A MATHEMATICAL MODEL FOR PREDICTING
TEMPERATURE IN RIVERS AND RIVER-RUN RESERVOIRS
Prepared by
John R. Years ley
Working Paper
No. 65
United States Department of the Jnterior
Federal "water Pollution Control Administration
Northwest Region, Portland, Oregon
1969
-------
e
The development and testing of jhe temperature prediction model
scribed in tnis paper has b^.ne^:i c^e irom che cooperation and contri-
butions of several individuals ; as well as various public and private
agencies .
Initial development ana prc0ra:;j.v_ng of che model was done by
Mr. William Morse " of the :-*.-.?CA_ ?ati.:ic Northwest Water Laboratory;
Corvallis., Oregon.
The hydraulic calculations ar^- c,c^pted rrcm techniques developed
by Dr. Bruce Tichenor er.^ /.r. A_een C.'.riseianson or the FWPCAj r'aciiiic
A o r >- nwes u ^vater ijaoorci^ory^ vjcrvc- ^,-t.^^ ^r^gon.
Mr. Robert Cleary, a Or^^.u^ce s^ucent in chetuical engineering
at che University o; Massecn^^cis ., i\;:.herst, Massachusetts, providea
technical support while working for tn^ Tivl-'CA., Northwest Regional Offiee^
?ortlanc3 Oregon, as a temporary employee during the summer of 1963.
A major part of cevclopi.-.g and .reccing the model has been the
ccrr.piia cion and organize cion or necejt-ary caca. The autncr wishes to
thank che US Army Corps of ^n^ineere; "leather Bureau; Bureau of Commercial
"fisheries ; US Geologeeal Survey; j'i_cev^l Aviation Agency; Acomie Energy
Commission; Cne^an., Granc ane Do^-^-^c. County VUD's and Bactelle Xorthwest,
for providing hydraulic, rr.et.-oro-o^it anc water temperature data.
Xr. Morse is presently ^ r;.^jl'.ir.^^^c_:*-;i v/Ich the Bonneville
-Jcv/oir .-**criTi.!Ln^.s ti^rs L^-on , - GIT c _o.r^c , uir^'_G.I *
-------
r* Gne~ a j-^an3 j-Ona L temperature ^r^.ai.C to-On mOGei. naS Deen develo
nd tested on the Columbia reiver. lY.a modal in its present form is
(i) Cross-occcicnal chc_^cccri.i t^c.s 3 including wacer surface
\-7idch and cross-3i.cc^.c,r.^_ area ^re available.
(2) Adequate raeceoro-O^ic ^aja ii available.
(3) Water surface e^ivacror. od.;a i3 available.
(4) Svaporacion c..ic sensible n^a^ flux can. be evaluated in
terras of rcj^on.,1 vr^ather aata and the results of representa-
tive field _,c^aioo.
(5) Lateral and -/artioal v^rlc-tions in temperature are negligible.
(6) The flow can be consiueru^. caasi-s teady.
Punched cards containing the 7 C. \1Iw-X statements shox>7n in Appendix C
can be obtained from the Y.-.-J^j Xortnv/cst Regional Office, Portland,
Oregon. The program can also ba ^xoe^ced directly from the FHTCA data
cell at the Departmenc of the Interior Computer Center in Washington, D.
-------
j. n e A. ma i. i o ac ^ irrt.^ o ^ ^^ u^^~- _ \ ^ r ^j ab a resuj.^ o A. ii.ec*. e rejection
.orn menu^accurm^ a^.^. p ^. .;>_.*, ^^.-^^-^-^ _^ -...(.^us^iT-^eSj j_rrt.poun^rnen«- ^y
*..tnD^ C*l.lU J.i.iJ_^TC*.ta,^-vJti. ^_^_V^. -^ ^/kt ^^i ', » ....^w.1 ^' »iii->l^A.C«.i_^-«-4.t^ Ctti. J-iilJ^X Wi-tLiU. ViOkL.^.*.
i ~ s " r a e "* o n
l\or thwes t Regional Ci~~lco_> is aev e,.^_jinj a coir.pueerized sys ten:.. 01
reservoirs.
^ ^
uor preeiecmg ceiuperajur^^ m riv _s ^.n^ea are reguiaced oy darris wj_a\
r^-.n or-cne^ri-ver^, or s,v.^._^ ^^^--'_<- ^ *_ore^ capacj-cy reservoirs j anc \.jnien
ruay also have reaches whlcn ar^. Jr^u-Jlov/ing.
In the river-run n;oe^^^ ^"_;._j^ vo-urr.cs 01 waeer, or water parcels3
c.r e r e i eas ec rror.i an up o - r c:a.v. ^ ^^.r - n ,^ pome at sp ec i r i ec m eervais or
.ownstrcam ehrou^h reservoirs and
open river reache
flow re^ula tion,
parcel ceiuperature due to weathe
are predicted .
-------
.i.ne rTiOce^ cove*.o~3ir.^t.t ~^r^
^xT j. cTJip CTTci. w U-j^
:orporatec, with cne 2;
.liver Thermal Zffecus S
program., is to pro vie a
crature rej^irr.e o^- tn^_ rr.c.m ^ ^.^
river's ecology, partic^ar-'/
-fishery.
Dur
;^ vv^^iccion efforts have been
n c/ic ScuGy3 2. uv-7 o y o ^ r
j- G IT c ^ e e v s. -^ ^j
-------
u.rpc s e or predicting t eruper ~
acures in rivers \vmen are r >^0v--acoa oy aa^s wien srn.3.11 storage c3.pc.cLzy,
1C IS dSSenClciJ. COr aO j j_ICa ClOn. a C ^.-.^ *AOC. i l. Uric4, u tt"l£ C^TOSS SGCul.Oi1C.J,
cnc*iTci.c L^X"j.s uxC6 on CiTic ^TL_\'^ir o^, r.,i-v^n . j u^r criC^'jrfkOjTc3 L.c nss bccn 3.ssur^uCi
chac:
(1) Til a prototype io ch^r.;^,ily w'cll-:rLixed vertically
ar^d laterally.
/o\ - ^ -» ' i_ ji _- --_"' * _ _: - v * 41
V,'-^ JLtUj.i^j.w'UG.^ttct^. C^ ^.^_-^_'.. ^. ,-^, >-^.L._v^:i.i>^wi.x v«c**i uC i^U^j-C^Ut-tCi
^ ji y j. n e w c. L. e r s "e r ^ ^ c < - ^ ~ j , ^ ^ > \ >' c c.. o c r.^ o s c*. i_ "i^n c . o i i
or iong3_C"J.G^-1"!^j, >-_.j cc.,~vv- o.\j_y. ^',',a cer surracG pro11. j.e reLers
lo tne s :i r" o e - c.-^ v^ ie"_" ^u r^ece as s cer; ^.n a lor.^i. tucinal
section ar.e eo^o no c ^.. p-_y a.": a^joiute suriace eievatron.
\,4; Weacner da ta ^rc:;. a rt^r^ocn j^txve s tation is adecua te lor
cescri.oxn^ ccr*u^ _!^ c_*s <_»j-on^/ a speci.jLj.eG porej_on or tn.e rj_vcr.
coer -j*-. :,.". j!J. can be used no eva j_uu tc- neat
ce.
Longi cucir.al van. a cicn^ la croo o-sec c ion a I character is tics
are imp or can o cn^v n z o ^zar as t ney e r r ec t t ne wa cer velocity.
Given the above asju;;,o clcns c^:..^>eracure changes can be predicted
wnere
water veoci
.c'-^^c neat o£ water
o£ wacer
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Y
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(2) Qr - shore \.-aYc refine j^ aolar radiacion
/ ^ \ ,-, ' - -
v.J; s;a - -£: ac^G^p^ir^c r^_;.i.cr.
(4) Q-u, - bach, or ^.v.l^.u.^ r^.:._^ciori from the wacer surface
^o^ QQ ~ ser.si-G^e a^a_ cGaa^cd-Gr^
Lin tac raoaelj che r.c^ haac ^-^x, C, caa i;e expressed as either
j.inear or parabolic iar.c^_G.". or c:.^ ,/^.cer ^err.peracure3 T. The r:\Gce
so C-^u-c e_^ticr ~< ac nca CuTi. oe cnosen, nov/ever.
inaica;es eaac a.^u ^_aear 1'unction. is adecuate .
r o
^ -., *«* o
(5.5)
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i."u.HC w. j_ C j.~.
0II "C n e V7 £ £ c: jf "i_ £ * CO c. iT C. C "v- -T >..
^e ncau iluxes
C O tTT^ C T ci C U j. c .
Finally, if cr;e
3XOOS UTc CLulC 3 ^ C~ Z - /
L c 1 e 3 U ( x j t) 5 is kr. ovra 5 en e
:roni the equation:
(3.3)
- e. cc.rrn.ineG i r ora e i en er
_C ac* ClOTt 3.1) OT o . / .
wiu'a the
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flux, Q, ar.d t
Ts aua so^.1- e cc...-i>
scation wich weather typical
01 the region in whicn en a r_\ v_r ^yoter;. is located.
yo. Tne inividua zuxes a
tne
V- i»O. -_ c^L
o che river system
w.i ^r.
a ci.cua cover as
a ^
(2)
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^Tci t LLl'C O £ uuc
A.;,
(4.5)
defined previously and
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cl j. 3.H Ji lV o L£
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_ irc.71 c^c Giz i or eric £ o e two en
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: ar;t c i. s charge anG p o o JL c ^ e va ci o
0 "jT V V/ ti u £ Z* S VI !ir -- C. C C; a_ i_ V c j>
t*"*rc Vc.iiri.cG ^jTu.Guaj_j.ya ^^G
ow^xi^ arc ci. cncr very Sma*
i. rfi^o c *_»n GOG.
Vic. jn
- ,' - ^ ^ !_*! CO ou.O Z" ci 3. C tfi C S O.J.LG
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^."..-';.c.-icr £i_evaciofi ana c.iScn<3.T':e
"* _~ ^ j ^_ c o o A. i_ L"I o s *j. ±j ^r o <_i c .1 i,
i ct _C ^ LI. o. S i_ n. ti cIV CITS-^C O.L i_i"
."."_ ,, wnz-CiTi con s ZA. cu c^ c r. G
o ^ c r. u s u b - r £ £. c h respectiv
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_c c^r. be shown that QDO is a
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:ow
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67
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ne res tr;.cc:ior:s
j u b r o u ^ i 7i e c a 1 c u 1 a c <.
continue.
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_scr.r.v0c .._*.,.;. ,_c./~j-~- ^o^'-cas, in ICOO cfs.
or
2.
u;i ca cemoerarure inou;
"' ^01^00irs.cu.ircs
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/veree oy a^a iy s i-S j eec.
'.:_^r_^. ..:. ._oar groups 5 cGrrespoaGinj co ur.ose
s j_s i^i a~/a .. -_*.. *.^^--j_..'v_. . -_-i^_» .». a o t^Vj-iTu-5o r o xn v-a »-i .*_ <_ .^._, <.^j. c. Oj_
NSECT - -iur;^^r of ^-c^ior.- -*.:co which river is divided
2>Z - z:.e "_" ei.j>. ^ ^;.c^^;:.^;~ ta ^'w- V*i V_-_i.- i ^ ^^v_lL--4-J~rf-OOi
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2,ec\/^^.r. ec.;..s ~1:^ i'.-/er ::u_y be civiaed int:o as raany seeeions
ao cir-^ 11 ee a^a, ^* e ^ .-^ - ^-c^a^-a c aoicns cr^ar.i eaais s InoVv'ever 3 cacr e
vaus e ->e a o eauaC'a a _ ja^ ups crearu race arid uuG covras "creaiTi. raee?
"tio ^.cao~L-a e n ^r _a _ ^ "' ^^- ea^u.n. oy cau earn..
are soeci^iea
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