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CEE 5134 - 1 -
CEE 5134
Deoxygenation – Reaeration
and the
The Streeter-hel!" E#$ation
Tho%a" &' (ri))ard
25 *+toer, 2007
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CEE 5134 - 2 -
Streeter-hel!" i"tory
. /odeling o oxygen dyna%i+" in loing ater" ha"
e+o%e a "o!hi"ti+ated di"+i!line in the 21"t +ent$ry
. /odern %odel" are $ilt on the or o Streeter and
hel!" on the *hio Rier in the ir"t #$arter o the 20th
+ent$ry 114 – 1256
. Deelo!ed a relation"hi! to !redi+t longit$dinal oxygen
!roile in loing ater" a" a $n+tion o
– Strength o degradale organi+ %atter
. Con"i"ting o %ix o a+gro$nd 8*D in "trea% and 8*Dintrod$+ed y a a"te di"+harge
– Rate o di$"ion o oxygen into the ater ro% the
at%o"!here
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9ho 9ere Streeter and hel!":
. 8eginning in 113, ;SS %aintained a laoratory in
Cin+innati dedi+ated to the "t$dy o <the %aniold!role%" o "trea% "anitation=
– Eent$ally %or!hed into the E> Cin+innati ?aoratory
. '9' Streeter Sanitary Engineer at the ;SS
Cin+innati ?a. E'8' hel!" roe""or o Sanitary S+ien+e at @n"tit$te o
$li+ ealth o the College o hy"i+ian" and S$rgeon"at Col$%ia ;nier"ity
. St$die" on the oxygen dyna%i+" o the *hio Rier ere+o%%i""ioned in 114 and 115 y S$rgeon (eneral9'' Fro"t
– @n 125, hel!" dedi+ate" hi" oo, Stream Sanitation, tothe %e%ory o Fro"t
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Aey i"tori+al $li+ation"
. $rdy, 9'C', <> St$dy o the oll$tion and Bat$ral $rii+ation o the
*hio Rier, ol' @, The lanton and Related *rgani"%",= PublicHealth Bulletin No. 133, U.S. Public Health Service, Washington,
D.C., 193.
. Streeter, '9' and 9'' Fro"t, <> St$dy o the oll$tion and Bat$ral$rii+ation o the *hio Rier, ol' @@, Re!ort on S$rey" and
?aoratory St$die",= Public Health Bulletin No. 1!3, U.S. Public
Health Service, Washington, D.C., 19!.
. Streeter, '9' and E' 8' hel!", <> St$dy o the oll$tion and
Bat$ral $rii+ation o the *hio Rier, ol' @@@, Fa+tor" Con+erned in
the heno%ena o *xidation and Reaeration,= Public Health Bulletin
No. 1!", U.S. Public Health Service, Washington, D.C., 19#.
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Ee+t o *rgani+ 9a"te" on Strea% E+o"y"te%"
. Streeter-hel!" /odel Di""oled *xygen Sag C$re
– The Streeter-Phelps Equation i" integral to %o"t o the
idely $"ed di""oled oxygen %odel" in $"e today
– >ddition o degradale organi+ %atter 8*D6 to a loingater+o$r"e +a$"e" a "lo de+rea"e in *2, +a$"ed y
heterotro!hi+ %etaoli"%
– *!!o"ing <deoxygenation= i" reaeration, hi+h !ro+eed" ata rate !ro!ortional to the +on+entration dei+it relatie to
the "at$ration +on+entration6
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CEE 5134 - -
*xygen Sag Ee+t" on 8iologi+al Co%%$nitie"
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Co%!onent" o the *xygen Sag C$re
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Deinition" or the D* Sag C$re
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Streeter-hel!" /odel >""$%!tion"
. Strea% ehae" a" an ideal !l$g lo rea+tor FR6
. Flo rate, "trea% +ro"" "e+tion and longit$dinal elo+ity
are +on"tant
. hy"i+al, +he%i+al, and io+he%i+al rea+tion" o intere"tare 8*D exertion and *2 tran"er a+ro"" air-ater
intera+e
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Streeter-hel!" /odel ?i%itation"
. Con"ider" only one <"in= or D* – Degrading 8*D
– /i""ing B*D, S*D, non!oint "o$r+e", algal re"!iration,degradation o %i+roial !rod$+t"
. Con"ider" only one <"o$r+e= or D* – >t%' Reaeration – /i""ing >lgal !hoto"ynthe"i"
. Don"trea% %oe%ent i" y ade+tion only ideal FR6'
– /i""ing Di"!er"ionDi$"ion
. elo+ity, de!th, 8*D exertion, and reaeration are
inariant ith di"tan+e
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1"t *rder 8*D Exertion Relation"hi!
. @t ha" een "hon that, $nder ex!eri%ental +ondition"
a!!roxi%ating tho"e !reailing in a "trea% +ontainingre"ere di""oled oxygen, thi" rea+tion an orderly and
+on"i"tent one, !ro+eeding at a %ea"$rale rate, and
a++ording to the olloing la
<$he rate o% biochemical o&i'ation o% organic matter is
(ro(ortional to the remaining concentration o% uno&i'i)e'
substance, measure' in terms o% o&i'i)abilit*.+
Phel(s, -arle B.
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G$antitatiely Stating the 8*D Exertion Relation"hi!
. >" long a" oxygen i" !re"ent, the rate o io+he%i+al
oxidation o organi+ %atter i" !ro!ortional to the a%o$nto organi+ %atter re%ainingH'
( )
d
d
-1
d
d?I rate o 8*D exertion I - ?
dt
here ? I organi+ %atter re%aining at ti%e, t, oxygen +on+entration $nit", %g ?
I ir"t - order 8*D rate +oei+ient, t
Se!arate ar
d
d
0 0
- t
0
d?iale" I - dt?
@ntegrate J a!!ly o$ndary +ondition" ? I ? K t I t 6
? I? e
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9hat doe" 8*D exertion loo lie:
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S$%%ary o 8*D Exertion
. >" long a" oxygen i" !re"ent, the de+line in
8*D re%aining ?6 i" ex!onential
. @ the "y"te% i" +lo"ed, "$+h a" in a 8*D ottle,
the D* "$!!ly i" ixed, and no re!leni"h%ent
ro% the at%o"!here +an tae !la+e
– Thi" i" the +a"e or laoratory 8*D %ea"$re%ent"
– >llo" +on"tr$+tion o a #$antitatie D* $dget eteen
"tart and the ini"h o the te"t
. 9hatL" dierent ao$t hat ha!!en" in <the orld:= – Sy"te% i" o!en
– 8*D i" exerted, $t D* de!letion i" o!!o"ed y
+ontin$o$" re!leni"h%ent ro% the at%o"!here
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*xygen Re!leni"h%ent y >t%o"!heri+ Reaeration
. Reaeration rate i" 1"t order ith oxygen dei+it
. Rate +oei+ient i" related to "trea% +hara+teri"ti+" – elo+ity, T$r$len+e, De!th
( )
r "
r
"-1
r
"
dCI rate o reaeration I M C -C6
dt
here,C I "at$ration oxygen +on+entration, %g ? I ir"t - order reaeration rate +oei+ient, t
Can al"o ex!re"" in ter%" o the "at$ration dei+it, D I C -C
(ii r
dDng I - D
dt
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9hat Doe" Reaeration ?oo ?ie:
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Reaeration *LConnor and Doin" For%$la
. 8a"ed on "$ra+e reneal theory
– /odel i" that <!ar+el"= o ater are ro$ght to air-aterintera+e or "o%e inite ti%e !eriod
– (a" ex+hange tae" !la+e only hile a ater <!ar+el= i" atthe "$ra+e
– >ter %oing aay ro% the "$ra+e, ater <!ar+el= %ixe"ith the li#$id $l
0'5 0'5
r r 1'5 1'5
-1 -1
r r
S@ Engli"h C$"to%ary
; ;
I 3'3 I12'
d 6, ; %!"6, %6 d 6, ; !"6, 6
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Reaeration Ch$r+hill et al. 126 For%$la
. 8a"ed on e%!iri+al "t$die" o reaeration o $nder-
"at$rated ater" don"trea% o da%" on the Tenne""eeRier
. Correlated %ea"$red reaeration rate" ith elo+ity ;6and de!th o lo 6
r r 1'7 1'7
-1 -1
r r
S@ Engli"h C$"to%ary
; ; I 5'02 I11'
d 6, ; %!"6, %6 d 6, ; !"6, 6
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Reaeration *en" and (i" 146 For%$la
. Cond$+ted "t$die" o 8riti"h "trea%" here oxygen a"
artii+ially de!leted y "$lite additione
. Co%ined 8riti"h and Tenne""ee Rier data
– Correlated %ea"$red reaeration rate" ith elo+ity ;6 and
de!th o lo 6
0'7 0'7
r r 1'5 1'5
-1 -1
r r
S@ Engli"h C$"to%ary
; ; I 5'32 I 21'
d 6, ; %!"6, %6 d 6, ; !"6, 6
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>t%o"!heri+ Reaeration
D e ! t h , 4 % 6
D e ! t h ,
4 0 t 6
. /ethod o Coar 176
. ;"e" or%$lae o
– *LConnor J Doin"
– Ch$r+hill
– *en"-(i". @n!$t "trea% elo+ity
and de!th o lo
. Sele+t r d-16 at
inter"e+tion o lo and de!th
+oordinate"
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Reaeration Coei+ient E"ti%ation ro% Strea% De"+ri!tion"
9ater 8ody De"+ri!tion r
day"-1 K 20 oC6
S%all !ond" and a+ater" 0'10-0'23
Sl$ggi"h "trea%" and large lae" 0'23-0'35?arge "trea%" o lo elo+ity 0'35-0'4
?arge "trea%" o nor%al elo+ity 0'4-0'
Sit "trea%" 0'-1'15
Ra!id" and aterall" N 1'15
So$r+e eay, Roe and T+hoanoglo$", 15
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Te%!erat$re Corre+tion" or Rate Coei+ient"
. ule o% $humb i" that io+he%i+al rea+tion rate"
do$le ith a 10 oC te%!erat$re in+rea"eanLt o R$le6
. >rrheni$" E#$ation %ay e $"ed to %orerigoro$"ly +orre+t rate +oei+ient" or dieren+e" in te%!erat$re
a2 a1
a2 a1
a2 a1
ET -T 6
RT TTa2
Ta2
o
a2 a1
E
RT T T2-T16
T2 T1
I
*3er the range o nat$ral te%!erat$re"
0 - 40 C6, the !rod$+t o T T i" O +on"tant,and allo" $" to deine 7
I e ''' and 1"t e#$ation e+o%e"7 Iθ θ
g
e
>rrheni$"
anLt o
S o $ r + e 7
1 1 1 ' 1
i 2 i ! e d i a ' + o %
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Deter%ine 8*D;?T in Strea% ater /ixing ith Di"+harge
. Con"tr$+t %a"" alan+e on rier lo and a"te
di"+harge to get 8*D;?T o %ixt$re
9a"te G,
@n!$t ?
Rier
Flo
Gr , ?
r
/ixed
Flo
Gr MG
, ?
0
r r 0
r
3
r
3
3
r
3
G ? MG ?? I
G MG
here,
G I Flo in Rier, ? t
G I Flo o 9a"teater Di"+harge, ? t
? I ;lti%ate 8*D o rier ater, % ?
? I ;lti%ate 8*D o a"teater di"+harge, % ?
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Si%!liied S+he%ati+ Re!re"entation o /odel
. >""$%e F and deine +ontrol ol$%e a" a $nit re+tangle
. Control ol$%e %oe" don"trea% at +on"tant elo+ity
. Deter%ine the initial oxygen +ontent ater %ixing ?06
. Co%!$te D* at any ti%e y "oling dierential e#$ationor 8*D exertion and at%o"!heri+ reaeration
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Dierential E#$ation or redi+ting ?ongit$dinal D* roile
. @dentiy a "ingle l$id ele%ent in the "trea%
– Con"tant ol$%e
– Con"tant elo+ity
. 9rite a %a"" alan+e or oxygen on the ele%ent a"
ae+ted y 8*D exertion and at%o"!heri+ rearation – >++$%$lation I in!$t – o$t!$t P rea+tion
– Sin+e the l$id ele%ent i" <inta+t,= there i" no lo in or o$t,and the %a"" alan+e e+o%e"
g gin in o$t o$t
dC I G C - G C - 48*D exertion6 M4Rearation6
dt
g gin o$t
dC I 0 - 0 - 8*D exertion6 M Rearation6
dt
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CEE 5134 - 27 -
deox reaer
d r "
"
d
Re%e%er the %a"" alan+e 7
dC dC dC I M
dt dt dt
S$"tit$te the rate ex!re""ion" 7dC
I - ? M C -C6dt
For %ath +on3enien+e, ex!re"" in ter%" o dei+it, D I C -C 7
dD dCI 0 - I
dt dt
g g
r ? - D
Streeter-hel!" Deelo!%ent, +ontin$ed
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CEE 5134 - 2 -
Streeter-hel!" Deelo!%ent, +ontin$ed
d
d
- t
d 0 r
- t
r d 0
S$"tit$te the 8*D Exertion Ex!re""ion
dDI ? e - D
dt
Rearrange to dD
M D I ? edt
The aoe i" o the "tandard or%
d)M) I G hi+h i" not "e!arale6
dx
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CEE 5134 - 2 -
Streeter-hel!" Deelo!%ent, +ontin$ed
r r
dr r r
r dr
dt t
- t t t t
r d 0
4 - 6t t
d 0
;"e the integrating a+tor
e I e
/$lti!lying ea+h ter% y the integrating a+tor gie"
dDe M D e I ? e e
dt
>nd''''
dDe I ? e
dt
∫
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CEE 5134 - 30 -
Streeter-hel!" Deelo!%ent, +ontin$ed
0
0
r dr
r dr
- 6t t
d 0
- 6t t d 0
r d
0 0d 0 d 00
r d r d
Se!arate ariale" and integrate
d De I ? e dt
(iing
?De I e M C
-
>t o$ndarie" o t I 0 and D ID , e hae
? ?D e I e M C '''' C ID -
- -
∴
∫ ∫
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Streeter-hel!" Deelo!%ent, +ontin$ed
r dr
d r r
- 6t t d 0 d 00
r d r d
- t - t - td 0 d 00
r d r d
-d 0
r d
>!!lying the integration +on"tant
? ?De I e M D -
- -
Soling or D
? ?D I e M D e - e
- -
8y gro$!ing lie ter%", "i%!liy to
?D I e
-
)d r r t - t - t
0- e M D e
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Co%!$te the Criti+al Dei+it
. The +riti+al dei+it D6 o++$r" here the rate o +hange o
D ith ti%e I 0 dDdt I06
. /ay e +o%!$ted ro% the original DE y "etting the 1"t
deriatie e#$al to )ero
d
d
- t
d 0 r C
- tdC 0
r
dD I 0 I ? e - Ddt
Rearrange to
D I ? e
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Ti%e to Rea+h the Criti+al Dei+it ?oe"t D*6
d r r
d r r
r
- t - t - td 00
r d
- t - t - td 0d r 0 r
r d
- t
dD >" noted, the +riti+al dei+it i" here I 0
dt ?dD d
I 0 I e - e 6M D edt dt -
Taing the deriatie" and "i%!liying
?0 I - e M e 6 - D e
-
Diide y e
0 r dr +
r d d d 0
, tae nat$ral log" on oth "ide" and "ole or t
D - 61t I ln 1-
- ?
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> Streeter-hel!" /odel Exa%!le role%
. 9a"teater %ixe" ith a rier re"$lting in
– 8*D I 10' %g? – D* I 7' %g?
– Te%!erat$re I 20 °C
. Deoxygenation +on"tant I 0'2 day-1
. >erage lo I 0'3 %"
. >erage de!th I 3'0 %
. 9hat i" the di"tan+e don"trea% o the %axi%$%oxygen dei+it:
. 9hat i" the %ini%$% al$e o D*:
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