Updated: 21 September 2017 - UMass Amherst · CEE 572 Lecture #4 9/21/2017 1 Lecture #4 (mass...

CEE 572 Lecture #4 9/21/2017

1

Lecture #4

(mass balance, loadings & steady state solutions)

Chapra L3

David A. Reckhow CEE 577 #4 1

Updated: 21 September 2017 Print version

Completely‐mixed lake or CSTR Often useful to assume perfect mixing

same concentration throughout system


Accumulation loading outflow reaction settling

C

VCin

Q

CQ

settling

reactionLoading

Outflow

CEE 572 Lecture #4 9/21/2017

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Accumulation


AccumulationM

t

cM

VM Vc

AccumulationVc

t

and

And if volume is constant:

Accumulation Vc

tV

dc

dt

So:

CV

Equals zero at steady state

Loading Point Sources

Municipal Wastewater

Industrial Wastewater

Tributaries

Non‐point sources agricultural

silvicultural

atmospheric

urban & suburban runoff

groundwater


Diffuse originmore transient

often dependent on precipitation

Well defined origineasily measuredmore constant

Loading W t Qc tin ( ) ( )

CEE 572 Lecture #4 9/21/2017

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Reported Values Of Selected Waste Input Parameters In The United States


Variable Unitsa MunicipalInfluentb

CSOc UrbanRunoffd

Agriculture(lb/mi2-d)

eForest

(lb/mi2-d)e

Atmosphere(lb/mi2-day)

f

Averagedaily flow

gcd 125

Totalsuspendedsolids

mg/L 300 410 610 2500 400

CBOD5g mg/L 180 170 27 40 8CBODUg mg/L 220 240NBODg mg/L 220 290Totalnitrogen

mg-N/L 50 9 2.3 15 4 8.9-18.9

Totalphosphorus

mg-P/L 10 3 0.5 1.0 0.3 0.13-1.3

Totalcoliforms

106/100mL

30 6 0.3

Cadmium g/L 1.2 10 13 0.015Lead g/L 22 190 280 1.3Chromium g/L 42 190 22 0.088Copper g/L 159 460 110Zinc g/L 241 660 500 1.8Total PCB g/L 0.9 0.3 - 0.002-0.02

(Table 1.3 from Thomann & Mueller)

Footnotes for T&M Table 1.3


aUnits apply to municipal, CSO (combined sewer overflow), and urban runoff sources; gcd = gallons per capita per day.

bThomann (1972); heavy metals and PCB, HydroQual (1982).cThomann (1972); total coli, Tetra Tech, (1977); heavy metals Di

Toro et al. (1978): PCB. Hydroscience (1978).dTetra Tech (1977): heavy metals, Di Toro et al. (1978).eHydroscience (1976a).fNitrogen and phosphorus, Tetra Tech (1982): heavy metals and

PC13, HydroQual (1982).gCBOD5 = 5 day carbonaceous biochemical oxygen demand

(CBOD); CBODU = ultimate CBOD; NBOD = nitrogenous BOD.

CEE 572 Lecture #4 9/21/2017

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Loading: Flow as a function of precipitation Non point sources are difficult to characterize

Empirical approach: export coefficients (see Table 3.1 in T&M)

Mechanistic approach: relate to meteorology, topology, etc.

Flow: use the rational formula: QR = cIA


Runoff flow [L3/T]

Runoff coefficient0.1-0.3 for rural areas (1 person/acre)0.7-0.9 for heavy commercial areas

Rainfall Intensity [L/T]

DrainageArea [L2]

Note:1 acre-in/hr 1 cfs

Runoff: Contrasting approaches Lumped model

Empirical

Built on a single rainfall intensity from rain gage data

Distributed model Mechanistic

Built on radar data for rainfall Spatial & temporal resolution

Combine with overland flow models Many computer codes

CASC2D, CUHP, CUHP/SWMM, DR3M, HEC‐1, HSPF, PSRM, SWMM, TR20


CEE 572 Lecture #4 9/21/2017

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Loading: conc. as a function of flow It is common for pollutant concentrations from uncontrolled sources (e.g. tributaries) to be correlated with flow


1

10

100

1000

1 10 100 1000

Flow (cfs)

Co

nc

en

tra

tio

n (

mg

/L)

establish a log-log relationship

c=aQb

Log(C) = log(a) + b*log(Q)

Loading Example: #3.1 from T&M Data: Runoff from 100 mi2 of agricultural lands drains to a point in a river

where a city of 100,000 people is located. The city has a land area of 10 mi2

and its sanitary sewers are separated from its storm drains. A sewage treatment plant discharges to the river immediately downstream of the city. The area receives an annual rainfall of 30 in. of which 30% runs off the agricultural lands and 50% drains off the more impervious city area.

Problem: Using the loading data from Table 1.3 and the residual fractions cited in the table below, compare the contributions of the atmospheric, agricultural and urban sources to annual average values of flow, CBOD5, total coliform bacteria, and lead in the river. Neglect any decay mechanisms for all parameters.


(at) (ag) (ur) Wastewater Treatment Plant Item Atmospheric Agricultural Urban Runoff Influent Resid. Fract.

Fow 30% precip. 50% precip. 125 gcd 1.00 CBOD5 40 lb/mi2-d 27 mg/L 180 mg/L 0.15 Total coliform 100/100 mL 3x105/100mL 3x106/100mL 0.0001 Lead 1.3 lb/mi2-d 280 g/L 22 g/L 0.05

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Solution to loading problem

Flow contributions


cfs

yrinmiagQ sd

dyr

inft

mift

3.66

3.0/30100)( 400,861

3651

121252802

cfs

MGD

dcap

galcapwwtpQ

MGDcfs

galMG

4.19

5.12

125000,100)(

548.1

101

6

cfs

yrinmiurQ sd

dyr

inft

mift

1.11

5.0/3010)( 400,861

3651

121252802

Solution to loading problem (cont.) CBOD5 loading


d

lb

dmi

lbmiagW

4000

40100)(2

2

d

lb

LmgMGDwwtpW LmgMGDdlb

2810

15.0/1805.12)( /*/34.8

d

lb

Lmgcfs

dlbLmgcfsurW

1620

/

/4.5/271.11)(

CEE 572 Lecture #4 9/21/2017

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Solution to loading problem (cont.) Lead loading


d

lb

dmi

lbmiatmW

13

1.03.1100)(2

2

d

lb

LgMGDwwtpW gmg

LmgMGDdlb

11.0

05.0/225.12)(310

/*/34.8

d

lb

g

mg

Lmgcfs

dlbLgcfsurW

8.16

10

/

/4.5/2801.11)(

3

Other Terms in the Mass Balance Outflow

Reaction

Settling


Outflow Qc

Reaction kM kVc

Settling vA c

k Vcs

s

k vHs V A HsSince:

J=vc

As

Sediment-water interface

CEE 572 Lecture #4 9/21/2017

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Combining all terms:

Dependent variable: c

Independent variable: t

Forcing function: W(t), the way in which the external world “forces” the system

Parameters: V, Q, k, v, As


Vdc

dtW t Qc kVc vA cs ( )

Steady State Case

Mass Balance

Solution

Assimilation factor

Where

The assimilation or “cleansing” factor


cW

Q kV vAs

a

Wc

svAkVQa

or

Vdc

dtW t Qc kVc vA cs 0 ( )

W

CEE 572 Lecture #4 9/21/2017

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Steady State Example


A lake has the following characteristics:

Volume m

dC

50 000 3

1

, Mean Depth = 2 m

Inflow = Outflow = 7500 mTemperature = 25

3

o

The lake receives the input of a pollutant from three sources: a factory discharge of 50 kg d-1, a flux from the atmosphere of 0.6 g m-2 d-1, and the inflow stream that has a concentration of 10 mg/L. If the pollutant decays at the rate of 0.25/d at 20oC (note: Ɵ=1.05).

a. compute the assimilation factorb. steady state concentrationc. show breakdown for each term

#3.1 from Chapra (pg.52)

Example 3.1: Solution


kd

0 25 0 25 1050 319

25 20 25 20

1. . ( . ).

First correct the decay rate for temperature

Now the assimilation factor

13454,23

)000,50(319.07500

dm

kVQa

CEE 572 Lecture #4 9/21/2017

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Example 3.1: Solution (cont.)


The surface area of the lake is:

AV

Hms

50 000

225 000 2,

,

The atmospheric and inflow load is then:

W JA g datmosphere s 0 6 25 000 15 000. ( , ) , /

W g dlowinf ( ) , / 7500 10 75 000Combining all loads: W W W W

g d

factory atmosphere low

inf

, , ,, /

50 000 15 000 75 000140 000

Example 3.1: Solution (cont.)


And finally, the concentration:

Lmg

dm

dga

Wc

/97.5

/454,23

/000,1403

CEE 572 Lecture #4 9/21/2017

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Transfer function & residence time


cW

Q kV vAQc

Q kV vA

s

in

s

c

c

Q

Q kV vAin s

Transfer function

E

E

dE dt w

V

Q

cs s

Vc

Qc kVc vA c

V

Q kV vA

Residence times

generic

water

contaminant

Pittsburgh


CEE 572 Lecture #4 9/21/2017

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CEE 572 Lecture #4 9/21/2017

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CEE 572 Lecture #4 9/21/2017

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Bromide in PAKelly D. Good and Jeanne M. VanBriesen, 2016 “Current and Potential Future Bromide Loads from Coal-Fired PowerPlants in the Allegheny River Basin and Their Effects on DownstreamConcentrations”, ES&T 50, 9078


CEE 572 Lecture #4 9/21/2017

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