urban water

Urban Water

Department of Hydro Sciences, Institute for Urban Water Management

Peter Krebs Dresden, 2010

1 Global water aspects

2 Introduction to urban water management

3 Basics for systems description

4 Water transport

5 Matter transport

6 Introduction to water supply

7 Water extraction

8 Water purification

9 Water distribution

10 Introduction to wastewater disposal

11 Urban drainage

12 Wastewater treatment

13 Sludge treatment

Urban Water Chapter 4 Matter transport © PK, 2010 – page 2

4 Matter transport

4.1 Introduction to transport phenomena

4.2 Transport processes

4.3 Reactor approach

4.4 Advection-dispersion approach


Peter Krebs

Urban Water


4 Matter transport






Peter Krebs

Urban Water


Passive solubles Travel ~ with water Often used to indicate velocity and residence-time distribution

Solids Transport decoupled from flow Suspended solids and gravel Sedimentation and Erosion

Reactive matter Can be solubles or solids Residence time and conditions in reactor importantReaction must be known for balancing

Characteristics of compounds


Quiescent conditions

Molecular diffusion

Stirring

Turbulent diffusion

Milk and sugar in a cup of coffee


L, t

Time

Co

nce

ntr

.

Transport with flow

Longitudinal extension of tracer cloud

Decrease of peak concentration

Tracer in a full pipe


0

4

8

12

0 1 2 3t /

Trac

er C

on

c.

Residence-time distribution in a clarifier


4 Matter transport






Peter Krebs

Urban Water


Transport with water flow no relative movement

Flux CvjAdv

Example: Transport of compound with constant concentration C in a tube with cross section A:

CQCAvAjAdv

Advection


Transport in the direction of decreasing concentration

1st Fick lawx

CDj M

Mmdmd

,

• 1D approach; it also applies in a 2D or 3D system

• Dmd,M is a specific value for a certain compound M

• Dmd,M is a function of temperature

Molecular diffusion


Process similar to molecular diffusion, but some orders of magnitude more efficient

• Dtd is dependant on flow and state of turbulence, not on the compound itself

• Concentration gradients decrease !!

C

xxC

Dj tdtd

Diffusive flux

Turbulent diffusion


Time

Co

nce

ntr

.

Dispersion is not transport relative to water, but inhomogeneous advection

In 1D formulation, dispersion “collapses on diffusion”

Dispersion


Sedimentation flux Cvj SSed v

vS

•Suspended particles have a transport component in gravity direction

• In reactors this effect is used for particle separation

• In transport systems, a sink or source term - depending on the operation conditions - is needed

Examples: - 1D clarifier model - Sewer sediments

Sedimentation


4 Matter transport



4.3 Reactor approach CSTR

Plug-flow reactor

CSTR in series



Peter Krebs

Urban Water


• Constant volume

• Immediate mixing

• Complete mixing no concentration gradients

• CReactor = COutlet

Q Q

Cin CCV

r

VrCQCQtC

V in

Mass balance

rCCtC

in

11

Continuously stirred tank reactor (CSTR)


Mass balance rCCin

11

0

0-order reaction 0kr 0kCC in

1st-order reaction Ckr 1

11

1k

CC in

0

0,2

0,4

0,6

0,8

1

0 5 10 15 20 25

Residence time

Co

nce

ntr

atio

n

0-order, CSTR1st order, CSTR

CSTR: steady state


Tracer pulse is introduced to the inlet tracer concentration is measured in the outlet

Mass balance CtC

1 No input, no reaction

tCC exp0

0

0,2

0,4

0,6

0,8

1

-1 0 1 2 3

t /

c/ c

0

CSTR: residence-time distribution (RTD)


4 Matter transport




Plug-flow reactor

CSTR in series



Peter Krebs

Urban Water


• Constant volume

• Constant cross section

• No mixing (ev. lateral)

• Concentration gradients along flow axes

Mass balance

x dxA

dxArdxxCQxCQtC

dxA

dxArdCQtC

dxA

rxC

vtC

Plug-flow reactor


Mass balance rdxdC

v 0

0-order 0kr vx

kCC in 0

1kCC in exp

Outlet concentrations: with x = L L/v =

1st order Ckr 1

vx

kCC in 1exp

0kCC in

0

0,2

0,4

0,6

0,8

1

0 5 10 15 20 25

Residence time

Co

nce

ntr

atio

n0-order, CSTR, plug flow

1st order, CSTR

1st order, plug flow

Plug-flow reactor: steady state


Tracer pulse is introduced to the inlet tracer pulse appears in the outlet unchanged !!

0

0,2

0,4

0,6

0,8

1

1,2

-0,2 0 0,2 0,4 0,6 0,8 1 1,2

t /

c/ c

0

Plug-flow reactor: RTD


4 Matter transport




Plug-flow reactor

CSTR in series



Peter Krebs

Urban Water


Q Q

Cin C1V1

r

C1

Q

C2V2

r

C2

Q

CiVi

r

Ci

Q

Ci-1

Q

CnVn

r

Cn

Q

Cn-1

iiii VCkCCQ 110 Reactor i

1st order reaction

QVkCC

ii

i

11 11

CSTR cascade


2 Reactors

2121111

212

21

11

11

1

QVkQVkQVkCC

CC

CC

totinin

Total volume

n

iitot VV

1

n = number of reactors

nV

V toti .const

n Reactors ntotin

n

QnVkCC

11

1

QVk

QnVktotn

totn1

11

1

explim 1kexp

CSTR cascade: 1st order reaction (i)


n Reactors ninn

nk

CC

11

1

0

0,2

0,4

0,6

0,8

1

0 5 10 15 20 25

Residence time

Co

nce

ntr

atio

n

0-order, CSTR, plug flow

1st order, CSTR

1st order, plug flow

1st order, 2 CSTRs

1st order, 3 CSTRs

1st order, 10 CSTRs

CSTR cascade: 1st order reaction (ii)


Initial condition in 1st reactor c0,1 as reference concentration

1st reactor1

1 Cn

dtdC

2nd reactor 212 CC

ndt

dC

i-th reactor iii CC

ndtdC

1

CSTR cascade: RTD (i)


Solving the coupled equations with Laplace transformation yields

t

nt

nn

CtdC nn

n exp!,

1

10 11

Mean valueQ

Vtot

Variance

0

2

10

22

ndt

CC

t n

,

Peak value at time n

nCt n

1max

2

2

n

CSTR cascade: RTD (ii)


0

0,5

1

1,5

2

2,5

3

0 0,5 1 1,5 2t /

C/C

0,1

1 CSTR 2 CSTRs 4 CSTRs10 CSTRs20 CSTRs50 CSTRsPlug flow

CSTR cascade: RTD (iii)


4 Matter transport






Peter Krebs

Urban Water


transportDispersive

disp

transportAdvectiveconcinChangex

CD

xC

utC

2

2

.

Analytical solution for a tracer pulse

2

2

2

12 t

utxtA

mtxC exp,

u = mean velocityDdisp = dispersion coefficient

m = total amount of tracer introduced A = cross-section areat = time from dosage

Advection-dispersion approach (i)


ux

Ddisp 2

huub

cD fdisp

22

fShgu

Standard deviation

Dispersion coefficient

Shear velocity

cf = Fischer coefficient = 0.011 (-)

b = width of water surfaceh = water depth Sf = friction slope

Advection-dispersion approach (ii)


0

0,1

0,2

0,3

0,4

1000 1500 2000 2500 3000

Time (s)

Re

lati

ve

co

nc

en

tra

tio

n

AdvectionDispersion, estimated by diffusion approach

Standard deviation

A-D approach: effect of dispersion/diffusion


0

0,2

0,4

0,6

0 1000 2000 3000 4000 5000

Time (s)

Rel

ativ

e co

nce

ntr

atio

n

x = 500 mx = 1 kmx = 2 kmx = 3 kmx = 4 km

A-D approach: tracer curves


0 50 100 150 200 250 300 350 400 450Time (minutes)

tracermodel

1

2

34

5

Boeije (1999)

A-D approach: dispersion in a river


Normalisation by length L of reactorLu

tt

t

*

Lx

x *

2

2

*** x

CLu

D

x

C

t

C td

Peclet number """"

PeDiffusionAdvection

LDu

DLu

tdtd

Pe large Advection dominant plug flow behaviour

Pe small Diffusion dominant CSTR behaviour

small < Pe < large CSTR cascade or A-D approach

A-D approach: reactor approximation (i)


Relation of turbulence/dispersion and standard deviation

PeexpPePe

112

22

2

Simplification for Pe > 100 (applies to conditions in sewers and rivers)

LuDtd2

22

2

Pe

Turbulence can be estimated from RTD (i.e. )

n1

CSTR approximation, „hydrologic model“

A-D approach: reactor approximation (ii)

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