 49200
~~ 10-000 ~~ 509000 694000 -128000~=S«—«S08000—
_ 10-000 49%000 634000 146000 50°000
10-000 494000 644000 20+000 499200
"2100600 490000 564060 ~~ 30000~”~—OHS8000002 OS
_ 6+000 499000 500000 146000 489200
4000 480000 484000 164000 48+200
es 50000 ~ 498000 "484000 86000 488500" ~
8+C00 540000 434000 94000 48700
10-000 520000 45000 96000 48700
10-000 S0e0cc 50000 ~ e000 486500 _
10+000 45000 74.000 beC00 489500
10+000 43000 76000 6000 484700
3-000 42+000 710000 44400 48+700 |
34000 420000 60+000 14.000 49000
34000 440000 474000 15+000 49+000
7 900 446000 404000 — 17-000 “496200
4 9690 42000 414000 126000 49°000
4.600 416000 404000 156000 49000

72


X, DX, DXPR, XEND: In 6E10.0 FORMAT (See STREAM for
description).

ERROR, DXMAX: In 6E10.0 FORMAT (See STREAM for description).

DT, U, AREA, EL: In 6E10.0 FORMAT. The initial temperature
error DT is read in at this point (other variables are
described in STREAM).

RHO, WMPH, TAIR, CLD, RH, ALBAR: In 6E10.0 FORMAT (See
STREAM for variable description).

PTBL(I), TTBL(I), HFGTB(I): In 6E10.0 FORMAT (See STREAM
for variable description).

DALP, DCLD, DRH, DTAIR, DWMPH: In 6£10.0 FORMAT. These
variables are the errors in the respective values of
the sun angle in degrees, the cloud cover in tenths,
the relative humidity in percent, the air temperature
in OF, and the wind velocity in MPH.

NTEMP: In 6110 FORMAT. This is the number of temperatures
given downstream.

STTBL(I), XTBL(I): In 6E10.0 FORMAT. These tables are,
respectively, the water temperature and the position
downstream in NTEMP sets.

The code then computes the error in the stream temperature
downstream from a given point. The output proceeds at each
print position with the position xX itself, the integration
step DX , the stream temperature TEMP, the error in the
stream temperature DT, and the derivative of the error in
stream temperature DDT .

Following this some auxilliary variables are also printed
out: EW, DEW, EWAP, HE, HFG, HYB, HC, HLYMN, and HD, which
are important for diagnostics, not for the solution itself,

MONT

The digital computer code MONT performs all the necessary
calculations to obtain the probabilities for meeting a given
temperature requirement. The code takes prepared data and
makes a series of trials, forming data differences, and
uses the sensitivity analysis code, SENSIT, to obtain the
stream temperature difference at a prescribed distance
downstream (in this case 25,000 feet [7620 meters]). The
code makes three studies:

30


LOL

189+000 4 BTU/SEC@FT21/)

181*900 79 FQORMAT(! OT ateta0st ODT stFiaeSa/)

182+000 WRITE(T8,80) DLYMN, OBTUS

1832900 890 FORMAT(! DLYMN at&taeSt DBTUS stE12.5a/)

184+#090 WRITE(18,90)

185.000 90 FORMAT (Axs60( tat ye//)

186+900 RETURN

187-000 END

183+900 SUBROUTINE DIFEQ

1894900 C3MMBN  QDT.0Ts XPRa XFNDs DXPRs Xs DXs

193+900 LINTLINTS, INs 18sELs AREAs Us ERROR» DXMAXs TEMPsHFGTB( 10) sCONSL
1912000 COMMBNRHA s WMD »HDs TAL Rs CaRHs ALBARs TWSa TAs EWs PAD HOSSHS
1924000 COMMBN HSRsHABsEBReEVAP SHE sHCaHLYMNs PTBL (10)4TT8L (10) sWMSsCONST
193000 COMMBN HEGsHECsCLO1sCLO2sCLO3s DAL Ps DALPRs DCLDsDRHsDTAIRs .
1944090 ADTAs Das DWMS 2 DWM s DEAs DEWs DLYMNs DBTUSANTEMPs STTBL (10) ¢XTBL (10)
1952000 C THERMAL FXCHANGF WITH ENVIRGNMFNTs

196+000 TEMP ® g7(XTRLagTTBLs XsNTEMP)

1974900 TWSs (TEMP#320) 450/90

1986000 OTWS 5 DT*5e0/9e

199°090 FTA = ST(TTBLsPTBLs TEMPs 10)

200900 TTW = TEMP + DT

2014000 FTB = SI¢TTBI sPTBLsTTWs10)

2024000 EW = (10130/1407)4F TA

203000 DEW = (10136/14e7)#(FTB = FTA)

2044000 C EFFECTIVE BACK RADIATION.

205-000 HOBs (14.38~609"TWSee046%RH) /69077

206+0900 ESR s HSA*CLOL

2072000 C EVAPBRATION HEAT TRANSFERe


90000 460000 716000 26000 = 52000

_ __.. 106000 450000 = 800000 = Se 000____ 522000 cence nen
u 10+000 496600 66000 66000 52200
Neos 100600 = 500000 SS 59.000 &.coo = 26500
oe 100000 510000 826000 = 60000. = 528700, ee
10000 486000 634000 4.000 52*500
10.000 460000 58000 3+C00 52500 ~
ee 102090 430000 764000. ___ 3000. 52200. a
10-000 420000 76000 2-000 52000
~ 106690 ~ 42000 89.000 34000 522000 oo
oe 10.000. __ 454000 _..770000..__ 60000, 520000,
10+000 474000 664000 7+000 52+000
~ “400000 = 456000 80.000 4000 520200 ”
een 100000 430000 892000 30000 = S2e200,
10000 430000 93.000 34000 52+000
——— 10600 436000 964000 ~~ 4eG00 ° ~=— $1" 700 ° °
109000 430000_ -93e000. = 50000, = 518500
10+000 456000 934000 3+000 51+500
~~ ~ 84000 53-000 62.000 100000  51#700 7
eee, 99000 556000 91.000 126000 520000 |
94000 560000 b1.C00 41.000 52+500
“e000 534000 97%000 66000 520200 7
ae ee 2000 _ 450000 654000 _. 49000 529000)
«000 372000 89.000 #000 51500
* 4000 32000 964000 +000 519200 7
. #000 390000 = 932000 000 10000
+0090 54.000 534000 114000 51+000
000 584000 446000 146000 549700

62000 62000 349090 102000 53000


ras)

#000 610000 78000 60000 §9¢000

46000 612000 78+000 7*000 58*700
VAs = 7718+000R WIOTHa 400s000CLD CavER® 2sO00TEMP AIR»
TEMP WATERS 60.0COREL HUM# 72eO000WIND SPEED= 86700

NUMBER UF CASESs 4
ALLOWED DTS
#100CGOVE O1
e200CCI0E CL
*300C0G0VUE O1
*4000000E O01
NUMBER GF DAYS AVERAGED® 2
CAYS AVERAGED
1
2
Prey. TTBL1 HFGTB4
HEM AH A He a ae EE AEE a8 A aE HEHE 0 SH ae EAE eae SHEE aE a HE a a HE HE a a a0 aH aH
*885E"01 #320E v2 +108E oF
e122E OC #40CE 02 +107E 04
©178E OS «500E C2 +107E 04
*256E 00 #600E 02 *106E 04
©363E CC +*700E 02 +105E 04
*507E OC +800E 02 ¢105€ 04
*698E GO *30CE 02 «104E 04
*949E 00 e100E 03 »104E OF
e127E O14 *110E 03 +103E OF
*169E 01 ¢120E 03 ¢103E O04

DAYS SINCE MARCH 21 75 eO000BETAS 452000

62.000


Lhe

1529+930
12090
cao
15232000
15746000
15?5+0G0
1526-900
18274090
1528-000

15286010

1528+920
1528+930
1532+900
1532
153
15344200
1535-000
1536-060
15374990
1538+090
1539+990
1549+000
15414000
15424090
15434000
15444000
15430990
15460030
1547.000

c

3) CONTINUE
NDAs NUMBER BF DAYS IN THE CASE
NDAst

NCLSNLN/R

FNDASNDASB

FNDAs1+/FNDA

REAQCING 1020)(T7( 1) eT m1 sNCL)

1020 FSRMAT(4F10+3)

D9 5 Te1,NC1
WRITE (TS, 2001) T7017)

2004 FSRMAT(/,2XsF 4497)

“5 C3NTINUE
IF(NDA*Nt1) 102999.999

10 CONTINUE

NOIFSNC1=NDA
ENUM206

DENsO+

DEND 200

D3 20 Taas1.NOIF
C5206

TAS20+

PHISs0.

W3206

TAS 30°

DS 30 1etsNDA
09.49 J2148
INTS8*(1204T #1) eUa8
Csecs + ClINT)
TASs¥AS_¢ TACINT)


The code then computes the errors in the stream temperature
downstream from a given point considering the days to be
averaged as requested. Three sets of input errors are
examined:

1. The airport meteorological data and stream temperature
against itself (simulating values at a site).

2. The time-averaged meteorological and stream tempera-
ture data against the site data.

3. The airport meteorological data and the remote site
water temperature data against the site data
water temperatures(simulating predictions from a
distance).

These are compared to the requirements on temperature for
each case. The number of cases for which the requirements
are satisfied are recorded and printed out in the form of
probability values.

33


peo

102000

106600 |

102000

“96000

82000
26000
+000
#600
#000
+000

#000
~ 46000

«000

36000
54000

#000

«000
«000

+000.

#000
40G00
34000
72000
8.000
4.000
56000
5-000
40000

2000

* 460000

65

964000 10*c00 534200

446000_ 1000000. _____76000.____. 53000
446006 1006000 66000 53-000
416000 1000000 ‘7eco0 =: 52#700
390000 894000____ e000 S500
430600 716000 96000 52+200
476000 560690 4.000 52+700
_494000 500000. #0000 «5 3e200
400000 864000 36cco 53+000
376000 894000 36000 52+700
340000 920000. Se000__._ 52+500_
33006 964000 42000 52+200
320000 1004000 3+000 526000
470000 _ 934000... 10600. 52*000
564000 644000 6+C00 520000
586000 60+000 7-000 $2+500
45.000 93.000  _ 56000 ___ 529500
446000 864000 56C00 52500
400000 966000 ~ 3000 ~~ S2e0N0
384000 1004000 30000 | 529000
35009 100.00 56000 519500
560000 724000 11-000 51*700
620000 58000 10.000 Ss S2e000
632006 54.000 6°C00 529500
55+000 694000 +600 526500
50000 864000 _3+Ggo __. 52200
470000 936000 36000 529000
448600 964000 36000 519500
390000 96 +000 36000 51200


z20L

208*000
299000
210000
211-000 C
2122000
2132000 C
2149900
215000 C
216+000
217+900
218+000
219000
229900
2714000
2222900
2234900
2742000
225000
226000
2774900
228+000
2299900
2302000
2312000
232000 CCCCC
233-000 CECCc
234000 CCCCC
2350000 CCCCC

1
1

EVAP = 625% (EWeEA)#(1e + ©009R« WMD)

HFG = ST(TTBI sHFGTBsTEMPs 10)

HE = EVAP*HFG*CANS1
CeaNvECT1aN HEAT TRANSFER
HC = 39e4( 026 + eO77#WKMS)#(TWS = TA )/14400
TSTAL HEAT TRANSFER TO WATERs 1.Y/MIN

HLYMN ® HS «
TSTAL HFAT TRANSFER,

HO
FL
F2
F3
Fa
FS
F6
F7

HLYMN # 00514
CLO *OCl D¥HeS
CLOR*HEC*#DALPR
©0765 *DCLD*HAR

(ERR + HSR + HC + HE)
BTU/SEC*F TA

~CLNL#(#eO9*#DTNS = 6046*DRH) /69072

@(F1+FF2)436¢/AL BAR
AS#3e¢#DALP/ALRAR##A
(Ce B5HHEG/144000) H(i te

s0098*OWMD*#(EW = EAD)
FS = =(390/1440+) 40 (626 + ©O77%WMS)*#(DTWS = DTA) +

0O77*DWMS#(THS = TAY)
DLYMN = FL + F2 + F3 + F4 + FS + F6-+ F7 + FB
DBTUS 2 DLYMN%s0614
ODT = CANST*NBTUS
RETURN

END

+ 60098#WMD)*#(DEW = DEA) «


SZL

4166000
4172900
418900
4196000
429-000
4215000
422.090
4230000
4240090
425.090
4260090
4274900
4236990
429-900
430000
4314000
432-000
433-000
434.900
4352000
43$+900
4376900
434-900
4339-900
4494900
4410900
4424900
4430960
444.900

ccccc
ceccc
cecccc
eccec

ccccc
ccccc

ERRAR1 Bent
DXMAX1#25000~
TEMP 18 TA CINVS)

W121 CINVS ) #6080«/3600¢
WMS1 aHl1 #6 304R

WMO1 21 ¥3600+#246/52806
WMPH1=NMNI/due
TAIRLETACINVS)
CLOLEC(INVS)
RdlaPHT(1Nvs)

ALBARL=AI NAP(OM@1,RETA)
ALBAR1SAL BAR1 #570296
TWS1 eS eet TEMP1=3201/98
TAL=S o#(TAIR1 0320) /90
DalPis5,

DALAR1L NALP1/57 +296
OcLo1=oC

ORH1 =P

DOTAIR1=NTA

DW1 20W¥6806 736008
OWMS1=D41 #63048
OWMO1=DA1 #36000 ¥246/52806.

CALL HUNNL


1-000

_, 24000

42000

“66000

___ 24000__

8+G00
7000
90000
7000G
20000

86000. |

Be000

66000 ©
__.100000 _

14000
5.000
_ 34000
14000
94000
_ 96000
10-000
10+000
10+000
10-000
10+000
_104000
10.000
106000
10-000

642000

660000

630000
65000

500000 _

50+00C
55+000
58*C00
620000
640000

_ 646006

55*C0G

“540000

516000
556000
606000
656000
686000
636000
57+000
54.000
624000
516000
549000
572000
58+000
6c+Gu0
616000
634000

652000

510000 _

65+0C0

802000

____ 83000

862000
754000
674000
586000
540000

580000 __

802000
754000
_80+000
75000
62.000
544000
494000

602000

724000.

80-000
902000
964000
100.000

930000

30000 _

330000

97000

972000
43

_49C00_

C00

34000

66C00
7CG0
_- 39900
66000
7COO
_ .7#000
5eC00
4eC00

7*COO
106C0O

__... 79000.

86000
15+C00
10.000
136000
+000
7.000
7.000
10+C00
106000.
116000
96000

106000,

13-060
102000

10-000

549000
54e000
540000

54*000

54000

54¢000
53°700
53*700
53°500
53*500

__53*500

53+500
53+500
530500
53500
53+700

_54*000

549500
54700
540700
549700
54°700
_ 55000
55°200
55+*500
§5+700
55e500
55200
95*200


Ss

200+000
201000
202000
2034000
2040000
205000
296900
2070900
208900
209-000
219000
211000
212+000
2139000
214000
2150900
216000
217000
218000
2194900
229+000
2710900
2720000
223+000
2242000
2254000
226000
227+000

210

220
254

ccccc
ccccc
c

ccccc
ccccc
ecccc

410

100

Ge Te 254

FIND X IN TARLE*S IN TAALE
DS 220 Iks2sN

Il » Ik

IF(XTBLOTK)©X) p20s 254s a54
CONTINUF

Xt = XTBI CT Ted)

X2 = XTALCITY

Yt = YTBC(1T=4)

Y2 = YTBL(IT)

Sta Yio VOqyt )#(X=X0) 70X29 X41)
RETURN

END

EULER AND MSDIFIED EULER INTEGRATION

SUBRSUTINE EULER YPs YsDTsNEs JoT)
DIMENSIGN YP(20)sY(20)2 SY¥(20),SYP(20)
GE T8(10,20)4 0

CeNTYNUF

De 109 t=tsNF

S¥(t)= YC)

SYP(I)= YP(T)

Yelya Yery+OT# YP (Ty

CONTINUE


SECTION IV
MODEL DESCRIPTIONS
THE GENERAL STREAM MODEL

The stream model used for the study is one-dimensional with
the independent variable being the distance downstream, x .
This model is simple enough that a general version of it
can be considered that encompasses all one-dimensional
models now in existence. A discussion of this concept

is given in Appendix D.

All one-dimensional models take a basic form as shown
in Figure 1 and are described by the equation:

vA aT _

“tdx = !paa ~ Snes ~ ack ~ Snvap ~ conv }

* ovis
(1)
where the variables are as defined in Section VIII, Symbols
and Variable Names, and the water is fully mixed at each
cross section. Any departure from the form of equation (1)
due to direct integration, approximations to the heat
fluxes Q , etc., can be shown by a Taylor's expansion to
be proportional to some power of the mesh size. (See
Appendix D)- Put simply, as the number of stations or
data points in the models are increased they will all
produce results converging toward the same solution, pro-
vided that total heat fluxes and the heat transfer coeffi-~-
cients are the same. The fundamental heat flux expressions
used in this work are taken from Laevastu [1960], where the
processes are linearized in T . Not all authors use
linearized expressions (See for example Jaske [1971]).
However, numerical experience shows that if the temperature
variations are small, i.e., small with respect to absolute
temperatures, and of course they always will be, it is
justified to linearize the expressions, including the
fourth power terms in the back radiation fluxes.

The concept of using a simple, general model was considered
essential in this work, since it reduced the study from

a huge data handling problem for many models to the study
of a single, general model whose error analysis applies
equally well to all models in question, provided a con-
vergent mesh size (station length) is specified.

The expressions for the heat fluxes from Laevastu [1960]
are:


CONTENTS

Section Page
I CONCLUSIONS 1
II RECOMMENDATIONS 2

III INTRODUCTION
PURPOSE 3
SCOPE 3

Iv MODEL DESCRIPTIONS

THE GENERAL STREAM MODEL 5
THE SENSITIVITY ANALYSIS MODEL 7
THE DATA ANALYSIS MODELS 8

V STUDY AREA AND METHODOLOGY

THE VERNON NUCLEAR PLANT 14
DATA BASE DESCRIPTION 1b
EXAMPLE PROBLEMS 7
VI DISCUSSION OF RESULTS 18
ORDER OF MAGNITUDE ERROR ANALYSIS 18
RESULTS OF DATA ANALYSES 20
VII REFERENCES 25
VIII SYMBOLS AND VARIABLE NAMES 26

IX APPENDICES

A. FORMAT SPECIFICATIONS 28
B. INPUT DATA 34
C. PROGRAM LISTINGS 77
D. THEORY OF THE SENSITIVITY ANALYSIS 150


6nL

1577+000
15784090
1579000
15995900
1531,.900
15324000
15334900
1584-000
15852000

15864090

1587-000
15386900
15392900
1590+000.
45914900
15921090
15934000
15942000
15956090
1595-900
15274900

1598-00.

15990909
1609+090
16916900
16914910
16315920
16019925
1601+930
16014040
1691.050
1691040
16024000
1603+090

72
69
20

73
2000:
999
4000

WiSewiSe.125
TWiSeTW1S#e195
OCCis-Cs
DTAsTALS~TAS
OPHIsPH11S*FHIS
DWEW1 Sows,
OTWaTW4SeTWS
STTSLI(tysTWIS
STT3L1(2y3TW1S
XTBL1 (1208
XTBL1(2)s25000%
NTEMPi ze
INVSEINTA 480%
CALL COMP (ENUMsDEN)

D2 60 ICASe1sNCAS

TF (ABS(Y4 (1). @DELTCICAS)) 65265270
CONTINUE

PRCTCAS) sPRIICAS) $16

C3NTINUE

CONTINUE

CANTINJE

DENP=NDIE

D9 75 IcaSe1,NCAS
PR(ICAS) sPR( CAS) /DENP

C3aNTINUE

WRITE (1822000) (PR(T)sT=1sNCAS)
FORMAT (/s2Xs STATION: 7 RESULTS1.10(F6*321X))
RETURN

CSNTINGE

WRITE(TSUs4000)

FORMAT (/+2Xa1T89 SMALL)

CALL Exit

RETURN

END


«G00

“000°

_ 36000

5.000
~ 94000
10.000
10+000
10+000
10+G00
10+000

~"" 109000

72000
2000
“4.000
1.000
16000

8+000
8.000
54000
7+000
62000
"109600
10000
10+000
10000
10-000
10.000

12000 "~

400000
372000
33+000
400000
56.000
55,000
520000

~ 49000

480000
46¢000
450000
470000

546000

“60+000

620000
5390000
55000
540000
540000
550000
64e00C
68000
92000
450000
430000
420000
420000

4#3e000

934000

~ 960000 ~~
_ 1004000

96 +000

64,000

83-000

966000
100+000
1004000

“934000

71.000

496000

“396000

382000
424000

47.000.

494000
552000
5746000
474000
39000
74.000 —
86000
93000
964000
100+000
83-000

40

+0000

“96000

“9000

2«co0

——F6607

39000

36000

10.000...
3+000

4«060

46000

7000
5.000
“436co00 |
10-6060
8-000
- 5558077
11-000.
10+000
10+c00
136000
10¢000
36000
96000
"Ss000
4e000

10c00

55500

_ 540500

54e500

*500,

.. 549700

55000

55000
549700

“54870000

54200
54500

~""55 «000 —

55200
55*000

“359000

559000
54700
“546500
540500
540700
55°000
54°700
540700
54*700
54¢700

54¢700

§5°000

“55900000


___ 102000
102000

102000

104000

10.000
10.G00

_. 104000

40000
26600
52000
8000
8e000
102600

9000

“' 10+G00
_., 100000

10-00”
8.000"
5000

444000

559000
500000

470006

440000

39000

50e000

“619000
624000

520006

~~ 430000

_ 54000

470000

~ #50000

440000
446000
406000,
38000
31.000
30000
27-000
39-000
426000
430000
37-000

410000

420000
400000
360000

69000

93000

” 964000

_..36+000

966000

676000

_. $7000

936000
100+000
962000
96s000
834000

804000 |

74000
742000

_.76+000

674000
92000
964000
96000
796000
634000
63+000
324000
73+000
71-000
624000

79000
71

860000 _.

$0000 52500
_. 34000. 52200
36c00 52+000
“30000 518500
32000 §1e200_
3-000 51*000
“6000 = 518500
84000, $2000.
36000 52+000
“30000-52000
_---499000___. _52¢000_
9.000 51+700
“10-000 “51500
429000 51#700__
13.000 522000
96000 529000
70000 512700
3*C00 51+500
3-000 51000
36000 512000
d6c00 50°70
4000 50500
89000 50+700__
96000 50*700
“36000 ~50+500
6000 _50#200 _
96000 50000
126000 50+000
36000 50+000



08

57000
58-000
59.900
60+000
614900
62+000
63+000
64900
65»900
65+000
67+000
68+900
69+900
72+090
712000
72*900
732900
74 +900
75+090
760090
77+000
78+900
739+000
894000
811000.
8240900
834000
84.000
854000

30
40

50
60

110
115

120
125

130
135

140
145
150

160

1

READ( IN» ?0) INTC

IFCINTG # 3) 30540250

INT a INTC

G8 TS 60

INT = 4

G3 Te 60

CONTINUE

CANTINUF :

G8 TH. (110s12011302140). INTC

WRITE (18.115)

FORMAT CHL wee  FULFR INTEGRATION weeHEN SZ)

G9 TS 150

CONT INUF

WRITE (18125)

FARMAT( 14 #eeee MBO. EULER INTEGRATION ##¥HHI//)
G8 T3 150

WRITE(182135) .

FARMAT(HG weeny 4 TH BRODER RUNGE*KUTTA ween //)
s8 T9 180

WRITE(18,145)

FaRMaAT(t4 weeee ADAMS MBULTEN INTEGe a eeennt //)
CaNTy NUE

INTEGRATTON CONTROL. PARAMETERS

READCINs 10) Xs#DXsDXPRs XEND

WRITE (18,160)Xs 9Xs0XPRs XEND

FaRMAT(' X etFAeA! OX StEinw3t DXPR stE1003
' “XEND StE10039/)

READ(IN»10) FRRARsDXMAX


EPA-660/3-75-037
JUNE 1975

Ecological Research Series



90L

372.000 YOT ESV CTIFCAKC Ts LI FAK (CT, b) +2 eH (AK (Te 2)4AK 6143) )) O08
3734000 400 CONTINUE

324000 RETURN

325000 END

326000 CCCCC
327000 CCccCc
3284000 Ccccc
3294090 €
339+000 CCCCC
331-000 CCCCC
332+000 CCCCC

ADAMS=M8ULTBN INTEGRATIEN

333-000 SUBRBUTINENGRD (Fs Te TLIMs Ye ERROR NE s Hs HMAXs UUMPs KSTPsKCENSCLIF SIG)
3342000 DIMENSIGN STARY(5)s¥(5)sSY(5)sSAVEY(5)2F (5) sFP(5) sDELTA(S) sDALTA(S
335+°000 1)4A(5)23(5)90(5)9005)2AA(5)2BB8(5)200(5)200(5)2SF (5)

3350000 IF (KSTP*22767)993s9912991

337-000 991 KSTPs28

3386000 C TEST FSR TYPF GF ENTRY

3390000 993 IF(JUMP)1s99Rs999

3422000 998 GB TB (1N00111s2128072803)s 1A
3412000 C JUMP PBSs REST@RE VALUES
3422000 999 TsSAveT

3430000 992 JUMP=0

3440000 08 901 le1sNF

3452900 F(I)=SF(1)

3462900 901 Y(I)sSAVFYCT)

3472000 G8 Te 107

348+000 C JUMP NEGs INITIALIZE
349+090 1 0G 5 [=4,NE

3500000 STARY(1)8Y(1)


OeL

539-900 1
$49e000 C
5412009 C
5$2.000
543*000
544-900
$635+900
555+900
557+900
558+500
559-500
5702000
5714000
5720000
573.990
5742000 C
575909
576+200 C
5778000
578+000
579+200
532e900
581000
582-0009
5835+900
5840000
535-6000
536+000

50

198

CONTINUE

INTEGRATISN CONTREL. PARAMETERS
CANST = FL/(RYS*#U¥ARZA)

CSNS4 = -(RHO¥D09037PB)/ (242 #36000 #000614)
OTA = OTAIRe5e/Fe

H8S = 1+9*SIN(ALBAR*RATIB)
HSC = 1+9*C3S(AIBAR*RATIO)
CLO1 #16 = 69755*CLO

CLD2 = =.001R#CLO¥¥2

CLD3 = te = 20096 #CLO¥#3

DALPR = AALP « 2ATIO

D-l = DWMPH#2AW/156

DYWMS = Dw # 63048

DHMD = DWMPH#A4s

INCIDENT SSLAQ RADIATION

dS 5 49S * C103

REFLECTEN SSILAR RADIATION

HSR zs HS * 34/ALBAR

FTA =-SI(TTBLsPTBLsTAIRs IQ)

EA = 2H¥(10136/14 708) #FTA
PAIRP © TAIR#OTAIR

FTAP = ST(TTRLsPTALaTAIRPs10)
DEA = (10130/14700)*(DRH#FTA + RH#(FTAP = FTA))
IFLAs1

IF (TEL A+1) 19891999198
CANTINUE

WRITE (18.200) OT


CAL REPORT DATA
(Please read Veations on the PORT before completing)

1, REPORT NO. 2. 3. RECIPIENT'S ACCESSIONNO.
EPA-660/3-75-037

4, TITLE AND SUBTITLE is. REPORT DATE

‘Improving the Statistical Reliability of

Stream Heat Assimilation Prediction 6. PERFORMING ORGANIZATION CODE

: » PERFORMING ORGANIZATION REPORT NO.
Pp AUTHORIS) Richard W. McLay i

Mahendra S. Hundal

1s
PERFORMING Gade NRA ON NAMERND "RUDRESS 10, PROGRAM ELEMENT NO.
Richard W. McLay, P.E. 1BA032
18 Redwood Terrace 11. CONTR. .
Essex Junction, Vermont 68-03-0439
05452
12, SPONSORING AGENCY NAME’ AND ADDRESS: 13. TYPE OF REPORT AND PERIOO COVERED
National Environmental Research Center inal
Office of Research and Development 14, SPONSORING AGENCY CODE

U.S. Environmental Protection Agency
Corvallis, Oregon 97330
16, SUPPLEMENTARY NOTES

[ie ABSTRACT

This work is an evaluation of existing, one-dimensional stream temper-
ature prediction techniques for accuracy and precision. A sensitivity
analysis of a general model is used in conjunction with statistical
methods to determine solution errors. Data taken in 1973 at the Vernon,
Vermont nuclear plant are used as a data base. These data are used with
Burlington, Vermont airport Weather station data to 1) gain insight ined
the orders-of-magnitude of the various errors and 2) carry out a de-
tailed data analysis to establish probabilities of meeting given error
requirements.

7. KEY WORDS AND DOCUMENT ANALYSIS
DESCRIATORS . Ib, IDENTIFIERS/OPEN ENDED TERMS |c. CosaTi Field/Group

Accuracy
Stream temperature prediction

18. DISTRIBUTION STATEMENT. 19, SECURITY CLASS (This.Report) 21, NO. OF PAGES

163
20. SECURITY CLASS (This page) 22. PRICE

EPA Form 2220-1 (9-73) # U.S. GOVERNMENT PRINTING OFFICE: 1975-699-072 14 REGION 10


ABSTRACT

In response to an increased interest in water quality

by the public, a large effort has been mounted to develop
mathematical models for predicting heat assimilation in
bodies of water. The accuracy of these models has recently
come under scrutiny due to the need for temperature
predictions within 1 °C of the ambient. This work is

an evaluation of existing, one-dimensional stream tem-
perature prediction techniques for accuracy and precision.
The approach is through error estimates on a general
model that encompasses all of the models presently used.

A sensitivity analysis of this general model is used

in conjunction with statistical methods to determine the
solution errors. Data taken in 1973 at the Vernon, Vermont
nuclear plant are used as a data base. These data are
used in conjunction with Burlington, Vermont airport
weather station data to 1) gain insight into the orders-
of-magnitude of the various errors and 2) carry out a
detailed data analysis to establish the probabilities

of meeting given error requirements. This report contains
the model descriptions for the general stream model, the
sensitivity analysis model, and the data analysis models;
a description of the Vernon, Vermont site; the data for
four problems from the Vernon nuclear plant; an order-
of-magnitude error study; and the results of the four data
analyses. The four appendices contain 1) a description

of the input FORMAT specifications, 2) the input data

for the four problems, 3) program listings, and 4) the
theory of the sensitivity analysis.

This report was submitted in fulfillment of Contract
68-03-0439 by Richard W. McLay, P.E., Essex Junction,
Vermont, under the partial sponsorship of the Environmental
Protection Agency. Work was completed as of May 1975.


LoL

3516200
3522000
353000
354-900
355000
356+000
3572000
338+000
359000
369+000
361+000
342.000
3634090
364.000
345000
366000
347000

368-000 C

369+000
3702060
371000
372-2000
373000
3748000
375+000
375000
377000
378090

1000
qaiii
c

600
601

605

ACT) 2000n1

Bil) 200ont

CilysOena1

OtleQv0on1

KsTP=0

KDELY=0

KCEN=0

XT=950/(PBBonbhe)
Us863+/(120#50406)
v=9500/2R8e0

R225 00/2400

Q535 00/7700

R=5.0/48.0

S#1.0/12000

TAst

JUMP 30

G8 T9 1104

BEGIN INTEGRATIAN STEP

D9 1121 TelsNE

SF(rysFcr)

SaVEv(1)e¥(T)

H TAS SMALL RETURN WITH JUMP NEGe
IF (ABS (T+H) ABS (T) 160516012605
JUMP a4

GS Te 14101

TaTeH

08 10 TetaNE

VOD RY CLI +HMCFCTIGACTI4BCDI SCOT EDIT) D



APPENDIX A
FORMAT SPECIFICATIONS

This appendix presents the procedures required for using the
three codes developed during the course of the project:

1. The general stream model code, STREAM.
2. The sensitivity analysis code, SENSIT.
3. The data analysis code, MONT.

STREAM

The digital computer code STREAM performs all the necessary
computations to predict the stream water temperature at any
distance downstream from a given initial station. The
principle of conservation of energy is applied to the stream
under steady-state, steady-flow conditions. Heat transfer
to and from the water in the various heat transfer modes is
computed. Numerical integration is used to compute the
stream temperature, with a choice of four algorithms
available to the user. A listing of the STREAM code is
given in APPENDIX C.

INPUT REQUIREMENTS TO STREAM
STREAM accepts the following information:

INTC: (1, 2, 3 or 4) in 6110 FORMAT. This parameter
respectively chooses Euler, Modified Euler, Runge
Kutta or Adams-Moulton integration routines.

X, DX, DXPR, XEND: in 6E10.0 FORMAT. These parameters are,
respectively, the initial value of the distance down-
stream in feet (usually 0.), the integration interval
in feet, the print interval in feet, and the final value
of the distance downstream in feet.

ERRGR, DXMAX: in 6E10.0 FORMAT. These parameters are used
by the integration routine only and represent the error
in the function in OF and the maximum integration step
size in feet allowed.

TEMP, U, AREA, EL: in 6E10.0 FORMAT. These parameters are,
respectively, the initial value of the temperature in
OF, the average velocity of the stream ig ft/sec, the
cross-sectional area of the stream in ft“, and the
stream width in feet.

28


L8

237+900 AK(t22) = DT#YP(T)

258.000 YC) = SYCT) + AK( TaD) e605

2590900 200 CONTINUE

2694000 RETURN

241000 30 CONTINUE

2624000 DS 300 1 = 1,NE

263+000 AK(1a3) = DT#yPtly

264+000 YCT) = S¥°CT) + AK(T53)

265+000 300 CANTINUF

246000 T = ST 4 OT

2670000 RETURN

268+000 40 CANTINUE

269+000 06 400 t s 4,NE

279+000 AK(Ta4) 5 DT. #YP(T) .
271.000 YCL)SSYCT DH CAK CT ot AK (Ta 4) 42 ow (AK OT 92) 4AK(193))) /60
2720090 400 CONTINUE

273+900 RETURN

2742900 END

275+000 CCCCC

278.000 CCcca

2776000 Cccce

2782000 C ADAMS*MBULTON. INTEGRATION

279¢900 CCCCC

280000 CCCCC

281000 CCCCC

282+000 SUBRBUTINENGRO (Fs To TLIMs Ys ERROR ONE e Hs HMAXs JUMPs KSTPa KCBNS CLIF S18)
2836000 DIMENST6N STARY (512 Y(5)4SY(5) sSAVEY(5)9F (5) FP(5) sDELTA(S) sDALTA(S
2342000 1)4A(5)48(5)00(5)20(5)2AA(5) 6 8B (5) 2CC(5)4D0 (5) sSF (5)


78

1144900
115000
1154900
1174000
118+000
119000
120+900
1714000
122000
1232000
124900
1254000
1264900
1?7+900
1284000
129000
130+900
131+000
132000
1334000
1346000
135+090
1366000
137+900
138-000
1392000
1406000
141+000
1422000

230
240
260
270
230

390

310

10
20

WRITE(18,230) TAIR»TAsCONST» CANS1

FARMAT(' AIR TEMPe s!F 501! DFG FIF8el!' DFG Cla

a! CONST B!1E1063! CONS1 SIE10+3+/)

WRITE(T9s 240) WeWMPHs WMS» WMD

FeRMAT(' WIND VELBC:TY afF 602! FT/SECIFBe2IMPHIF8e2) METERS'y
A't/SEC'FR.2t MILES/DAYI/)

WRITE( 18.260)

FORMAT (/sT201TTSL'T4O!PTBLIT6OIHFGTBI/)

08 270 T #1910

WRITE(16,280) TTBL(T)s PTBL(I). HFGTB(I)

FaRMAT(10Xs3F 2065)

WRITE(18,300) EAsH8S,HS,HSR

FARMAT(//1 FA at E1204! HOS at£i2e4! HS elEl2e4s
11 ASR st'E1204!  LY/MINI)

WRITE (18,310)

FaRmaTclit)

RETURN

END

SUBRBITINE QUPT

COMME8N OTEMP, TEMP» XPRsXFNDsDXPR»XsDXe INTs INTC
4 INS 18sE) 2 ARF As Us ERRARSDXMAX

COMMSNRHBs WMDs HDs TATR»CLDs RHs ALBARS TWSaTAs EWsEAsHOSAHS
C3MMON HSRsHAS8sF BR» EVAP.HEs HCaHLYMNs PTBL (10) TTBL(10) aWMS»CBNST
CaMMEN HFGsCLD1sHFGTB(10)sCB8NS

WRITE(T8s10) Xs9X

FORMAT (! X ate7att FT OX atFbels /)
WRITE( 10.26) TEMPsOTEMP
FORMAT(' STREAM TEMP s!F6e2' OFG F OTEMP stE1003%

11 DEG F/FTI,)


Put simply, for given errors in the initial mean temperature
and the meteorological data, it is possible to determine

the probability of meeting a given error requirement. This
can be done by using the sensitivity analysis model, which
appears to be a very realistic approach to assessing the
effects of the introduction of a heat load.

The methods employed in the data analysis are best visualized
by using Figure 2, where the six primary variables are il-
lustrated. The effect of errors in these variables is shown
schematically in the figure, where the initial error é6T is
seen to decrease with increasing x. By collecting a data
base of these variables over a period of several months, it
was possible to make use of the sensitivity analysis to
compute the error in temperature for a number of trials,
based on moving averages. With an increasing number of
trials the probability, P , that |éT| is less than some
given error in temperature can be estimated from:

+ Number of trials with |éT| < given value
~ Total number of trials .

Figures 3 and 4 illustrate the methods for utilizing the
data base to evaluate equation (5). Figure 3 shows
schematically the data base at an airport weather station
remote to the site, in this case the Burlington, Vermont
weather station. These data are collected together with the
stream temperatures at the site every three hours, eight
points per variable per day. These data are averaged over

a specified number of previous days, then compared with the
respective averages over the day following to form the errors
in the input variables, 6¢ , etc. The sensitivity analysis
is then used to predict the errors in temperature down-
stream, These are then compared with the given temperature
error requirement and the components of equation (5) com-
puted. It should be noted here that this model simulates
the errors of an observer at the site attempting to predict
the temperature several days ahead.

P

(5)

Figure 4 shows schematically the second data analysis model,
where the data base includes 1) the remote weather station
data, 2) the stream temperatures at the site, and 3) the
average daily stream temperature above the site (See

Figure 5, station 7). In this case the daily average of

the station 7 temperature is compared with the following
day's daily average temperature at the site. Meteorological
data are averaged in an analogous manner, i.e., daily
averages compared for two consecutive days. These form the
errors in the input variables. The sensitivity analysis

is then used to predict the errors in temperature downstream,
these are compared with the given temperature error require-
ments, and the components of equation (5) computed. This


ADDITIONAL DATA FOR STATION 7 ABOVE THE DAM
TEMPERATURES AT STATION 7, CF, DAILY AVERAGES, MAY 1-JUNE 21

48.9, 50.7, 52.3, 52.4, 52.0, 51.8, 51.9, 52.9,
53.0, 53.1, 54.5, 55.2, 55.0, 54.5, 54.2, 54.2,
53.6, 52.4, 51.4, 50.8, 51.0, 50.6, 52.3, 53.9,
53.7, 54.1, 53.8, 54.3, 55.3, 56.3, 57.9, 59.0,
59.4, 58.8, 58.9, 60.0, 60.9, 62.3, 63.3, 64.6,
65.2, 66.8, 69.2, 69.3, 69.0, 67.3, 66.9, 66.1,
62.4, 59.3, 59.9

52


16

456*000
4574990
4584000
4594000
4692900
461000
4624000
463+000
4642000
4654000
4660900
4672000
468000

335

135

1101
1150
1151

H=2.0*H
09 135 tetsNF

A(T) =200#A(T)

B(l)s4s0eB(1)

C(1)28eOeC (1)

D(Tyel6.n*D(1)

KDELY#0

G3 Te 1000

CANT INUE

FORMAT (/+2XatHH ts 2XsF 1006s t4XstKSTP ©142Xe13910Xs 'DALSA( I) '5/)
FORMAT (5(2Xs1291XsE1407))

RETURN

END


_ 84000
10+000
10+G00
_ 106000
7000
94000
106000
106000
106000

74060,

7600
82000
86000

400000

_ 396000

370000
360000

_ 360000.

38¢000
399000
370000
330000
312000

326000 _

312000

“340000

35¢C00
386000
35¢000
340000
32-000
31-000
30000
280000
31000
320006
29+000
280000
27«C00
250000
270000

300000

394000 14000
_ 38000. 189000
394000 12.¢00
“400000 444000 ©
400000 37-000
374000 12-00
36+000 116000
__. #44000.

446000
“4740006000 ©
4340007000.
436000 74000
420000 - 42 ec00
379000, .9#000.,
314000 106000
324000 414000
374000 7+000
334000 11.000
36000 10+C00
416000 126000
534000 144000
40+000 10+C00
366000 176000.

58 +000 8+000
614000 124000
550000 7000.
60+000 96000
61+000 «000
584000 42000

73

48500
48500

482500
48e200
489200 _

489200

48*200

479500...

47000
47000
464700.
46*700
472000 —
472000
464700
460500
_ 45#700_
459500
454200
45*000
45000
45+000
459000 __
446700
446500

_ 449500

44 +500
440500

449500



APPENDIX B

INPUT DATA

This Appendix contains the input for the four example

problems.

EXAMPLE PROBLEM INPUT

The input for the four example problems has the following

format:

1.

2.

The number of lines of data, 8 lines per day
(3 hour intevals).

The time averaged data logic parameter (1 means
time averaged data appears, 0 means no time
averaged data appears).

The data now appears by line as follows:

a) Cloud cover in tenths.

b) Air temperature in OF.

c) Relative humidity in percent.

d) Wind speed in knots.

e) Water temperature OF.

The average data (over a day) for the plant site
then is:

a) The flow volume, UA, £t3/sec.

b) The river width, EL, ft.

c) The cloud cover at the site in tenths.
d) The air temperature °F

e) The water temperature OF.

f) The relative humidity in percent.

g) The wind speed in knots.

34


8000

6#000 |

106600
10+000
"9,000
8.000
8.000

~ 106000
10.000
7.000
72000
104000

72000

“74000

10.000
8000
~ 102000
86000
72000
46000
#000
+000
+000
1-090
8000

* 8.000
8000

72000

652000

77000

800000
760000

760000

684000
67+000
716000
742000
780000
85-000
710000
724000
724000
726000
70+000
70800C
76000

79+000

~ 800000

738000
61000
569000
95°000

680000

“770000

81-000

84e000

"9639007 7

100-000

65000
760000

874000
874000

“"814000

76 +000
714000
61+c00
90+000

84.000

76.000

824000

84e000

84.000 |

66000
52+000

410000 ©

454000
754000
934000
934000
684000
566000
614000
462000

46

74*000

32000

~ Beco0

12C00

4-000

-- 2000
82000

~ 408000 —

72000
136000

“42e¢00

7000
50000

“106000

13-000

16+000

14.coo

13000
8000
136000 -
86000

50000

“36C00 |

3+000
10-000

116000

12+c00
10000

“7000”

61¢000

“61000

619500
61°700

~~ 61°700

62°000

62°000

"620200

62200

62*700

63*000

639100

63°500

. 62°700
62*700

“629500.

_62*700

63*000

~ 64000

649500
64°700
64*500
649200
641000
64*0U0

. 649700
65¢200

“63*000°



OLL

85.990
B769C0
880000
890050
32.960
91-9090
92.990
934050
34-050
93-990
95+990
974999
98-990
994900
109+090
191-090
1024900
103+990
1940950
195+990
105-000
197-050
108+990
10342999
1122050
111-6090
11¢2000
1132000

2092 FSRMAT(1HOs2xX55(F100301X))
10 CONTINUE

c PLANT DATA

c UATVELs FLERTVER WIOTHs CLsCL3U9 CAVERy TAIR=88VIGUS

c THAsWATER TEMPs RHeREL HUMIDITY, WSP2N4IND SPEED(KNGTS)
READCING 1005) UarEL + COs TaTR
WRITE (T8Us2003)U42ELeCCy TAIR

2003 FARMAT(1HOsAXe MASTS TXeF10032IR WIOTHE IS LXSF10¢3,'CLO COVER='s

LLXsF 41903, 'TEMD AIR=191X2F 1003)
READ (INs 1000) TWAs RHs wSP
WRITE (13112004) TWAs Ry NSP
2004 FARMAT( "ON s2xet TEMP WATFREMe1XeF 10030 'REL YUME'a 1XeF 10032! WIND
1LSPEED# 4 4XaF 5203)
TF (LA31)15s14015
15 CANTINUF
READC INS TOOL)NTA
WRITE( I 30,2005) NTA
2005 FORMAT (1HOs2Xs'NUMRER OF TIME AVe DATA POINTS«41Xs15)
WRITE (18U+2605)
2006 FSRMAT('0's2xXsICLD C3VER TA ATR TEMP TA REL HUM TA WIND TA
1. TEMP wATER Tat)
D3 20 TsteNTA
READ( INS 1000) CTA(T)sTATA(I) sPHITA(T) TACT) a TWTA(T)
WRITE (19a 2007 CTACT a TATACT a PHITACT)aWTACI) a TWTACL)
2007 FARMAT(1HOseKsS(F100301X))
20 CANTINUE
16 C3NTINUE
READCINs 1001)NCAS


eo

NUMBER OF CASES= 4
ALLOWED DTS
*100C000E O1
*200C000E O41
*3000U00E O01
*400CG00E O1
NUMBER 8F DAYS AVERAGEDe 2.
DAYS AVERAGED
1 ne nS
2

PTBLI TTBLI HFGTB1

OES tCri il rire errrrectererervecrrvrerertrrrcrerirerrirery
*S85E-01 *S2cE 02 *108E of
#122E OC #40CE 02 +107E OF
©178E 00 *SU0E C2 e107E OF
*256E OC +*60CE 02 +106E 04
*363E 00 *700E 02 +105€ OF

*507E OC »*800E 02 *105E 04
*698E 00 *SOCE 02 *104E O04
eS4SE OC +10CE 03 #104E OF
e127E O1 911GE C3 +103E OF
e169E O1 12GE 03 +103E o4

DAYS SINCE MARCH 21" = 41,000BETA= 45.000"


LEL

7386000 G9 Te(t0.20s20s40.eJ
759-900 10 CANTINUF

769+900 D8 100 TstyNr
761-990 SY(T) ® ¥(T)
7520300 AK(Tal) 3 OTHYPCTY
»300 YCD) © YET) #AKC Tat ees
2000 190 CaNTINUE
7652969 Ste T
74$*500 Te T+ need
74675900 RETURN
753.000 20 C8NTINUF
7530900 D8 200 TetsNF
7729+000 AX(1a2) = OT#YP(T)
771-901 YOL) = SYCT) # AK(T22) #05
200 CaNTINUE
RETURN
30 CONTINUE
7734900 03°309 tf = 1,NE
7760990 AK(Ts3) a DT#YP(L)
7774580 Yel) = S¥ (I) + AK(Is3)
778+090 300 CONTINUE
7735090 T = ST + OT
739+C00 RETURN
7314969 49 C3NTINUE
7821060 D3°409 I = 4,NE
783-000 A<(Ta4) = DT #YO(T)
7342900 VCD) sSY (TY HCAKC Ta LI FAK (Ta 4) +2 ow (AK (192) 4AK (193) )) /60

785+000 400 C3NTINUE
7360500 RETURN


8000

q 36000
5 e000
__10+000

10000

92000
88000
7eCoc
2.000
22000
5e000

__42000

86000
6+000
10.000

72090
100000
100000
10-c00

BeG00

26000

62000
10-000
102000

20000

—

* 89000
46000

_ 2000

2.000

79000

_719000

730000
712000
719000
72+000
796000
820000
772000

72.000

670000.

640000

“646000
716000_

760000
79«000

708000 _

634000
646000
664000
724000
736000
864000
860000
78+000
67+000
660000
669000
734000

524000 7+c00 65*500
640000 8000 «65 e 700
574000 125000 65*700
14000-13000 65700
662000___ 109000. 65500.
68.000 116300 65500
626000 —- 12000 66*200
430Q00__ 100000. 67#000
502000 8+C00 67500
“$0s000— 8+co0 = 7 #500
8300007200067 #000
674060 5-000 67+000
“68.000 “8.000 66«500
49400062000 66#700.
43.000 86000 67+500
414000 «6000 68+500
(744000 44000 69#000_
934000 3+C00 69#000
“902000 —«5*#000°»=— 68500
78.000 32000 68 #000
664000 86000 67500
626000 10+000 679700
554000 5000-68 +500
80+000 8-000 69*700
624000 +000 60#200
90.000 4000. 70*200
93.000 3eco 699700
934000 5*000 69°700
822000 86000 69+500

47


OL

84000 572000 424000
6+000 480000 634000
8000 450000 712000

UAz 7640«¢0COR wWIOTHe
TEMP WATER= 41¢700REL
NUMBER 6F CASES® 4
ALLOWED DTS
*1O0O0COO0E 01
*20CCOO0E O1
*300C000E O01
*400CU00E O1
NUMBER 6F DAYS AVERAGED®
CAYS AVERAGED
i
2
PTBL1 TTBLI HFGTB1

12000
4-000
4.000

400*000CLD C@VER=

HUM@

68«000WIND

420000
429000
429000
1+OOOTEMP AIR#
SPEED 46400

ee eee e eee ee er rire rrr ere rerio irrirr ry

*885E-04 +32cE 02 *108E oF
e122E 00 e40CE 02 +107E 04
©178E 0060 *500E 02 +107E 04
«256E 00 *6U0E 02 +106F 04
©363E 0G +*700E 02 +105€ o4
*DO7E GO *8UCE G2 *105E 04
*698E OC *90CE 02 s104E 04
*949E OC #1UCE 03 *104E O4
o127E 01 110E 03 +103E OF
°169E O1 120E 03 +103E 0+

50000


98

228000
2294000
239+000
2312000
2322000
233950
2340000
2254000
2249000
237000
238-000
23392000
2499000
2410000
242000
2439000
2442000
2452000
246+090
24742000
2482000
249000
250000
2512000
2526000
2534050
254000
2554000
2562000

20

200

ccccc
ccccce
ccccc
ccccc
ccccc
cccce

10

100

20

TaTsoT

RETURN

CANTINUR

D9 200 TelsNF

YOT y= SYCTI#CYP(T)4SYP(L) HDT / 26
CONTINUE

RETURN

END

FOURTH SRDER RUNGE-KUTTA INTEGRATIBN

SUBRBUTINE RK4(YPsYsDTsNEadaT)
DIMENSION YP(5),Y(5)sAK(524)sSY(5)
G8 16(10.20220240)s5

CONTINUE

09 £00 relsNF

Sy(t) © y(t)

Ak( Tet) s DT*YP(1)

YUL) = CT) AK (Tad aed
CANTINUF

sts T

TET # ATeed

RETURN

CONTINUF

08 700 faisNr


10+600

___ 166000

106000
“10%600

__..100000

10-090
10000
10-000
106000
10+000

_10+000
10+000
10.000

10-000
6000
__ 5400
94000
104000
104000
10+000
104600
106000
10+000
106600
10+000
10.000
106600
104000

106000 |

480000

490000

51-000
49000
420000
420000
410000
420000
450000
460000

460000

460000
472000
492000
482000
50000
60.000
672000
639000
60+000
530060
$1000
512000
50000
52e0c0
51000
490000
480000
470000

864000
_ 892000
77000
83000

964000.

100+000
100-0C0
964000
936000
934000
93000
934000
86000

. 774000.

802000
80000
624000
492000
500000
65+000
30+000
100+000
100+000
100-090
100+000
96000
36+u00
368000

100+000
441

4ec00

5eco0
96Co0

106C00
10eCOO
10«C00
13-000
10000

--.8#000 __

9*960
7+c00

- 7000...

52000
36000

3400,

40000
10-6060
36000
32000
96000
6000
1l.cco
136000

_144¢00

15cG0
14000
136000

_ 49000 |

72000 |

540700
540700. _
549500
546000
_53*700
539000
52+500
$2°500
52500

520500

__.52¢500

52500
52500
519700 __.
51¢500
519200

810000.

51000
51000
51000
51*000
51000
51*000
50°700
50°500
50*200
50200
509200
50+200


BCL

5924000
593-000
594500
595-009
5962000
597-090
593000
599-090
5122000
511-900
5120000
5132009
514.900
5150000
516-000
517-000
518-000
5196090
5202900
5214000
522.050
523-000
524 +990
525-000
5262000
§27+900
528-000
529.009

110 CANTINJE
XPR = X 4 DXPR

120 0 C9NTINUF
CALLNIRD( YP» XsXPR» Ys ZRRORa NES OXs DXMAX2 JUMPS KSTPsKCONSCLIF S18)
TF (JUMP )13021402150

130 WRITE( 18,131) X

1310 FARMAT(//1 0X s'E14e5!) INTEGRATION FAILURE!)
RETURN

140 9 C3NTINUE
CALL DIFF
G3 Ta 12

150 CONTINUE
IF(X = XFND) 1102300.300
300 CONTINUE

RETURN
END

cecce

ceccc

ccccce

cccce :
SUBROUTINE INPUT
DIMENSION PTAL(10)/TTBL(10)
CAMMEN YP(5)s OT, DUM( 4)» XPRs XFNDsOXPRs XsDXsNEsINTSINTCy
1 TNs 19a E1 sARF ASUS ERRGRs DXMAXs TEMPS HF GTB(10)»CONS1
CSMMSNRHAs Ws WMD-HDs TAIRs CLD#RHs ALQARs TWSa TA, TWSASEWs EAs HOSS HS
COMMON HSRsHO8sEBRVEVAPsHEsACeHLYMNSPTOL » FTBL 2s WMS» CONST
COMMBN CLIFsHFG»H8CsCLO4sCLO22CLD3sDALP,DALPRsDCLDsDRHsDTAIR»
1DTAsDWs WMS» DWMDs DEAs DEWs DLYMN, DBTUSSNTEMPs STTBL (10) + XTBL (10)


LLL

4652000 G8 8 78
4664000 100 KDELY®KOFLY+4

4674000 C WILL NEXT STFP MgvF PAST TLIM
468-000 102 IF (ABS(TI IM=T)=ABS(H) 110321032110
469+000 C YESevesveeseSAVE T AND Ys INTEGRATE TO TLIMsRETURN®
4702000 103 ENOHSTLIM-T

4714000 03 105 T=1,NF

4724000 AACy)SENDH#A(T) /H

4734000 BB (1) sENDH##2 4B (1) /Hawe

4740900 CC(T) SENDH##R CCT) /Ha wd

4752000 105 DO(T) SENDH##ueD CT) /Hawd

476000 SAVETST

477000 D3 200 TelsNF

478 +000 SF(T)SF(T)

4792000 B00 SAVFY(I)sY(T)

4890000 806 T=TLIM

481+000 O08 106 telsNr

4324000 YCD) VY Cp) +ENDHRCF CDI +AACT) +BB C7 I4CC( 1) 40D 7))
4332000 106 FRUIT SF CT) +2cO#AAL] 43 008BB (1) +4eMHCC (1) +5 e0HDD (1)
4384-000 TAsy

4854000 G8 Te 1104

4362900 802 D3 805 IutsNF

4874000 805 S¥(r)e¥(1)

488-900 D@ 107 TalaNF

4894000 DELTA (t)SFCT)WFP(T)
4992000 107 YCTys¥(1)+V¥DELTA (1) #ENDH
4910900 TAG -

4924000 G@ Te 1101


LZL

228-600
223-6000
2394000
231-009
222-960
233-000
234.009
235-000
236+900
237000
2384900
239+900
2496900
241090
2424900
2434960
2444000
245.900
2464300
2474990
2436900
2496000
235.050
31.509
2524090
2534900
254.0900
255+000
256900

65

79
69
20

THLSsTWIS#e125

OceciS-cs

DTAsTALS@TAS

DPHT=PHTtS-PHIS

OWEy1S—46

DTWaTHLS-THS

STTALI(1)=TWIS

STT3L1(2)=2TW1S

XTBLI(1) 206

XTBL1(2)=250006

NTEMP1 =?

INVS5 INT? #694

CALL COMD (ENUM, DENS

09 50 ICASe1,NCAS

IF (ABS(Y1(1))=DELT(ICAS)) 65265470
CANTINUE

PRP( TDA, TCAS) PRPC IDASICAS) 446
CANTINGe

CONTINUE

CANT INGE

DENP ENDIF

03 75 ICaS=1,NCas

PPC IDA TCAS) PRP (IDAs CAS) /DENP
CANTINUF

IF (L831)700. 2012200

CONTINUE

ENUM29«

DENeOs

DENP =e



9tL

4454000
4452000
447%000
4482090
4436000
459+000
45180900
090
53-909
43542900
4556990
4356200
457+000
4536900
4590000
4504000
4612990
4620590
4434900
464-090
#65*500
4560000
457.000
468-000
4690000
4724000
471»900
4720000

cecce

10

20

ccccc
ccccc
ccccc
ccece

ccccc
cccce

IrLAGso
TFCABSOY4 (12) *DT)10210270
CANTINUE

ENUMBENUM+41

IFLAG=1

CONTINUE

sDENete

FUNCTISN ALNAP(DMALs BETA)

SNL=SIN(RETA) .

THis Se28*DMAL/ gem.

DEL ASIN(SIN( (23045/570296) )¥SIN(THL))

PHY SACSS(TAN(QEL) #TAN( BETA) )

TOS? Ke PHY /6 028

TLEDbeeTy

AVE (TI-TOV*SIN(DELI#SNI Abe .
Ay#AyaCOS(DEL )eCOS( SETA) #CSIN( Ho QRxT1L/2he ) =SIN(6e28%TO/246))/6028
ALNAPSASIN(AV)

RETURN

END



APPENDIX D
THEORY OF THE SENSITIVITY ANALYSIS

In this appendix the development of the equation (1) is
discussed together with the problems of convergence and
averaging of the data. The sensitivity analysis is further
discussed with the exact solution being developed for the
constant coefficient equation.

THE GENERAL STREAM MODEL

The equation (1) is developed from the energy equation
heat fluxes shown in Figure 1. The development of this
equation is based on a Taylor's expansion of the dependent
variable, T , about the position, x , downstream:

T(x + ax) = T(x) +2 (x) ax. (D-1)

Use of this Taylor's expansion together with the conserva-
tion of energy for the control volume in Figure 1 yields:

vA T(x) - uA [T(x) + 22 (x) ax] +
dx

+ £2 dx [Qpaq ~ Opes ~ Back ~ Evap * Qconv * Qmisc! = 9 -
(D-2)
The first term is the enthalpy transport into the control
volume; the second term is the analogous transport out of
the control volume; the third term is the total heat flux
at the air/water interface. Some minor algebra on equation
(D-2) yields equation (1), assuming that the quantities u,
A, and & vary slowly along the stream.

The equation (1) holds between points at which flows and/or
heated effluents are added to the stream. Many authors
directly integrate equation (1) over a station length, Ax ,
assuming the fluxes and u, A, & to be constant over that
distance. The result is simply

xtAx xtAx

J dx =
x

ole
ite)
Hw

lOpaa ~ SRnet ~ pack + 2gvap
x

+ Qconv + Misc! dx,

150


DATA AVERAGING

For small changes in temperature the equation (1) will be
quasi-linear. Thus, the theory of linear equations can he
applied to the problem. This is very important since
average data will produce average output to such a model,
while a nonlinear model will not yield results of this
type.

To prove this it is necessary to examine the theory from a
different viewpoint. For example, the temperature at any
point x downstream from the disturbance can always be
represented from the linear equation in the form

x
ro = > fre 5) 0, (6) (-15)
i 0

where the integral occurs over the domain of interest and
the kernel function Kj;(x, €) relates the effect of the
i'th heat flux Q;(£) “at the position &€ to the temperature
T(x) at the position x ; the sum produces the effects of
all of the heat fluxes. Since the expression is linear , it
can be summed over several sets of heat fluxes to obtain the
sum of temperatures as follows

n n x
YS yw - VV fe £) Oy5(8) ae
j i ojo? (D-16)

Suppose that equation (D-16) is rearranged and both sides
of the equation divided by the number of temperatures n :

n n

1 955

o> 00 = > fees ove iw a.
3 io j (D-17)

This proves that the effects of the heat fluxes can be
averaged and the equation used to compute an average tem-
perature. This would not be true for a nonlinear equation
and is a significant result.

154


EPA-660/3-75-037
JUNE 1975

IMPROVING THE STATISTICAL RELIABILITY OF

STREAM HEAT ASSIMILATION PREDICTION

by

Richard W. McLay
Mahendra S. Hundal
Kathleen R. Lamborn

Contract 68-03-0439
Program Element 1BA032
ROAP/Task No. 21 AJH/35

Project Officer

Bruce A. Tichenor
Pacific Northwest Environmental Research Laboratory
National Environmental Research Center
Corvallis, Oregon 97330

NATIONAL ENVIRONMENTAL RESEARCH CENTER
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CORVALLIS, OREGON 97330

For tale by the Superintendent of Documents, US, Government
Printing Office, Washington, D.C. 20402


9584000.

959-000
969+090
9614000
9624000
9639000
564000
943900
$65+900
967+900
968.000
9694990
979+900

9714000

272+000
973-000
974.900
975+000
9756000
9772900
978069
973-090
9890990
9312900
932900
9434900
9342000
909

110
141
120
124

125

130
335

135.

1101
1150
1151

ccccc
cceccc
ecccc
ccccc
cccce

IF(ABS(TI IM=T)+ABS(2004H) 11000,1000s141
TF (KDELY*4)1000»1202120

IF (AgS ¢2.0#H) *A3S(HMAX) 1121212141000
D3 125 TatsNF

IFCABS(DALTA (1) )SERRBR/(128+0%ARS(H)) 125.1254 1000
CANTINUE

IF (VadeCl IF "000625)130s 10002 1000
C3NTINUE

HsZe0*d

08 135 IstsNe

A(T) e2008A(TY

BCL) s4e0eB(1)

Col) 23e0eC(1)

O(1)s1660*D(1)

KDELYs90

G3 78.1000

CONTINUE

FORMAT (/s2Xe 14H sts 2XsF 10060 14Xs1KSTP ts 2Xe13s10X2 1DALSA(T) 9 /)
FORMAT (5 (2X 12a 1XsElbe7))

RETURN

END”

SUSREUTINE DFBUG( AIAN)
19Us102


HOL

265+000
2661900
2576000
268+000
269+000
272+000
2714000
2722090
2730500
2741000
2754090
276+000
277000
278*900
273+000
2392000
281+900
2824000
283-000
234+000
285+900
286+000
2376200
2884000
2290000
299°000
2914000
2924900
2934000

cecccc
c

ccccc
ccccc
ccccc

10

190

20

200

ceccc
ceccc
ccccc
c

ceccc
cccce
cccce

EULER AND MONIFTED EULER INTEGRATION

SUBRBUTINE EULER (YPsYsOTsNEsdsT)
DIMFNSISN YP(20)s,Y(20)s SY(20),SYP(20)
G5 Te(10s20)ed

CANT INUE

D3 100 IsisNF

Sy(ry= Yer)

SYP(T)=s YP(I)

Yely= Ytry+DT#YPry

CSNTINUE

TsT+0OT

RETURN

CANTINUE

D8 200 IstsNF

YOD)= SYCL)#CYPCL)4SYP(T)) #OT/36
CONTINUE

RETURN

END

FAURTH BRDER RUNGE-KUTTA INTEGRATI8N


10.000

“100000

10.G00
10.000

“100000

16.000
10,000

~ 100000

96000
8.000
19.000
10.000

10000

“102000

1-000
34600
52000
1.000
5600
1¢.000
10000
10-000
1c.000
10.Gou
10-¢G00
10.000
10.000

10.000

45000
450000
450000

472000

~ 450000

450000

436000

420000 —

380000
460006
47000
460000

450000

430000-

370006
320000
32¢000
4346600
480000
50+000
446000
450000
446000
446000
440000
44000C
460000

460000

684000

774000

804000
56000

586000

600000
734000
76000
824000

666000

618000

669000
71+000

~ 73.000

894000
964000
100000
100000
636000
50-000
684000
58«000
80#000
862000
860000
100090
100.000

100000
64

10+C00
“10-000
_ 146000
106000

~~" 72000

72000
42000
~ “49000
5-000

70000

6000
4eCOO0
34000
36000
3-000
3000

000
3-000
5-000
66C00
8000

17-000

11-000
7+000
3*CU0
66000
7000

6.600 -

54500

55°000
55+*500

55+500

55+500
556000
55«000
549700
549700
546500
54 +500

54200

“$40200

542000
54000
536700
53700
53°700
54200
54000
54°000

$3+700

53*500
53200
53°0U0
53000
53*500


os

102000 712000 66+000 3000 59000

102000 744000 60*000 ~— 6#000.~——s«¥§ 9#000
4,000 724000 64.000 36000 59+000
3,000 63,000 84,000 5,000 594000
72000 59.000 934000 30000 58*700
100000 646000 90+G00 4000 58*500
106000 684000 874000 106000 58 +200
2.00 762000 662000 + 126000 = 58 #500
8000 830060 46+000 166000 596500
8000 836000 514000 116000 60+000
34000 78s000 624000 ss Be COD 60#200
34000 73-000 76 +000 76000 600200
UA= 13503+000RK WIDTH= 400e000CLD CeVER# BeOO0TEMP ATR® 75000

TEMP WATER® — 62eOCOREL  HUM® 71sO000WIND SPEED 8700


66

1234000
124900
1250000
125°000
1274000
1284000
129+900
1302900
1314000
1324000
133-000
134.000
135+900
135+900
137+900
1384900
139+090
1496000
141000
142.000
143900

1452900
146-000
147.000
148+000
1492000
15909000

200
210

220

230

240

260

270
280

300
310

320

330

FoRMAT(! INITIAL DT #tE1204

WRITE (165210) Us AREASEL

FORMAT(' STREAM VELOCITY o!Féea! FT/SEC

1F1002' |SQe FTe

WRITE (18,220) RH8,CLD,RH, ALBAR
FORMAT() RHEstr5e2! LB/CUeFT.

ae RHetFSett PCTIyst

ALBAR atFuet!

WRITE(16,230) TAIRsTAsCANSTsCONSL
FORMAT(' AIR TEMPe se! Fool!
i CONST stE1O03! CANSL stFine3s/)

WRITE (18,240) WeWMPH,WMSs WMD

' DEG F'/)

cLostFuet!
DEGRFES!/)

OFG FIFgelt DEG Cts

CRe SECe AREA e!
SURFACE WIDTH atF6eqt FTI/)

TENTHS ta

FORMAT(' WIND VELOCITY s!F6e2t FT/SEC'F8e2tMPHIFRe2! METERS!
LI/SEC'FRe2! = MILES/DAYI/)

WRITE (182260)

FORMAT (/sT2OITTBLITSOIPTBLITAOIHFGTBI/)

O08 270 1 ©1410

WRITE(16,280) TTBL(I)s PTBL(I)s HFGTB(T)

FaRMAT(10Xs3F 2005)
WRITE(18s300)0CLOsDRHsDT
FORMAT(/1 DCLD s!F6.3t
WRITE(182310) DALPs OWMPH
FaRMAT(/! DALP =!FGe2!

WRITE (Tgs320) H9SaheCsHSsHSR

FORMAT(/1! HASEIE12 64!
IN')

WRITE(18,330) EAs DEA
FARMAT(/1 EA. alEi2ea!
WRITE(18s340)

AIR

DRH s!F5.31

DTAIR alFée

DWMPH =tF6e2)

HOCHIEIP at

DEA

eteieady

HSatE12e4!

2)

HSRetEl2e4t LY/MIN')


approximate stream temperature within an error of 2 OF
(1.11 °C) between 90 and 100 percent of the time. He would
compute an approximate stream temperature within an error
of 1 OF (0.56 °C) between 60 and 77 percent of the time.

Table II shows that the same observer at the site making
the same prediction of an average temperature as made in
Table I but using the average of two prior days' data
would compute an approximate stream temperature within an
error of 4 OF (2.22 °c) between 92 and 100 percent of the
time. He would predict within an error of 30°F (1.67 °C)
between 83 and 100 percent of the time. Similarly, a 2 °F
(1.11 °C) error would be obtained between 69 and 96 percent
of the time. Finally a 1 °F (0.56 °C) error would be ob-
tained between 47 and 53 percent of the time.

Table III shows that an observer using data from station 7
above the Vernon dam, attempting to predict stream tem-
peratures at the Vernon site by averaging a day's station
7 water temperature would compute an approximate stream
temperature within an error of 4 °F (2.22 °C) between 92
and 96 percent of the time. He would predict within 3 °F
(1.67 °C) between 88 and 92 percent of the time, within

2 OF (1.11 °C) between 80 and 92 percent of the time.
Pinally, he would predict within 1 °F (0.56 °C) between

52 and 85 percent of the time.

Thus, it is seen that predictions made at the site (with a
daily average) a short period of time into the future are
accurate over 90 percent of the time for a 2 °F (1.11 °C)
allowable error. Averaging data over a two day period of
time reduces the probability of meeting the 2 OF (1.11 °c)
error to around 80 percent, which appears to be a primary
influence of the travel time through this portion of the
Connecticut river system, one to two days. These studies
indicate that it is important to have good records of up-
stream temperatures from a site. With these it will be
possible to predict an average temperature within 2 OF
(1.11 °c). It is important to note, however, that short
term fluctuations are not predictable by these methods as
used in the project. Finally, the effects of season do
not appear to make appreciable differences in predictions.

23


i}

6C (Error in cloud cover) 2 tenths ,

6a (Error in sun angle) = 5° ,

6 (Error in relative humidity) 29 percent ,

5.2 MPH ,

6W. (Error in wind speed)

6Tayy (Error in air temperature) 16 OF . (8)
Thus, the problem will yield approximate errors for predicting
the water temperature on June 21, 1973 given the meteoro-
logical data from May 31, 1973. Of interest are the solutions
Cy of equation (6), which are due to the respective errors

in equation (8). They are found from equation (4) and
Appendix D to be

1, STop = 10.2 OF ,

2. 6T_ = 1.71°F  ,
aP

3. 8T yp = 0.357 °F ,

4. 8Tyop = 1.33 °F ,

5. 6p = 19.63 °F . (9)
airP

Similarly, from the same source the coefficient a can be
found:

a = 0.947 x 108 , (10)

The corresponding value of x where the value of 6T falls
below 5 percent of the original value at x= 0. is

x = 3,170,000 feet ,

(966,000 meters)

600 miles .
(11)
(966 kilometers)

This order-of-magnitude study indicates important facts
associated with the Vernon site:

19


€OL

236 +000 FUNCTION SI(XTBLs YT8L2 XaN)

Cc LINEAR INTERPOLATION GR EXTRAPALATIGN GF SINGLE VARIABLE FUNCTION
c XTBL = INDEPFNOFNT VARIABLE TARLE
239000 C YTBL = OFPENDENT VARIABLE TABLF
2400000 C IND s INDICATOR 8F EXTRAPGLATIBN
2414000 C BaNg EXTRAPGI ATIGNs LSLAWER EXTRAPOLATION, PebPPER E73RAPOLATION
2422000 DIMENSIBN XTAL(4O)s YTBL (40)
243.000 C CHECK Te SEE IF EXTRAP@LATION 1S NEEDED.

2446000 IF (X*XTBL (1) 12021300150
245-000 120 IND = 1

2460000 130 If + 2

2476000 G8 Te 254

2480000 450 IF(XTBLIN)#X) 14021802210
2492000 160 IND 5 2

253+000 180 tlen

251000 G3 TR 254
2524000 c FIND X IN TARLE*S IN TABLE
210 DA P20 IkKs2aN
It = IK

IF(XTBLOTK)*X) 920s 2542 254
220 CANTINUE
2570000 9 2B4 Xt = XTSI (Tt)

2584000 X2@ = XTBI CIT)

259-000 Yl = YTBI (TIe4)

2698900 Ye a YTBL CIT)

2610000 St_& Yie(V2"¥t) #(X9X1) 70 X2"X1)
2622000 RETURN

2632900 END

2642000 CCCCC


If LOG1 = 0 the next two sets of data are not necessary. If
LOG1 # 0 the next two sets of data for the dates of past years
prior to the date in question (the time-averaged data).

NTA: In 6110 FORMAT. This value is the number of lines of
time averaged data.

Following this NTA lines of data are read in as follows:

CTA(I), TATA(I), PHITA(I), WTA(I), TWTA(I): In 6F10.3
FORMAT. The variables are 1) the cloud cover in tenths,
2) the air temperature in °F, 3) the relative humidity
in percent, 4) the wind speed in knots, and 5) the
water temperature in °F.

Following this, the data describing the analysis are read in:

NCAS: In 6110 FORMAT. This is the number of cases of
temperature to be used in the trials.

DELT(I), I=1, NCAS: In 6E10.3 FORMAT. These are input in
sequence, the first six values of allowed temperature
on the first line, the second six on the second line, etc.

NDAS: In 6110 FORMAT. This is the number of day cases to
be tried.

IDAS(I), I=1, NDAS: In 6110 FORMAT. These are the number of
days of data to be averaged in each day case. They are
input in sequence, six values per line.

Following this, 10 lines of data are input for the sensitivity
analysis program, which is a subroutine in this case.

PTBL(I), TTBL(I), HFGTB(I): In 6E10.3 FORMAT (See STREAM for
variables description).

The next data to be input are 1) the number of days since
March 21 (the vernal equinox) which is required to estimate
the coefficient involved with the sun heating effect, and 2)
the latitude of the plant in degrees:

DM21, BETA: In 6E10.0 FORMAT.

The last data to be input are the daily average of the water
temperature at the remote site:

T7(I), I=1, Number of Days in 6E10.3 FORMAT.

32


tLe

29-900 CSMMEN UAsELsCCeTAIRs TWAs RHs WSp

34-900 CAMMEN INT2s INTQ220C2 DTA OPHI s DWeOTws INVSs TFLAG
320060 COMMSN DM214RETA

33.990 COMMEN LAG1sNDAS

34.900 CAMMEN TCASsPRP (50210)

350020 CCCCC
34900 CCCCC
37.900 ccecc
3+200 CCccc

394500 CALL 2FFAD

404500 OTet.

410009 DA 720 IDAF1sNDAS

426000 _ CALL PRSR

434950 20 CSNTINUE

444900 CALL PRS

432000 CALL VERNSN

454990 CALL EXIT

472090 END

48000 C SUBROUTINE FSR QEADING IN DATA

4360390 CCCCC
59.090 cece
51000 CccCcc
52.900 CCCCC
536090 CCCCC
9 SUBROUTINE RFEAD

COMMON YP1(5)2Y1(5)4XPR12XEND1sDXPRIisXL,DX1sNELs INTIs INTCLs INI s 181
56900 4eEL1,4REA19 U4



SEL

7910090
72020900
7330090
7946900
7954900
796+990
737-909
7986909
7394990
7194990
711-9900
7126900
7134900
7140990
71359000
719+990
7170909
7184950
7196990
7256900
7210900
7226090
7234900
72449900
7294000
7284900

¢
c

A anano

c

lade
cece

120

130

150
160
189

210

FUNCTI8N SI(XTBLaYTBLs XN)
LINEAR INTERPBLATIGON OR FXTRAPALATION OF SINGLE VARIABLE FUNCTION
XTBL = INOEPFNOENT VARIABLE TARLE
YTSL = OFPENDENT VARIABLE TABLE
IND = INDICATIR OF EXTRAPSLATIAN
3=N9 EXTRAPOLATION, 1*LEWER EXTRAPBLATION, 2=4PPER E73RAPGLATION
DIMENSI8N XTBL (40) YTBL (40)
CHECK 79 SEE TF EX7RaPALaTION 1S NEEDED,
Ip (X=XT31 (1)) 12021302150
IND = 4
Il = 2
63 Te 254
TF (XTBL(N) #=X1.° 16021809710
IND = 2?
Il N
G3 Te 254
FIND’ X IN TABLE’S IN TABLE
D9 229 Ike2sN
It 2 Ik
TF(XTBLOTK) =X) 22022544254
CONTINUE

~ XL = XTSL (TI-1)

X2 = XT3 CIT)

Yt = YTS (Tle4)

Y2 a YTAl (IT)

SIs: YL4 0201) CX" XL) 0X29 X1)
RETURN


1.000

66000
ho 9000
8+000

mo O99
10.000
102000
“409000
10*000
10000
"192000
10,000

10-000

ee 400 000
10.000

10000

oo #000
6000
62000
4000
5000
10-000
10.000
94000
4.000
2-000
‘ #000
10.000

690000

"760000
77000 |

69*000

"632900

616000
626000
646006
65000
75+000
76000
726c0c

70*C00

~ 68000

664000
67°C0C
790000
84e000
866000
81-0000
78*000
77000
630000
68+000
760000
830000
85+000

774000

68000

48000 —

482000
736000

974000
934000
90000
93000
69000

~ 66.000°

842000
90+000

"90.000 —

93.000

100000
974000

77*000 —

65+000
63-000
79000
88+000
886000
93000
100+000
974000
744000
53s000

646000
56

6000 73°200
“9.000 74000
_ 86000 749000
56000 73*500
"4,000 73.000°
46000 729700
4.000 72+700
~ 86000 Fee700
$6000 73*000
6«C00 74000
~~ "6s000 748000

46000 = «730700
46000 73500
56000 «736500 ~
4600 73*500
46000 73+500
86C00 ~~ “74*000
46000 744500
40000 75+000
4000 75*000
96000 75 +500
96000 75 +500
"96000 = 75 #200
4.000 75+000
46000 75 +500
“150000” ~=—- 76000
96000 76 +500
5000 774000



OLL

436000
4379000
4382000
439*000
442#000
441+000
4424000
4432000
4440000
4452000
4460090
4472900
448+000
4499900
459900
4514000
4320000
453000
454000
4554000
456-000
457.2000
458000
455-900
4600000
4614000
4622000
4634000
4642000

79

86

87

88

89

390

gt

08 79 T=stsNE

YeT)eSTARY(L)

Aclys-A(1)

c(lyeectr)

G8_T8 1000

16TH STEP» HaLVP Hs aPPLY TEST 1
HsH/200

D8 87 LatsNE

ACL) 2A(1)/200

B(1) 2311) 7400

C(LyaC( 11/800

D(Ty29(19/1600

09 88 LatsNE

IF(ABS(DALTA (1) )°ERRBR/ABS(H) RRs 88289
CONTINUE

PaSs TEST GO FAWARO WITH HALVFO H
G3 T8 78

FAIL TEST BEGIN AGAIN WITH HALVED H
H="H

D8 92 1=1sNE

YCT)SSTARYCT)

Ge Te 1

24TH STEP» OBUBLE Hs STARTING SFQUENCE ENDS
HsHe200

D9 94 I=4sNE

ACL BACT) #200

B( 1) aB(ti#4e6

CclyaC(1y#8ed

D(1y2D( 1) #1600


72000

~""94000

9600
10-000
10+600
10-000
10.000

100000

10-000
92000
10000
7«C00
92600

“9.000

64000
72000
"7000
96000
92000
106600
92000
10.000
10-000
10-000
8.000
72000
5.000
10.000

36+000

380000
38+000

360000
36000

“330000

~~"360000

36000 ~

38+000
402000

~ 410000

38+000
366000

~ 350000

320000
326000

346000

332000
316060
29000
264000
254000
254000
25000
309000
336000
316000
316000

52000

46.000

462000
53000

“874000

62000
624000

“574000

$0000
45000

“wB3600 7°

57000
424000
42.000 |
452000
434000
“386000
454000
434000
45-000
462000
46000
53.000
664000
56000
42.000
436000

452000
74

6«C00

--.400CQ0
7000

7000
10000

_ 456000
14.000

8000
122000

~ "74000 |

7000
12-000

12.coo”

126000
156000
13.C00
14.000
12.000
12.000
402000
92000
“94000
10000
116000

“T2ecoo

~~Bec00

~~ $5+600 —

13-000

44°700

440700 —

44200
449000

““aheQ00

440000

430500

"434500 —

43e500

434500

43500

430200

"439000

43000
434000
~ 43000
43000
429500
“420200
420000
41700
"410500"
410500
410500
“449200
410200

41000

“434500


SECTION VIII
SYMBOLS AND VARIABLE NAMES
VARIABLE NAMES

a - General integration limit.

A - Stream cross-sectional area.

A,A - Amplitude of exponential decay function.
Cc - Cloud cover.

Cy - Constants.

c - Specific heat.

DT - Allowable error in stream temperature.
f - Function of several variables.

£ - Derivative of £( = = a£/dx)/

h - Integration step.

I,, Ig - Integrals.

Kj - Kernel function.

&, EL - Stream width.

n - Integer number.

O(-) - Of order (Order of magnitude).

P, Py - Probability of occurence.

Q - Total heat flux.

Qi, Qij3 - Particular heat fluxes.

Qpack - Back radiation heat flux.

Qconvr Oconvection ~- Convective heat flux.
QEvap, Snvaporation ~ Evaporative heat flux.
OmMis(c) ~ Miscellaneous heat flux.

Qrad, Qsolar - Radiative solar heat flux.
Orefr QReflective - Reflective heat flux.
T - Stream temperature.

T> ~ Initial stream temperature.

U,u - Average stream flow velocity.

W, W2 - Wind speed.

x - Distance downstream.

Xj - Particular position.

26


26

3994000
4002000
4919900
4024000
4031900
404000
4054000
4062000
4074000
498-900
499000
419.000
4412000
4429000
4130090
4142000
4154900
4160000
4170090
4186000
419*000
4292000
421+000
4222000
4234000
4242000
4254900
4262000
4274000

88

89

92

390

91
100
102
103

105

CONTINUE
PaSs TEST G8 FAWARD WITH HALVFD H

G8: TS 78
FAIL TEST BEGIN AGAIN WITH HALVFD H
HeeH

06 92 Is1sNE

YCL)sSTARY(1)

Ge_T8 1

24TH STEP, DAUBLE Hs STARTING SFQUENCE ENDS
Hshe2eO

D8 94 I=4sNE

ACT) SAC 1) #200

B( 1) 2B (Ty #400

Cily eC Ty #800

D¢(1y29(1) #1600

Ga Te 78

KDELyskorLy+4

WILL NEXT STFP MByE PAST TLIM

IF (ABS (TIIM=T)*ABS(H) 110341034110
YESeeseeeeeeSAVF T AND Ys INTEGRATE TS TLIM,RETURNe
ENDHsTLIMeT

09 105 1te1sNF

AAC T) SENDH#A CT) /H

BB( 1) SENDH##P BIT) /HaHe

COLL) SENDH¥*R HC (1) /HHH3

DO(T) SENDH##4HD (CLI /HE HE

SAVFT3T

O03 800 TsisNF

SFC1)=F(1)



1 Orca t Qret # Qeccx t evap 4 Q conv 4} Qwise

Figure 1. Control volume for stream model


10+000
10000
94000
"8000
___8 9000.
10-000
“100000
104000
10.000
10-000

10+C00

16.600

10,000 ~

109000
102000
10.000

334000

290000

272000
29000

(336000

36000
342000

34¢000
340000

354000,

37+000
390000

__. . 399000

_..10+000 —

40.000
~ 10+000
_ 106000
104000
10000
_ 306000
104000
10+G00
109000
10-000
10000

86000

396000
410000

#40000

414000
442000
430000
450000
460000
462000
4698000
48e000
512000
540000
$50000
612000

~ 810000
_ 270000 |

~ 66000
550000 62000 |
536060 6-000
494000 ~~ 96000
_..#8 000 #0000.
424000 7000
4446000 6600
_.. 500000. 000.
520000 7000
644000 7+000
594000 106000
594000 126000
93.000 144000 ©
.84+000_.___156000_
48 6000 146000
“436000 144000 ©
434000. 136000 _
434000 13-000
45000 10coO0
470000 136000
494000 10000
£04000 84000
52.000 e000.
526000 8c00
48.000 446000
48.000 146000 _
510000 14-000
974000 10-000
464000 434000

75

__429500_

1-000

__. 416000

419000

419000"

424000

422000
41¢500

_ 410200

410200
410200

419200

412500

410500 7
449500

410500
410500

415500
41+500
410500
410500

419500

419500

419500

410500

419500

419500
419500
429000



LB

850000
874900
886000
39+000
904000
912900
924900
932000
942000
954000
960000
974000
98000
99000
199+000
101 #000
1022000
103+090
194000
105*000
104000
1072000
198*000
1094000
1102000
1412000
1126000
113000

180

185

200
210

220

WRITE (10,180) EQRERsDXMAX

FORMAT(? “ERRGR stE100e3! = DXMAX BIELOe30/)
READ(INs10) TEMPsUsAREASEL

READ(IN, 10) RH8+WMPH, TAIRsCLDsRHs ALBAR

08 185 Is1s10

READ(IN, 10) PTBL(T)s TTBL(I)s HFGTBIT)

TA a (TATR = 3261#50/9e

Wo? WMPHw220/15¢

WMS « Wo# 03048

WMD = WMPH * 24

CONST » FL/(RHS*U*AREA)

CB8NS{ = (RH8#0+00328)/(P4e#36000#000654)
SINA * SIN(ALBAR#3+14159/1800)

CLD1 = 16 = eO7K45¥CLO

INC{DENT SOLAR WADIATION

HES = 169*SINA

HS = HOS#(1e = «0006*CLD##3)

REFLECTEN $81.AR RADIATIBN

HSR = HS#3e/ALBar

EA = RH(19130714700)*SI(TTBLsPTBLs TAIR« 10)
WRITE(18.200) TREMP

FeRMAT(! INITIAL STREAM TEMPERATURE ='F602!
WRITE(1@s210) Us AREASEL

FSRMAT('| STREAM VELSCITY ='F603" FT/SEC Re SECe AREA at

DEG FI/)

1F1002' SQ» FTe SURFACE WIDTH atFéeit FT'/)
WRITE(18s220) RYGsCLOsRHs ALBAR
FORMAT(! RHBalF5e2t LA/CUFT. Clos'Fael!

“TENTHS! a

it RHetFSey! PCTty!' ALBAR stFuei! DEGRFES'/)


SECTION II
RECOMMENDATIONS

The results of this study indicate that, in this case,
water temperature at the site is the most important
variable in the prediction of the downstream Connecticut
river temperature. In order to minimize the errors of
prediction, it is recommended that temperatures be
monitored upstream of the site and predictions be made
over as short a period of time as possible, i.e., average
the previous day's data to predict the following day's
average temperature. This is motivated by the need to
know the initial temperature in the analysis for the entire
travel time of water through that portion of the stream
being simulated.

These predictions are useful to a plant operator making
day-to-day decisions as opposed to a planner predicting
conditions to occur several years in the future using an
historical weather data base.

This work indicates that for the Vernon plant such pre-
dictions will have error within 2 °F (1.11 °C) 90 percent
of the time. Smaller streams may have greater effects from
meteorological parameters, which were not found from the
Vernon problems. It is also recommended that similar
sensitivity analyses be performed for two-dimensional models
used for predicting temperatures in lakes, estuaries, and
cooling ponds.


60L

408-900 De 56 ImtsNE

4994900 ACT) aA(TY/200

4102000 BCL) e801) 7400

4110000 CUT yal (1/800

4124000 FCI) sSF(1)

4134000 VCD) sSAVFYCT)

4144000 56 OLT)89(1171600

41520000 G8 Te 1000

4160900 C PASS TESTS, CORRECT Av BsCsD
417900 60 KSTPsKSTP41

4180000 D8 65 TatsNE

419-900 ACT SACTI+3004B(T)FHe0#C(T)41000*#D(T)+P@DALTA (1)

4294900 62 BCL) S301) +40 O¥C(1)410204D (1) +Q#DALTACI)
4214900 64 CCLYSC(1)+5 008001) +R*DALTA(T)

422+000 67 D(Tysd0 TY +S#DAL TACT)

423.000 65 CONTINUE

424.000 C IF IN STARTING SEQUENCEs BRANCH

425.000 TF (KSTP=24)702902100

4769000 70 GE TS (1000s 10002 10002 74s 10003 100s 1000s 78s 1000s 1000, 1000274210004
4274000 11000210002 86210002 1000210002 74510002 10002 1000)sKSTP
4284000 C 4TH, 12TH» 20TH STEPs G@ BACK

4296000 74 Hae

4302000 08 75 IetsNE

431000 A(T) ee Att)

4320000 75 C(liseCit)

433000 G3 T6 1000

4340000 C 8TH STEP GB FBWARD

435000 78 Hae


6L

?9*900
324000
41-900
322900
332000
34-000
354000
36000
374000
334000
39+000
492000
414000
420900
434000
4469900
454000
46+000
474000
480900

“494000

526000
51-000
526000
§3+000
54900
550000
560000

105”
110
120
130
134
140

150
300

10
20
c

XPR = XPR + AXPR

IF(X = XFND) 523002300

CANT INUF,

JUMP = <4

CONTINUE

XPR 3 X 4 DXPR

CONTINUE

CALLNSRO (OTEMPs Xs XPR» TEMPSERRSR st sD Ks OXMAXs JUMPaKSTPaKCs 1 eB @ 6s 10)
IF (JUMP) 13021402150

WRITE( 18,131) X

FgRMAT(//! XM s!€14e5!  INTEGRATIQN FAILURE?!)

CALL EXIT

CONTINUE

CALL DIFFO

G3 TS 120

CALL SUPT

IF(X = XFND) 1104300,300

CALL EXIT

END

SJBRBUTINE INPUT

“CAMMBN DTEMP,TEMP, XPR»XENDsDXPR»XsDX4 INTSINTCe

1 IN, TSsF1 s ARFASUs ERRSRs DXMAX

COMMENRHSs WMDZHDs TATRsCLOsRHALBARsTWSs TAs EWsEAHESAHS |
COMMEN HSR»HASsEBReEVAPSHEsHCsHLYMNe PTBL (10) sTTBL(10) »WMSsCONST
CaMMEN HFGsCLO1sHFGTB(10)sCONS4

FARMAT(6F 1000)

FORMAT (6110)

CHOICE SF INTEGRATISN SUBRGUTINF


sot

2942000
295-000
235000
2974000
2984000
299000
399+000
301000
302900
3030000
2044000
305000
306900
3074000
398+900
399-000
3196000
3116000
3120000
3136000
314600
3156000
315*000
3174000
318000
3194000
329+000
321-2000

10

100

20

200
30

300

40

SUBRBUTINE RK4(YPsYs0TsNEs ds T)
DIMENSTIAN YP(5)s¥(5)2AK(524)sS¥(5)
GB 18102202302 40)5

CgNT NUE

De 100 tetsNe

Sy(r) ® ¥{T)

AK(tad) = DT#YP(T)

YCL) 3 YC) FAK Cra teed
CONTINUE

sta T

TS T + DT#e5

RETURN

CANTINUE

DS 709 TalsNF

AK(122) 2 DT#YPCT)

YCL) = S¥(T) + AK(Ts7) #05
CONTINUE

RETURN

CANTINUE

03 300 1 « i4NE

AK( 123) = DT#YPET)

Yer) © SY (1) + AKOTS3)
CONTINUE

T * ST + DT

RETURN

C8NTINUE

De 400 1 = 44NE

AK(ts4). 2 DT #YP(T)


06

342.000
3432000
3442000
345-000
346*000
3476000
348+000
343+000
352-000
351000
3522000
353-000
354000
355000
356*000
357000
358-000
3390000
3602000
361000
3622000
363000
364+900
365-000
366000
367+*000
368000
3694000
379s000

29
30

31
35
40
45
50
58

223

56

60

62
64

YCL)sS¥(T)+VHDALTA (1) #H

C8NTINUE

TEST FSR STARTING SEQUENCE

IF (KSTP#?8) 35240540

APPLY TEST 2 8N ZERSTH STEP

IF (KSTP)50s50260

HALVING TESTS

D8 45 TetsNE

IF(ABS(DALTA (1) }*ERROR/ABS(H) 45245955
CONTINUE

TF (V#H#el TF 001295 160260s55

TsTeH

FAIL TESTS» HALVE H

H=H/200

KDELY=0

D8 56 1e1sNE

ACT) sA(T)/200

Bt lys3¢1y/4ea

CtlysC (1/80

FclysSFir)

YCI)sSAVFYCT)

D(1)=D(1)/1600

G9 Te 1900

PASS TESTSs CBRRECT AsByCaD

KSTp=KSTp+l

D8 65 letsNE

ACL) GACT) 4300uB(1)4600eC (1) 410004011) +P4DALTA
BC Ly eS (Ty +4eO%C(1)41000*¥D(1)+QxDALTA(T)
C(T) eC t) 45 ¢0#D(T)+ReDALTA(T)

(1)


‘
: ~

102000
fo,

;

“ 100000

10,000

"100000

46000

95000

5-600

9+000

42000

162000

10.000

10-000
10000

102000
10+000
10-000
10000
10.000
106000
10.000
100000

8000

92000

aoe

10+000

106000

96000

“100000 —

10+000

62+000

“629000

60000
566000
“548000
510000
510000
619000
636000

649006

~ 576000

520000
512000

"500000

510000

596000

~ “59e000

55¢000
54*000
540000
54000
540000
54000
550000
57000
59000
57°00
576000

486000

“§00ecoC

58000
734000

~~~834000—

90000
93000
980096
462000
474000

~~ 694000

832000
86+000

“B6e000-

90+000
70+000
75000
904000
934000
96000
964000

96000

96000

1002000
364000
932000
393°000

96+000
58

9+000

6#000
32C00

“~"3eC00 |

3+000
3000
100000
9eC0O
8000

3000
6+000
4.000 —
7+000
82000

50000
8+000
46000
66000
46000
—_ 66000
3*C00
8000
5eCOO
9¢C00
7+000

86000

“3000070

56000

700500
70500
70*500
70500

"700200 0°

70000
69500

699500

70200
70500

70500
70200

“699700
_.. 69#500

69°500

~ 689500

68*200
68000
68000
68000
67700

“ 67«500

67°700
67*700
67700
679700
67*700

7190000—~—CO


30000

_ 86000
94000
“104606
8.000
3000
86600
56000
72000
54000
99000
94000
32000
34000
#000
+000
40000
2.600
1.600
«G00

#000
#000
4000
8000
8+000

~ "8.000-

46000 —

“9000°

660000

 76«C00

85*00C

85+000

674000

660000
65000

~ 639000

63+000

740000

~ 8C+000

69¢000

720000

* 670090

630000
60«000
630000
749000
739000
81000
730000
67+000
630000
610000
626000
75000
80+000

800000

397 000

740000

57+000
51000

87.000

93.000
93-000
97000
100+000
84.000
674000
846000
34,000
"936000
974000
976000
97000
744000
60-000
544000
694000
84s000
30+000
934000
97+000
69+000
494000
566000

54

3°C00

“69000

4000
9¢C00
12.000
56000
5«C00
66000
_4«C00
4000
“46000 —
9000
5-000
5000
34000
+000

46000

102600
5-000
5.000
64900
36000
6+000
7*C00
36000

11-006

10+C00

10000

76#700

769700

779200
77000

77460000

764700
76500
76*500
75#700

76#000

76*700

77200

77*200

76200
75700

75*700 0

76+000
76200
76+700
77*200
76 +500
76*200
760200
75*200
75*500
769000
764500

“760500



Qo vw 2
——==—a
SCALE IN MILES

RETREAT
MEADOWS

NEW HAMPSHIRE

VERMONT

CHESHIRE CO.

WINOHAM CO.

FRANKLIN CO.!

MASSACHUSETTS

Figure 5. The Vernon nuclear plant-temperature
sampling stations

15


9+U00
100600 _
10-000

2.000

_ 69600

3.000

2+C00

1-000
+00
+000

#000

+000
“4600
+000
+000
+600

_ 46000
8.000
64000

__ 6+000

10.000
106000
106000
10-000
102000

104000.

16.000
10.000

106000

669000
566006
530000

~ 470600
416000 _

470000
510000
57+000
592000
56000
430000
390000
369000

_ 4746600

620000
684000
702000
650000
$3*000
49000
536CO0
586000
979000
60+000
640000
630000
620000
600000

600000

_ 964000.

65-000
90+000
964000
83000

68000

46000

446000

454000

_ 834000 _

93000

* 964000

_. 834000

41.000
33-000
364000
522000
862000
89.000
864000
84.6000
966000
100+000
970000
100000
100+000
100+000

100+000
45

440000.

__. 38060 __

eco00

-_- 8 0C0

66000

be000
_#9000_

12+c00

10+000

_126coo |

10-6000
52000

36000
000

30000

16.000

5000

22000

32000

30000,

4000
9eC00
6+0C0

#000
C000

46000

36000
36000

32000

86000 _

61°700

619500,

61+000
61*000
_ 61#000_
60*7C0
60700
619500
61+700
614500

3. 1619200

61#000
~ 609700

61*000
“@4200

61*500

“619200

619000
61000
_ 619200
619500
~ “629000

61+500
619200

619200

619500

608700,

619000

629000



100000 51000 80°000 26000 68500

100000 | 490000 894000 #000... 689000.

: 106600 480000 366000 4.000 679500
"==" = 9999 508000 «964000 7ecer  — 67#000
_ _..106600_ 520000 964000 7e000_ «79000
10+090 52.006 100000 5.000 67500

~ 400000 = 538000 100-000 ° 6000 689500
oe 10+000 52000 100-000. 126000 = 68 +500
1oecoe 52.000 100-000 12+c00 68+500

~ 10.000 49+000 100-000 © 46000 67*700
cee — -5 9000. 450000 960000 _____ e000... 67 #500
34000 51-000 30+0C0 8-000 67*000

64000 570000 724000 «#000 7#000

cence ene, 44000 629000 | _ 886000 __ 124000... 67.9200,
3+000 65-000 50.000 10+000 67+700

a 34000. 624000 542000 — 86000 67«500
54000 490000 934000 == 3#000. 66 9500.
72000 454006 1004000 26000 65+700

. ~ 8.000 434000 £00000 34000 65*000
. 106000 = 50*000 934000 | 46000. 63#500
10-000 61-000 754000 1U+co0 62+200

10-000 626600 734000 11+c00 62¢200

- _ 89000 654000 614000 126000 ___ 62#700
10-000 59000 754000 106000 62*700

21000 554000 904000 ©6000 62#200

a 2000 55000 «930000 e000... 649200
' 106000 559C00 966000 2ecoo 60#200
: 10-000 572000 934000 3000 599500
10+000 654000 78.000 52000 59*000

49


BL

1.000 ¢

29090 C

32900 C STREAM

46000 C

5#900 C

60000 C STREAM THERMAL MSDFL

7-000 C

86900 C

9-900 COMMBN DTEMP,TEMPsXPR»XENDsDXPRsXs0Xs INTe INTCoINs LO,ELsAREASU
100000 CaMMEN ERRBR,OXMAX

11-6000 COMMENRHAs WMDsNDsTATRsCLDsRHs ALBAR,TWS, TAs EWS EAs HOSSHS
12+900 COMMON HSRsHABsEBRsEVAPsHEsHCoHI.YMNs PTBL (10)s TTL (10) 2 WMSs CONST
13.960 COMMEN HFGs CLD1sHFGTB8(10)s.CENS1

142000 INES

150900 19*6

16900 CALL IN°UT

17+0900 IF(CINTC © 4) 451052105

18000 4 XPR_ a DXPR

190000 5 CBNT1NUE,
20060 08 100 uF= 1,INT
21+000 CALL OLFFQ
22000 G8 TS (1019.20). INTC
p3*900 19 CALL EULFR(OTEMPs TEMPsDXo 1s JES Xd
242900 GS TS 100

254000 20 CALL RK&COTEMPs TEMPsOXa ta dba X)
26+900 190 CANTINUE

27.000 IF(X © XPR ) 52502250
28-000 250 CALL SUPT


Lae

8726000
8736000
8746000
875+900
8762000
8774500
878.000

8739+900

8396000
83812900
8322000
883.000
884+000
8854000
83864000
8874900
8384000
889+000
8994000
8916000
8926000
893-0000
8949090
8954060
8964000
8972000
8984000
8994900
909000

36

60

62
64
67
65

70

74

75

78

79

YO) sSAVEY(T)

D(1) 8301171600

Ge Te 1000

PASS TESTSs CBRRECT AsBsCsD

KSTPeKSTP41

D9 65 LatsNe .
ACTISACTY+36O#B (TI +G0O#C( 1) 410008007) +PHDALTA (1)
BCLs Cry +b eO¥C UT) +40004D (1) 4QeDALTA(T)
CELTS CT 45 eO#O( 1) eREDALTACT)

Del) aD (Ty+S*DALTAIT)

CONTINUE

IF IN STARTING SEQUENCE, BRANCH

IF (KSTP=24) 702905100

G9 TB (100041900 10002745 1000410002 1000278 1000s 1000110004741 10004

1100041004864 1050. 1000s 10002 74s 1000 1000s 1000)sKSTP
4THs 12TH» 20TH STEP, GA BACK
HseH
O08 75 TeqsNE
ACT) s=acr)

Cilyeec(r)

G3 TS 1000

8TH STEP G8 FBWARD

Heer

03 79 TatsNE

VOT) eSTARY(T)

A(T) seA(t)

Cilyesc(r)

GS 16 ton0

‘6TH STEP, HALVE H» APPLY TEST 14


hel

673+900
674.000 C
675+500
676-000
677+900
6736000 C
679+900
6392099 C
6814000
6824900 C
633050
634-090
6454930
68454900
6874000
688-000
6394950
6990900
691+950
692+900
6936060
63949090.
6954009
636+900
6976590
638-099
699-6090 CoCcc
709.090 Ccecce

1

1

E8R = H9R*CLO1

EVAPBRATION HEAT TRANSFER®

EVAR 3 e25% (EWMEA)#(Le + ©O09R* WMD)
HFG a SICTTAL sHPGTAs TEMP, 10)

HE

EVAP #HFG*CINS1

CONVECTIAN HEAT TRANSFER

HC a 3904026 + eO77#WMS)¥(TWS = TA )/14400
TSTAL HFAT TRANSFER TO WATERs |.Y/MIN

HLUYMN s nS = (E3R +.HSR + HC + HE)

TSTAL HEAT TRANSFER» STU/SEC-FTD

4D
Fa
F2
F3
Fa
FS
F6
F7

4LYMN *® «Q4514

CLOA*DC. O¥4ES

CLOR*HOC*DALPR

s07465*0CLO*HAR

TCLOL# Ce sO09*DTWS = 6 O046#DRH) 69072

= (FV4F2)*36/ALBAR

AS #2+*DALP/ALRAR* HD

= (eo RS*HEG/144000 HC (Le + eQO9R*WMD)*(DEW = DEA) -+

*Oo93* WMO (EW = FA))
FS = =(39e/14HO+) #6 (026 + ©O77HNMS)*(DTWS = DTA) +
eO77#DWMS#(THg = TA)) 7
DLYMN = FL + F2 + FQ + F4 + FS + FE + F7 + FB
D3TUS =» ALYMN¥* 614

DOT

= ‘CANST#N38TUS

RETURN

END


C ( dieu)

pees
ry ty

Vit Tair

@ (Arcus)

W (wind)

Initial

(nt rare) xX>
1

T To E - eel

Stream Ter ( stream remperoture )
(ase)

Figure 2. Six primary meteorological and site variables

10


9EL

7394039
7412990
732+509
733-900
7242900
73354900
738°090
7376390
7384900
73396590
7492099
7412900
74263500
7430050
7446900
7454950
7460900
7472000
7432990
7490999
7529900
731-990
590
7339990
734.099
7332090
756+G00
7572000

ccccc
cccec
ceccc
cceccc

19

190

20

200

cecce
cecce
cccec

cecce

SUBROUTINE EULER(YPs YsDTsNEs daT)
DIMENSISN YP(20)2Y(20.)4 SY(20),SYP(20)
G3 Te(10,20).9

C3NTINUE

09.100 I=tyNF

SY(1Ty= YCT)

SYP(1)= YP(1)

YcL)=s YT) +DT#YA(T)

CSNTINUE

TsTs0T

RETURN

C3NTINUE

D3 200 TstsNF

YCL) 2 SYCT)#CYPCI)4+SYP(1) )#DT/ Pe
CANT INUE.

RETURN

END

SUBRBITINE RKGCYPsYsOTsNEs Ue T)
DIMENSION YP(5)sY(5)sAK(5a4)sSYV(5)


~ SEPTEMBER 27, 1973
AUGUST 15 to 31 and SEPTEMBER 12 to 26

___NUMBER 6F DATA LINES=

256TIME AVe LOGIC®

CLO COVER AIRTEMP REL HUM WIND SPD

52000

_5:000

10060
3100600

90000
102000

~~ 102000
_ 49000

#000
92000

106000

102000
4-000

_ 44000

#090
+000
+000

16000

8-000

8.000

84000

74090

94000

9.c0u

5+CO0

5.000

666CO00

64000

66#000
700006

_ 784000

750000
712000
662000
62+000
622000
62¢000
67+000
779000

_ 739000

75+000
65000
62000
59000
62¢600
750000
82000
836000
790000
682000
669000
64000

874000.

8720000

900000

90000
844000

-694000.__
664000

76000

934000
974000

100-000

934000.
624000
474000
584000
874000
934000
974000
974000
714000 |
55000
914000
654000
936000
934000

974000
53

O. .

WATER TEMP

“e000 78000
600CO. 779500
7°Cod 77°200
86000 77500

106000 77200

10-000 77000

66000 77000
_..32000_ _77#000

36CoO 76*700
32000 76+700

4000 762000
4.000 75+700

~ 8.000 75+700

34000 #500
3000 77000
35000 76+500
36000 76*200
46000 76+500
46co0 76#500

_ «C00 __ 76#500
56000 76 +500
7#COO 77500
4eC00 77*700
36000 77#200
3+C00 76*700
3-000 76#700



Zz
uewn =|o

FIGURES

Control Volume for Stream Model

Six Primary Meteorological and Site Variables
Model Using Site Water Temperature Data

Model Using Site and Station 7 Temperature Data

The Vernon Nuclear Plant-Temperature Sampling
Stations

iv

Page

10
11
12

15


10.000
w ~ ~""106000
a we ee 109600
10,000

a “10,600
106060
106000
92000
8-000
2.600

~ “4000
+000
+000
+000
+000
+000
“4.000
+000
34000
5+000
+000
000
+000
+000
+000
45000

36000

7+G00

440000

44.000

464000
46,000

~~" 46,000

442000
44000
41«C00
390000

43000

~ 474000

49000
400000
372000
34+000
33-000
326000
47000
560000
58*0UC
45000
446000
400000
38+000
35000
560000
620600

630000

360000

100.000

160.000
100,000

~ 96,000 —

100000
100-000

~ “100-000

894090
71.000

~~ 564000

50000
86000

~~ 894000

326000
360000
100+000
936000
646000
602000
93%000
864000
36000
1002000
100+000
722000
58000
946000

70

7«C00

~~" 36000 ©

66600
7,C00

10,c00 —

7+€00
66000
“7eC00
+000

9+000

~ 460087

42000

3*000

36000

96C00
4.000
“36000
10000
6000

7000

5000

3+000
5000
116C00
106C00

6000

3000° 7

“ §30000

53°200

530000
532500

“53.200

53000
53+*000

520700

529500
52+¢200

529700

53200
53000

‘520700

529500

52e200

“52000000

52000
52000
520500 —
52+500
52+500

~ 52*000

52000
51+500
510700
52000
52+500


SECTION VII
REFERENCES

Aquatec, Inc, 1974, Ecological Studies of the Connecticut
River Vernon/Vermont, Report r'ti Prepared for Vermont Yankee
Nuclear Power Corporation, Rutland, Vermont.

Asbury, J.G. 1970. Effects of Thermal Discharges on the
Mass/Energy Balance of Lake Michigan Argonne National
Laboratories, Argonne, Illinois Report No. ANL/ES-1.

Hogan, C.M., L.C.Patmore and H. Seidman 1973. Statistical
Prediction of Equilibrium Temperature from Standard Meteor-
ological Data Bases U.S. Environmental Protection Agency,
Corvallis, Oregon. Report No. EPA-660/2-73-003.

Jaske, R.T. 1971. Use of Simulation in the Development of
Regional Plans for Plant Siting and THermal Effluent Manage-
ment Paper 71-WA/PWR-3, Presented at the Winter Annual Meeting
of the American Society of Mechanical Engineers.

Laevastu, T. 1960. Factors Affecting the Temperature of the
Surface Layer of the Sea Commentationes Phvsica-Mathematical
XXV, 1, Societas Scientiarum Fennica, Centraltryckeriet
Helsingfors, Helsinki.

Policastro, A.J. and J.V. Tokar 1972. Heated-Effluent Disper-
sion in Large Lakes: State-of-the-Art of Analytical Modeling
Part 1, Critique of Model Formulations Argonne National Lab-
oratories, Argonne, Illinois Report No. ANL/ES-11.

Rosko, J.S. 1972, Digital Simulation of Physical Systems
Addison-Wesley: Reading, Massachusetts p.2.

Salvadori, M.G. and M.L. Baron 1961. Numerical Methods in

Engineering Prentice-Hall, Inc.: Englewood Cliffs, New
Jersey.

25


ADDITIONAL DATA FOR STATION 7 ABOVE THE DAM

TEMPERATURES AT STATION 7, CF, DAILY AVERAGES,
OCTOBER 1-3, 17-24, AND 27-29

60.4, 60.0, 59.8, 55.8, 55.0, 54.5, 53.6, 53.4,
53.0, 52.6, 52.9, 52.7, 51.9, 51.0

68


Ghl

946.900
987-900
933-6000
9394900
$99»900
991-000
93920000
993999

9975050

9936000

999-2990
10902090
10014000
10026090
1993%000
1094-930
1095+930
109$+030
1997+990
£0386009
10994960
10198000
10112900
1012*090
10134900
1014000

2000
2001

WRITE (13021009) 4aN
1000 FORMAT (1HSs2Xs tall Lhe 75 1X0 tNatsTe)

RETURN
ENO

SUBROUTINE PRAU

CaMMeN

YPL(5)9¥1(5)¢XPR14XEND1 se DXPR1sX44DX14NELS INTIe INTCL9 INL 104

1LsEL1sARFAL.UI

caMMEN

CaMMaN

ERROR + DXMAX1s TEMPL/HFGT31(10)sCANS14
RHS1+WiaWMDLsHO1s+TAIR1+CLD1.RH2s ALBAR1sTWS1 TAL, TWSALZEWL)

1£41,H9S1,HS1

caMMeNy

HSR12H9B1,EBR1sEVAP Ls HE4 SHC »HLYMNG »PTBL1(10).TTBL1(10)0

1WMS1,C8NATL

caMMeN

Cl IFAsHF34+HOCL4CLO114CL 921+ CLD31,DALP1sOALPR1»DCLD1sDRH1s

LOTAIRISDTALs N41 6 DWMS1 + OWMD » DEA sDE412OL YMN1,DBTUSI »NTEMP La
@STT3L1(1m)4XTBL1(10)

C3MMBN
CaMMav
COMMBN
c3MMsN
caMMaN
COMMEN
CaMMsN
CaMMSN
C8MMBN

C¢1000)TA(1000)4PHI (1000) »W(1000),TW( 1000)
NCASs NDAs DELT(SO)s pOagt5O)

NLNa TNs T9UsDTsNTAs IDA

CTA(5O)s TATA(50) PHITA(5O) »WTA(50)4 THTA(5Q)
UAsELsCCsTAIR, TWAsRHaWSP

INT2s INT 3s DCs DTAsDPHI,5W,DTWs INVSs IFLAG
DM21sRETA

LAG1LsNDAS

TCAS» PRP (50916)

WRITE( TSU, 2000)

FSRMAT (1H072Xs *PRABABILITY CHART! /s2Xs60( tet) )
WRITE (T9t22001) (DELT(I)sTetsNCAg)

FORMAT (1HOs10Xs 10(E100341X))

D8 10 IstsNDas


70000

_. 104000

14000
~ 100000

_.. 409000

106090

106000
104000.

10-000
10-000

_ 104000

106000
10.000

_. 100000

5-000
74000
__2000
14000
36000
3.000
7000
86000
44000
+000
#000
8.000
2.000
72000
7.000

562000
532000
616000
642000

572000

55«000
520000

_509000

50000

502000

500000

51.000
530000 |
53000
50+c00
49006
460000
430000
56e000
634000
65000C
$4e000
500000
470000
440000
45-000
60e000
669000
68+000

1006000
100000
974000
75+000

_..93+000 _

964000
100+000
100000
1004000
100+000

.. 1002000. _

100-000
100-000

_100+000

930000.
93000

_,.89+000

100+000
78000
652000
594000
83000
892000
962000
960000
1004000
874000
68000

634000
61

36C00
36000
96000
66000

_.7eT00

6000

8000

___ 8000

7000
6¢0C0

79000 _

72000
46000

_... #9600

6*c00
46600
44000
36000
74000
10.000
10.000
7000
5+000_
56000
36000
4.000
146000
12.000
106000

60*700
61000
61+500
62¢000
619500
61500
61200
61*200_
610200
61°000
. 600500
60*000
59500

59200

59+500.
59«500
_59¢500
60°000
609700
60¢+700
60+000

599500

_._. 59200

59000
“59000
589700
59°200
59+500
59000


EXAMPLE PROBLEMS

Four example problems were constructed from the data base
previously discussed. They represented spring, summer, fall
and winter conditions at the Vernon Nuclear Plant. The days
studied were June 21, September 27, October 30, and November
15, 1973. The dates of the data blocks were as labeled a),
b), ¢c) and d) above in the previous section "Dates of Avail-
able Data." Computations were carried out to a point 25000

feet (7620 meters) downstream from the initial point in the
model (x = 0).

17


5. If time-averaged data appears the following then
is required:

a)

b)

6. The
a)
b)

ce)

a)

7. The

The number of time-averaged data lines.

The lines of data as follows

1) Cloud cover in tenths.

2) Air temperature OF.

3) Relative humidity in percent.

4) Wind speed in knots.

5) The water temperature in OF.

control data for the data analysis is next.
The number of cases of error in temperature.
The allowed errors in temperature.

The number of cases for days of data to be
averaged.

The actual number of days of data to be
averaged.

tables of vapor pressure and latent heat (no

changes except for very unusual conditions).

8. The
9. The
10. The

the

PRINTOUT FOR

days to the date in question past March 21.
latitude of the site in degrees.

daily averages of the water temperatures at
remote site. In this case station 7.

THE FOUR EXAMPLE PROBLEMS

35


phe,
-3 -
Sie (0.35) (1.0 + 9.8 x 1073 W,) (bey - 6eq)

§Qnvaporation
ohsg 1073 6W,) (e, - e.)
+ Fe (0.35) (9.8 x 9) (ey - ea
5
8Qconvection = 39-0 (0.077 SW) (T - TA) (>) (0.061)

+ 39.0 (0.26 + 0,077 Wo) (8T - 67) (B) (0.061)

(D-20)

Note that small changes in the heat fluxes imply like changes
in 6T. Thus, the justification in linearizing the analysis.
The equation (D-18) is solved in this investigation using
numerical analysis procedures. However, for constant coef-
ficients in equation (D-18), the equation can be solved in
closed form. Symbolically the equation can be written as:

M6E = Lo p-cy'6T + Cp + Cg + ~~ Cel {D-21)

where the Cj; are constants. The solution to this equation
is found in two parts, a homogeneous solution decaying with
distance and particular solutions that are constant. The
homogeneous solution is found by assuming it in the form

6r = Ae , (D-22)

where A is an initial amplitude and a is the decay rate.
Substitution into equation (D-21) yields

-(0) Ke = & [I-c; Re), (D-23)

which finds a to be

ao = > Cy . (D-24)

156


100000

_ 106000

10-000
10000

100000 |

10+C00
10+000
164090
8-600
1.000
24000
+000
~~ 4000"
_ 9000 |
94600
“72000
99000
106000
10-6000
10-000
1-000
000
+000
+000
16000
_ 9G00
#000
+0900
+000

70+000
686000,
696000

67+000

__ 664000

652000
610006

_ 614000

612000
540000

540000 _

472000

50000

_ 648000

680000
702000
670000
67+000
65+000
642000
609000
640000
68+000
740000
656000
55e000
520000
49e000

$1000

792000

__. 34000

76000 *
732000

812000

844000
934000

. 200000

754000

~ 900000
934000

100-000

100+0900

654000

55000

~~ 494000

662000
61000
73000
754000.
814000
$8000
49000
48.000
61000
90+0C0
93-000
964000
100+000
55

_. 600

__ 49000 _

6#C00

86000
136000
132000.
11-000
10+C00
82000.
46000
46000
_3+000
34000

e000

_..- £9000

9000
52000

32000
82000

_. 29000

7000
8-000
6000
84000
46000
5+000_
34000
46000
56000

769700

_ 769500

75+500

75+200

752000

75000
75¢200

76*000_ __

76000

76000 |

74*500

73+500

739500 _

74*000

” 75s000

5000
749500
73700

_ 734500

73+200
73*200

_ _... 73#500

74 +500

75*000

_ 746500

739700
73*500
739200

759500



1. Modeling the Connecticut river, which has a large
flow rate in comparison with its width, produces
a solution that will have errors decaying over a
very long distance. This means that the error in
temperature will remain virtually constant near
x=O0O.

2. The computations are far less sensitive to the
environmental factors than thought previously-

3. Since the error in temperature is nearly constant
near x = 0 , the variable of primary concern is
the initial error in temperature 6T(o) . Thus, it
appears that records of upstream temperatures in the
Connecticut river would be useful for projecting
average temperatures downstream.

4, Since a is inversely proportional to the volume
flow Au it is possible to compute the value of
a for the various flow rates, given a known flow-

rate Aj,U4 and associated a , 04 :
a =a AU
1 “Au

This expression will yield the stream flow for
which a given decay rate will exist.

RESULTS OF DATA ANALYSES

The results of the four data analyses are presented in the
form of probabilities for meeting given error requirements
with the variables averaged over a given number of prior
days. Tables I, II, and III contain results from all four
problems. It should again be emphasized that these results
relate to a plant operator making decisions as to the
operating conditions of a given plant as opposed to a site
planner predicting conditions several years in advance.

As indicated on the tables, the river temperatures and the
meteorological parameters are averaged over one and two
days respectively.

Results in Table I indicate that an observer at the site
attempting to predict the average stream temperature the
following day at a position 25,000 feet (7,620 meters)
downstream by averaging the previous day's meteorological
data and stream temperature data would compute an

20


enk

9294990
9390000
931-900
2322000
933-900
934.990
935-000
936-900
9374000
9382090
939-950
949+900
9414000
9424900
943-000
9442500
9454900
946+000
94740c0
948.900
9496300
9392000
951-900
9524000
9534000
9540300
9552000
956°900
957-000 C

103

105

800
806

106

802
805

107

803
108

ENDHsTLIM=T

DB 105 TelsNF

AACT) SENDH#ACT) /H

BB (1) #ENDH##2 48 (1) /HeH?

CCl yy SENMH## RECT) /H#HS

OD( TS *ENDH#*OeD(T) /Hewd
SAVETST.

DS 800 TetsNF

SF(T) SF 1)

SAVEY( Lys ¥(T)

TsTLmTM

C8 105 IetsNr

YT) SY CT) FENDH# CE CT SAACTI+8B CT ecC(1) +DD(1))
FP(T) F(T) +2 00#AA (1) 4300488 (1) 44 00eCC(1)+500#DD( I)
TAS4

Ga TS 11n1

D3 805 te1,NF

Sy(tysycr)

DS 107 tstsNF

DELTA. “C1)SFCT)=FP(I}
YOD)sYC1)+V*DELTA (1) #ENDH
TAs5

GS TS 1101

D8 108 I=1sNF

DALTA (T)SFCI)=FP(T)
YCL)sSYCP)+V#DALTA C1) *ENDH

JUMP at
S39 T8 1104
NB eee eTEST FAR DBUBLINGs IF ‘@Ks BEGIN NEST STE= AFTER D8UBLING



88

235000 IF (KSTP#32767)99329942994
286«900 991 KgTP=28

2874000 C TEST FSR TyPF OF ENTRY
288000 993 IF (JUMP)1,99R1999

23899000 998 GI TS (1000211271,8022803)21A
290-000 C JUMP P8Se RESTORE VALUES
291+000 999 TeSAVET

2920000 9592 JUMP=90

293+000 D@ 901 [e1sNF
294000 F(lysSFcr)
2950000 904 YC(I)=SAVFY(T)
296+000 G3 Te 10?

2974000 C JUMP NEG. INITIALIZE
2984000 109.5 [=1.NE
2994000 STARY(1)sY(1)
300+000 A(1)20e001
3015000 B(1ys0e001
302+900 C(l)ed0enn1
303000 5 Dt1l)s0-oni
304+000 KSTP«0

305000 KOELY#0

3064000 KCEN«O

307+000 XTE95¢/ (PBB ewbbe)
308+000 Us8530/( 120450400)
3092000 V89500/28800
319+000 Pm25 40/2400
3112000 Q535.0/72+0
3124000 R*5+0/4860

3132000 Sle0/17000


106000
~~" 100000
a _... ., 400000

10.000
76000
6+000

ome 9600

+000

#000
2+000
8+C00

B00

9+C000

9+000

_ 9000
36000

4,000
3,000
32000
34000
24000

16600

ae

10.000

3+000

““100000 °° |

600 -

#000 —

“4e000 ~

470006
~ 474000
470000
50.000

580000
560000
“43600
390000
374000
~ 454000
60+000

674000

~~ 68.000

63¢600

52000

470000

420000
51.000
600C00
67,000
68,000
634000
520000
450000
460000
472000

540000

“540000 —

1002000

~ 1006000

100000
864000

~~ 754060

652000
624000

"960000

100000
1002000

“1000000

70000
402000

3830007

434000
744000
894000
969000,
90+000

©" 70«C00°

25,000
51,000
50000
834000
964000

100+000

100+000
864000

42

8ecCO 50°200
“7.000 ~«S0e2000—”t”—t—=‘“‘i‘CS*
_. 99000 ._... $0000
7.000 50+000
~"8eC00 «50200 ~
_. 106600 . 50«500
4+C00 516000
“2eco0 = SLe000— tt”
36600 50¢500
26000 50+000
“3.000. «~5090000—~C~—~CS~TM~S*
36C00 50000 -
60000 50¢500
“7.000 ~=«51e2000—~C~S
7+c00 51+500
46000 510700
7-coo. 8 °§»=—540500—~—~—~CS~S~S
2000 51500
6000 51¢500
"86060 “516700 —
12.c00 520000
13.000 52700
"e006 5385000
36000 530500
2c00 53500
“$0000 53700 ~
beCo0 53700
86000 53700


€8

1434000
1444000
1452000
1462900
147.000
148+000
1496000
1592000
151-000
152000
153+000
1544000
155000
1542000
157000
1584000
159+000
169+000
161-000
162+000
1634000
164+900
165-000
1662000
167+000
168-000
169-000
4700000

30

50
60

70

WRITE (16430) EWs EVAPAHES HFG

FORMAT(t EW alFiaeSt

1612.5! 1 Y/MIN HFG stE12e5s/)
WRITE(18,50) H83,EBR,HC

FaRMAT(t HOR atetPeS! EBRetr 1205!
WRITE(18s60) HLYMNs HD

FSRMAT(! TOTAL HEAT TRANSFER «11205!
1!" BTU/SECSFT21/)
WRITE( 18470)
FORMAT(////)
RETURN

ENO

SUBROUTINE DIFEQ
COMMSNDTEMPs TEMP,
LIN, TSF sARFAsUs ERRSRs DXMAX

EVAPsMMPO «'£12e5! HE s!

HC wtE1205' LY/MINI/)
LY/MIN'E15 05s

XPR»sXFENOsOXPRsXsDXa INTs INTC

C8MMBNRHAs WMDsHDs TATRsCLOsRHs ALBARs THSaTAs EWs EAs HOSZHS

CIMMBN HSRsHA3sFBReEVAPSHE SHC oHLYMNa PT
COMMBN HFGsCLO1,HFGTB(10)sCONS1
THERMAL FXCHANGS WITH ENVIRONMENT.
TwSs (TEMP232 0) 650/90

Eds (10136 1407) #SI(TTBLsPTBL» TEMP +10)
EFFECTIVE BACK RADIATISNs

HOBw (14038 609" TWS=6O46#RH) /69072
EBR , HAS * CLO1

EVAPBRATION HEAT TRANSFER«

EVAR = 235 (EWeEA)#(1e + 2009R* WMD)
HFG » SI(TTBI sHEGTB, TEMP 10)

HE © EVAP*HFG*CANS!

BL(10)sTTBL (10) aWMSsCONST


26000
“4.000
6«C00

10.000

“102000

10.000
10-000
10-000
94000
5+000
~ 4000

#000
5-000
000
24000
5-000

“4eG00-

36600

+000
+000
G00
10-000
102000
10-000
102000
10-000
10-000
96000

63-000

650000

55¢000

546000

“54.00

54e000

55e000

“550000

S1e0u0

470000

35000

332000

~ 34e006

486000
53+000
54.000
444000
38+000
354000
346000
416000
44.006
486000
516000
546000
57«00¢

580000

~ 432000”

60+000
“69.0000
624000
644000

“676000 —

694000
75000

“800000.

30¢000
74000

°73+000 —

85-000
924000

~ 964000

664000
45 +000
“42.000
742000
864000
894000
92000
864000
80.000
936000
93+000
100+000
100000
100000
60

“74000 ~

13-000

12.C00 .
11.000

~~ 13.060

166¢00

146000
86000.

122000

35000

"60000

40000
44000

W607

4000
6000
7.C00 -
5-000
50000
4e000
4+ca0
10-000
70000
12-000
23+000

“176900 7

10-000
8000

646200
” 63*700
63+700
63+500

63500
64+000

649200

649000
63*500

63000
62700

63*500
63200

“620700

62*500
62*200
62+200
62#000
61700
616500
619500
61+500
61*000
61000
60700

6365000

“"639000,—COTM”

“6300000



18

1712000 C CONVECTISN HEAT TRANSFER

172+000 HC = 395e4#( +26 + 6 O77*WMS)#(TWS = TA 1714400
1736000 C TSTAL HFAT TRANSFER TO WATERs LY/MIN
1742000 HLYMN = HS = (ESR + HSR + HC + HE)

175000 C TOTAL HEAT TRANSFER, BTU/SECeFT?

1762000 HO = HLYMN # 20614.

1776000 C DERIVATIVE. CALCULATION

178+000 OTEMP = CONST#HN

179900 RETURN

180+000 END

1812000 Ccccc
182+900 CCCCC
183-6000 CCCCC
1846990 Ccccc
1834000 FUNCTISN SI(XTBLs YTBLs XaN)

18646900 C LINFAR INTERPSLATIGN BR EXTRAPALATISN OF SINGLE VARIABLE FUNCTION
1872000 C XTBL = INDEPFNOENT VARIABLE TARLE

128+900 C YTBL = DFPENNENT VARIABLE TABLF

13892900 C IND = INNICATAR OF EXTRAPSLATIAN

199+000 C 82N8 EXTRAPGILATIONs 12L8WER EXTRAPSLATIGN, pe4PPER E73RAPOLATION
191-000 DIMENSIAN XTRL(40)sYTBL (40)

1922000 C CHECK T@ SEE IF EXTRAPGLATIGN 1S NEEDED,

1934000 IF (X=XTBI-¢1)) 12021307150
194e000 120 IND = 1

195¢000 130 II = 2

196+000 G9 TS 254

1970000 4150 IF(XTBLIN)=X) 16021802210
1986000 160 IND =?

1990000 180 Tl =N


OZL

1996290
2092990
2016000
2929000
293+000
294.950
293°099
205+000
2974900
298+5G0
292+900
210-909
2114000
2120090
213.900
2140999
215+500
216+5900
2172000
2182009
2196960
2296990
221-0090
2724900
223+900
2246500
2254900
2762000
27742990

a

40
30

59

WS3VS + WCINT)
TwS2 TNS + TWCINT)
CONTINUE

CONTINUE

CSs CSefNDA

TAS. TAS#FNDA
PHIS=°HTS*F NDA

WS WS*FNDA

TNSs TWS*FNDA
INT? F31LEWING DAY
INT2= LPO+NDA
CiSecs

TALS=06

PHIiS=Oe

WiS=0+

INT3= THE ACTUAI PASITIA8N IN THE NAY
INT3SINTP*8-R41
C1SsC1S+C( INT3)
TalSsTatS+TACINT3)
PHLIS=PHTIS+pHI (INT3)
WiSswiSew( INT3)
TwlSaTW1S+TwCINT3)
CANTINUE

CiSsCiS¥.125
TALS=TALS #0125

PHI 4S=PH11S*0125
WAiSaWiSne125


eZb

285900 D3 90 ICASe4,NCAS
2360950 PR(TCAS) EPROTCAS) ZFNTA
2876900 CALL SUTT(ISUsPR(TCAS)sIDAY
2386090 90 CONTINUE
22242900 20L CONTINUE
299-990 RETURN
231+3090 999 CANTINUF
2924000 WRITECISUs4009)
22346050 4000 FARMAT(1HO,2Xs.'T8A SMALL)
2940090 CALL EXIT
END
3 cccce
2374900 CCCCC
2984000 Ccccc
233-000 ccccc
399-039 ccccc
375.200 SUBROUTINE CUTT(13UsP2T)
3764990 WRITE (1322000) 12P
3770990 2000 FARMAT(1H9s2xXs CASE, TysPXs PRASABILITY'2E 1407)
378+090 RETURN
379-09) END
3896900 -cccce
3A1+390 CCCCC
3326000 CCC
343-000 CCCCc

3842000 SUBREUTINE COMP (ENUM, DFN): .
385-000 COMMON YP1(5)e1(5)¢XPRisXEND1sDXPRLsXL,DOX1aNELsINTiSINTCLs INL e184
336.990 TsEL1sAREALs Ut

387+000 CaMMgN ERRBR1sOXMAX1s TEMP1,HFGTAL(10) 4 Cgnsit


RHO, WMPH, TAIR, CLD, RH, ALBAR: in 6E10.0 FORMAT. These
are, respectively, the density of the water (62.4
4/23 , here), the wind speed in MPH, the air temperature
in OF, the cloud cover in tenths, the relative humidity
in percent, and the average sun angle in degrees.

The final information is read in a loop of 10 sets:

PTBL(I), TTBL(I), HFGTB(I) in 6E10.0 FORMAT. These tables
are, respectively, the water saturation pressure table
in PSI, the water temperature table in °F, and the
latent heat of evaporation table in BTU/pound, all
corresponding and in order. This information is
extracted from standard steam tables for the temperature
range 32 to 120 OF.

The code then computes the stream temperature downstream
from a given point. The output proceeds at each print
position with the position X itself, the integration step
DX , the stream temperature TEMP, and the derivative of

the stream temperature with respect to the position variable
DTEMP. Following this, other auxilliary variables are
printed out: EW, EVAP, HE, HFG, HG, EBR, and HC. These are
only important for diagnostics, not the solution itself

SENSIT

The digital computer code SENSIT performs all necessary
computations to predict the error in the stream water
temperature (due to the difference between the actual and
approximate input data) at any distance downstream from a
given initial station. The variation in the original
equation is coded to examine the effects of errors of the
input variables. Numerical integration is used to compute
the error in stream temperature, with a choice of four
algorithms available to the user. A listing of the SENSIT
code is given in APPENDIX C.

INPUT REQUIREMENTS TO SENSIT
SENSIT accepts the following information:

INTC: In 6110 FORMAT. This variable is the integration
routine to be used (See STREAM for description).

29


THE SENSITIVITY ANALYSIS

The fundamental equation for the sensitivity analysis is
developed by varying both sides of the solution equation

ast L
“ax = Fy 62 , (D-18)

where 6Q is the change in the heat fluxes and 6T the
resulting change in the stream temperature. The initial
condition for the equation is

6T(0) = 6% , (D-19)

i.e., an initial error in temperature 61g is known at a
point in the stream, which allows the computation of the
error at points downstream.

On closer examination of 6Q the linearity of the solution
emerges. The individual terms are

Q 1.9 sin @ (-0.0018 Cc? 6c) (0.61)
Solar
+ (1.0 = 0.0006 c3) (1.9 cos G 63) (0.061) ,
8Q 3 60 (0.061) - &. 9 8a (0.061)
Reflected = “Solar “2 “solar ’
a a
SQpacy = (1-0 ~ 0.0765 C) (gg Haq) (- 0.0918 61) - 0.046 86)

- 0.0765 8C (gg iggy) (14.38 - 0.09(E(T - 32) - 0,046 9)

155


2. There is usually a lack of data from which to
compute the evaporation and other heat transfer
rates for a stream.

3. The temporal variations of the data cause large
variations in short-term predictions.

In addition to these, we find that the spatial and temporal
variations of temperature in the environment without the
addition of a heat load may be greater than either the
requirements set by a regulatory agency or the errors in-
herent in the model, i.e., there is a question as to what
the ambient values really are. Local variations in topog-
raphy and/or tree cover can shade the stream and cause
effective incoming and outgoing radiation areas to be
reduced. The accuracy of instruments used to measure

the various physical quantities is always subject to review.
Ground water advection can be an influential transport
process, but it is almost impossible to measure. Finally,
the large amount of data taken for any site provides a

good probability for human error in recording or trans-
position.

This study considered the three principal problems listed
above by using a sensitivity analysis of a general, one-
dimensional stream model. This was combined with data
analysis techniques to compute the probabilities of meeting
given error requirements. Data from the Vernon, Vermont
nuclear plant and the Burlington, Vermont weather station
were used to form a data base, from which four example
problems were obtained and evaluated.


SEL

7274500
738-900
759.000
7390900
731«900
732900
7939930
7942900
7936090
796+000
7979630
7284090
7992090
899-000
8912090
8929090
893+000
8040000
8056900
8952000
897+060
898900
899-090
8192000
8114000
812.900
813.000
814-000

cccce
ccccc
ccccc
ceccc

991

993
998

999
992

go1

END

SUBRBUTINENSRO (Fa Te TLIMs Ys ERROR «NE» Hs HMAXs JUMPs KSTPs KCENs CLIF 2 18)
DIMENSTBN STARY(5)s¥(5)sSY(5) sSAVEY(5)9F (5) FP (5)sDELTA(S) sDALTACS,
14405) 4805) 90(5)20(5) 9 Aa (5) #BB (5) 1CC(5)s00(5) sSF (5)
Ip (KSTP=32767)9939910994

KSTP=28

TEST SBR TYPE OF ENTRY

TF (JUMP) 149983999

G9 TE (1000111,21,802803)s1A

UMP Pes. REgTARE VALUES

TeSAVET

JUMP =O

D8 901 T=tsNF

F(L)aSF1)

YCP) sSAVEYOT)

G8 Te 10?

JUMP VEG. INITIALIZE

D8 5 I=t,NE

STARY(T)eY(1)

A(T) 200004

B(1) 20.001

Clty s09004

D(1)20s004

KSTPsO



Ont

40435+000
10164090
10174900
101$+990
10194000
1552990
15091.090
1592+900
15931000
1594050
1595.900
1596-900
1537-900
15984000
1599-9900
15136000
1511+900
1512900
15132000
1514-060
1515-990
15162000
1517+900
15186000
1519+900
1520«000
15292910
15294920

2002
10

WRITE (TStis 2002) TDAS (1) 2 (PRP (Te.1) 9 JeqaNCaS)
FORMAT (1HOs 2x2 1501X25(E100301X))

CONTINUE

R

URN

evD

SUBRSUTINE vERNSN

COMMEN

YP41(5)4Y1(5)4XPR1(2XEND1s,DXPR1sX1L4DX1sNELe,INTIsINTCLo INL, TOL

TsEL ts AREAL U1

C3MMgV
caMMsn

FERRARI ADXMAX1s TEMP124FSTAL(10)4CQ9NS1t
RHOtsWtsdMD12401.TAIR12CLD12RHis ALBARIsTWS1s TALs TWSALZEW 1s

1E41,H3S1,HS1

C3MMeN

HSR1+HOBL,ERRIZEVAPLsHE 1 HCI sHLYMNGSPTBLA (10) sTTBLA(SO)s

1WMS1sCONSTL

caMMsN

1DTAIQI,

ClIF AHF S4,H9C1,CL011,5C1.921,CL031s0ALP1sDALPR1,DCLD1,DRH1,
DTAL20W16D0uMS1 2 DWMD1bDEA1,DEW120LYMN1sDBTUS4 sNTEMP 1

@STT3L1(10)sXT3L1(10)

CaMMgN
CcIMMeN
COMMSN
CcaMMsN
COMMEN
CcaMvaN
C3MMEN
C3MMBN
C3MMEN

C(1000)%s TA(1000)sPHI (1990) +W(1000)2TW( 1000)
NCASs NOAs DELT(50)2 TDAS(50)

MENa TNs L8UsDTaNTAs IDA

CTA(SO)s TATA(SO)sPHITA(SO) sWTA(5QO) » THTA(SO)
YArELsCCeTAIRs TWAs RAs WSP
INT221NT920C29TAsDPHI 2 DWeDTWs INVSs IFLAG
OM21,RETA

LAGLsNDAS

IPAS PRP (5010)

DIMENSI8N PR(10)sT7(100)
03 3 T81,NCAS
PR(T)206


Yy

£(x, y) ,
y(0) = yo - (D-5)
If Euler's method is taken as an example the form for the

integration is

Yior = Yu t Elxp , yy)ho oo, (D-6)

which is seen to be identical in form to equation (D-3).
From p. 91 in Salvadori and Baron

x
y(x) = fro dz '
a
¥(x) = f(x) ’
¥(x) = £(x) 1
(n)
y(x) = £1 (xy, (D-7)

A Taylor's expansion about x yields:

2. 3"
y(xth) = y(x) + qBeGe + no g(x) Bp f(x) #---
. . . (D-8)
From this the following expressions can be evaluated
xth
1 = f(z) dz = y(xth) - y(x) 1

x

h-
= h{f(x) + oT E(x) + ’ (D-9)

and

152


10.000

~ 106000

102000
8+000

7eG00

10,000
106000
100000
10.00
10.000

“10+000°

92000
10,000
34000
10.000
10.000
10.000
10000
102000
102000
10-000
106000
3+000
12000
+000
96009
92000
10-000

624000

619000 ©

620000
69eC0C
720000
60.000
580000
57¢GG0
550000
56000
62¢000
68¢000
64«000
639000
58000
56000
$6000
56000
59+000
6090006
640000
622000
58¢000
560000
54000
609000
640000

620000

974000
100+090
100000

730000

“87000

874000
932000
93*000
100000
964000

46600

~ “86000

750000

55+000
54600
65 +000
90+000
964000
100.000
100*0C0
100+000
84.000
73«0CO
734000
814000
804000
804000
70+000
684000
75000

4y

6+060
96000
“8sC00,
56000
6#C00
*coo”
*000

36c00

“20000

5°00
5»C00

+000

3+000

36000

4.000 —

000
46C00
16*«C00
4e0CO

4 C00

~ 66000

10000
8000

“136000-

146000

9+000

55°500

559500

55+500
56¢500
'57«200
579700
57+700

57700

57*700
57500

570700,

58*000

58*200

“'58+200—

58°500
58500

58500 —

59*000
59°000
699500
609000

60¢000

~ 60000

60*200
60200
60700
619500
62¢000


SECTION III
INTRODUCTION
PURPOSE

Stream temperature is an important factor in water quality.
The temperature of the stream directly controls the types
and amounts of plant and animal life native to it. In
recent years there has been an increased interest in

water quality by the public. In response to this interest
a very large effort has been mounted to develop mathe-
matical models for predicting heat assimilation in

bodies of water.

Models have been developed for predicting temperatures

in streams, cooling ponds, reservoirs, estuaries, and
large lakes (See, for example, the review by Policastro
and Tokar [1972]). Usually, a stream model assumes a
one-dimensional problem with uniform mixing and with
various modes of heat transport at the air-water inter-
face simulated by semi-emperical expressions. In addition,
the problem is generally treated as one in a steady-state
condition, i.e., the formulation involves a relationship
between the flow rate and the position downstream from

an initial point, x , allowing the independent variable
to be chosen as the variable x while the time variable
becomes implicit to the problem. While the development
of stream models is straight-forward, their use is
subject to a great deal of interpretation and judgement
(See, for example, Asbury [1970]).

The purpose of this study is to determine the effects
of variations in initial temperature and meteorological
data on results of mathematical models for predicting
stream temperatures.

SCOPE

Three principal problems inherent in model use to predict
heat assimulation are:

1. The extrapolation of weather station meteoro-
logical data to the site under study is subject
to considerable variance.


556000 47.000 4.000 53*200

“43.000 83,000 2.000. «520700
380000 = 89000 9000... 520500
46000 35-000 926000 36000 52+000
oo 54000 «© «536000 9400G.—=—«<‘CSHCs*=“<«é«“CSCRs—“( <;eCS
a 10-000 630000 _ 484000 14ecco . .5@e200,
10+000 686000 414000 16000 52+700
—— 106000 ~~ 63eG00 ~~ 404000 16ec0O——~*~S BHO —_
102000 57000 570000 .. 69000. . 53*#700
10.600 54000 694000 126060 53+500
~~ ~“4G0000 «498000 © 9Be000—“—~C<‘—“‘<‘é«~CHODC*C“‘C (‘CCO®#”®#
. 10.000 50«c00 964000 32000 53000 ae
10+000 50+000 96+C00 46000 53000
~~ 402000 —-55e00C 934000 7e000S~S—=C<~SBW FOODS
_ 10+000 564000 93000 86G00 _ _52*700 a
10+600 57000 83.000 9-000 536000
oe" 100000 “552000 866000000 538200
. 10-000 536000 934000 .. 69060 _. 539200 _
10+000 54.000 90+000 5-000 53*200
1020000 ~ 559000 ~ 4-000 7000 539000
7+000 58e000 876000 86000 52¢500
10.000 649000 754000 13-000 52+500
“106000 716000 640000 128000” ~~ 53000
10-6000 750000 50.000 126600 54000
: 10-000 686000 636000 5000 54000
7 10+G0C ~~ 620000 674000 18.000” ~ S4e000 © °
y. 10000 57000 100-000 8-c00 54000
10+000 576000 934000 126000 546200

38


6LL

171-900
1722060
173-000
174-900
1752900
176-000
1774200
178.090
173.050
185-000
1a1.9D0
152900
183.900
1244050
185-000
1856090
187-900
1236200

159-6900

19160

192.990
133-900
194-900
1956900
1961900
1574900
1986990

CaMMeN OM21,RETA
CAMMEN LAGL»NDAS
CAMMEN ICASs PRP (50,10)
DIMENSTAN PRO10)

NOAs \UMBER aF DAYS JN THE CASF
1D81DA

NDASTIAS(IO)

NCLSNLN/R

FNDASNDA,3

FADAGL/FNOA
IF(NDANC1) 1929992999
CONTINUE

NOTF=ENC1-NDA

ENUMz36

DENsCe

DENP=06

D3 20 1PnstsNDIF

CStae

TASs9e

P4ISs)s

WSE%.

THSe0e

O03 30. 1=1sNDA

TAS=TAS 4 TACINT)
PHISsPHIS + PHI(INT)


The data base variables and dates for the Vernon plant
example problems are as follows:

1.

Burlington, Vermont Weather Station data and Vernon,
Vermont water temperature data:

Variables

Cloud cover, air temperature, relative humidity,
wind speed, water temperature from stations 3 and 7.

Dates of Available Data

a) May 1, 1973 to June 20, 1973

b) August 15, 1973 to August 31, 1973 and September
12, 1973 to September 26, 1973.

c) October 1, 1973 to October 3, 1973, October 17,
1973 to October 24, 1973, and October 27, 1973
to October 29, 1973.

d) October 17, 1973 to October 24, 1973, October 27,
1973 to October 29, 1973, and November 1, 1973
to November 14, 1973.

Vernon, Vermont (and vicinity) site meteorological
data:

Daily Average Variables

Cloud cover, air temperature, relative humidity,
wind speed.

Dates of Available Data

June 21, 1973
September 27, 1973
October 30, 1973

November 15, 1973

16


Process moves through ail of data

—|[-—|-—[ —]

TO be
Average Over Compare With ‘Airport Data and
Several Days Average Over Day Stream Temperatures

Following

Average Over Several Days
Average Ower Day Following,

COMPUTE AND
PRINT PROBABILITIES

Figure 3. Model using site water temperature data


TABLES
No. Page

1 Probabilities of Meeting Given Error Requirements 21
by an Observer at the Vernon Site Averaging One
Day and Predicting One Day Ahead at a Distance of
25,000 Feet(7620 Meters) Downstream.

2 Probabilities of Meeting Given Error Requirements 22
by an Observer at the Vernon Site Averaging Two
Days and Predicting One Day Ahead at a Distance of
25,000 Feet(7620 Meters) Downstream.

3 Probabilities of Meeting Given Error Requirements
by an Observer at Station 7 One Day Ahead at the
Vernon Site(Same Position as for Table 2). 22


vel

5374000
588.900
5896900
599 +990
591-000
592 +900
593-909
5944990
595 +000
535-090
597 +000
593+009
5994000
6094990
691 +900
692 +999
693+000
694-900
695+000
695+000
6976990
6.8 +000
6999999
61249090
6114090
6124000
613+000
6140959
615 +930

290
210

220

230

240

260

270
230

330.
310
320

330
B49

FORMAT(! INITIAL DT =tF 1204 ' DEG F'y)
WRITE (192210) JeARFASEL
FARMAT(1 STREAM VELSCITY ='F a3! FT/SEC CRe SEC+ AREA 8!

1F10.2' Se FT SURFACE WINTH s'F6et! FTI/)
“WRITE (18% 2201 RH8,CLD2RHs ALBAR
FORMAT(1 RHO='F5e2 | LB/CUsFT. CLos'F4ed! TENTHS's

a RHetPSeit PCTI/! ALBAR wtFuedt DEGREES'/)
WRITE (134230) TAIRs TAs CINST2CSNSL

FORMAT(? AIR TEMPe e!F Sei" OFS FIF8elt DEG Cle
ut CANST =!E1903! CONSL =tF1003s/)

WRITE (162240) WeWMPHs WHSs KMD

FARMAT(! WIND VELACITY atF6eat FT/SECIF8e2IMPH'FBe2! METERS's
LI/SECIFR62! MILES/DAYI/)

WAITE (14.260)

FSRMAT (7s T201TTSL ITSO! PTBLITEONHFGTSI/)

D3 279 1 #1419

WRITE(19,280) TTBL(1)s PTBL(T). HFGTB(I)

FSRMAT (10X23 2065)

WRITE (19.4300) DCLDs 08H» OTAIR

FARMAT(s1 OCLD =1F603!, ORH -2I1F 523! DTAIR s'F 602)
WRITE (182310) DALPs DWMPH

FORMAT ( DALP stFGe2!. DWMPH = 1F602)

WRITE (194920) “H9SsHIC*HSAHSR

FORMAT(/? HOSStE1204! HSCatEtPeat HSa'E1ae4! HSRatE12e4t LY/MIN')
1Nt)

WRITE( 19.336) EAsDFA

FORMAT(/1 EA stEtAey! DEA st 1204)

WRITE( 134340)

FORMAT (7+ T20'ISTTSLIT&OIXTALI/)



BOL

379-900
380+000
331.090
322-6000
3332000
384-000
3854000
386.0900
3872900
388+000
3a90000
3922000
3912000
392+C00
393000
334-900
395000
396-000
397000
3982000
3998000
4092000
4912000
4022000
4034000
4942000
405%900
406+000
4072000

10

ti
12

at
29
30
31
35
40
45
55
723

FRULYSF CT) +20O#ACT) #30O¥B (1) +4 eneC( 1) +5.0#D(1)

TAe2

G9 T8 1101

D6 12 Tat,NE

Sy(tysy(r)

09 70 Tst,NE

DELTA (T)BFCT)FP(T)
YCLysY(Ly+V"nELTA (1) #H
TA®3

KCONeL

G3 TS 1101

KCENSO

DB 30 LstsNE

DALTA (T)SF(T)*FP(T)
YCTyeSYCr)+VHDALTA (1) #H
CANTINUE

TEST FOR STARTING SEQUENCE
IF (KSTP#P8) 35240940

APPLY TFST 2 @N ZERATH STEP
IF (<sTP) 5050260

HALVING TESTS

09 45 Te1sNE

IF(ABS(DALTA (1) )ERRER/ABS(H) 145. 45295
CONTINUE

TF (Vala Ch IF 900125 60260055
TeTeH

FAIL TESTSs HALVE H
HsH720

KDELY =O


Lg

102000 472000
UAs €240¢000R WIDTH=
TEMP WATER® 50*QOOREL

NUMBER OF CASES® 4
ALLOWED OTS
*100CO00E O41
*200CO00E OL
*300C000E O01
*4OCCOOUE O1
NUMBER GF DAYS AVERAGED=

DAYS AVERAGED

FA AEA EI IA IT FEIT A FETE SE ETE ITE ETI LEHR TEE aE TEE

1

2
PYBLI TTBLA HP GTB1
*885E-01 +3206 02 +108E
e122E 00 +400E 02 +107E
e178E OC #SOUE U2 +107E
e256E OC +600E 02 +106E
©363E OC *700E 02 +105E
*507E OO *8V0E 02 *105E
e698E OC #900E 02 *104E
*949E 00 *e100E 03 +104E
e127E OL e110E 03 +103E
e169E 01 *12CE 03 +#103E

DAYS §,NCE MARCH 21*

933000 8000
400e000CLD C8VER®

HUMS 6B*°0COWIND

o4+
O4

108 eQOOBETA® 454000

509000

1sOOOTEMP AIR®

SPEED®

#000

45.000


LZ

Table 1, PROBABILITIES OF MEETING GIVEN ERRO'R REQUIREMENTS BY AN OBSERVER AT
THE VERNON SITE AVERAGING ONE DAY AND PREDICTING ONE DAY AHEAD AT A

DISTANCE OF 25,000 FEET(7620 METERS) DOWNSTREAM

Error Requirement °F (°c)

19°F (0.56%) 2° (1.11%C) 33°F (1.67%)

yor (2.22%)

June 21, 1973 0.6 0.9 0.94
September 27, 1973 0.613 0.935 0.963
October 30, 1973 0.769 0.923 0.923

November 15, 1973 0.75 1.0 1.0

1.0
0.968
0.923

1.0



rans

901-000
902.000
$03+000
9940000
9052000
9962900
9972000
998-990
99946050
919000
9114900
912.000
9130090
214.000
913000
9154990
917900
9186000
942+000
9290000
9216000
9226590
923000
924-900
925560

926.590 C

9274000

928.000 C

86

87

88

89

92

390

94
100
102

Heks290

O39 37 IstaNE

ACT) SA(T)/200

Bt lyeB(1y/heo

Ctly sl 19/800

D(T)eD(1)/1600

O3 88 Te1sNE

IFCABS(DALTA (1) )"ERRBR/ABS (H) 182s 88s 89
CONTINUE

PASS TEST G8 FEWARD WITH HALVFO H

Ge Ts 78
FAIL TEST REGIN AGAIN WITH HALVED H
HeeH

03 9? LetaNE

Y(T)sSTaRY(T)

G3 Te 1

24TH STEPs DAUBLE H» STARTING SEQUENCE ENDS
HeH*200

OS 94 T=4qaNE

ACT) SA(T1#200

B(Lye3( ry e400

Clyec( ry #8en

D(1) (1) #1600

G3 Ts 78

KDELysKOFLy+1

WILL NEXT STEP MAyF PAST TLIM

IF (ABS(TI IM*T)=48S(H) 110321034110
YESeeeeesee*SAVE T AND Ys INTEGRATE TO TLIMsRETURNe


Q = 1.9 sin @ (1 - 0.0006 c3) (0.061),

Solar
=3 0.061
reflected z Ssoiar ( ) ’
= - - 5 -
QB ack = (1.0 0.765 C) (14.38 0.09 G (T 32)

- 0.0469) (gam),

ph

£2 (0.35) (1.0 + 9.8 x 1073 w. -
‘Evaporation aioe a * 2) (ew - 4)

u

Q

39,0(T - T,) (0.26 + 0.077 Wo) 3 (0.061).

(2)
The definitions are given in Section VIII. It should be
noted here that the coefficients from these terms are
largely based on empirical data and in many cases will vary
considerably. However, as is seen in Section VI, the
sensitivity analysis reveals little change in temperature
decay rates with considerable change in these flux expres-—
sions for the Connecticut River. For smaller streams, this
would not be the case.

convection

THE SENSITIVITY ANALYSIS MODEL

In order to study the errors in stream temperature preduced
by the errors in input data, it is necessary to develop
certain error expressions from equation (1). These are
developed in view of the two fundamental definitions from
Rosko [1972]:

1. Accuracy is defined as conformity of fact.
2. Precision is defined as sharpness of definition.

In the case of this work then, accuracy refers to the con-
vergence of the solution to the actual stream temperature.
Similarly, precision refers to the rate of convergence.

The sensitivity, or error, analysis required for the work
was developed from equation (1) by making a small change,
or variation, in the variables. Equation (1) can be
rewritten symbolically as

at &
m7 we (3)


Onl

8444000
8454000
8460000
8474000
8484000
84920900
859000
8514000
8524000
853000
8542000
855+000
8552000
857000
858+000
859-000
8604000
8614090
8524000
8532000
8640000
85542000
8662500
8474000
868-000
853-000
879-000
8714000

20.

21
29
30
31
35
40

45
50

55

223

DELTA (t)SFCT)-FP(T)
YCLSYCLI#VADELTA (1) #H
TAs3

KCENet

G3 TS t1in1

KCENsO

DS 30 Ta1sNE

DALTA (1) eF(T)=FP(T)

VelysS¥cr)+VeDALTA (1) #H
CANTINUE

TEST FOR STARTING. SEGUENCE
TF (KSTP#28)35940840

APPLY TEST 2 GN ZEROTH STEP
IF (KSTP)50250260

HALVING TESTS

D9 45 T=tsNE

IF(ABS(DALTA (1) YeERRSR/ABS(H) 14545055
CONTINUE

TF (V#4eC) IF 902175 160260255
TaTeH

FaIL TESTS» HALVE H
Hakh/2+0

KOELY#0O

D8 56 [etsNE

A(1)sA(1T1/200
BUlysB( ty 7400
C(1)sl(1)/8+0

Fc lysSF (1)


R - Particular value of x.

Xmax ~ Maximum value of x.

y - General function.

y - Derivative of general function.

y(0), Yo - Initial value of general function.
Yi ~ Value of function at xj.

z ~- Dummy variable of integration.

GREEK SYMBOLS

a - Decay rate coefficient.

a, @ ~ Sun angle.

6C - Error in cloud cover.

6Q - Error in total heat flux.

6T - Error in stream temperature.

6T(0) - Initial error in stream temperature.

6Tair - Error in air temperature.

8Tairp - Particular solution for air temperature error.
6Tcep - Particular solution for cloud cover error.

6Tj - Particular solution.

6Tw2p - Particular solution for wind speed error.

6Top
6Tpp - Particular solution for relative humidity error.

- Particular solution for sun angle error.

6W,6W. - Error in wind speed.

6% - Error in sun angle.

6¢ - Error in relative humidity.

Ax - Increment in downstream distance.
$ - Relative humidity in percent.

p - Density.

€ - Position.

27


96

370900
38+0900
39000
42+000
412000
424000
434000
446000
452000
452000
472000
43900
492900
50000
51000
52-000
53+900
542000
55000
5$+000
57+900
58+000
59+900
699000
61+900
624000
632000
642000

120

130
131

140

150
300

10
20

c

30

CANT INUE

CALLNSRD (DDT, Xs XPR» DTV ERRBRe 120Xs DXMAXs JUMP A KSe KC dE w 6s 18)
IF ( JUMP) 1302140+1850

WRITE(I6,131) X

FaRMAT(//!) XX wIEL4e5! | INTEGRATTSON FATLURE!)

CALL EXIT

CONTINUE

CALL DIFFO

G3 Te 170

CALL SUPT

IF(X = XFND) 11023002300

CALL EXIT

END

SUBROUTINE INOUT

C8MMBN ODT, OTs —XPR»XENDsDXPR»X,DXs INTs INTCy
1 IN, 19sE1. 2 ARF As Us ERROR» DXMAXs TEMP» HF GTB (10) ,CONSL
C8MMANRHBs WMDs TATRs CLDs RH ALSARs TWS2 TAS EWs EAs HOSZ HS
CO8MMSN HSReHS3sF BR, EVAP,HE dHCaHIYMNs PTBL (10}4TTBL( 10) s WMS» CONST
CAMMEN HFGsHACICLO1sCLO2.CLO3%DALPs DALPR» DCLDsDRHsDTAIRs
ADTAs OWs WMS» NWMD» DE As DEWs DLYMNs NBTUSANTEMPs STTBL (10) sXTBL(10)
FSRMAT(6F 1000)

FSRMAT (6110)

RATIO = 3°14159/1808

CHOICE 8F INTEGRATION SUBRSUTINE

READ(INs?0) INTC

IF(INTC = 3) 30040250

INT = INTC

G8 Te 60


6EL

815-090
816.900
817-000
818-000
8196000
8294090
8214000
822¢000
8234000
€24.000
825900
8264000
8274000
8282900
829+900
839°000
831.000
8326900
&33+900
8344900
8352900
834900
8374000
8382000
8392900

8494900-.

8412000
8424000
8430900

100
aat

60
60

Q
1

0
4

KOELY#0

KCONSD

XTEISe/(PBBewbbe)
URnB63%/(120#50400)

V295 00/2800

Pa2500/2400

G235+0/7200

R35 00/4800

Sale0/17000

Tast

JUMP ad

G3 TS 1104

BEGIN INTEGRATISN STEP.

OF 1141 feteNne

SF(ryer(r)

SAVEY(T)aY¢T)

H T86 SMALL RETURN WITH JUMP NFGe
TF (ABS (T+H) =ABS(T1)6052 6015605
JUMP eed,

Gs _Te tint

5 Tate

09.10 IstsNE

YUL SY CD HHaCP CTI SACT SBC DIFC (TI ¢D(1))

PECTISECLT#B OMA CTI #30048 (11440 08C(1) #5 004D(1)
AaR

G9 TS ttot

D9 12 T=tsNE

SY(T)#YC1)

DS. 20 LstaNé


68

314-000
3154000
3162000
317%000
318-000
319-000
3292000
3214000
322-000
323+000
324-000
325-2000
326-000
3274900
328+000
329+000
330050
331-000
3324000
333.000
3342000
335*900
3364000
337-000
338-000
339+000
3494000
3412000

1000
q1ad
600
601
605
10

ai
12

20

21

TAst

JUMP 290

GS Te 1104

BEGIN INTEGRATION STFP

De 1111 T=1sNE

SF(r)eF CT)

SAVEY(T)eY(L)

H Tag SMALL RETURN WITH JUMP NFGs
IF (ASS (T+H) ©ABS(T)) 60526019605
JUMP se4

Ge_Te 1101

TeT+H

De 10 TstsNE

YD SY C1 HAC FCT FACT SBC) 4CCTY4D001))
FPL) SF (1) +20QA(T) +R0OHB lL) +4e0HC(T) +500" (1)
TAs?

G3 Te 1191

03 12 TetaNne

sy(ryey¢t)

DS PO TstaNE

DELTA (T)aFCT)-FPCL)

VCD) sYCL)+VSDELTA | (1) 4H

TA®3.

KCONetL

G8 Te 1101

KCENsO

Da 30 IstsNE

DALTA (T)eF(L)=FP(T)


A variation of both sides of this equation yields

a6T _ L

Se * om 6Q 4
where it is assumed that the flow rate and stream width are
known and that the process described is linear in its varia-
tion. If the error in temperature, 6T(o) , is known at
the initial point, it is apparent that, given the errors in
the heat fluxes, 6Q , the error at any given point down-
stream can be computed from equation (4) by a direct
computation.

It is important to note that, if approximate values of the
coefficients in equation (4) are obtained as constants,
the equation can be solved in closed-form,

6T = A exp[-ax] + 2 Cy .
i

Results of an analysis of this type are presented in
Section VI.

THE DATA ANALYSIS MODELS

Previous sections have described the mathematical model
studied and the sensitivity analysis model. The first of
these models, or close variations of it, has been used
extensively to predict the mean temperatures of streams.
The sensitivity analysis model has the capability of pre-
dicting the error in the mean temperature at a point down-
stream, given the corresponding errors in the mean initial
condition for temperature and the mean meteorological con-
ditions. Thus, the mathematical models were available for
a study, setting the stage for the data analysis.

The philosophy of the data analysis emerged from the con-
cepts of probability. As has been shown by Hogan et al.
[1972], it is unrealistic to establish the error in the
estimate of the mean temperature deterministically.
Rather, the statement must read to the effect:

If the errors in the heat fluxes |6Q| and the
error in initial mean temperature |[|6T(o)| are
constrained in size to given values, then the
probability that the error in mean stream tem-
perature Yor < 1/n °C will be Py, |6T] < 1 °C
will be P9(P2 > Py) . 7


sé

SENSIT

SENSITIVITY ANALYSIS

ananganann

CSEMMBNODTs0T sXPQs XENDsDXPRa XaOX, INTs INTCs IN, TBZEL 2 AREASU

CeMMEN ERRBR,DXMAXs TEMPsHFGTB(10) .CANS1

COMMBNRHBs WMDs HDs TATRa CLDsRHe Al BARs TWS2 TAs EWs EAs HOSZHS

CAMMBN HsRsHeBsEBR»EVAPsHEsHCoHl. YMNePTBL (10). TTBL(10) sWMSaCONST
COMMBN HFGsH@CsCLO41sCLDP,CLO3sDALPs DALPRs DCLDs DRHsDTAIR»

1D TA, Ds DWMSsOWMD,s DEAs DEWs DLYMNs DBTUSsNTEMPs STTBL (10) sXTBL (10)

INaS

1936
184900 CALL INPUT
194900 IF(INTC = 4) 49108. 105
290000 4 XPR a DXPR
215000 S CONTINUF
22.000 DS 100 JF= t,INT
734900 CALL DIFFO
244000 G3 TS (10410,20), INTC
250000 10 © CALL. EULFR(DDTsDTsDXs 1a JEaX)
26+000 Ga Te 100

272000 20 CALL RK&(DDTsDTsDXs 45 JEeX)
28-000 100 CANTINUE

2394900 1F(X = XPR ) 322502250
324090 250 CALL SUPT,

312900 XPR = XPR + DXPR

32+900 IF(X + XFND) 543002300
33.990 105 = C8NTINUF

342000 dUMP = 44

35-000 110 CaNTINUE
36+000 XPR = XK + OXPR


OOL

1514000
152-000
1532000
154000
155-900
1356+0900
157+000
153-000
159-000
160°900
1614000
162+900
163090
164000
1652000
166+000
16740900
168+090
149+900
179+000
171+990
1722000
1732000
174e%000
175900
1764090
1774000
1782090
179+000

340

350
360

10
20

30

50
60

FORMAT (/,T2O'STTBLIT&OIXTBL'/)

REAO(INs 20) NTEMP

D3 450 I s1sNTEMP

READCIN«10) = STTBL¢1)sXTBL(T)

WRITE(18.280) STTBL(7)sXTBL( 1)

CANTINUE

WRITE(18s360)

FARMAT( HA")

RETURN

END

SUBRBUTINE @UPT

CaMMEN NDTs DTs XPRs XEND.NXPR» Xe DXs INTS

LINTCs INs Taf Ls AREAS Us ERRER2 DXMAXs TEMPs HFGTB( 10) s CONS
COMMBNRHBs WNDeHDs TAIR*CLO2RHs AL SARs TWS2 TAs EWs EAX HOSA HS

CBMMBN HsR» Hees EBRsEVAPsZHEsHCsHL YMNs PTBL (10), TTBL( 10) sWMSeCANST
CSMMBN HFGsH8CsCLD1sCLD2,CLD3s DAL Ps DALPRs DCLDsDRHsDTAIR»
1DTA, DW, OWMSs DWM, DEAs DEWs DLYMN, OBTUSZNTEMPs STTBL (10) sXTBL(10)
WRITE( 18,10) XsnX

FORMAT (1 X StE7e4t FT DX str eel a)

WRITE( 18220) TEMP

FARMAT(' STREAM TEMp atE12e4! NEg FIt/)

WRITE (18270) OT»DDT

WRITE (16.30) EW,DEWsFVAPs HED HFG

FORMAT() EW atF41205) DEW =tF1205+ EVAP stEi2e51 HE =!
161265" TF Y/MIN HFG s!€12052/)

WRITE(18,50) H8RsEBRsHC

FORMAT(1 HBR F1E1205! EBRetr 1? a5! HC stEl2e5!' LY/MINIZ)
WRITE (18.60) HLYMNsHO

FORMAT(1 | TETAL HEAT TRANSFER st&{2e5!' LY/MINIEL5 05s


LLL

1149990
1151950
1154990
1174960
41184000
1196900
1294990
121.090
1224000
1234900
1244990
175.000
1264939
1272960
1284990
123-95)

139-900
431-900
1326900
433+0c0
134.900
135-990
1464000
1376000
1384090
133+900
442000
1416000
1424000

2008
3000
3010
2009

2012
1002
185

2013

WRITE (1811s 2008)NCAS

FORMAT (1HOs2Xs NUMBER, OF CASES ste tXaly)

READ( TN» 3000) (OFLT(y)e tTalsNCAg)

FORMAT (6F1003)

WRITS (1823010)

FORMAT (1HOs2Xs !ALLAWED DTS!)

WRITE (1822009) (DELT(I}sT=12NCAS)

FORMAT (1HOsAX E1407)

READ CIN» 1001)NDAS

WRITE (1812 2010)NDAS

FSRMAT('G's 2X2 1NUMBER OF DAYS AVERAGEDa ty 2X5 14)
READCIN« 1001) (T9AS(I)s Tet, NDAS)

WRITE (TS923011)

FARMAT(1HOs2x2'NAYS AVERAGED!)

WRITE (1811s 2059) (IDASCI eT s42NDAS)
FARMAT(1HOs2xs15)

WRITE (TeU22511)

FSRMAT (1HOs 2X9 1PTSLA TTBLA HFGTBLt0//s2Xs60(1#1))
D185 Tsts10

READCIN, 1002) PTBL4CI)sTTBL1(1)HFGTB: (1)

WRITE 1922012 97EL1(T eT TBL ACT sHFGTBL(T)
FORMAT (SF 1003)

FORMAT (6F1003)

CONTINUE

READ( IN» 1000) OM21,3ETA

WRITE (19t1s2013)DM219 BETA

BETA=3ETA/57 0294

FORMAT (1HOs 2X2 NAYS. SINCE MARCH 2121s2XsF 1003s 1 BETAS 's2X9F 1003)
O08 30°1DAs4sNDAS


CLD COVER AIRTEMP REL HU”

e000

+000
+000
_ 34000
8.000

52000

5.000
30000

10+000
10-000
_. 102000
10.000
10+000
10+600
10-000
10-000
106000
10+000
10-000
10.000
10+G00
10.000
10.000

2600

5-000,

104000

__NUMBER OF DATA LINES®

38+000

370000

38¢00C
560000
65000
66*000
59*000
55000
540000
54e000

539000

58000
60000
58+000
58+000
59°000
61000
600000
620000
65*000
642000
654000
620000
580000
440000
450000

_.... OCTOBER 30 , 1974 . a _
OCTOBER 1 to 3, 17 to 24, and 27 to 29, 1973

112TIME_AVe LOGIC®

63

0.

WIND SPD WATER TEMP

“60000
_60+000
60900
609700

_. 649000.

60°700
60°500

-60+500

60*200

60°200

60°500

60°500

60°500

60°5U0

_ 609500

60°500
609500

_..60*700

960000 46000
_ -.400+000._ 30000.
1006000 3-000
724000 6000
nee 47 9000... 89. 000_.
37000 8-000
56 +000 96000
_.-. 679000 80000.
750060 116000
754000 “ 7+000
740000. 98000
724000 10-000
846000 10.000
960000. __ 99000. _
100+000 86000
974000 7+000
974000 _ +000
100+000 4000
100+000 46000
974000 _ 66000_
100000 7000
1004000 36000
1000000 99000
100+000 36000
654000 96000
654000 66000

61°000
60°700
60°5u0
60°50
540700
540700

60#200__

609700,


NOVEMBER 15, 1973
OCTOBER 17 to 24 and 27 to 29, 1973
NOVFMBER 1 to 14, 1973

_..NUMBER 6F DATA LINES® 200TIME AVe LOGIC# Oo.

CLO C6VER: AIRTEMP REL HUM wIND SPD WATER TEMP

10.000 440000 ~ 65000 ~~ 96000 5 4*700
10000045000 654000 64000 Bee700
102000 454000 684000 106000 540500
a 104000 450000 774000408000 = S$ 4# 700 ~ -
100000 #52000 804000. 149000. = 589000,
10000 474000 564000 10+c00 55 +500
mm = 400000 450000 584000. «= 7*000~S—s«S 5 #5 es
_ 100000. 4500 602000 _-559500

10+000 434000 73.000 46000 55+000
_ "10.000 420000 76.000 ~ 4e000~~——« 55 #000 ee
cee, 91000 389000 820000 82000. 549700 aS
84000 462000 664000 74000 54*700
~ 10.000 470000 ~ "61.000 6000 540500
uw. 409000 460000 664000 0000, 542500 oe ee
10600 450000 71000 42000 540200
10.000 43000 732000 36000 54e200
1.600 378000 «== 89000 3-000 549000 .
36000 320000 964000 36000 542000
54000 329000 100+000 3600 53°700
wee nee, 40000 430000 1000000 ___. #000. §3*700.
5+000 486000 636000 36000 53700
° 10.000 500000 500000 5.000 = S4##200 7
ec... 100000 440000 684000 000. 540000 _
10.000 450000 582000 86000 54000
10.000 440000 80.000 11.000 53*700
10.000 446000 86.000 116000 53+500

69


model simulates the errors of an observer remote to the
site attempting to predict the site stream temperature
one day ahead.

13


L6

651000
660000
67.000
684000
891000
724900
710000
722900
734000
746000
754000
76200
77900
782900
734900
899900
816900
824900
832000
846000
85+000
864900
874000
88+000
89*000
99+000
912000
924000
934000

40

590
60

110
115

120
125

130
135

140
145
150

160

180

INT 2 4

G8 Te 60

CONTINUE

CONTINUE

GS TS (110217001300140). INTC

WRITE(I 9.115)

FARMAT(N4 wean FULER INTEGRATION He eHET SS)

G3 Te 150

CaNTINUF

WRITE (18,125)

FORMAT (14 wewee MED. EULER INTEGRATION ####*T//)
G8 TS 150

walte(19.135)

FARMAT( M4 #eeee 4 TH GRDER RUNGE*KUTTA HHH HHENS ZY)
Ge T3 150

WRITE(185145)

FARMAT( 14 Perret ADAMS MBULTAN INTEGs s#eawewel//)

CaNTINUE

INTEGRATION CONTREL PARAMETERS

REAN( IN, 10) X#0X+DXPRs XEND

WRITE (18s160)Xs9XsDXPRs XEND.

FORMAT(' X s!tF4e?! OX 21E10631 DXPR stE1003
1t XEND s1E10039/)

READ(INs 10) FRRAR»DXMAX

WRITE (18.180) ERRBRsDXMAX

FaRMaT(! — ERR8R etE1063' = DXMAX BIEL003s/)
READ(INs 10) OT» Us ARFAZEL

READ(INs 10) RHO,WMPHs TAIRsCLDs RH, ALBAR


SECTION VI
DISCUSSION OF RESULTS

In addition to the analyses described in Section IV, the
Data Analysis Models, an order-of-magnitude error analysis
was carried out using the closed-form solution developed
from equation (4).

ORDER-OF-MAGNITUDE ERROR ANALYSIS

In order to gain insight into the solution of equation (4),
it was logical to simplify the form of the equation and
obtain a closed-form solution

éT = Ae + Fo cy, (6)

where the C; are the respective particular solutions
associated with the errors 6 , etc., and the constant A
is related directly to the error in water temperature 6T(o)
at the site, x = 0. The coefficient a then reveals the
rate of decay of the error in temperature with distance.
The June 21, 1973 averaged data were taken as an example
problem:

uA (Rate of flow) 13,503 ft3/sec ,

El (River width) 400 ft ,

Cc (Cloud cover)

8 tenths ,

el

(Sun angle) = 59° ,

o (Relative humidity)

71 percent ,

W2 (Wind speed) 10 MPH ,

Tair (Air temperature)

75 OF . (7)

These data were compared with the averaged data of May 31,
1973 to obtain the errors in the variables between these
two days. This was done arbitrarily to examine the
order-of-magnitude of the resulting errors in temperature.
The input errors are:

18


—

5.000
22000
10.000
100000

~~ " Be000

5000
10*000
100000
10-600
10.000
~ 60000

42000

#000
86000
8+090

22000

36000

8-000

34000

54000

#000
0000
26000
92000
102000
10+0090
10.000
10.000

732000
"850000 ©
700000
686000
“660000 —
640000

64000

680000
71.000

664000
706000
71000
61.000
610000
576060
62+000
70000
739000
756000

~ 70s000

61+000
530000
50¢000
55000
550000
57*000
580000

550000

48

79000 11.000
61.000. 116000
936000 66000.
100+000 6«C00
““$oo.000. SC CO
1002000 3000
100+000 4.000
“930000 coo.
90+000 . 5000.
1002000 60000
760000 —«86000-
68000 beCOO |
90+000 3-000
"842000 = 5eCG0
96*000 36C00
874000 3.000
“530000. 7000
414000 116000
344000 146000
384000 134000
33C00 136000
644000 8-000
“74.000 ~ = 8ea00
61.000 126000
552000 8000
60+000 66C00
554000 76000
594000 6000

70¢200

710000
... 7829500
729200

“729000

71700
719700

“729000

729000
72«0G0

~~ 749500

729000

72200

“72900000

72°000
719700

“74070000

719700
71*500
716500:
70°500

70+000

68700
68+500
68°700
68¢500
68500

699500


£6

4286900
429+900
439+000
4314000
432000
433°090
4340900
435000
436000
437+000
4384900
4396000
4499000
4410000
4426000
4436900
44446000
4456900
4460900
4472900
4486000
449+000
4590000
451+000
452.000
4534000
4540000
4550000

800
806

106

802
805

107

803

108

110
111
120
124

125
130

SAVEY(T)s¥(1)

TaTLTM

DS 106 TelsNe

VCD) SY CL #ENDH# CECT D+AA(T)+98¢ 1 4C0(1) 4D0( 19)
FPCTI SECT) +2 ,O#AALT +3 00#BB (1) +4eMH¥CC(1)+5eQ4¥DD(1)
TAS,

GS Te i101

D9 805 tstaNr

sYy(r)sy(1)

OS 107 IelsNF

DELTA (T)SF(T)=FPCT)

VCD s¥(1)+V#OELTA (1) #ENDH

TASS

G3 Te 1101

DA 108 TslyNr

DALTA (T)aF(T)-FR(T)

YCT)sS¥¢1)+V#DALTA (1) #ENDH

JUMP a1

G8 Te 11014

N@eaeeseeTEST FSR OSUBLINGe IF SK, SEGIN NEST STE= AFTER DOUBLING
IF (ABS (TI IM=T) "ABS (260#H) )1000,1000, 111

IF (KDELY=4) 100001201120

IF CABS (2 60#H) ASS (HMAX) 112121211000

Da 125 TeleNr

TFCABS(DALTA (1) )*ERRAR/(128eO"ABS(H) 1912541252 1000
CONTINUE

TF (V#H#CL TF 00062531302 1000s 1000

CONTINUE


1. Computes the probability of meeting given error
requirements given in both initial water tempera-
ture and meteorological data at the site.

2. Computes the probabilities of meeting given error
requirements given errors from water temperature
and meteorological time averaged data at the site.

3. Computes the probabilities of meeting given error
requirements given errors from water temperature
and meteorological data at a remote site, in the
case of this study, at station 7 above the nuclear
plant. A list of the MONT code is given in
APPENDIX C.

INPUT REQUIREMENTS TO MONT

MONT accepts the following information:

NLN, LOG1: In 6110 FORMAT, The first variable is the number
of lines of data (taken every 3 hours, 8 data points per

day, each day always complete); the second variable is
0 if no time averaged data appears, not zero if time
averaged data appears.

Following these integer variables the major part of the data

is read for the remote airport and the site water temperature

in a loop of NLN lines as follows:

C(I), TA(I), PHI(I), W(I), TW(I): In 6F10.3 FORMAT. The
variables are: 1) the cloud cover in tenths, 2) the
air temperature in °F, 3) the relative humidity in
percent, 4) the wind speed in knots, and 5) the site
water temperature in OF,

Next, the average data for the actual site and the particular

day are input:

UA, EL, CC, TAIR: In 6F10.3 FORMAT. These variables are 1)
the flow volumes of the river in ft3/sec, 2) the river
width in ft, the cloud cover in tenths, and the air
temperature in OF.

TWA, RH, WSP: In 6F10.3 FORMAT. These variables are 1)

the water temperature in °F, 2) the relative humidity
in percent and 3) the wind speed in knots.

31


Zeb

6164309 199 CANTINUF
6174990 RETURN
480099 END
619-599 ccccc

629-590 Coccc

6214900 CCCCC

622.900 Ccccc

6235900 SUBRSJTINE 6UPT
624.990 C3MMON DOTS YPC 4), DTsDUMCH) s XPRs XENDZDXPRsXsOX¥NEDINTS
6250050 LINTC) IN, T8sF Ls AREA, Us ERRGRD OXMAX s TEMP? HFGTB( 10) sCONSL
6260590 CAMMA NRHA» Wo MO + HDs TATReCLDs Rs ALRARs TWGa TAs THSADEWS EAs HOSPHS
627-990 CIMMEN HSQaHASs 7 BRsEVApsHEs 4CeHLYMNe PTBL (10) TTBL( 10) »WMSsCENST
6234090 C3NMBN Cl IFs HFG» H8C,CLO1,CLI2s;0L02, DALPsDALPR»DCLDsDRHsDTAIRs
623,000 LDTA,D% 5 DWMSsD 4M 94 DF As DEWs DLYMN, DBTUSYNTEMP, STTBL(10)sXTBL( 10)
6399000 WRITE(1S.10) xX99X
631-000 10 FSRMAT(! x StF7ei!t FT Dy s'F60el1 af)
6322000 WRITE(13,26) TEMP-
633.050 20 FSRMAT(' STREAM TEMP sStEL2e4! NEG FI/)
6740990 WRITECISs70) OTDDT
6350090 WRITE( 13230) EWeDEWsEVaPa Hes HG

2005 FSR4AT(t Ey ath1p.5! Dew e1F1Ae5' EVAP st£1205! HE =!

2EL265" 1 Y/MIN HFG stE129Ss/)
WRITE(13s50) H88sERRsHC
FARMATC! HSR s161205! EBRstr12e5! HC stEl20e5' LY/MINI/)

6494990 WRITE (T3160) HLYMNs HD .
6416900 60 FORMAT(! TOTAL HEAT TRANSFER etF12«5! LY/MINIE15 65s
642%000 1! BTU/SECeFT2I/)

6434000 79 FORMAT(! DT stEiae5! ODT stF1re5e/)


SECTION V
STUDY AREA AND METHODOLOGY
THE VERNON NUCLEAR PLANT

A program of ecological studies of the Connecticut river in
the vicinity of Vernon, Vermont was initiated in 1967, prior
to the operation of the Vermont Yankee Nuclear Powerplant.
The preoperational studies were enlarged in scope in sub-
sequent years and were continued after the plant became
operational in October 1972. The location of the plant is
shown in Figure 5, where the positions of short-term and
long-term sampling stations are shown. The particulars of
these sampling stations are as follows:

Station No. Location Relative to Vernon Dam Type
1 6.45 Miles (10.4 Km) South Short-term
2 4.70 Miles (7.56 Km) South Short-term
3 0.65 Miles (1.05 KM) South Long-term
4 0.55 Miles (0.89 Km) North Short-term
5 1.25 Miles (2.01 Km) North Short-term
6 4.10 Miles (6.60 Km) North Short-term
7 4,25 Miles (6.84 Km) North Long-term
8 8.70 Miles (14.0 Km) North Short-term

Stations 3 and 7 are permanently emplaced below and above
the site respectively. Stations 3 and 7 yielded the water
temperature data used in the study while meteorological data
were obtained from instruments at the plant and in Keene,
New Hampshire.

DATA BASE DESCRIPTION

The data base used in the project was taken from four
sources:

1) Measured water temperatures at the site
(See Aquatec [1974]),

2) Measured meteorological data at the site
(Personal communications) ,

3) Meteorological data from Brattleboro, Vermont
and Keene, New Hampshire (U.S. Weather Service),

4) Meteorological data at the Burlington, Vermont
weather station (U.S. Weather Service).

1a


10+000
_ 10000
10-000

“" 94000
99000 |

96000
69000
_ 96000
10+000
“" 9000

#000 |

46000
8000
64000
10.000
~ 100000
_10+000
104000
10000
10-000
16.000
10600

__ 106000

2.000
8.000

____. #9000

96000
20000
+600

58e00c

58000 __

560000

542000

_ 542000 _

536000
516000
51.000
50s000
426000
_ 414000
524000
55+000

56+000

500cQ0
480000

500000.

50+000
520000
52e000
54e000
510000
512000
470000
410000
39+000
402000
54000

612000

930000

. 904000

83.000
“754000

—.. .69#000

720000
69000

.710000

714000

“86000
_ 964000

55000
534000
_ 512000
740000

934000 ©
_,_ 938000

964000
962000
960000
96 e000
96000
93-000
932000
960000

100+000

100«000
900000

720000
59

116000 67°500

126000 _ 670200. |
126000 679000
"136000 679000

ww 129C00_ 679000
126000 67°00
96000 669700
146000. 660700
10000 669500
34000 66000

_ #0000, 654500
86000 669000
116000 666700
56000. 66+700
4600 669200
7.000 669000
_ 106000 ___ 65#700_
86000 65+700
126000 65+000
46000 65000
66000 65+500
136000 65*200
132000 64#700__
46000 64200
36000 64*000

_ 30000 6385000
30000 63*700
84000 646500
10-000 646700



671

530-900
531-900
532.000
533.000
534.000
535+000
5350090
§37+590
538-000
539ed00
542+000
541-900
542005

543-000
544e500
545000
545-0000
5472000
548-090
549900)
5520000
531-900
5524990
553+000
554e990
5554000
5562000
557-000
5382000

19
20

30
40
$0
60

65

110
115

120
125

130
135

140
145

FSRMAT (KE 1060)

FORMAT(6110)

RATIO. = Re14159/1R00

NUMBER BF EQUATIONS: AND. CHBICE SF INTEGRATI8N SUBRGUTINE
NESt

INTCal

IF(INTC = 3) 3024050

INT = INTC

Ga Te 60

INT 2-4

GS. TR 60

CANT INUF

C3NTINUE

JFL Ast

TF (SFL A€4 16521502465

CONTINUE

G9 .TS (110417021302140)s yNTC

WRITE(TS2115y

FSRMAT( IY wees EULER INTEGRATION eeeeel//)

G3 TE 130

CANTINUE

WRITE (19,125)

FORMAT NS #eeee MBO. EULER INTEGRATION *eeeet//)
36 T3 150

WRITE (184135)

FaRMaT('4 weeee 4 TH OROER RUNGESKUTTA ‘###ewRT//)
G6 79 150

WRITE(16s145)

FARMAT('4 ane ADAMS MAUI TSN INTEGe. x#aewuet 7)


eLb

14900
26990
36930
460C0
3-060
6900
7-399
8-0c0
96500
19-900
11+990
12990
134900
14-000
198999
16+990
170900
18*900
420999
790000
24.000
224000
23+909
24.000
250009
POS!
276920
P6e090
794900

CcCccC C(1n00)=rLeun CAVER
CCCCC TACLOS0)sAIR TEMPe

nNaNNAANNAAN|AADA

P4I(1900)8 REL HUMIDITY

w(10C9)= WIND SPEED

TW(1000)= WATER TEMP

NCAS= N&y OF CASES

NDAs NS» SF MAYS TA BE TRIED

DELT(S0)2 ERRSR CRITERIAR TF BF SarISFIED

TDAS(50)= Ne SF DAYS IN TRIAL

NUN= Ne AF LINES

OTe FRNSR CRITERIGR FAR THE SPECIFIC CASE

NTAz N3. SF TIME AVERAGED DATA PRINTS

UASEL2CCaeeeesNSP= ACTUAL DATA’ AT VERNEN

INT2a0NTR2= Data PAINT NUMBERS

OCANTAseesesMTya FRROR IN VARTARLES

CINMBN YPL(5)9¥1(5)2XPR12XE 101, DXPRIsXL,0X4sNELs INTs INTCLAING 184
1sEL1s ARE Atet

COMMSN FRROR1DXMAK 1, TEMPLe4FGTI1(10)sCANS1L

COMMEN RAStswieWMO1sdD1s TAIR1sCLD1sRHis ALSARI,TNS1s TAL, TWSALZEWLs
TEAL, HSS1,HSt

C3MMSN HSR12HO81s FBRIZEVAPL2HE1 HCL a HLYMNIsPTELA (10) sTTBLI(10)5
twMSt,Canstt

COMMSN Cl IF1sHFS1.HOC1,CLD41.C1 921,CL0342DALP1, 0ALPR1,0CLD1,0RH1>
ADTATQL,0TAL,DW4 DMS1sDWMD1s DEAT DEWNLSDLYMN1sOBTUSLINTEMP Ls
2STTBLL(19)sXT3L1(10)

COMMBN C¢1000)2TA(1000)sPHI (1900) 2w( 1000) 4 TH( 1000)

C3NMBN NCASs NDAs DELT(50)s TOAS(SC)

C3MMEN NI-Ns INe TAUs OT NTAs IDA

COMMEN CTA(SQ)4TATA(50)/PHITA(RO) NTA (SO). TWTA (50)



GLE

572290
332900
596400
59-901

$1-000
624000
4490090
640950
63.900
66-030
67.900
686060
632930
79-059
714000
726990
73-000
749900
735000
752090
776990
784900
734500
326009
81.900
824000
334030
84.900
85.900

1030
1001

2009
2001

CEMM9N ERROR12DXMAY Ls TEMP1sHFGT31(10)2CANS11

C3MMBN RHOL4W12WMDL es HOLs TATR1 CLI 2 RHi2 ALBARL ST WSL a TALs TWSALEWL>
1EA1,H9S1,HS1

COMMON HSR1,HOS1 LER EVAPLOHES HCLeHLYMNG sPTBL 2 (10) ¢TTBLA( 1005
LWMS1,CENSTI .

CAMMEN Cl IFLsHFGLsHOC1,CLO114Cl D24,CLO31sDALP1sDALPR1s0CLO1,DRH1>
LDTAIRY,DTAL + DNL» DWMS1sDWMO1s DEAL» DEW Ls OLYMN1sDBTUS1»NTEMP Ls
2STTSL1(10)sXTALL (10)

C3MMEN C1000) sTA(1000),PHI (1909) 4W(1000)4TW( 1000)

CSMMEN NCASs NOAs DELT(5O)s IDAS(RO)

COMMEN NINZ INS T3UsDTINTAs IDA

C3MMSN CTA(50)sTATA(50) sPHITA(5O) sWTA(50)s TWTA(SO)

COMMON UAZELsCCs.TAIRs TWAsRHe WSP

CAMMBN INT2, 1NT3sDCs DTA, DPA, OWeDTWs INVSs FLAG

COMMEN OM21 RETA

C3MMBN LAGLsNDAS

C3MMBN IC.ASsPRP (50,10)

INF101

13Ua19P

FORMAT (661003)

FORMAT(6T10)

READ (INJ1001) VLNsLaG1

WRITE (18,2000 )NLNsL8G1

FORMAT (4HO22Ka tNUMRER OF DATA LINFSsteIS5etTIME AVe LEGICs!s 15)

WRITE (T8uUs P0601)

FORMAT (1HOs2X,1CLD C8VER AIRTFMP EL HUM. WIND SPD WATER TEMP!)

D3 10 T= tsNLN

READIN, 1000) CeLysTACLYsPHE CTV sWtT TW (LD

WRITEC Sue 20020 (ps TACT SPAT Cp VawC Te TA CLD


SECTION I
CONCLUSIONS

One-dimensional stream temperature models are effective
for predicting average temperatures over a short period
of time.

Predictions made using site data averaged over a day
will be within 2 OF (1.11 °C) of the actual average

temperature of the following day over 90 percent of

the time.

The water temperature at the Vernon site is the most
important factor in the analysis. The Vernon study
indicates that meteorological factors affect the
Connecticut river predictions results very little.
Travel time thus appears to be a major factor since a
knowledge of the initial temperature is essential to
accurate predictions.

Seasonal effects on prediction accuracy are minor.


16

3712000 67 D(1)aD(T)+S#NALTACT)
3722900 65 CONTINUE

3734000 C IF IN STARTING SEQUENCE, BRANCH
3746000 IF (KSTP=?4)702990100

375+000 70 GI TA (1000210094 10002745 10004 100s 10004 78s 10007 1000s 1000274210008
3764900 11000, 1000286410001 1000s 1000474, 1000 1000s 1000) sKSTP
3774900 C 4TH, 12THs 20TH STEPs GS BACK
378+000 74 HaeH

373-000 be 75 TateNe

380+000 Atl) aeACT)

381+000 75 Ctlys=Ccr)

332.000 G8 TS i000

33.000 C &TH STEP GB FSWARD

334-000 9 78 HseH

385+000 DS 79 TstsNne

386+000 YCD) #STARY(T)

387-000 AclyasAct)

3384000 79 Ctlyaectr)

3396000 G8 Te 1000

3902000 C 16TH STEP» HALVE Hs» APPLY TEST 14
3910900 86 HsH/200

392#000 Da 87 12t4NE

393+000 ACTIFACTI/200

394+000 Be ly=B ty Zhen

395+0900 CilyeC( 1/890

3964000 87 Di ly=2(1) 71600

397 +000 De 88 I=14NE

398°000 IF CABS(DALTA (1) ) *ERROR/ABS(H) 188s 88289


;
~ some 21, 1973

+ MAY 1 to JUNE 20, 1973
i)

' RUN

“TNUMBER GF DATA LINES= GORTIME AV. LOGIC= 0
\ _.. CLD COVER AIRTEMP REL HUM WIND SPO. WATER TEMP | a

+000 354000 924000 46000 48000
momen ByGoO ~~ 33e000 96000—~*C*S CDBG
86600 394000 9346000. 42000 . 48200
96090 56000 60000 $000 48500
~T  1G0000 S6e00G 574000 °°-—O 00 HBC
ae 10-000 61000 = 520000. «44.000 4807000
106000 53000 80000 46000 49+000
oo "108000  S4000 964000 e000 OGD
10+000 56000 _ 909000... 7#000 490200 |
105000 554000 90000 8000 49+200

106060 ~~ 664000 © 0004905
oo. 5000 660000 684000 446000. = 50«000 .
10.600 726000 574000 14.000 50«500
omen" "4000007 ——- 730000 856000 9e000~S*~*~C~S DD
oe 100000 680000 684000 96000... 540000 .
106000 616c00 902000 46000 51+000
° “400000 640000 84e0G0 4000S Ste200
10+000 650000 784000 10000 51+500
86000 634000 30+000 116000 51700
oe 100000 = 70eG00 «734000 Be 000 SBe209
10.000 754000 546000 106000 526700
10+000 554000 100000 7000 53+000
ooo 10.c00 50+000 93000 100000 52e500 ~~
¢ 10+000 50000 80.000 4.C00 S29200
10-000 486000 684000 5000 52000

36


The particular solutions are found from
Si
mr (D-25)

where i= 2, ---,m. It is on this basis that the
sensitivity analysis is completed.


nZb

3884900
3294900
299-000
391+900
3924000

397°G09
338-090
399+990
4392090
491-000
4926000
“4932090
& 900
4954900
4960000
4974590
4082000
4996500
449+0U00
41146090
4126900
4435000
4142900
4150000

caMMeN
1€41,49S
CaMMsy
1WMS14C09
CaMMSN
1DTAIR1s
2STTBLI¢
CaMMaN
CaMMsN
CaMMeN
caMmen
CaMueN
COMMBN
CaMMeN
C3aMMeN
caMmeN
R4E1260
X19

Y1(1)80
XEND1 22
DXPR1is
0x12190
NElst

INLaIN
1972180
ELise€L
AREALSU
Ute.

RHO. Wis MOL HDL es TAIR LS CLDIsRHi2 ALBARL es TNSLa TAL TWSALZEWLs
1484

HSR1sHSB1 ss ERRIsEVAPLAHEt HOS sHLYMNIsPTEL) (10) sTTBL1(10)s
NSTL

Cl IF1,HFS1sHOC1+CLO11sCl D212 CLO31,DALP 1s DALPR1»OCLD1,0RH12
DTALsNW1 »DYIMS1,OWMD1,DEA1sNEW1+OLYMNisDBTUSL »NTEMP 1,

1014 XTBL1 (10)

C(1000)4TA (1009) + PHI (1090) W( 1000), TW( 1000)

NCASs NDAs DELT(S0)s 10Ag(50)

NUN INs T5UsDTsNTALIDA

CTA(S0)2 TATA(5O)sPHITA(SO) sWTA( SO) TWTA(5O)

UAr EL, CC. TAIRs TWASRH WSP

INT2s INT92DCa OTA, DPHI DW sO TWs INVSs IFLAG

DM21/RETA

LAGLaNDAS

TCASs PRP 50910)

“4

Tw

5006+

NDA
Oe

A


86

942000

95-000

96.900

974000

98-000

99000
109+900
101+900
102000
103+000
1040900
105+900
106000
107900
198+990
109+900
119+000
111+000
1124000
1132000
1142000
115+900
1166900
117*000
118-000
119000
1209000
121+000
1222000

185

08 185 151210

READC(IN» 10) PTBL¢(T)s TTBLOD)s HFGTB(I)
TA = (TATR = 326)*50/9e

Wo® WMPH¥220/156

WMS 2 Wo # 03048

WMD 6 WMPH * 246

CONST & FL/(RHS#U*ARFA)

CANS1 = (RHB#D0903781/ (P40 ¥36004#00 0644)
READIN» 10) DALPsDCLOsDRHs DTAIRs DWMPH
OTA @ DTAIR#*5s/9e

H9S = 169*SIN(ALBAR#RATIQ)

H3C = 169*COS(ALBAR*RATIS)

CLD1 = 16 = 20765*CLD

CLDA s -.001RHCLDax?

CLDR & te = 90N6*CLD¥#3

DALPR = DALP #* RATIA

DW a DWMPH¥2p0/45¢

DWMS = DW: #* 03048

DWMD = DWMPH#?4.

INCIDENT S8LAR RADIATION

HS _= Hes * C193

REFLECTFD S81 AR RADIATION

HSR = HS * 34/ALBAR

FTA = ST(TTBU sPTBL»TAIRs 10)

EA = RH¥(10130/14700) #FTA

TAIRP & TAIR4OTAIR

FTAP = ST(TTRLYPTBLaTAIRPs 10)

DEA =..(10130/14700)*#(DRH#FTA # RH¥(FTAP = FTA))
WRITE(18,200) OT


106000
106600
72000
8-000

_..... 99000

70000
10-000
92000
96000
10.600

10.000
72000

_ 99000

10.000
10.000
_ 106090
10600
10+000
10.000
10000
86000
7+000
96000
10-000
_ 106000
10-000
10.000
84000

100000 _

56000

_ 879600.

52e000
560000

_ 564000.

540000

520000

_ 460000

49000
572000
55+000
51-000
530000
492000
48e000
452000

466600

50-000
52e000
51000
51000
480000
450000
420000
460000
50«000
600000
510000

490000

939000

779000.

93-000

836000
83000

-80+000

724090
690000
80000
80600
692000

100-090

934000
864000
89.000
864000
806000
694000
294000
644000
27.000
664000
744000
834000
77.000
694000
694600
644000

714000
39

—-— #9000

_. 49 C00__,

11-000

~~ 109000. _

8eC00
~ $4000

42000

“56000”

36000
94000

100000 |

13+000
134000
10000
62000
5-060
4.000
5000
32000
10*C00
7000
8000

46000

54500
54500
540700
55000
.55#000
55000

55200

_.... 55#000

55+000
95500

____56*200_

56*700
56*500
56200
56000

~~ 55700

_55*500_
56#000
56*200
56700
56500

569000

__ 86#000

§5+500
55200
55200
55700
56000

56*000