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CHAPTER FOUR Fundamentals of Water Quality
This chapter discusses the fundamental relationships for water quality. The basis for all water quality analyses is conservation of a constituent mass that was developed in Chapter 3. In this chapter, it will be applied to mixing at a node and transport in a pipe. Constituent balance in a storage tank is discussed in Chapter 7. A critical aspect of constituent mass balance is the constituent reaction. Reaction relationships and estimating decay and growth coefficients are also presented in this chapter.
4.1 CONSERVATION OF A CONSTITUENT MASS
As presented in Chapter 3, conservation of constituent mass (Eq. 3-18) is:
dt
)VC(ddt
dmQCQC c
outflowsoutout
flowsininin =+− ∑∑ (4-1)
C is the volume concentration [Mc/L3], V is the volume of water in the control volume and Q is the flow rate.
The right hand side is the rate of change in the mass of the constituent within the control volume. The left hand side is the transport of constituent mass into or out of the control volume. The first two terms are transport with fluid. The last LHS term, tddmc = [Mc/T], is a reaction term that accounts for growth or decay transformations of the constituent in the control volume.
The concentration used in this equation is the mass of constituent per unit volume of water or:
⎥⎦⎤
⎢⎣⎡== 3L
MVmC
Units of C in SI are typically given as g/m3 or mg/l where l is liters. Conversion factors from g to mg and m3 to l are the same so the concentration value for both units is the same. 1 g/m3 is also equivalent to 1 ppm (part per million). Fluoride injections and chlorine are typically on the order of mg/l. Salinity or total dissolved solids may be measured on the order of grams/liter (g/l). The units of kg/m3 and ppt (parts per thousand) are identical to g/l. Bacteria are
4-1
4-2 CHAPTER FOUR
commonly in the range of micrograms/liter (µg/l). Equivalent measures to µg/l are mg/m3 and ppb (parts per billion).
The constituent mass added to water per unit time is the loading rate, = m/t. In a distribution system, one method for adding constituent is injection of a disinfectant mass. The loading rate will result in an increase in concentration, ∆C, in the volumetric flow rate, Q or:
cm&
⎥⎦
⎤⎢⎣
⎡==∆== 3
3
LM
TL
TMCQ
tdmdmc&
Example 4.1
Problem: The chlorine concentration of water flowing at a rate of 0.3 m3/s is 0.03 mg/l. The desired concentration is 0.2 mg/l. What loading will be needed to cause this increase? If the chlorine is injected continuously for a month, what is the chlorine mass required?
Solution: The increase in concentration is 0.2 – 0.03 = 0.17 mg/l. The loading rate is then:
hrmgsmgmgsmCQmCl
/184/051.0/)03.02.0(/3.0 33
==−⋅=∆=&
The mass per month is the loading rate times the duration of injection or:
mo/gmo/mgmo/dd/hrhr/ss/mg.tmm ClCl
132132000302436000510
==⋅⋅⋅=⋅= &
4.2 TRANSPORT AND MIXING IN THE PIPE NETWORK As discussed in Chapter 3, the flow distribution in a water distribution network is defined by conservation of mass and energy. We assume that any constituent in the water does not affect the terms in these relationships. Thus, a hydraulic analysis to determine the pipe flow velocities and nodal heads can be completed without regard to water quality. However, since water quality transport is directly related to the flow velocities a strong understanding of the network hydraulics and a well-calibrated hydraulic model are necessary to perform and evaluate a water quality analysis. All available water quality models first
WATER QUALITY FUNDAMENTALS 4-3
perform a hydraulic analysis then provide the resulting flow distribution to a water quality module to transport a constituent through the system.
The result of a water quality analysis in a pipe network is the constituent concentration at all junction nodes. A temporal record or unsteady analysis is generally desired. Time invariant or steady conditions are simpler to analyze but are less realistic. In steady and unsteady analyses, two processes must be considered when modeling water quality in a pipe network; mixing at nodes and transport in pipes. Both processes are described by conservation of a constituent mass (Eq. 4-1).
Nodal mixing accounts for the mixing of waters with different constituent concentrations. No storage is provided at a node and water is assumed to pass through the node instantaneously. If additional constituent is added to the water system, it is assumed to occur at a node. Several types of injectors can provide constituent.
Transport of water through a pipe introduces time dependency to water quality analysis. The pipe travel time provides the time lags in the flows between junction nodes. In addition as water travels through a pipe, a constituent concentration may change due to decay, reactions, or growth. The remainder of this section discusses the mixing and transport processes and their mathematical formulations.
4.2.1 Mixing at Junctions
4.2.1.1 Simple Junctions
At junction nodes within a pipe network, water quality changes due to dilution and injection. Conservation of mass is applied at junctions to determine the effect of combining flow with different constituent concentrations. Full and complete mixing is assumed to occur resulting in concentrations that are uniform across the downstream pipe section. Since a node cannot store water, the mass of constituent at the node is constant and the left-hand side of Eq. 4-1 equals zero. Also since pipe lengths at the junction are very small, no time is spent at the node so no constituent growth or decay can occur. For a simple junction, no constituent is supplied at the node. The last two statements imply that dtcdm equals zero. Under these conditions, Eq. 4-1 becomes:
0=−⇒+−= ∑∑∑∑ outinc
outinQCQC
dtdm
QCQCdt
)VC(d
(4-2)
Consider a junction with three pipes carrying flow to the node, two carrying flow away from the node and a lumped withdrawal at the node (Figure 4-1).
4-4 CHAPTER FOUR
Applying conservation of a constituent mass with no storage, we can write Eq. 4-2 as:
05544332211 qC - Q - C Q - C Q C Q C QC outout =++
Figure 4-1: Junction node with three inputs and three withdrawals.
The signs correspond to inflows and outflows from the node. Pipes 4 and 5 carry flow away from the node. During a water quality analysis, these flow directions and the flow rates are known from a prior hydraulic analysis. The concentrations of constituents input at the node are also known.
Assuming that complete and instantaneous mixing occurs at the node, the concentrations in pipes 4 and 5 and the nodal withdrawal are the same, Cout, or:
054332211 qC - Q - C Q - C Q C Q C QC outoutoutout =++
Since the inflow concentrations are known, this equation can be solved for Cout or:
.C q Q Q
Q C Q C QCC outoutout 923
1 2 3(2) 5 (2.5) 3 (1.5) 4
54
332211 =⇒++
++=
++
++=
Eq. 4-2 can be written for a general node as:
WATER QUALITY FUNDAMENTALS 4-5
Q q
qC QC
C
out
in
Jll
out
inin
Jlll
out ∑
∑
∈
∈
+
+
= (4-3)
Here, flow enters the node through Jin pipes and as an externally supplied flow, . Each of these flows may have a different constituent concentration, C
inq
l and Cin, respectively. Outgoing flow consists of the nodal withdrawal, qout, and the flow in the set of Jout pipes carrying flow from the node. With the complete mixing assumption, all outflows have the same concentration, Cout. Equation 4-3 shows that the outgoing concentration is a flow weighted average of the incoming concentrations.
4.2.1.2 Junctions with Injected Constituent
In most models, a constituent can be introduced to the system in five ways. First, for a dynamic simulation, the initial constituent concentrations, C0, must be defined for all locations. Second, constituents may be added to the system with water entering the network as source contributions. The last three methods are injections of a constituent at a node without water, such as a tracer or disinfection booster injection. Three types of boosters are commonly used: mass, setpoint and flow paced. For these cases, dtdmc in Eq. 4-1 is not equal to zero.
A common injection approach is municipal chlorinators (Figure 4-2) that can be used within the water treatment plant or at booster locations. Chlorine is available in gas, liquid or solid forms. For safety reasons, tablets are most often used away from the plant. In the injector, water is drawn from a distribution system pipe into the unit. Water flows through the contact chamber containing chlorine tables. The flow rate and chamber size affect the dosage and multiple contact chambers may be used for larger flows. Depending upon the design, flow may be sent to a small reservoir before being returned to the pipe. A gas or liquid feeder works under the same principle with the disinfectant typically fed directly to the flow with a device to insure proper mixing.
4.2.1.2.1 Source Concentration
In most water quality models, the concentration for all external flow entering a node can be specified. The flow is typically a negative demand at a junction node or a flow from a reservoir or water treatment plant. The input water concentration, Cin in Eq. 4-3, is described as the source concentration.
4-6 CHAPTER FOUR
Figure 4-2: Tablet feeder chlorinator system. Flow is drawn into the system by the pump in the lower left. Tablets are added to the cylinder on the top of the unit.
4.2.1.2.2 Mass Booster Injection
A mass booster injects a fixed mass rate ( ) of constituent to the junction inflow. This option is useful when analyzing a tracer study or when modeling the possible impacts of an unwanted intrusion due to a backflow event or intentional contamination. The inflow can represent the mass of tracer injected or contamination entering the system over time. In Eq. 4-1, the mass rate of injection is:
inm&
∑∈
==inJl
linjin
cin QC
dtdm
m& (4-4)
where is the effective concentration resulting from injecting the defined constituent mass. Substituting this term in Eq. 4-1 and solving results in:
injinC
WATER QUALITY FUNDAMENTALS 4-7
∑
∑∑
∑
∑
∈
∈∈
∈
∈
+
++
=+
++
=
out
inin
out
in
Jll
outJl
linjin
inin
Jlll
Jll
out
inin
inJl
ll
out Q q
QCqC QC
Q q
mqC QC
C
&
(4-5)
Example 4.2
Problem: A 2.5 lb (1.1 kg) tablet that is 16% Cl2 dissolves in one hour in a tablet feeder chlorinator. Three percent of the 800 gpm (50 lps) supply pipe flow is passed through the unit. Determine the increase in concentration in the water being returned to the pipe and in the water in the pipe downstream of the chlorinator return.
Solution: The 1.1 kg provides 0.176 kg of Cl2 (16%*1.1) to the flow. In the main pipeline, the mass rate of 0.176 kg/hr is dissolved in the flow of 50 lps. Thus,
lmgC
CslhrskgmghrkgCQm
pipe
pipec
/98.0
*)/(50)/(3600/)/(10 )/(176.0 6
=∆⇒
∆==∆=&
The concentration in the feeder unit has the same mass added but the flow is only 3% of the 50 l/s or 1.5 l/s. Thus, the concentration is:
lmgC
CslhrshrkgCQm
feeder
feederc
/6.32
*)/(5.1)/(3600/)/(176.0
=∆⇒
∆==∆=&
Example 4.3
Problem: A mass loading rate of 10000 units/hr of the constituent modeled is added to the node shown in Figure 4-1. Determine the effective injection concentration and the resulting concentration in flow leaving the node.
Solution: The effective injection concentration, , can be computed from Eq. 4-4.
injinC
)()/(3600/10000 321 QQQCsunitsQCm injin
Jll
injinin
in
++=== ∑∈
&
4-8 CHAPTER FOUR
333 /46.0/)25.25.1(/ LunitsCsLLunitC injin
injin =⇒++=
The concentration of flow leaving the node is determined by Eq. 4-5.
3/38.46
8.25.23
)32(13600
100000)255.235.14(
Lunits
Q q
mqC QC
C
out
in
Jll
out
inin
inJl
ll
out
=+
=
++
++⋅+⋅+⋅=
+
++
=∑
∑
∈
∈
&
The injected mass increased the outflow concentration by 0.46 units/L3 from 3.92 to 4.38 units/L3.
4.2.1.2.3 Setpoint and Flow Paced Boosters The last two injector types supply constituent on the outlet side of the node to increase the concentration of the nodal withdrawal and the outlet pipe flows. Based on that definition, the added constituent mass rate is:
⎟⎟⎠
⎞⎜⎜⎝
⎛+= ∑
∈ outJll
outinjout
c Q qCdt
dm (4-6)
Substituting this term in Eq. 4-1 and assuming no nodal storage gives:
⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟
⎟⎠
⎞⎜⎜⎝
⎛+−+=
=+−
∑∑∑
∑∑
∈∈∈ outoutout Jll
outinjout
Jll
outout
inin
Jlll
c
outflowsoutout
flowsininin
Q qCQ qCqC QC
dtdm
QCQC 0
that can be solved to show:
CQ q
qCQC
C injout
Jll
out
inin
Jlll
out
out
out ++
+
=∑
∑
∈
∈ (4-7)
WATER QUALITY FUNDAMENTALS 4-9
A flow paced booster adds a user defined fixed concentration of to all outflows. A setpoint booster fixes C
injoutC
out and determines the required to meet the setpoint. Using different logic, both boosters can be modeled in a water quality computer model with Eq. 4-7. If the setpoint concentration is less than the concentration without injection, i.e., the first term in Eq. 4-7, the concentration is not reduced. Rather, is set to zero and the C
injoutC
injoutC out is set to the
available concentration.
Example 4.4
Problem: Determine the constituent concentration for the node in Figure 4-1 if a flow paced booster injects 0.5 units/L3 to all flows leaving the node. Solution: The concentration without any constituent injected at the node was determined in the text to be 3.92 units/L3 using Eq. 4-3 which is equivalent to the first term on the left hand side of Eq. 4.7. The injected concentration, , is given as 0.5 units/L
injoutC
3. Eq. 4-7 is then applied to determine the outlet concentration, Cout.
LunitsCQ q
qCQC
C injout
Jll
out
inin
Jlll
out
out
out 3/42.45.092.3 =+=++
+
=∑
∑
∈
∈
Example 4.5
Problem: A set point booster is used to increase the constituent concentration of the flow leaving the node in Figure 4-1 to 4.25 units/L3. What is the injection concentration and loading rate of constituent injected? Solution: A setpoint booster defines the outlet concentration at the node, Cout in Eq. 4-7. As noted, the concentration without the addition of injected constituent is 3.92 units/L3 which is the first LHS term in Eq. 4-7. The required injection concentration, , can be determined by substituting the known values in Eq. 4-7 giving:
injoutC
injout
injout
Jll
out
inin
Jlll
out CCQ q
qCQC
C
out
out +=++
+
==∑
∑
∈
∈ 92.325.4
4-10 CHAPTER FOUR
3/33.0 LunitsC injout =⇒
As defined in Eq. 4-6, the loading rate is the flow rate leaving the node times the injection concentration or:
[ ] ( )( ) [ ] sunitsLLunit
QqCmoutJl
loutinj
outc
/2/321/33.0 33 =++=
⎟⎟⎠
⎞⎜⎜⎝
⎛+= ∑
∈
&
4.2.2 Advective Transport in Pipes
Transport of a general fluid property can occur by five mechanisms: advection, molecular diffusion, turbulent diffusion, dispersion and radiation. Radiation is restricted to energy transport by electromagnetic waves and is not considered here. Under most conditions, the dominant mechanism for transport in the pipe network is advection. Molecular diffusion and turbulent diffusion are normally neglected in water distribution networks since the flow is generally turbulent with a relatively high velocity. Most water quality models represent advection only. Even if multiple mechanisms are modeled under conditions of laminar flow, advection must be included. So we begin with advection and develop relationships for advective transport. Extensions to other transport mechanisms are then discussed.
Advection is the movement of the constituent with the water in the direction of flow with the magnitude of the main velocity component. In other words, transport by carrying a constituent along with the flow of water. An example of advection is the movement of a cleaning “pig” in a pipe. To remove encrusted material from a pipe wall, one practice is to insert a “pig” into the line. The pig is a bullet-shaped object that is covered with rough material, often a metal mesh, and acts like a scouring pad (Figure 4-3a). It is inserted at a hydrant and moves with the water’s velocity to a withdrawal point downstream (Figure 4-3b). This movement is advective transport. In Eq. 4-1, advection is represented by the first two terms on the left hand side. Each term is related to the mean flow velocity through the discharge rate.
In pure advective transport, a mass of some constituent injected into a pipe will move in a similar manner to the pig and a pipe will act as a plug flow reactor (PFR). Flow passes through a PFR in the sequence that it enters. In other words, the pipe is first in-first out reactor. Consider a pipeline with a fluoride injector. To begin the injector is off, flow is steady and no fluoride is in the pipeline (Figure 4-4). At time t, the injector is instantaneously turned on. At time t + ∆t, the injector is turned off. The fluoride injected in the ∆t time step is
WATER QUALITY FUNDAMENTALS 4-11
carried downstream in the water in a pulse (like the scouring pig). Assuming other transport mechanisms are negligible, the pulse length, Lseg, remains constant and the time required for the front of the pulse to move from the injector to the end of the pipe can be computed given the flow velocity as the distance divided by the velocity or:
VL
=τ (4-8)
Figure 4-3a: Polyester foam pipeline pigs can be bare foam or coated with a polyurethane material. Coated pigs may have a spiral coating of polyurethane, various brush materials or silicon carbide coating to improve wall scrubbing. The pigs above are medium density (5-8 lb/ft3) open cell polyurethane foam with a polyurethane coating. (Courtesy of Girard Industries Incorporated (http://www.girardind.com/redpolly.htm). See also Pigging Products and Services Association at http://www.piggingassnppsa.com/).
V
Figure 4-3b: Movement of pig with mean flow velocity through pipeline.
4-12 CHAPTER FOUR
where L is the pipe length and τ is the travel time. For example, if the pipe length is 1000 ft and the velocity is 1.25 ft/s the travel time is 1000 ft/1.25 ft/s = 800 seconds. The entire pulse will move with this velocity through the pipe. Note that travel times through pumps, valves and other appurtenances are very small so they are not considered as components in a water quality analysis.
Following or tracking the pulse of water with fluoride and identifying its location over time is a Lagrangian analysis and is one approach to modeling water quality in networks. The alternative Eulerian approach is to partition the pipe into discrete volume elements (i.e., control volumes) and monitor the concentration within each element as the pulses of constituent move through the system. The series of elements in an Eulerian approach corresponds to a cascade of PFR’s. Both approaches can provide identical results and are discussed further in this chapter and in Chapter 6.
Figure 4-4: Advective transport of pulse of constituent injected at left beginning at time t. The input pulse duration is ∆t hrs and results in the gray block. The fluid mass moves with the same length through the entire length of the pipe. The front of the pulse reaches the downstream end of the pipe in τ hrs where τ is the travel time in the pipe.
4.2.2.1 Advection Equation
The first governing constituent mass balance for a pipe describes advective transport in a pipe. Eq. 4-1 can be written for a pipe filled with water (V is constant) with no external withdrawal or supply under steady flow conditions as:
WATER QUALITY FUNDAMENTALS 4-13
dt
dmQCQC
tCV c
outin +−=∂∂ )()( (4-9)
If we write the inflow-outflow terms in a single differential form for a pipe segment of length ∆x and divide both sides by the pipe volume. The inflow-outflow terms are:
( )xCV
xC
tx
xAC
txA
VCQ
VQCQC outin
∂∂
−=∆∆−
∆∆
=⎟⎟⎠
⎞⎜⎜⎝
⎛∆∆−
⎟⎠⎞
⎜⎝⎛
∆∆
=∆−
=− )()()(
(4-10)
Substituting this term in Eq. 4-9 and defining )(1 Crdt
dmV
c = results in the final
differential form of conservation of constituent mass for a pipe element:
)(CrxCV
tC
=∂∂
+∂∂ (4-11)
This equation represents advective transport in a plug flow reactor with reactions. The terms represent changes in concentration over time, longitudinal transport along the pipe, and reactions, respectively. The units are concentration per time (e.g., mg/l/t). Eq. 4-11 can be applied to an entire pipe or a pipe element. The reaction relationships and their parameters are described later in this chapter.
Example 4.6
Problem: Water flows at a rate of 2.0 ft3/s through a 900 foot long 8-inch diameter pipe. Determine the velocity in the pipe and the travel time.
Solution: For a flow rate of Q = 2 ft3/s, the flow velocity is:
( ) ( )( ) m/s/π
.Dπ
.AQV 73.5
4)128(02
402
22 ====
The travel time is:
minutess.V
L 6.2157735
900====τ
4-14 CHAPTER FOUR
Example 4.7
Problem: Three grams of fluoride are injected at a rate of 0.1 g/s into a pipe that carries water at a flow rate of 0.3 m3/s beginning at time 0 s. The pipe has a length of 100 m and diameter of 800 mm.
Determine the concentration, the length of the pulse, and the time that the pulse reaches and leaves the pipe outlet.
Solution: The 3 g are injected at a rate of 0.1 g/s. The duration of the pulse is:
tinj = 3 (g) / 0.1 (g/s) = 30 s = 0.5 min
The concentration of fluoride in the water is:
l/mg.m/g.s/m./s/g.Q/mC c 330330)(30)(10 33 ==== &
The velocity is:
( ) ( )( ) m/s..π.
Dπ.
AQV 600
4)80(30
430
22 ====
Since the fluoride is injected over a 30 s period, the length of the pulse is:
ms/m.sVtL injseg 1860030 =⋅==
The front of the segment begins at time 0. The travel time through the reach is:
minutes.s.V
L 782167600
100====τ
where L is the length of the pipe. Thus, the front edge of the segment will reach the end of the pipe at time 2.78 minutes. The 18 m segment takes 30 s to develop with the injected fluoride. Thus the back edge of the segment will reach the outlet at time 2.78 minutes + 0.5 minutes = 3.28 minutes.
Example 4.8
Problem: For the pipe conditions in Example 4.7, track the segment of fluoride through the pipe. Specifically show the location of the segment at times 0 s, 30 s, 90 s, 2.78 minutes, and 3.28 minutes.
WATER QUALITY FUNDAMENTALS 4-15
Figure E4-8 (a-e): Time history of segment affected by fluoride injection at the left pipe inlet beginning at time 0 s. in the 100 m long pipe in Example 4.8.
Solution:
a) Time 0 s. – The fluoride has just begun to be injected so the water has a zero concentration for the entire length of the pipe (Figure E4.8a).
b) Time 30 s. – The fluoride has been injected for 30 s and, as computed in the previous example, the distance the water at the injector at time 0 has traveled 18 m. which is the length of the segment (Figure E4.8b).
c) Time 90 s. – During the time period of 30 to 90 s, the water will move 36 m = 0.60 m/s * 60 s. Therefore the front of the segment at time 90 s that was at 18 m into the pipe travels to a location that is 54 m (= 18 +
4-16 CHAPTER FOUR
36) from the inlet. The back end of the segment was at the pipe entrance so it travels to a location 36 m into the pipe (Figure E4.8c).
d) Time 2.78 min (167 s) – The travel time for the pipe is 2.78 minutes. Therefore, water at the entrance at time 0 s traversed the entire pipe length. This water corresponds to the front of the segment. The back of the segment is 18 m to the left of the front. The back of the segment entered the pipe at time 30 s. so it has traveled through the pipe for 137 s (=167 s – 30 s) and a distance of 82 m (=0.60 m/s * 137 s) (Figure E4.8d).
e) Time 3.28 min (197 s) – The front of the segment reached the outlet of the pipe at time 167 s. After 30 s (=197 - 167s), the remainder of segment reaches the downstream end of the pipe so the back end of the segment is just leaving the pipe at time 197 s. (Figure E4.8e)
4.2.2.2 System of Equations for Advective Transport
In summary, the primary mechanisms governing water quality in a water distribution system are advective transport in a pipe and complete turbulent mixing at a node. Mixing at a node without injection is represented by Eq. 4-3:
Q q
qC QC
C
in
in
Jll
out
inin
Jlll
out ∑∑
∈
∈
+
+
= (4-3)
One equation of this form can be written for each node. Pipe transport is described by Eq. 4-11 or:
)(CrxCV
tC
=∂∂
+∂∂ (4-11)
One equation of this form can be written for each pipe. Example 4.9 demonstrates how the advection and node balance equations can be combined to determine downstream concentrations. The approach shown is the basis for steady state and dynamic water quality analyses.
Since flow is unaffected by water quality, the system hydraulics including tank flows can be determined by a standard hydraulic analysis prior to considering water quality. Given the flow distribution, these equations can be solved to determine the constituent concentrations throughout the distribution
WATER QUALITY FUNDAMENTALS 4-17
system for steady conditions or over time. Tanks are modeled with the relationships described in the next chapter.
Water quality in a water distribution system can vary over time (described as dynamic or unsteady) or reach a constant time invariant condition (steady state). Steady conditions are not typical for most systems as nodal demands will change faster than the time needed to reach constant conditions. However, a steady state solution may provide an initial assessment of problem areas in a system, requires less information regarding demands, and can be solved more quickly than a dynamic model. Chapter 5 presents a formulation and solution for a steady state modeling of a general constituent.
Dynamic simulation is more detailed and will track how the network conditions will change over time with variations in demands and pump (on/off) and tank (filling/draining) operations. More information is needed than a steady state simulation but the majority of that information is demand related. A tank water quality model is also necessary (Chapter 7). Several methods (Eulerian and Lagrangian) for dynamic water quality modeling are discussed in Chapter 6. These methods have been successfully applied to model water quality (conservative and reactive species) in simple as well as very large and complex water distribution systems.
Example 4.9
Problem: A conservative constituent is injected at time 0 at the source nodes 1 and 2 with concentrations of 2 and 5 mg/l, respectively (Figure E4-9a). The injection remains on and the flow rates do not change. The water initially in the pipe contains no constituent (i.e., C = 0)
a) Calculate the outflow from node 3 (q3) b) Compute the flow velocities and the travel times for each pipe. c) Calculate node 3’s constituent concentration, C3, at times t= 0, 12,
30, and 45 minutes. Solution: a) Based on the steady flows, the outflow from node 3 is the sum of the flows from the two pipes (Eq. 3-7) or:
s/m...q q Q Q 33321 902070 =+=⇒=+
b) Given the flow rates, the velocities are computed by V = Q/A and travel times are found using τ = L/V.
4-18 CHAPTER FOUR
Figure E4-9a: Two-pipe system with fluoride injection.
0
1
2
3
4
5
6
0 10 20 30 40 50 60
Time (minutes)
Con
cent
ratio
n (m
g/l)
Pipe 1 Pipe 2Node 3
Figure E4-9b: Concentration versus time for Example 4-9. Concentrations are shown for flows at the downstream end of pipes 1 and 2 as they enter node 3 (Pipe 1 and Pipe 2) and the resulting weighted concentration leaving node 3. Although pipe 2’s concentration is 5 mg/l, the pipe only contributes 2/9 of the nodal flow. So the node 3 concentration does not increase dramatically when the constituent arrives in that pipe. Pipe 1:
sm/..π.
mDπ/sm.
AQ
V 482460
7004
700222
1
3
1
11 ====
)(
WATER QUALITY FUNDAMENTALS 4-19
min.sm/s.
mVLτ 715943
4822340
1
11 ====
Pipe 2:
m/s..π.
Dπ.
AQ
V 591440
2004
20022
22
22 ====
)(
min.s.V
Lτ 5332010591
3200
2
22 ====
c) To determine the concentrations at the node 3, the movement of constituent laden water is tracked over time. Since the initial constituent concentration is zero, the concentration at the node 3 is equal to zero. It is calculated by applying the nodal concentration equation (Eq. 4-3) or:
( ) ( )0
90)20()0()70()0(2211
0,3 =+
=+
== ...
QQCQC
C3
t
The concentration remains equal to zero until the first constituent reaches
node 3. This holds until flow from pipe 2 that contains constituent reaches the node. Since the travel time in pipe 1 is 15.7 minutes, node 3’s concentration at time 12 minutes is equal to 0 mg/l.
This concentration will change when water that contains constituent from the upstream nodes reaches node 3. This occurs at time 15.7 minutes due to the inflow from pipe 1. A second change in concentration at node 3 does not occur until time 33.5 minutes (= τ2). Therefore, the concentration at node 3 at time 30 minutes will only be affected by node 1 injection. The outflow concentration at node 3 at time 30 minutes is computed using equation 4-3 or:
( ) ( )l/mg.
...
QQCQC
C mint, 56190
200702
3
2211303 =
+=
+==
)()()()(
Finally, after water containing constituent in pipe 2 reaches node 3 the concentration changes and is computed by Eq. 4-3 for all times greater than 33.5 minutes. Therefore, the concentration at time 45 minutes is:
( ) ( )l/mg.
...
QQCQC
C mint, 67290
205702
3
2211453 =
+=
+==
)()()()(
Figure E4-9b is a plot of the concentrations at the outlet of each pipe and the flow-weighted nodal concentration as a function of time. The steps occur when constituent-laden waters reach node 3.
4-20 CHAPTER FOUR
4.2.3 Other Transport Mechanisms
Advective transport is the dominant transport mechanism in most distribution system pipes. However, other transport mechanisms may be important in conditions that are not fully turbulent. The non-uniform velocity distribution occurring in laminar flow causes longitudinal mixing or dispersion that does not take place in turbulent flow. In addition, radial mixing must be understood to correctly account for reactions between waterborne constituents and the pipe wall. Mixing decreases with the level of turbulence. Research has progressed to model radial and longitudinal mixing to represent and characterize the other transport mechanisms that are described in the following paragraphs.
In 1883, Reynolds reported on experiments that led to development of the first laws describing laminar and turbulent flow. The experiments also provide a visual picture of transport in a pipe. Reynolds’ apparatus was a tank connected to a glass pipe that contained a valve (Figure 4-5). The tank included a thin tube to inject dye into the pipe. By manipulating the valve, he created different flow regimes that were visually apparent in the motion of the dye. With low flow rates and laminar conditions, the dye continued in a near straight line (Figure 4-5a). When the valve was opened slightly further, the dye began waving as it moved through the pipe and may have completely colored the downstream water (Figure 4-5b). If the valve was opened further causing higher velocities, turbulent flow resulted and the dye was rapidly spread across the full pipe section (Figure 4-5c).
Figure 4-5: Reynolds’ experimental setup (left) and spread of dye for (a) laminar, (b) transition and (c) fully turbulent flow (right).
4.2.3.1 Molecular Diffusion
The spread of dye across Reynolds’ pipe was caused by random motion of molecules and parcels of fluid. Movement due to random motion is described as diffusion. Molecular diffusion, also termed conduction, is mass transport caused by the movement of molecules (Figure 4-6a) known as Brownian
WATER QUALITY FUNDAMENTALS 4-21
motion. Molecular diffusion can be very small. In Figure 4-5 (a), no diffusion is shown in the short pipe.
Figure 4-6: Flow in a pipe showing (a) molecular diffusion as demonstrated by a droplet of dye at three different times. The droplet expands as it travels with flow (advection). (b) Pipe flow at an instant in time with eddies in flow resulting in turbulent diffusion.
An example of molecular diffusion is a cup of hot water. A tea bag is
slowly placed in the cup causing little motion. The water will slowly turn brown as the tea is mixed with the water by the random motion of molecules in the fluid. This molecular effect occurs very slowly. Redistribution of a constituent in a tank or in a pipe with still or very slow moving water is also caused by conduction. Since this mechanism can occur when the water is not moving, it is accounted for in the last term on the left hand side of Eq. 4-1. Molecular diffusion occurs as a result of the nonuniformity of constituent concentration throughout the fluid. The rate of conduction is related to the magnitude of the concentration imbalance by Fick’s First Law. Fick’s law states that the rate of mass transfer is related to the mass concentration gradient (Chapra, 1997) or:
2
2
yCD
tC
m ∂∂
=∂∂ (4-12)
where Dm is the molecular diffusion coefficient [L2/T], which is on the order of 10-5 cm2/s. The direction y is arbitrary since conduction occurs in all directions.
The velocity of water in a pipe is on the order of feet or meters per second while molecular diffusion is on the order of feet/day. Thus under most conditions, the additional spreading in the direction of flow (longitudinal spreading) due to molecular diffusion is not detectable unless flow is very slow.
4-22 CHAPTER FOUR
It may become important in low velocity conditions that occur in dead-end pipes under constant or intermittent conditions.
4.2.3.2 Turbulent Diffusion
Turbulent diffusion is transport caused by the random movement of fluid parcels due to turbulence versus fluid molecules in molecular diffusion. Again consider the cup of hot water. Instead of placing the tea bag in and leaving the cup, now the tea bag is lifted in and out of the water. This action will cause eddies (turbulence) in the water and increase the rate of mixing and coloration. Parcels of water with high tea concentrations mix in parts of the cup that have low concentrations until the tea is uniformly distributed throughout the cup. In tanks like the tea cup, mixing is critical to understanding constituent distribution. A jet of water entering a tank may cause turbulence and mixing.
Turbulent diffusion also occurs in a pipe during turbulent flow. At the pipe wall under laminar flow with low velocities, water will pass over and around the imperfections on the pipe wall. As velocities increase, water essentially runs into the bumps and bounces away from the pipe wall forming eddies. This process distributes a constituent through the water across the pipe section more rapidly than molecular diffusion since the fluid is mixing (Figure 4-6b). The momentum and velocity are mixed in the same way resulting in a relatively uniform velocity distribution across a pipe as shown in Figure 3-4. In Reynolds’ experiments, as the level of turbulence increased with the flow velocity, the dye mixing was more rapid (Figures 4-5b and c).
The formation and size of eddies is a random process so the transport can be modeled as a Fickian diffusion process. Thus mathematically, turbulent diffusion is also described by Eq. 4-12 but with a turbulent diffusion coefficient, Dt [L2/T]:
2
2
yCD
tC
t ∂∂
=∂∂ (4-13)
where Dt is typically in the range of 100-105 cm2/s.
4.2.3.3 Dispersion
Advection is the transport at the mean fluid velocity. In turbulent flow, the velocity is nearly uniform across a section and nearly equal to the mean value (Figure 3-4) so spreading of a constituent laden mass in the axial direction is small. At low flow rates and laminar flow, the non-uniform velocity distribution (Figure 3-3) causes variations in axial transport across the pipe. As shown in Figure 4-7, the center of the pipe has a velocity greater than the mean.
WATER QUALITY FUNDAMENTALS 4-23
If only advection is considered, the additional transport above the mean velocity would not be considered.
Axial (also termed longitudinal) spreading of a constituent mass due to non-uniform velocities is known as dispersion. It has been shown that dispersion can also be represented by a Fickian diffusion process (Eq. 4-12) or:
2
2
yCD
tC
disp ∂∂
=∂∂ (4-14)
where Ddisp is a dispersion coefficient that is usually in the range of 106 cm2/s.
Figure 4-7: Laminar velocity distribution showing average velocity. The higher velocities at the center of the pipe will cause constituents in those waters to arrive at downstream locations before constituent reaches the locations near the pipe wall.
4.2.3.4 Impacts of Diffusion and Dispersion
To summarize the effect of the transport mechanisms introduced in this section, diffusion affects mass transport in the radial (across pipe) and axial (along pipe) directions. Dispersion is a laminar flow transport mechanism that only affects axial transport. Table 4-1 lists conditions in which each transport mechanism is applied and its associated parameter.
Molecular diffusion occurs under all flow conditions although it is generally negligible if the fluid is moving. Turbulent diffusion only occurs in turbulent flow but as the level of turbulence increase its impact in the longitudinal direction diminishes since momentum and the constituent is uniformly distributed across the pipe.
The advective transport equation (Eq. 4-11) can be extended to account for diffusion and dispersion to the two-dimensional advection-dispersion equation for turbulent conditions as:
4-24 CHAPTER FOUR
)()()( 2
2
2
2
CrxCDD
rCDD
xCV
tC
tmtm =∂∂
+−∂∂
+−∂∂
+∂∂ (4-15)
where x is the distance along the pipe and r is the radial distance from the center of the pipe. The first term on the LHS is the unsteady concentration, the second term represents advective transport in the axial direction, the third term is radial transport due to molecular and turbulent diffusion, and the final LHS term is longitudinal diffusion. As noted, for laminar conditions Dt is dropped and dispersion is added so the resulting advective–dispersion equation for laminar flow is:
)()( 2
2
2
2
CrxCDD
rCD
xCV
tC
dispmm =∂∂
+−∂∂
−∂∂
+∂∂ (4-16)
Alternative formulations of Eqs. 4-15 and 4-16 are discussed below and solution methods and results are presented in Chapter 6 for longitudinal transport and Section 4.3.4.3.2 for radial transport.
Table 4-1: Impact of alternative transport mechanisms in radial and longitudinal directions.
Molecular diffusion
Turbulent diffusion Dispersion
Cause Movement of molecules within
fluid.
Eddy transport in turbulent flow
Variation of velocity across a pipe section
Longitudinal transport
Little impact in moving fluid.
May be important in static water.
No impact since turbulence causes uniform velocity
profile ( 10000>ℜ )*
Significant in laminar flow ( 2300<ℜ )* No impact in fully
turbulent flow Radial
transport Little impact in moving fluid.
May be important in static waters
Significant impact causing complete
mixing across section
No impact
Parameter Molecular diffusivity, Dm
Eddy diffusivity, Dt Dispersion coefficient, Ddisp
Coefficient magnitude
(cm2/s)
10-4
100 – 105
106
*The range in which turbulent diffusion and dispersion fully occur is shown in the Table. With mixed flow will occur that is not fully laminar or turbulent. Clear
distinctions of transport in this range are difficult to define since flow conditions vary in time and space. Radial transport has been modeled more accurately for these conditions and discussed in Section 4.3.2.4.2.
100002300 <ℜ<
WATER QUALITY FUNDAMENTALS 4-25
4.2.3.4.1 Dispersion Effect on Axial Transport
Axial transport above advection is caused by molecular diffusion and dispersion and for laminar flow regimes can be modeled by:
)()( 2
2
CrxCDD
xCV
tC
dispm =∂∂
+−∂∂
+∂∂ (4-17)
Eq. 4-17 can be solved step-wise. First, only the first two LHS terms are considered and advective transport is resolved. Then, the impact of diffusion and dispersion are modeled by an equation including the first and third LHS terms. Finally, constituent decay is represented using the concentrations resulting after transport.
If the magnitude of the advective term is large compared to the diffusion/dispersion term, the latter term can be dropped. The dimensionless Peclet number can provide an indication under what conditions dispersion will be important. Axworthy and Karney (1996) and Lee and Buchberger (2001) developed relationships between the mean velocity and the dispersion coefficient to identify when the dispersion effects would be negligible and advection dominates. In most water distribution system pipes this condition holds. Dead end or slow moving laterals pipes may be the exception (Lee and Buchberger, 2001). Based on experimental and computational studies, Lee and Buchberger found that dispersion can be a significant portion of the lateral transport at low Reynolds numbers. These modeling approaches are not available in first generation water quality models but research work is discussed in Chapter 6.
4.2.3.4.2 Diffusion Effects on Radial Transport
Radial transport is important when examining the reaction of water-borne constituent with material on the pipe wall. The radial advection-diffusion equation for general flow conditions is:
)()( 2
2
CrrCDD
xCV
tC
tm =∂∂
+−∂∂
+∂∂ (4-18)
One example where radial transport is important is the effect of the interaction of chlorine with biological material (biofilm) on the pipe wall. Biswas et al (1993) and Ozdemir and Ger (1998 and 1999) evaluated the effect of radial diffusion on chlorine decay through two-dimensional modeling. Biswas et al examined fully turbulent conditions while Ozdemir and Ger focused on conditions with less turbulence. Rossman et al (1994) developed a
4-26 CHAPTER FOUR
mass transfer relationship for modeling the wall-bulk fluid interaction. For fully turbulent flow all three methods gave similar results. These methods and results are described in detail in Section 4.3.2.4.2.
4.3 REACTION KINETICS
As discussed in Chapter 1, constituents in the distribution system react with materials in the water and on the pipe and tank walls. These reactions must be represented in the conservation of constituent mass relationship (Eq. 4-1). Substances react according to different relationships and rates. Reaction kinetics are used to describe these relationships and include parameters that relate the reaction rate to system conditions (Connors, 1990).
The net change of constituent in the distribution system is dependent on the time spent in the system. Longer detention times can cause or worsen water quality problems for a number of constituents (Table 4-2). As such, water age is often used as a water quality surrogate indicator. Beyond water age, efforts to date have primarily focused on modeling disinfection decay and disinfection by-product. Microbial transport has been studied to a lesser degree. This section provides background on reaction kinetics, their mathematical description and determining reaction coefficients.
Table 4-2: Water quality problems associated with water age (from EPA/AWWA white paper
(http://www.epa.gov/safewater/tcr/pdf/waterage.pdf)).
Chemical issues Biological issues Physical issues
Disinfection by-product formation
Disinfection by-product biodegration
Temperature increases
Disinfectant decay Nitrification Sediment deposition
Corrosion control effectiveness
Microbial regrowth/recovery/shielding
Color
Taste and odor Taste and odor
Conservative (inert) constituents, like fluoride, are not reactive. Others, like
chlorine, react with other constituents in the water and are reduced in the system. Most system models assume that, within the pipe network, the rate of reaction for chlorine decreases exponentially with time and is not related to the amount of chlorine present. This relationship is described as a first-order reaction. Trihalomethanes (THMs) and other constituents may increase in water during travel through the pipe network. The reactions are also usually described
WATER QUALITY FUNDAMENTALS 4-27
by first order kinetics. In some cases, due to the availability of a co-constituent, the amount of a constituent may be bounded. If the reaction rate is dependent upon the amount of the constituent, the relationship is described as second order. Finally, kinetic models representing the interactions between multiple species have been recently developed.
Reactive constituents are affected by the other chemicals in the water in so-called bulk flow reactions and by materials on pipe surface in wall reactions. Bulk reaction relationships occur in pipes and tanks and their rate constants can be estimated by laboratory jar tests. Wall reactions only occur in pipes and are more difficult to quantify.
4.3.1 Reaction Relationships
The kinetics or rate of reactions is assumed to be a function of the time and/or the reactants concentration. The reaction term is introduced in conservation of a constituent mass through the term, dtdmc . In developing conservation of a constituent mass for an unsteady system with only advective transport and constituent reactions, dtdmc was defined as V )(Cr . With this assumption the resulting relationship is Eq. 4-11 or:
)(CrxCV
tC
=∂∂
+∂∂ (4-11)
A simple reaction relationship is a first order relationship in which the reaction is linearly related to the concentration or:
CkCr =)( (4-19)
where k is the reaction constant. This section focuses on conditions within pipes that act as plug flow
reactors. As will be shown, the reactions are time dependent and are related to the flow velocity and length of the pipe. Before moving to pipes for a clearer understanding of reaction relationships, a closed tank will be studied first. Consider chlorine in a tank that acts as a continuously stirred reactor (CSTR), i.e., the tank is completely mixed and C is uniform through the tank. The pipe to the tank is closed so no flow enters or leaves. In this case, the second term on the LHS of Eq. 4-11 is zero and assuming chlorine decays following a first order relationship the equation becomes:
CkCrtC
==∂∂ )( (4-20)
4-28 CHAPTER FOUR
where k will be negative since chlorine is a decaying substance. Eq. 4-20 is solved by separating variables and integrating:
∫ ∫ =⇒= kteCCdtkCdC
0 (4-21)
where C0 is the initial concentration in the tank. Thus, the chlorine concentration decreases exponentially from C0 at time t = 0. The rate of decay is defined by the rate constant, k [1/T]. If the initial tank chlorine concentration is 3 mg/l and the decay constant is -0.125/hr, the concentration after 3 hours will be 2.06 mg/l (= Co ekt = 3 e(-0.125*3) = 3 * 0.687 = 2.06 mg/l).
An interesting interpretation of k is that if the absolute value of k is less than 0.5, k * 100% is approximately equal to the percentage of constituent lost in each time increment defined by the time units. For example, if k = -3 1/day we can convert k to hr-1 to reduce k less than 0.5 or k = -3 (1/day) * 1/24 (day/hr) = -0.125 (1/hr). Since k is less than 0.5, this implies that 12.5% of the remaining constituent is lost each hour regardless of the remaining concentration. This interpretation for k is only valid for a first order reaction.
The general form for for decay and growth processes are: )(Cr
(4-22) 1)()( −−= cnC*CCkCr
(4-23) 1)()( −−= cnCC*CkCr
respectively, where C* is the limiting concentration or non-reactive portion of the constituent, k is the reaction constant, and nc is the reaction order (e.g., nc = 1 defines a first order reaction). Common mathematical forms for alternative reaction types are listed in Table 4-3. In the previous chlorine decay example, the final chlorine concentration, C*, was 0 and nc equaled 1 so Eq. 4-22 became kC.
The future direction of distribution system water quality modeling is to account for and simultaneously represent multiple components since the growth or decay of a substance may be related to the reactions with other constituents in the system. For example, THM production is linked to the reaction of chlorine with organics. A joint relationship has been proposed by Clark (1998) and extended by others. Others have (e.g., Lu et al, 1995; Munavalli and Kumar, 2004) developed a multi-constituent reaction/transport model for distribution systems and linked the changes in chlorine, organic substrate and bacterial growth. Research has progressed in this area in the past decade and it is expected that practical models will soon introduce these higher-level representations.
WATER QUALITY FUNDAMENTALS 4-29
Table 4-3: Reaction types and mathematical forms.
Reaction type nc C* k Rate units r(C) Example constituent
Conservative - - 0 - 0 Fluoride Zero order growth 0 0 1 Mc/(L3 T) k Water age First order decay 1 0 <0 1/T kC Chlorine
First order saturation growth
1 C* >0 1/T k(C* - C) Trialomethanes (THM)
Second order decay
2 0 <0 L3/(Mc T) kC 2 Initial chlorine reactions
Second order reactions for dependent
constituents
2 0 L3/(Mc T) kCACB Chlorine-THM (where CA and CB are the
concentration of the two constituents)
The focus of the remainder of this chapter will be on pipe transport and
reactions. For a pipe, Eq. 4-11 can be solved analytically for different cases of for steady conditions. Recall that )(Cr tC ∂∂ in Eq. 4-11 is related to the rate
of change of constituent concentration within the differential element. Steady hydraulic and water quality conditions imply that this term equals zero. This assumption states that the difference of concentration over time is zero. It does not require that the inflow and outflow concentrations are the same; only that each value is constant over time.
4.3.1.1 Conservative Constituents
Conservative substances do not react. Salt, possibly measured as dissolved solids, and fluoride are examples of non-reactive or conservative species. Unless additional conservative substance is added or dilution occurs, conservative constituent concentrations remain constant. As seen in Table 4-3,
for conservative constituents equals zero since k = 0. So Eq. 4-11 becomes:
)(Cr
0=∂∂
+∂∂
xCV
tC (4-24)
Under steady conditions, ⎟⎟⎠
⎞⎜⎜⎝
⎛=
∂∂ 0
tC ,
0=∂∂
xCV (4-25)
4-30 CHAPTER FOUR
Considering a pipe segment (Figure 4-8), if we separate variables and integrate Eq. 4-25, the left hand side equals zero or:
1212 000 2
1
2
1
2
1
CCCCCdxdC CC
x
x
C
C=⇒=−⇒=⇒= ∫∫ (4-26)
where C1 and C2 are the pipe segments’ inflow and outflow concentrations. Eq. 4-26 states that, under steady conditions, a conservative substance will not change in the direction of flow and have the same inflow and outflow concentrations.
Figure 4-8: Pipe section with inflow and outflow concentrations.
4.3.1.2 Zero Order Decay/Growth Kinetics
For zero-order decay, nc=0 and is then: )(Cr
(4-27) kCCkCCCkCr cn =−=−= −− )10(1 )0()*()(
Note the dimensions of k for a zero order reaction are (Mc/L3)/T. Thus, the constituent mass decreases/increases by k units per unit mass per unit time. Under steady state conditions, Eq. 4-11 reduces to:
kxCV =
∂∂ (4-28)
Separating variable and integrating Eq. 4-28 yields:
2112
12 −=−
=− τkV
xxkCC (4-29)
WATER QUALITY FUNDAMENTALS 4-31
The fraction on the right hand side is the travel time, τ1-2, for flow to move from section 1 to 2. Thus, the final right hand side is the rate of addition of constituent times the time step or the total constituent addition.
A useful special case of zero order kinetics is for a constituent representing water age (i.e., Mc = T). Water age can act as a surrogate for first order reaction constituents since their concentrations are directly related to the retention time in the network.
For water age k equals 1 [T/L3/T] representing an increase of one unit of time per unit time. With this definition in Eq. 4-29, the difference in concentrations (water age) is the travel time. Thus, the difference in water age, C2 – C1, in Eq. 4-29 is equal to the travel time in the pipe.
Water age can identify regions of long travel times that may indicate potential poor disinfectant levels. An advantage of using water age as a first level indicator of water quality over other parameters is that no water quality calibration is necessary. Water age is only based upon the flow distribution in the pipe network and the resulting pipe travel times. This clearly demonstrates the relationship between the flow distribution and water quality and reinforces the need for a well calibrated hydraulic model.
4.3.1.3 First Order Growth/Decay Kinetics
As noted earlier, decaying constituents often approximately follow first order reactions with nc equal to 1 and C* equal to 0. Chlorine and other disinfectants in the distribution network fall in this category. Substituting these values for
in Eq. 4-22 gives: )(Cr
(4-30) kCCCkC*CCkCr cn =−=−= −− )11(1 )0()()(
For steady state conditions,
CkxCV =
∂∂ (4-31)
where a k value less than zero denotes a decaying constituent. For a pipe, Eq. 4-31 can be solved by separating variables and integrating along the pipe length or:
( )VxCklndx
VkCCddx
VkCCd x
xCC
x
x
C
C2
1
2
1
2
1
2
1
11=⇒=⇒= ∫∫ (4-32)
After substituting,
4-32 CHAPTER FOUR
( ) ( ) ( )τ=
−=−
VxxCklnCkln 12
12 (4-33)
Note that the pipe segment length (x2 – x1) divided by the flow velocity is equal to the travel time in the pipe segment, τ. The ln denotes log base e. Rearranging and raising both sides to the power of e gives:
(4-34) τkeCC 12 =
With k < 0, this relationship states that the downstream constituent concentration decreases exponentially with the length of the pipe and the travel time. This equation is very similar to Eq. 4-21 for the decay in concentration in a closed system over time. Eq. 4-34 makes an important extension showing that the change can occur in a moving fluid. In steady state, the downstream concentration has decreased relative to the inlet concentration.
This relationship can be used to estimate the concentration at the end of the pipe given the inlet concentration. In addition, the profile within the pipe can be computed for steady condition by recognizing the travel time to different locations in the pipe. Finally, the concentration in a pulse of water moving through the network can be computed given the travel time between points under steady or unsteady conditions.
4.3.1.4 First Order Saturation Growth Kinetics
First order saturation growth is an exponential growth model similar to first order decay except that the sign of k is positive rather than negative and we assume that the amount of constituent that can be produced is limited. The concentration is bounded by a maximum of C* due to the inability of the system to sustain a larger concentration or that the precursors that form the constituent are limited. An example is the formation of trihalomethanes that is constrained by the initial chlorine concentration. As demonstrated by Elshorbagy (2000), the bounding concentrations can be estimated from empirical equations and field data, lab studies and/or from stoichiometry for multiple species of THM’s.
For steady conditions in a pipe segment, substituting the growth relationship from Table 4-3 results in:
( C*CkxCV −=
∂∂ ) (4-35)
where k is a positive growth coefficient. Eq. 4-35 has the same form as Eq. 4-31 and can be solved in a similar manner. The result is:
WATER QUALITY FUNDAMENTALS 4-33
( ) τkeC*C*CC −−−= 12 (4-36)
Since k is positive, the second term on the left hand side goes to zero as t increases and the concentration at the pipe outlet approaches the maximum concentration. Similarly for a very short travel time (t ~ 0), the exponential term equals one and the concentration at the downstream section is close to C1.
4.3.1.5 Second Order Reaction Kinetics
4.3.1.5.1 Special Case: Single Reactant Species
Second order reactions (nc=2) relate the reaction rate to the present level of the constituent. Thus, at higher constituent levels reactions are more likely and the rate of change in the constituent level will be higher. During disinfection in the water treatment plant, a rapid initial loss occurs corresponding to high disinfection levels. It may be more appropriate to model initial disinfection using a second order model. After some time disinfected waters are released in to the pipe system, the decay rates are reduced and a first order reaction rate fits in many cases or the second order may remain appropriate.
For the special case of a single constituent with second-order kinetics, nc is 2. With C* equal to 0, Eq. 4-22 reduces to:
(4-37) 2)( CkCr =
For a decaying substance k will be negative. Substituting in the steady state form of Eq. 4-11 yields:
)(Cr
2CkxCV =
∂∂ (4-38)
This equation can also be solved by separating variables and integrating along the pipe length.
τk
Vxx
kCC
dxVkC
CddxVkC
Cd x
x
C
C=
−=−⇒=⇒= ∫∫ 12
2122
1111 2
1
2
1
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⇒=−
⇒τ
τ1
1221
12
11Ck
CCCkCCC
(4-39)
4-34 CHAPTER FOUR
For the second-order decay kinetics with C* not equal to zero, Eq. 4-22 reduces to:
C*CCkCr )()( −= (4-40)
Substituting in the steady state form of Eq. 4-11 yields: )(Cr
CCCkxCV )*( −=
∂∂ (4-41)
The above equation can be solved for the general form for the downstream constituent concentration:
( )
τ)( *CCk*CC
*CC−+
−+=
1
12 1
(4-42)
Note that Equation 4-39 is a special case of this equation with C* equal to 0.
4.3.1.5.2 General Case: Multiple Competing Species
A general second order model for two constituents can be developed based on the chemical reaction:
aAr + bBr ⇒ pP (4-43)
where Ar and Br are the two reactive constituents with concentrations in mg/l of CA and CB, respectively. Their reaction constants are kA and kB, respectively. P is the resulting product. This model has been applied in distribution system modeling with chlorine and natural organic matter as the reactive constituents and trihalomethanes as the resulting product (Clark, 1998). However, most water quality models to date have not included this level of reaction kinetics complexity. The reaction rates of the two constituents are the same and can be written as:
BABB
B
BAAA
A
CCkdt
dC-Cr
CCk dt
dC- Cr
==
==
)(
)(
where kB = (b/a) kA. Following Clark and Sivaganesan (1998), the general solution for this formulation is:
WATER QUALITY FUNDAMENTALS 4-35
))1((00
00000)(1
)()( ττ
,BA,B,A CkCa/Cb,A,B
,B,AA eCb/Ca
b/CaCC −−−
−= (4-44)
Defining u = N (1-KAB), KAB = a CB,0/b CA,0, and N = kA b CA,0/a this form can be rearranged to:
ττ uAB
AB,AA eK
KCC
−−
−=
1)1(
)( 0 (4-45)
The two parameters, N and KAB provide flexibility in the shape of the resulting relationship and result in better fits to field data. Two special cases of this model for chlorine reactions are presented in Section 4.3.2.2.
4.3.1.6 Michaelis-Menton Reaction Kinetics
A special reaction relationship outside the forms described thus far is the Michaelis-Menton (M-M) rate equation. The M-M kinetics equation was derived from rates of chemical reactions of enzymes and is given by:
)(
)(CC
CkCr M −= (4-46)
where CM is the Michaelis constant. The denominator becomes CM + C for
growing constituents. The additional parameters and alternative mathematical form provides flexibility in the shape of the function (Figure 4-9). As discussed in the next section, CM and k can be determined from a Lineweaver-Burk plot.
Microbial activity and bacterial growth is often modeled by Monod equation that has the same functional form as M-M kinetics. In biological applications, the Monod equation is strictly empirical for microbial growth.
As shown for the previous reactions, the concentration of a constituent following MM kinetics in the pipeline example can be determined by integrating Eq. 4-46. The result is:
(4-47) τkCClnCCClnC MM =−−− )( 1122
A direct solution for C at a given location/time is not available and must be solved iteratively by successive substitution or a Newton-Raphson type scheme.
At concentration extremes, simplifying approximations result in more easily solved conditions. Examination of Equation 4-46 shows that when C is large (C >> CM), CM can be neglected and the reaction rate approaches a constant value. The constant value implies that the growth/decay follows a zero
4-36 CHAPTER FOUR
order relationship. When C is small the denominator equals CM and the reaction relationship approaches kC/CM or k’ C. This relationship has the same form as a first order relationship.
Figure 4-9: Schematic of Michaelis-Menton rate of growth as function of concentration.
4.3.1.7 Multi-Species Models
Multiple dependent constituent models have also been proposed (Lu et al, 1995; Dukan et al, 1996; Bois et al, 1997; and Munavalli and Kumar, 2004). The constituents in these models include terms representing chlorine, organic matter, and biomass/biofilm growth. Reactions are caused by interactions between constituents. Thus, reaction equations describing growth/decay and constituent concentrations are functions of the concentrations of the other constituents. As a result, a set of coupled equations must be formulated and solved. For example, Monod kinetics are commonly assumed for biofilm growth. This assumption is common for bacterial growth models and results in a model the growth constant will vary with an available substrate or:
bdbb Xk
dtdX
)( += µ (4-48)
where Xb is the bacterial concentration and µb and kd are the bacterial growth and death rates (T-1). µb is often modeled with M-M kinetics using Eq. 4-46 with the C being the concentration of a supporting constituent (substrate). This approach is similar to the competing model described in Section 4.3.1.5. However, rather than relating production of one constituent to the decay in a
WATER QUALITY FUNDAMENTALS 4-37
second through a stoichiometric relationship, here the growth and decay constants are linked and form a set of differential equations.
The formulation presented by Zhang et al (2004) is representative of linked multi-species models. Munavilli and Kumar (2004) developed a similar but somewhat more complex formulation. Zhang et al defined the substrate as biodegradable dissolved organic carbon (BDOC) and considered a balance of the terms in the four solid boxes in Figure 4-10. The following relationships were applied in their model:
bdepbdhadetbbb XkXkR/VXkX
tX
−−+=∂
∂µ (4-49)
hbdepadadetaaa RXkXkVXkX
tX
+−−=∂
∂µ (4-50)
( bbhaag
XR/XYt
S µµχ
+⎟⎟⎠
⎞⎜⎜⎝
⎛−=
∂∂ 1 ) (4-51)
hwb R/kClkt
Cl−−=
∂
∂ (4-52)
where Xb, Xa, S, and Cl are the concentrations of free bacteria in the bulk water, bacteria attached to the pipe wall, substrate measured as BDOC, and chlorine, respectively. Equation 4-49 is the reaction rate of free bacteria. Its first and third terms correspond to the growth and death in the bulk water with growth rate and mortality constants, µb and kd, respectively. The second RHS term represents the rate of attached bacteria that detaches and enters the bulk water while the fourth RHS term is the rate of free water bacteria that deposits on the wall. The rates that these physical processes occur are defined by rate constants, kdet and kdep, respectively. V and Rh (=R/2) are the flow velocity and hydraulic radius, respectively. The reaction terms for the attached bacteria (Eq. 4-50) are similar as the free bacteria but growth and death coefficients and signs on the detachment and deposition are changed.
BDOC concentration is reduced as a function of the growth of bulk and attached bacteria (Eq. 4-51) that are scaled by the growth yield coefficient of bacteria, Yg, and the number of cells in the cell biomass produced per milligram of organic carbon, χ. Finally, although recognizing that the rate of chlorine disappearance is likely related to the substrate concentration, first order decay for bulk water and zero order for wall reactions models are applied (Eq. 4-52).
µb is represented by a Monod reaction and, most importantly, is related to the BDOC and chlorine concentrations by:
4-38 CHAPTER FOUR
tiopt
opt
c
t
sbb ClCl
TTTT
ClClCl
KSS
>⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−⎟⎟
⎠
⎞⎜⎜⎝
⎛+
=2
max, expexpµµ (4-53a)
tiopt
opt
sbb ClCl
TTTT
KSS
≤⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−−⎟⎟
⎠
⎞⎜⎜⎝
⎛+
=2
max, expµµ (4-53b)
where µmax,b is the maximum bacteria growth rate in the bulk water, Ks is the half-saturation constant for substrate uptake, Clt and Clc are the threshold and characteristic chlorine concentrations, respectively, and Topt and Ti are the optimal temperature for bacterial activity and a temperature parameter, respectively.
Figure 4-10: Schematic of processes and interactions between constituents. Research models have represented the solid lines. DBPs have not been included in formulations to date.
Since the constitute concentrations are dependent upon each other, these
reaction relationships form a system of differential equations. Given the
WATER QUALITY FUNDAMENTALS 4-39
complexity and dependency of these relationships, general analytical solutions are not available. Zhang et al and Munavilli and Kumar solved for the reaction effect on constitute concentrations using a Runge-Kutta fourth order scheme. Most numerical analysis texts present the Runge-Kutta method (Press et al, 2002 - http://lib-www.lanl.gov/numerical/index.html).
4.3.1.8 Summary of Reaction Relationships
Analytical solutions for the reaction relationships described in the previous section are summarized in Table 4-4. A half-life column is introduced to provide context on the rate of change of a constituent. The number of parameters describing the reactions increases with reaction order and provides more variability in the shape of the function.
Table 4-4: Summary of reaction relationship solutions.
Reaction type nc r(C) Half-Life*Concentration profile
Conservative - 0 - Constant
Zero order growth/decay
0 k C0/(2k) C0 – kτ
First order decay 1 kC 1/(1.44k) C0 e-kτ
First order saturation growth/decay
1 k(C* - C) 1/(1.44k) C* - (C*-C)e-kτ
Second order decay 2 kC2 1/(kC0) C0 /(1+k C0τ)
Second order competing reaction
2 kCACB - Eq. 4-45
* Half-life relates to decaying substances. It is equal the doubling time required for growth reactions. Note C0 is the initial concentration in time or at the beginning of the pipe (i.e., C1 in the pipe results.)
The concentration profile can be related to the profile along a pipe where the distance is related to the travel time to that location (see Example 4.10). It can also describe the concentration over time for a closed well-mixed system such as a tank. In this case, rather than defining travel time, τ is the time in the tank.
From Table 4-4, it is seen that a zero or first order reaction is only a function of time and system characteristics. In a second order reaction, the concentration is multiplied by a term that includes the travel time but it also contains the initial concentration. With a positive k, for a growing constituent, a higher initial concentration will result in a larger downstream concentration for the same travel time. The reverse holds for a decaying constituent. Thus, the decay/growth of a constituent following a second order reaction will be more
4-40 CHAPTER FOUR
dramatic than a first order reaction. Higher order reactions will result in more pronounce differences. Like the first order reaction for a decaying relationship, it asymptotically approaches zero or a defined limiting concentration. Figure 4-11 shows the shape of the decay functions for four different single species models.
Figure 4-11: Illustration of various reaction kinetics orders. A second order competing reactant model was also introduced in this
section. Its application to chlorine decay and TTHM development is presented in the next section. Complementary work in this area has followed extending the model for both chlorine and DBP subspecies of chlorine and developing parameter relationships. Finally, microbial growth is often defined using the Michaelis-Menton reaction kinetics. This relationship has a steep portion followed by a flat region approaching a maximum rate.
The following two examples demonstrate reactions in pipes. The first example shows a steady state system and examines the variation of concentration along the pipe. In steady state, conditions within the pipe are constant and mixing at a node is a function of concentration at the pipe outlet. In the next chapter, the relationship for the pipe outlet concentration is linked with nodal mixing (Eq. 4-3) for computing steady state concentrations throughout a network. The second example extends Example 4.4. In this example, the variation of constituent concentration in the water is shown. Tracking the movement of water parcels in a pipe is the basis for a class of dynamic simulation methods (Chapter 6).
The third example returns to Example 4.9 and shows the variation of a concentration at the junction of two pipes but in this example constituent decay during travel in the pipe is included.
WATER QUALITY FUNDAMENTALS 4-41
Example 4.10
Problem: An 18,000-foot long pipeline transports water with a velocity of 1 ft/s. Compute the concentration at 1,800 ft increments for a constituent in a water that: (a) is conservative (k = 0), (b) follows a first-order decay reaction with k = -0.5/hr, (c) follows a zero-order growth reaction, and (d) follows a first-order saturation growth reaction with k = 0.5/hr and C* = 10. Assume that the concentration at the source is 10 except for first order growth where C0=0.
Figure E4-10a: Pipe segment for constituent transport Example 4.10.
Solution: The concentration at 1800 ft corresponds to a travel time of 0.5 hours (= L/V = 1800 ft/1 ft/s) and is computed for each reaction below. All results at 1800 ft (0.5 hr) intervals are listed in Table E4-10 and plotted in Figure E4-10b.
a) Conservative constituent: Since k=0, the concentration is constant for all time, Ct = C0 by Eq. 4-26. So Ct = 0.5 hr = C0 = 10. b) First-order decay: The first-order reaction results in an exponential decay relationship that is represented by Eq. 4-34. With k = -0.5/hr, the relationship is:
τττ
5.00 10 −
= == eeCC kt
For L = 1800 ft, τ = L/V = 0.5 hrs, the resulting concentration is 7.8. c) Zero-order growth model: The zero-order growth is equivalent to the time spent in the pipeline. So by Eq. 4-29, the concentration at t = 0.5 hrs equals 0.5.
hr..CC 'L'L 50500180001800 =+=+= == τ
d) First-order saturation growth: First-order saturation growth is described by Eq. 4-36 or:
( ) ττ
kt eC*C*CC −= −−= 0
4-42 CHAPTER FOUR
where C0 is the initial concentration (=0). With k = 0.5/hr and C* = 10, the concentration at a location 1800 feet from the source or 0.5 hours travel time in the pipeline is:
( ) ( ) 228710010100 )50(5050 ..ee*C*CC ... =−=−−=−−= −− ττ
Results for all locations are shown in Table E4-10 and in Figure E4-10b.
Table E4-10: Results for concentration as a function of pipe location for different reaction relationships for Example 4.10.
Distance (feet)
Time (hrs)
Conservative constituent
First order decay
Zero order growth
First order growth
0 0.0 10 10 0.0 0 1800 0.5 10 7.8 0.5 2.2 3600 1.0 10 6.1 1.0 3.9 5400 1.5 10 4.7 1.5 5.3 7200 2.0 10 3.7 2.0 6.3 9000 2.5 10 2.9 2.5 7.1
10800 3.0 10 2.2 3.0 7.8 12600 3.5 10 1.7 3.5 8.3 14400 4.0 10 1.4 4.0 8.6 16200 4.5 10 1.1 4.5 8.9 18000 5.0 10 0.8 5.0 9.2
0
2
4
6
8
10
12
0 5000 10000 15000 20000
Distance along pipe (ft)
Con
cent
ratio
n
Conservative First order decay Zero order growth First order growth
Figure E4-10b: Plot of constituent concentrations versus pipe distance for different reaction kinetics for Example 4.10.
WATER QUALITY FUNDAMENTALS 4-43
Example 4.11
Problem: For the pipe conditions presented in Example 4.10, consider four conditions of constituent tracking for a fluid mass that is moving through the pipe. First, water age is to be monitored as a flow moves through the pipe. Next, a conservative substance is to be evaluated. Finally, first and second order decaying substances are to be modeled.
Assume that the constituent is continuously injected into the 18000 foot long pipe for 1000 seconds. The pipe flow velocity is 1 ft/s. The initial concentration of all substances except water age is 10 units. The decay coefficient is -0.5 in all cases. Determine the pulse location and constituent concentrations of the pulse at 1, 2 and 5 hrs. Solution: Conservative constituent - A conservative substance is identical to Example 4.4. The injected mass begins at time zero and moves into the pipe. After one hour the material that was injected at time zero has moved 3600 ft (=1 ft/s * 3600s). This material is at the front of the segment and is called the front. The back end of the segment was injected at time 1000s. The segment length is 1000 ft. Locations of the segment for all three periods are shown in Figure E4-11. Note that the location of the segment is the same for all constituents since it is driven by advective transport.
Figure E4-11(a-c): Time history of segment affected by constituent injection at the pipe inlet (left side) beginning at time 0 s. Pipe is 18000 ft long.
4-44 CHAPTER FOUR
Time 1 hour – At 3600 s, the front of the segment has traveled 3600 ft while the back of the segment, which entered the pipe at 1000 s has traveled for 2600 s to a location 2600 ft into the pipe. Since the constituent is conservative the concentration for its entire length is 10 units. Time 2 hours – The front of the segment has moved another 3600 ft from hour 1 to 2. The new location is 7200 ft into the pipe. The back of the segment is now at location 6200 ft. (=2600 ft + 3600 ft). As before the concentration at all locations within the segment is 10 units. The concentration at all other locations is 0 units. Time 5 hours – The total travel distance for a point beginning at the pipe inlet at time zero is 18000 ft (=5 hrs * 3600 s/hr * 1 ft/s). This location is the pipe outlet and the location of the front. The back of the segment is at location 17000 ft. Locations within the segment have a concentration of 10 units and elsewhere the concentration is 0 units.
Water age – Time 1 hour - The water age is the time parameter necessary for evaluating temporal variations in concentrations. After one hour the front of the parcel has moved from the entrance to a distance 3600 feet into the pipe. This water had a water age of zero at time 0 s. Thus, its water age at 1 hour is 3600 s (= 0s + 3600s). The back of the segment entered the pipe at time 1000 s. At time 1000 s, its water age was zero. Thus at time 3600 s, its age is 2600 s (= 3600 s – 1000 s). Time 2 and 5 hrs – After the segment is fully within the pipe, it ages with the clock time. So the water ages for the front at 2 and 5 hours are 7200 s and 18000 s, respectively. The water ages at the back of the segment are 6200 s and 17000 s for times 2 and 5 hrs., respectively, since they did not enter the pipe until time 1000 s. Water age varies linearly in the segment with the oldest water at the front and youngest at the back of the segment. First order decaying constituent – Time 1 hour – First order decay is a function of time and the decay constant. Thus, at time 1 hour, the water at the front is 1 hour old and its concentration decreased from 10 units to:
units..eeCC .kt 166101010 1500 ==== −− )()(
The water at the back of the front had a concentration of 10 units when it entered the pipe at time 1000 s. It has spent 2600 s in the pipe and its concentration only decreased to:
WATER QUALITY FUNDAMENTALS 4-45
units..eeCC /.kt 077001010 36002600500 ==== −− )()(
Times 2 and 5 hours – At these times, the water age can again be substituted in the first order relationship for constituent concentrations of 3.7 and 0.8 units for hours 2 and 5, respectively, for the front and 4.2 and 0.94 units, respectively, for the back of the segment.
The concentration varies in the segment following the shape of the exponential decay function with higher concentrations in the back of the segment with the lowest values at front.
Second order decaying constituent – Time 1 hour – To model second order decay we apply Eq. 4-39 with τ equal to the water ages. For the front at time 1 hour, C1 hr (front) = C0 /(1+k C0 τ) = 10/(1+0.5*10*1) = 1.7 units. The back of the segment is slightly younger with τ = 2600/3600 = 0.72 hrs and the concentration is C1 hr (back) = C0 /(1+k C0 τ) = 10/(1+0.5*10*0.72) = 2.2 units. Times 2 and 5 hours – Similar calculations as for 1 hour are made using the water age at these times. The concentrations at times 2 and 5 hrs for the front are 0.91 and 0.38 units, respectively. The segment back concentrations are 1.04 and 0.41 units for times 2 and 5 hours, respectively. As seen in Figure 4-11, concentrations for constituents following this higher order relationship are lower than those for first order decay.
Example 4.12
Problem: (a) A fast reacting constituent that follows a first order reaction is injected in the Example 4.9 pipe system at the same times as that example. Calculate the node 3 concentrations at time 0, 12, 30, and 45 minutes if the decay coefficient is -0.5/hr.
(b) If the injection in pipe 1 is stopped after 45 minutes and the pipe 2 injection is ended at time 1 hour. Determine the concentration at node 3 at 60, 75, and 120 minutes.
Solution: a) Here the mixing relationships will be the same as in the previous examples. The node 3 inflow concentrations will be decreased by decay during travel in the pipe. The concentrations at the pipe outlets are found by:
lmgeeCC kinpipeoutletpipe /75.12 )60/7.15()5.0(
111 === −− τ
and
4-46 CHAPTER FOUR
lmgeeCC kinpipeoutletpipe /78.35 )60/5.33()5.0(
221 === −− τ
As in Example 4.9, node 3’s concentrations at times 0 and 12 minutes are 0 mg/l since no constituent has reached that node. At time 30 minutes, the concentration is altered due to the input from pipe 1. As in Example 4.9, the outflow concentration is computed using Eq. 4-3 with the pipe 1 concentration after decay or:
( ) ( )l/mg.
.QQCQC
C mint, 36190
(0.2)(0)(0.7)(1.75)
3
2211303 =
+=
+==
The concentration at time 45 minutes the outflow concentration is again altered with inflow from pipe 2 (Figure E4-12) or:
( ) ( ) l/mg..Q
QCQCC mint, 20290
(0.2)(3.78)(0.7)(1.75)
3
2211453 =
+=
+==
b) Since the injector is shut off, the constituents now move as pulses through the system as in the previous example. Schematics of the pulse movement can be drawn for each pipe as in Figure E4-11a-e.
The pulse injected into pipe 1 reaches node 3 at time 60.7 minutes (tend + τ1 = 45 + 15.7). Water without constituent in pipe 2 reaches node 3 at time 93.5 min. (tend + τ 2 = 60 + 33.5). Time 60 min. - As shown in Figure E4-12, the concentration at time 60 minutes is the same as at time 45 min. or 2.20 mg/l. Time 75 min. - At this time, the concentration has dropped since constituent is only supplied from pipe 2 until time 60.7 min. The node 3 concentration from time 60.7 to 93.5 minutes is found by substituting a concentration of zero for pipe 1 and 3.78 for pipe 2 in Eq. 4-3 or:
( ) ( ) l/mg..
....Q
QCQCC mint, 84090
)20()783()70()00(
3
2211753 =
+=
+==
Time 120 min. – After time 93.5 min., all constituent has passed through the system from both pipes. Thus, the concentration at node 3 after this time is equal to zero (Figure E4-12).
WATER QUALITY FUNDAMENTALS 4-47
0
0.5
1
1.5
2
2.5
3
3.5
4
0 20 40 60 80 100 120
Time (minutes)
Con
cent
ratio
n (m
g/l)
Pipe 1 Pipe 2Node 3
Figure E4-12: Concentration versus time at node 3 for Example 4-12. Concentrations are shown for flows in pipes 1 and 2 as they enter node 3 (Pipes 1 and 2) and the resulting weighted concentration leaving node 3.
4.3.2 Estimating Reaction Order and Coefficients
Given an understanding of some reaction relationships, the next issues are estimating the reaction order for a specific constituent and the coefficients in that relationship for a specific constituent in a particular system. Reaction coefficients and, possibly the reaction order, can be different for the same constituent in different waters.
Reactions are impacted by the surrounding conditions due to the availability of reacting substances. Reactants are present in the bulk water and may also occur at high concentrations on surfaces of pipes or tank walls. For example, chlorine reacts with organic carbon in the moving water or material held to the pipe wall in a biofilm or as corrosion products as illustrated in Figure 4-12. Here, free chlorine (HOCl) is shown reacting with natural organic matter (NOM) in the bulk phase and free chlorine is also transported through a boundary layer at the pipe wall to oxidize iron (Fe) released from pipe wall corrosion.
In tanks, the wall mixing zone is small relative to the tank size as a whole and this effect is not considered separately from the bulk reaction. In pipes, however, these so-called wall reactions can be significant and theory to model those reactions has been developed. Due to their complexity and variability
4-48 CHAPTER FOUR
between pipes, wall reactions are typically represented by first or zero-order reaction kinetics.
Bulk Fluid
Boundary Layer
HOCl NOM DBP
Fe+2 Fe+3
kb
kw
Figure 4-12: Reaction zones within a pipe (adapted from EPANET, 2000).
In most models to account for both bulk and wall reactions, a total reaction constant is used:
wallb kkk += (4-54)
where kb and kwall are the bulk and wall decay coefficients, respectively. Bulk decay coefficients can be estimated from jar tests. Procedures for
conducting and analyzing jar test results are described in the next section. To generalize jar test results, several researchers have taken data describing water quality from a range of locations and developed equations for estimating bulk reaction coefficients. These equations are presented in the next section.
Wall reactions are more difficult to estimate since they can vary with flow conditions and the availability of reacting material. Biofilm and pipe conditions, however, are rarely, if ever available at the individual pipe level. As a result, a single coefficient describing reaction efficiency is usually defined for the entire distribution system. A general mixing theory based on the level of mass transport is applied to account for waters contact with the wall in individual pipes. This theory is also presented below.
4.3.2.1 Methods for Estimating Reaction Order and Coefficients
Reactions of the constituent of concern with other substances in the moving water are described as bulk reactions and the corresponding reaction order and coefficients can be estimated from laboratory jar (bottle) tests. These coefficients vary by constituent and possibly other factors for the given substance. For example, chlorine bulk reaction coefficients depend on the
WATER QUALITY FUNDAMENTALS 4-49
amount and type of organic matter in the water and on the temperature. Ground waters will typically be less reactive than surface waters (unless there are significant amounts of inorganic reductants, such as iron). The coefficients usually increase with increasing temperature. Running multiple jar tests at different temperatures will provide more accurate assessment of how the rate coefficient varies with temperature.
4.3.2.1.1 Jar Tests
Jar tests consist of taking a sample volume of water in a series of non-reacting glass bottles and analyzing the contents of each bottle at different times. It is best to obtain the water samples at the distribution system inlet, such as from the treatment plant clearwell or from the pump station feeding the pressure zone. This water should be stored at the temperature that is expected to be experienced in the pipe network. An adequate sample volume should be taken since analysis for some constituents may require removal of a portion of the sample water. The sample volume should also be taken to ensure adequate water is available for the experiment duration. The experiment should last at least as long as the expected maximum travel time in the network or until the initial concentration has been reduced by more than half.
Samples are periodically tested for the constituent of concern and the measured values are recorded with their measurement time. After the experiment duration is completed, constituent concentration, Ct, or a permutation of it, are used to estimate the proper reaction order and coefficients as described below. Vasconcelos et al (1997) describe this procedure in more detail for chlorine analysis.
4.3.2.1.2 Analysis of Data
Several approaches can be applied to estimate the reaction relationship order and determine the rate coefficients (Chapra, 1997). Through plotting, the Integral Method graphically examines the data fit to various relationships. To test for a reaction order, the constituent concentrations to the nc-1 power are plotted versus time. For a first order relationship, ln Ct is plotted. If the correct order is chosen the data will plot as a straight line. Figure 4-13 shows plots for zero, first and second order relationships. The reaction coefficients can then be estimated as listed in Table 4-5 and are sufficient to apply the equations discussed in Section 4.3.1. For example, the resulting linear relationship for a first order reaction is:
0ClntkCln b += (4-55)
4-50 CHAPTER FOUR
Table 4-5: Parameters determined from plots of concentration data for various reaction orders.
Order Dependent value Intercept Slope
Zero C C0 -kb
First ln C ln C0 -kb
Second 1/C 1/C0 kb
Second order competitive
ln (CA/CB) ln (CA,0/CB, 0) (CA, 0-CB, 0) kb
ncth cnC −1
cnC −1
0 (nc -1)kb
Note: If a portion of the constituent, C*, is non-reactive or the constituent is growth limited, the term in dependent value is (C - C*) rather than C and the intercept term will be (C0 – C*).
As shown in Figure 4-13b, if the reaction is first-order, plotting log(Ct) against time should result in a straight line, where Ct is concentration at time t and C0 is concentration at time zero.
In all reactions, the linear fit can only provide three parameters for the model, k, C0 and nc. The reaction order is determined by confirming the data follow a line on the appropriate plot and the equation of the line gives two other equations that are used to compute k and C0. Zero order relationships (Eq. 4-29) and the equations for constituents that decay completely only use these three parameters (Eq. 4-34 and 4-39).
However, when a constituent is either growth limited or has a recalcitrant component that does not decay completely, C* is not equal to zero in Eq. 4-36 and Eq. 4-40. Therefore, a fourth unknown, C*, must be determined by plotting concentration, C, versus time. C* is then the estimated limiting concentration that is asymptotically approached over time. For example, assume that the data for the first order decay and growth constituents in Example 4.10 (Table E4.10) was measured and provided to a modeler. By plotting the C vs. t data, as shown in Figure E4.10b, the modeler could assume that the decaying constituent would decay completely ( = 0) and the relationships above would be applied directly. The growth saturated constituent, on the other hand, would be limited at about 10 ( = 10). Given that estimate, the dependent variable in the plots defined in Table 4-5 would be modified to be C – C* = C - 10. With this change, the appropriate plotting relationship would be linear and provide k and C
*decayC
*growthC
0. A second approach, the Differential Method, provides the reaction order
and reaction coefficients directly but requires plotting the derivative of the concentration with respect to time rather than the concentration directly. An equation of the form:
WATER QUALITY FUNDAMENTALS 4-51
ClnnklndtdCln cb +=⎟
⎠⎞
⎜⎝⎛− (4-56)
The derivative terms are numerically estimated by (Ci+1-Ci-1)/(ti+1-ti-1) where the i+1 and i-1 subscripts correspond to non-consecutive data points. Numerical derivatives may introduce errors so an approximate dC/dt versus time curve may be fit to the discrete points (Figure 4-14a). Data points from the approximate curve for log (-dC/dt) and log C are then plotted (Figure 4-14b). A best fit line of the form of Eq. 4-56 is made to the approximate points to estimate nc and kb where nc is the slope and kb is found from the y-intercept. As in the Integral method, the procedure is slightly modified for constituents that are saturated growth limited or do not decay completely. C* is again estimated by plotting C vs. t. Then to determine kb and nc, ln (C – C*) is plotted on the x-axis in Figure 4-14b.
Figure 4-13(a-c): Plots to test if reaction follow (a) zero order, (b) first order, or (c) second order relationships.
A differential type method can also assist in identifying reactions that
follow Michaelis-Menton kinetics. A Lineweaver-Burk plot is used to confirm the model and determine the reaction coefficients (Figure 4-15). In this case, 1/(-dC/dt) is plotted versus 1/C. If a straight line is obtained, the slope is equal to kb/CM and the y-intercept is equal to -1/CM.
4-52 CHAPTER FOUR
Figure 4-14(a-b): Graphical representation of differential method for reaction order and coefficient determination. Computed numerical derivatives are plotted versus time as a histogram (a). A curve is drawn to the data to smooth the results while ensuring that the areas under the drawn curve and the histogram are the same. The log transforms of the fit gradients and the concentration are plotted in (b). A best fit line is determined via regression.
Figure 4-15: Lineweaver-Burk plot for verifying Michaelis-Menton reaction and determining reaction coefficients.
WATER QUALITY FUNDAMENTALS 4-53
Numerical Curve Fitting is the third method for determining reaction order and coefficients. In most cases, a reaction form must be selected and the analysis scheme solved several times. Most spreadsheets can minimize the least squares errors between data and a general function form. Full exposition of this method is beyond the scope of this text. Several texts cover data analysis methods and using spreadsheets for data analysis.
Example 4.13
Problem: The data below (Table E4-13) is chlorine data taken from a jar test. Apply the Integral method to determine the reaction function and its rate constants.
Table E4-13: Chlorine concentrations over time from a jar test.
Time (hrs) Concentration (mg/L)
0 9.79 12 8.13 24 6.53 36 4.26 48 3.34 60 2.94 72 2.14 84 1.26 96 1.04 108 1.45 120 1.28 132 0.57 144 0.99 156 0.42 168 0.17 180 0.28
Solution: To determine the reaction order, the concentration and inverse concentrations are plotted in Figure E4-13a and b. Neither set of data can be represented by a straight line suggesting that the reaction is neither zero nor second order. From Figure E4-13a, it appears that the substance does decay completely (C*=0) so no adjustment is needed in plotting concentrations. Figure E4-13c shows a plot of the log base 10 of concentration versus time. This data appears reasonably well fit by a straight line and the data likely follows first order kinetics. This fit is consistent with the concentration plot (Figure E4-13a) that appears to be an exponential decay function. The functional equation is then:
4-54 CHAPTER FOUR
tkClnClneCC bkt +=⇒= 00
From the semi-log plot, the y-intercept is near or slightly below 9.79 (the value at t = 0) and the slope can be roughly estimated by the first and last data points or:
kb = (ln 0.28 – ln 9.79)/(180-0) = (-1.28 -2.28)/180 = -0.0198
By this approximation, the resulting equation is:
C = 9.79 e-0.0198 t
The equation can also be computed using linear regression analysis that is
available in most spreadsheets. Using the Excel trendline option, the best-fit kb is -0.020 hr-1 and the intercept (C0) is 9.63 or:
(rt.e.C 0200639 −= 2 = 0.9376)
where t is in hours. These coefficients provide a better fit to the data and would be used in the model. Note that this data was generated from with randomly introduced errors.
t.eC 0208010 −=
y = 9.6281e-0.0202x
R2 = 0.9376
0
2
4
6
8
10
12
0 50 100 150 200
Time (hrs)
Mea
sure
d co
ncen
trat
ion
(mg/
l)
Figure E4-13a: Plot of jar chlorine concentration versus time for Example 4.13 data.
WATER QUALITY FUNDAMENTALS 4-55
0
1
2
3
4
5
6
7
0 50 100 150 200
Time (hrs)
1/C
(1/(m
g/l))
Figure E4-13b: Plot of jar inverse chlorine concentration versus time for Example 4.13 data.
-2-1.5
-1-0.5
0
0.51
1.5
22.5
0 50 100 150 200
Time (hrs)
ln (M
easu
red
conc
entr
atio
n)
Figure E4-13c: Semi-log plot of jar chlorine concentration data for Example 4.13.
4.3.2.2 Empirical Bulk Chlorine Decay Coefficient Relationships
Chlorine modeling was the primary impetus for analyzing distribution system water quality. As such, a large amount of work has been completed to develop
4-56 CHAPTER FOUR
reaction relationships and decay coefficients. The observed trend for chlorine decay is that a large drop in concentration occurs just after chlorination (for about four hours) then decay continues but at a slower rate.
Powell et al (2000) compared six kinetic models for the decay of free chlorine in bulk waters using two in-situ pipes. They found that for network modeling purposes a first-order decay model for bulk and overall decay was appropriate, particularly after the initial steep post-chlorination drop. Vasconcelos et al (1997) reached a similar conclusion with respect to bulk decay based upon actual distribution system data. They also recommended that reactions at the pipe wall be characterized by first or zero-order mass transfer limited reactions. A wide range of bulk decay coefficients for first-order reaction kinetics has been reported in the literature (Table 4-6).
Bulk chlorine decay has been shown to be related to temperature, initial chlorine concentration, the number of times the water has been chlorinated and the water’s total organic carbon. The relationships are usually shown as linear equations or power functions. For example, using data from a treatment plant in the UK, Hallam et al (2003) fit:
( )[ ]27314106210
61095 +−−−= wIA TRE.Cl
.b eNTOCCx.k (4-57)
where the bulk decay coefficient is in (1/hr), C0 is the initial chlorine concentration, TOC is the total organic carbon (mg/l), NCl is the number of times that the water has been chlorinated, EA is the activation energy (J/Mol), RI is the ideal gas constant (8.31 J/Mol οC), and Tw is the water temperature in οC. Hua et al (1999) varied a number of these parameters for two waters and fit linear equations for kb as functions of TOC, 1/ C0, and T. Kiene et al (1998) also fit a relationship for first order chlorine decay:
(4-58) ))273(6050(61081 +−×= wT/b eTOC.k
where kb is in 1/hr, TOC is in mg/l and Tw is the temperature in oC. Chlorine decay has been modeled using Michaelis-Menton kinetics (Eq. 4-
46) and Koechling (1998) found that kb and CM could be related to the water’s total organic content and ultraviolet absorbance for this reaction model as follows:
DOC
UVAUVA.k .b
)100(320 3651= (4-59a)
(4-59b) DOC.UVA.C M 911984 −=
where DOC is the dissolved organic carbon concentration in mg/l.
WATER QUALITY FUNDAMENTALS 4-57
Table 4-6: Reported bulk chlorine decay coefficients.
Reference/water source Bulk decay coefficient, kb (1/day)
Vasconcelos et al (1997)
Bellingham water treatment plant 0.833
Fairfield treatment plant 1.16
Harrisburg Oberlin pump station 0.232
North Marin Russian R. aqueduct 1.32
N. Marin Stafford L. treatment plant 17.7
N. Marin 50/50 blend of aqueduct/treatment plant water
10.8
N. Penn Keystone tie-in 0.082
N. Penn Forest Park treatment plant 0.767
N. Penn 50/50 blend of tie-in/treatment plant
0.264
N. Penn well W17 0.355
N. Penn well W12 0.102
Rossman et al (1994)/Cherry Hill & Brushy Plains
0.55
Boulos et al (1996)/Azusa, CA 0.301 (full test), 0.82 (1st 4 hours after chlorination)
Kennedy et al (1993)/Akron, OH 0.26 – 0.39
El-Shorbagy (2000)/Abu-Dhabi, UAE 1.68
Powell et al (2000)/32 locations in Severn Trent region of the UK
0.24 – 7.7 with ~90% less than 3.6 and ~50% less than 1
Zhang et al (1992)/Macao 1.15 - 2.3
Hua et al (1999)/UK final treated and tap
0.48 – 5.4
Using the second order competing reaction formulation (Eq. 4-45) in
Section 4.3.1.5.2, Clark (1998) proposed a joint chlorine decay-THM production model with CA being the chlorine concentration. Later, Clark and Sivaganesean (1998) developed the following regression relationships to determine the equation parameters.
(4-60a) 1402906304400
320 .w
....AB TpHTOCCleK −−=
pHT.T.pH.TOC..Nln ww 010070140190462 +−−−−= (4-60b)
4-58 CHAPTER FOUR
where TOC and Tw are the total organic carbon (mg/l) and water temperature (Co), respectively, and Cl0 is the initial chlorine concentration (mg/l).
Boccelli noted that these regression equations are inconsistent with the parameter definitions (i.e., K should be linear with the 1/ Cl0 and M should not be independent of Cl0). As a result, Boccelli et al (2003) considered a slightly different form of Eq. 4-45 with parameters independent of Cl0:
)
)(τ
τ **AB uK/Cl*
AB
*AB
eCl/KKClCl
)1(0
00)(1 −−−
−= (4-61)
where = a C*ABK B,0/b and u* = kA CB,0 where CB,,0 is the initial reactant
concentration which, in this case, is the organic matter concentration. This model could be evaluated over a larger range of initial chlorine concentrations and applied to rechlorination conditions.
To better represent the observed chlorine decay, that is, initially rapid and later gradual decay, Clark and Sivaganesan (2002) extended their second order model for competing reacting constituents to account for the two types of organic materials that react with chlorine. Rapidly reacting organic materials were associated with part of the free chlorine (component 1) and the remainder of the free chlorine (component 2) was associated with slow reacting organic components. The resulting derived relationship for the change in overall chlorine residual was:
τττ2211 )1(
2
20)1(
1
10
1)1()1(
1)1(
kRkR eRRZCl
eRRZClCl −− −
−−+
−−
=)( (4-62)
where the subscripts 1 and 2 refer to the chlorine component and k1, k2, R1, and R2 are positive fit parameters. Z is the proportion of total chlorine that is associated with component 1, i.e., Z = Cl1/Cl0 where Cl0 is the total initial chlorine concentration. Note that if Z = 1, Eq. 4-62 reverts to a single chlorine species second order decay model (Eq. 4-45). Regression equations based on 36 data sets were developed for the five equation parameters.
0604512530
637254
6625861 )1()1()1()1( ...... BrpHClUVTOCek ++++= −−
6900516810
943254
6815631 )1()1()1( .
w..... TpHClUVTOCeR −− +++=
( ) ( ) ( ) ( ) 1403203106330
717254
4328342 1111 ....... alkBrpHClUVTOCek −−−−− ++++=
wT.... eClTOCeR 0308210
8114802 )1()1( −++=
7901615700
892254
944 )1()1()1( .w
.... TpHClUVeZ/Z −−−++=− (4-63a-e)
WATER QUALITY FUNDAMENTALS 4-59
where UV254, Br and alk are the initial ultraviolet spectral absorbance (cm-1), bromide ion concentration (mg/l) and alkalinity (mg/l), respectively.
4.3.2.3 Bulk Reaction Relationships for Disinfection By-Products
Within a pipe network, chlorine reacts with organic materials, such as humic and fulvic acids, and forms disinfection byproducts (DBP) including trihalomethanes (THM) and haloacetic acids (HAA). Either the amount of free chlorine or organic precursors may limit DBP formation.
Empirical regression type models have been developed to estimate total THM (TTHM) and its individual compound formation within a distribution system (e.g., Amy et al, 1987). These equations tend to be site specific and include factors such as total organic carbon, temperature, chlorine dosage, bromide concentration, reaction time, and chlorination pH (Vasconcelos et al, 1996). Sung et al (2000) fit the following equation to data from the Massachusetts Water Authority’s surface water systems:
[ ] ( )( ) ( ) ( ) 0870470254
5205306 11022 ...ko
.algaeUVeCHO.TTHM −−− −×= τ
(4-64)
where TTHM concentrations are in µg/L, is the hydroxide concentration that incorporates temperature and pH, UV
−HO254 is the ultraviolet absorbance at a
wavelength of 254 nm (1/cm), algae is the algae concentration in ASU/mL, and C0(1-e-k τ) is the amount of chlorine reacted from the point of chlorine addition with concentration C0 to the point of concern during its travel time, τ. To predict TTHM using models of this form or with a direct time in the distribution system term, a water quality model would be run for the chlorine concentrations or travel times and the results substituted in Eq. 4-64.
To model the spatial distribution of TTHM generation in a system, a first order saturation growth model may be appropriate with the saturation level being based on the limiting chlorine or precursor concentrations. Alternatively, using the second order competing reaction formulation (Sect. 4.3.1.5.2), Clark (1998) proposed a second order TTHM formation model that linked DBP precursors with chlorine demand and availability. Based on a balanced reaction (Eq. 4-43), TTHM production, P, is a linear function of the chlorine demand as such he proposed:
0)1(0
0 1)1( TTHM
eKKClClT)t(TTHM tuK
AB
ABTTHM AB
+⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎦
⎤⎢⎣
⎡
−−
−= −− (4-65)
where TTTHM is a calibrated parameter and TTHM0 is the initial TTHM concentration. Clark and Sivaganesean (1998) fit a regression equation to identify TTTHM from water characteristics and field conditions.
4-60 CHAPTER FOUR
(4-66) 2809601804800
491 .w
....TTHM TpHTOCCleT −=
where KAB and u (= N (1-KAB)) are estimated using Eqs. 4-60a and b. Boccelli et al (2003) computed TTHM production with a similar
relationship with their modified parameters. Both Clark’s and Boccelli’s groups demonstrated that the second order model outperformed a first order kinetic reaction model but at the expense of more model parameters. Boccelli’s work further examined rechlorination conditions but did not evaluate sufficient sites to provide guidance for the general model application. Clark et al (2001) further extended their disinfection by-product relationship to multiple species of chlorinated and brominated compounds. They also produced a set of regression equations for estimating model parameters.
A difficulty in multiple source systems is that the THM formation is not a simple additive response during mixing. El-Shorbagy (2000) developed a general stoichiometric-based approach for modeling TTHM and the four primary THM compounds: trichloromethane (chloroform), bromodichloro-methane, dibromochloromethane, and tribromomethane (bromoform). The model is based on first order kinetics for chlorine decay and DBP growth. It requires chlorine levels, several calibrated parameters and the solution of a small optimization problem at each modeling time step.
4.3.2.4 Pipe Wall Reactions for Chlorine Decay
To estimate constituent reactions with the pipe material or material on the pipe wall, two issues must be addressed. First, the actual decay coefficient must be identified. Some recent studies provide general guidance in this regard and they are described in the next section. The second issue is radial mixing in the pipe to determine the amount of contact that the waterborne constituent will have with the wall material. The level of mixing depends on the flow regime. Several models for radial mixing have been developed. The second subsection presents those models and their parameters.
4.3.2.4.1 Reaction Coefficients
The range of impacts of wall reactions is quite large for chlorine decay. Hua et al (1999) examined three test pipes and found the kwall was only 10% of kb resulting in a minor change in chlorine demand. Clark et al (1993), however, found that kwall could exceed kb. Clearly, conditions vary between networks and field calibration is necessary.
Based on several published studies, Rossman et al (2001) reported that a kwall value on the order of 3 day-1 (0.125 hr-1) was reasonable for cast and ductile iron pipes. Hallam et al (2002) completed field and laboratory studies and determined wall decay constants for a range of pipe materials. From tracer
WATER QUALITY FUNDAMENTALS 4-61
tests, wall decay constants ranged from 0 to 1.64 hr-1 with 70% of the values being less than 0.4 hr-1. Cast iron mains had the highest values and the most variability. Cement lined iron (CICL), polyvinyl chloride (PVC) and medium density polyethylene (MDPE) pipes were examined in the laboratory and found to have decay constants of 0.01 to 0.78 hr-1. Based on these results they classified, cast and iron (CI – wallk = 0.67 hr-1) and spun iron (SI – 0.33) pipes as reactive and CICL (0.13), PVC (0.09) and MDPE (0.05) as non-reactive. The value in parentheses for each type is the average in-situ wall coefficient. From laboratory data, the coefficients were the same except the CICL was 0.12 hr-1. Studies also indicated that kwall increased linearly with velocity in iron pipes but a clear relationship was not apparent for the non-reactive pipes. Finally, kwall was inversely related to the initial chlorine level. No other definitive relationships could be discerned.
4.3.2.4.2 Radial Flow Transport
Pipe wall reaction coefficients are affected by three factors; (1) the reactive ability of the bioflim layer, (2) the wall area available for reactions, and (3) the movement of water to the wall. Rossman et al (1994) developed a theoretical mass transfer approach for estimating kwall that accounted for these factors. The reactive nature of the wall material is measured by a wall reaction rate constant, kw [L/T]. The surface area per unit volume of pipe section accounts for the wall area available to interact with water. For a circular pipe, this term equals 2/R ( )LRLR2 ππ= 2 where R is the pipe radius and L is the pipe length.
Water mixing is more complex. Mass transfer between the bulk water and the pipe wall is represented by the Sherwood number (Sh) in a mass transfer coefficient, kf, as:
⎟⎠
⎞⎜⎝
⎛=D
DShk m
f (4-67)
where Dm is the molecular diffusivity of the transported constituent and D is the pipe diameter. The dimensionless Sherwood number differs with the flow regime. In laminar flow ( <2300), molecular (conductive) scale mixing dominates:
ℜ
( )( )[ ] 3
20401
06680653ScL/D.
ScLD..Shℜ+
ℜ+= (4-68)
where Sc is the dimensionless Schmidt number ( )mDν (Edwards et al 1976).
4-62 CHAPTER FOUR
More mixing is expected in turbulent flow compared to laminar flow as turbulent diffusion causes significant movement of water masses through the fluid. For this condition, Notter and Sleicher (1971) proposed the following empirical relationship for the Sherwood number:
3188001490 Sc.Sh .ℜ= (4-69)
For a first-order reaction, the terms that describe the three factors are combined to form:
( )( )fw
fwwall kkabsR
kkk
+=
2 (4-70)
where abs is the absolute value operation. kwall is then substituted in Eq. 4-54 to compute the overall decay coefficient for the reaction relationship r(C) = k C. Note that the Sherwood number and the resulting kf and kwall can be different for each network pipe accounting for variable flow conditions.
If the wall reaction is modeled as a zero-order reaction, nc equals 0 and r(C) is a constant. The reaction rate cannot be higher than the rate of mass transfer so r(C) equals the minimum of kw (2/R) or an apparent first-order reaction coefficient kf C (2/R).
All terms in Eq. 4-70 are available from the physical data or hydraulic analysis except the wall reaction rate coefficient (kw) and the constituent’s molecular diffusivity (Dm). The only means to determine the reaction rate constant is to calibrate the water quality model to measured field data. The molecular diffusivity is a physical property of the constituent. For example, Dm for hypochlorous acid (HOCl-), the dominant species of free chlorine, is 1.44 x 10-5 cm2/s (0.00112 ft2/day) while Dm for methyl ethyl ketone (a representative trihalomethane precursor) is 9.8 x 10-6 cm2/s.
The wall reaction coefficient can depend on temperature and can also be correlated to pipe age and material. As metal pipes age their roughness tends to increase due to encrustation and tuberculation of corrosion products on the pipe walls. This increase in roughness produces a lower Hazen-Williams C-factor or a higher equivalent sand roughness coefficient (e), resulting in greater frictional head loss in flow through the pipe.
Some evidence suggests that the same processes that increase a pipe's roughness with age also tend to increase the reactivity of its wall with some chemical species, particularly chlorine and other disinfectants (Vasconcelos et al 1996, 1997). Based on their work, a pipe's kw can be expressed as a function of the coefficient used to describe its roughness and a wall reaction correlation coefficient F as listed in Table 4-7.
WATER QUALITY FUNDAMENTALS 4-63
Table 4-7: Wall reaction coefficient relationships for different headloss equations.
Headloss Equation Wall Reaction, kw
Hazen-Williams F / CHW
Darcy-Weisbach -F / log(e / D) Chezy-Manning F nm
F has a different meaning depending on the head loss equation. The
advantage of using this approach is that it requires only a single parameter, F, to allow the wall reaction coefficient to vary throughout the network. Both kw and F are system specific. Vasconcelos et al (1996, 1997) reported values for kw ranging from 0.1 to 5.0 ft/day and values for F ranging from 10 to 650 for the four distribution systems that they studied.
Ozdemir and Ucak (1998) solved the radial flow relationship with an effective diffusion coefficient that varied with the Reynolds’ number and was available from the literature. For Reynolds numbers greater than 10000 (nominally fully turbulent) the results matched the Rossman and Biswas’ models. For lower ℜ ’s, a variable bulk decay coefficient and the modeled kw improved the match between experimental and modeled results compared with the other two models. For first order reactions a correction factor, FC, was then developed based on two-dimensional flow modeling as:
11000
95380100
−⎟⎠⎞
⎜⎝⎛ ℜ
−=⎟⎟⎠
⎞⎜⎜⎝
⎛log.
LkRF
logw
C ν (4-71)
FC is inversely related to ℜ and its effect is negligible with ℜ greater than 30000. FC can be used to adjust the total decay constant by:
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=C
tdifft F
CC1
(4-72)
where Ct is the chlorine concentration at time t using a first order decay relationship and decay constant computed using the estimated bulk and wall decay coefficients and Ct
diff is the chlorine concentration after correction for diffusion.
Example 4.14
Problem: Determine the total reaction coefficient for water at 50o F with a bulk decay coefficient of -0.15/day in a 10-inch diameter, 1000-ft long pipe that is
4-64 CHAPTER FOUR
carrying 0.5 cfs. The wall coefficient, kw, is -0.5 ft/day and the molecular diffusivity of the chlorine species of concern is 0.00112 ft2/day. Assume that a first-order decay model holds.
Solution: The total k value equals:
wallb kkk +=
where kb is given at -0.15/day. kwall must be computed for the pipe’s flow condition using Eq. 4-70. R and kw are given as 5 inches (0.42 ft) and -0.5 ft/day, respectively. The mass transfer coefficient, kf, is based upon the Sherwood number that varies with flow regime. The Reynolds’ number for this pipe and velocity (V = Q/A = (0.5 cfs)/((π(10/12)2/4) ft2) = 0.92 ft/s) is:
⇒×=×
==ℜ−
45 1002.7
1009.1)12/10()92.0(
νDV flow is turbulent
The Sherwood number for turbulent conditions is estimated by Eq. 4-69 with the Schmidt number equals to 839 (Sc = ν/Dm = (0.94 ft2/d)/(0.00112 ft2/d)). Note that the kinematic viscosity is converted to ft2/d to be consistent with the diffusivity so that Sc is dimensionless. The dimensionless Sherwood number is then:
2581)839()1002.7(0149.00149.0 3/188.043188.0 =×=ℜ= ScSh
and the mass transfer coefficient is computed by:
[ ]day/./
.D
DShk mf 1473
)1210(0011202581 =⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟
⎠⎞
⎜⎝⎛=
Finally, kwall is:
( ) ( ) day/...abs/
..kkR
kkk
fw
fwwall 12
473)50()125()473()50(22
−=+−
−=
+=
Interestingly, the wall reaction constant is quite large compared to the bulk reaction term of –0.15 due to the high level of turbulence. Finally, the overall decay constant is:
day/...kkk wallb 252)102()150( −=−+−=+=
WATER QUALITY FUNDAMENTALS 4-65
The sign on the term is negative since it is a decay reaction. If the chlorine concentration entering the pipe is 10 mg/l, the outlet concentration under steady conditions is:
l/mg.CeeCC outlet/..k
inoutlet 72910 )24300(252 =⇒== −τ
where τ is the travel time in the pipe (τ = L/V = 1000 ft/(0.92 ft/s * 3600 s/hr * 24 hr/day) = 0.0126 day = 0.3 hr).
Table E4-14 lists k, the travel time and steady state outlet concentration for different flow rates for an inlet concentration of 10 mg/L for the 1000 ft long pipe. In this example, the outlet concentration decreases with decreasing flow rate. Although the level of turbulence decreases causing less reaction at the wall, the travel time in the pipe increases faster than the reaction coefficient reduction.
Table E4-14: Decay coefficients, travel times and outlet concentrations for various flow rates in the 1000 ft long pipe described in Example 4.10.
Q (cfs) k (1/T) Travel time (days) Coutlet (mg/L)
3 -2.48 0.0021 9.948 2 -2.45 0.0032 9.923 1 -2.38 0.0063 9.851
0.5 -2.25 0.0126 9.720 0.25 -2.05 0.025 9.496 0.125 -1.76 0.051 9.148 0.05 -1.30 0.126 8.490 0.025 -0.95 0.253 7.874 0.02 -0.85 0.316 7.657 0.015 -0.27 0.420 8.926 0.010 -0.25 0.631 8.517
Flow rates less than 0.016 cfs result in laminar flow. In these instances, the
Sherwood number was computed using Eq. 4-68 while Eq. 4-69 was used in flows greater than 0.016 cfs. The transition between laminar and turbulent regimes is not smooth. Although the travel time increases, Coutlet actually increases because k drops dramatically when switching equations from Eq. 4-69 to Eq. 4-68. The magnitude of the jump to laminar flow conditions is related to the flow conditions and the pipe length. Note that the turbulent flow equation does not consider pipe length. So as L decreases, the time and the contact area both decrease.
4.3.3 Water Age and Source Tracing
The initial driving concern that led to water quality modeling was chlorine residuals. If decay coefficients are not known, a surrogate measure for chlorine
4-66 CHAPTER FOUR
decay is water age. As noted earlier, a zero-order growth model can be used to represent water age. A water age analysis may suggest useful more detailed studies.
Water age is a function of the pipes’ volumes and flow rates. Pipes are plug flow reactors and, in effect, act as tanks in transmitting flow from one node to the next. The pipe flow rate and velocity is directly related to its travel time and water age. A pipe that was designed with a large diameter to reduce energy losses or was oversized in expectation of future growth can adversely affect quality as water may remain in the pipe for long periods of time. A tank can have a similar effect. Since it is purely hydraulically dependent, a hydraulic analysis provides all of the information necessary to compute the average age of water withdrawn at all nodes.
A second useful analysis can identify the source of water for a particular withdrawal point. If multiple supplies of varying quality provide water to a network, the final mixed water may be of concern with respect to taste, odor and, possibly, water quality. The utility of source tracking is also helpful in identifying vulnerabilities and contamination sources. For example, if a field meter detected a change in water quality, one could identify possible contributing locations and other nodes that those locations are impacting.
To identify the contributions of the Ns sources under dynamic conditions, (Ns – 1) water quality forward simulations must be run. The individual runs are made with a fictitious conservative substance supplied from one source node (ns) with concentration of 100. This water is distributed through the network and a node’s withdrawal concentration is its percentage contribution from source node ns or p(ns). Under unsteady demands, this percentage will change over time. The conservative tracer analysis is completed for all but one of the other source nodes. The percentage contribution for the last source node (source node Ns) is the unaccounted withdrawn water and computed by:
(4-73) ∑−
=
=1
1
)(100)(s
s
N
nss np - Np
Backtracking algorithms can be more efficient in determining source locations, particularly for individual nodes, under dynamic conditions. These approaches (Chapter 6) begin their analysis at the withdrawal points and work back through the system using the hydraulic conditions to identify sources and when water was supplied to contribute to the withdrawn flows. Problems
Problem 1. Determine the concentration in pipe 1 in Figure P4-1 if the outflow concentration in pipe 3 is 3.5 mg/l.
WATER QUALITY FUNDAMENTALS 4-67
Figure P4-1: Junction node with two inputs and three withdrawals.
Problem 2. A 1.4 kg tablet that is 17% Cl2 dissolves in 1.5 hours in a tablet feeder chlorinator. Four percent of the 62 lps supply pipe flow is passed through the unit. Determine the increase in concentration in the water being returned to the pipe and in the water in the pipe downstream of the chlorinator return.
Problem 3. A mass loading rate of 50,000 g/hr of a constituent is added to the node shown in Figure P4-3. Determine the effective injection concentration and the resulting concentration in flow leaving the node.
Figure P4-3: Junction node with two inputs and three withdrawals. Problem 4. Determine the constituent concentration for the node in Figure P4-3 if a flow paced booster injects 0.25 g/m3 to all flow leaving the node. Also compute the mass injection rate.
4-68 CHAPTER FOUR
Problem 5. A set point booster is used to increase the constituent concentration of the flow leaving the node in Figure P4-3 to 1.94 g/m3 (= 1.94 mg/l). What is the mass injection rate? Problem 6. A 300 m long, 150 mm diameter pipe carries 0.05 m3/s of water. Determine the pipe’s velocity and the travel time. Problem 7. Determine the velocity and travel time for a pipe flowing with 0.07 m3/s of water. The pipe has a diameter of 250 mm and is 250 m long. Problem 8. A mass of 0.015 lbs of fluoride is injected at a rate of 3.0 lb/day into an 8 inch diameter, 2500 ft long pipe. The pipe’s flow rate is 1.5 ft3/s. Determine the concentration, the pulse length, and the time that the pulse reaches and leaves the pipe outlet (starting at an arbitrary time zero).
Problem 9. For the pipe and injection defined in problem 8, track a segment of a conservative constituent through the pipe. Specifically, show the location of the segment at 0, 7.2, 8, 9.69, 12, and 16.89 minutes.
Problem 10. Water flows through a 6,000-meter long pipeline with a velocity of 0.45 m/s (Figure P4-10a). Under steady state conditions, compute the concentration at 600 m increments along the length of the pipe for the following conditions. (A) A conservative constituent (k = 0), (B) Water age (zero-order growth reaction), (C, D and E) A constituent following first-order decay reaction, a first-order saturation growth reaction and a second-order saturation growth reaction. Each has a reaction coefficient of 0.5/hr with appropriate sign, (F) A constituent that follows a Michaelis-Menton reaction (CM=15). The limiting concentration, C* , when applicable is 12 mg/l. Assume that the concentration at the source is C0 = 12 for the decay reactions and 0 for the growth reactions.
Figure P4-10a: Pipe segment for constituent transport Problem 4-10.
Problem 11. For the pipe in Problem 10, track a constituent pulse through the pipe for the four reaction conditions listed below. Assume that constituents are injected at the upstream end of the pipe for 1000 seconds. Determine the pulse
WATER QUALITY FUNDAMENTALS 4-69
location and constituent concentrations at the front and back of the segment at 1, 2 and 3.7 hrs. Compute the water age and the concentrations for conservative and first and second order decaying constituents. Use the same parameters as in Problem 10. Problem 12. Assume that a conservative constituent is injected into two pipes beginning at time 0 sec. as shown in Figure P4-12.
a) Calculate the outflow q3 b) Calculate the outgoing fluoride concentration at t = 0 minutes. c) Calculate the outgoing fluoride concentration at t = 15 minutes. d) Calculate the outgoing fluoride concentration at t = 20 minutes. e) Calculate the outgoing fluoride concentration at t = 30 minutes.
Repeat the analysis with a constituent that decays following a first order relationship with decay constant of k = -3.0 /day.
Figure P4-12: Two pipe, three node configuration. Flow enters through nodes 1 and 2 toward node 3. Problem 13. The data in Table P4-13 is chlorine data taken from a jar test. Determine the reaction function and its rate constants using the integral method. Problem 14. Using the differential method determine the reaction order and its decay constant for the constituent for the data in Table P4-14a.
4-70 CHAPTER FOUR
Table P4-13: Chlorine concentrations over time from jar test. Time (hrs) Concentration (mg/l)
0 11.50 12 10.37 24 7.27 36 6.39 48 6.18 60 4.23 72 4.20 84 2.95 96 3.22 108 2.25 120 2.73 132 1.45 144 0.80 156 0.66 168 0.78 180 0.81
Table P4-14a: Measured concentration over time from jar test. Time (days) Concentration (mg/l)
0.00 6.01 0.25 4.89 0.50 4.14 0.75 3.60 1.00 3.15 1.25 2.82 1.50 2.58 1.75 2.35 2.00 2.16 2.25 1.97 2.50 1.87 2.75 1.72 3.00 1.60 3.25 1.55 3.50 1.47 3.75 1.38 4.00 1.30 4.25 1.24 4.50 1.17 4.75 1.16 5.00 1.07 5.25 1.04 5.50 1.02
WATER QUALITY FUNDAMENTALS 4-71
Problem 15. A 14-inch diameter, 2000-ft long pipe has a 0.1 cfs flow rate. Determine the pipe’s total reaction coefficient for water if kb = -0.30/day. The wall coefficient, kw, is -0.25/day and the molecular diffusivity of the chlorine species of concern is 0.00112 ft2/day. Assume that a first-order decay model holds. Use ν = 1.09 x 10-5 ft2/s. Solutions
1. All outflows have the same concentration as pipe 3 so the constituent mass balance for the node is:
C1 Q1 + C2 Q2 – Cout(Q3 + Q4 + qout) = 0
Substituting the known values gives:
C1 (4) + (1.5) (3 ) – 3.5(3 + 2.5 + 1.5) = 0 ⇒ C1 = 5 mg/l
2. The 1.4 kg tablet provides 0.238 kg of Cl2 (17%*1.4 kg) to the flow. In the main pipeline, the mass rate of 0.159 kg/hr (=0.238 kg / 1.5 hr) is dissolved in the flow of 62 lps. Thus,
lmgC
CslhrshrhrkgCQm
pipe
pipec
/71.0
*)/(62)/(3600/5.1/)/(238.0
=∆⇒
∆==∆=&
The concentration in the feeder unit has the same mass added but the flow is only 4% of the 62 l/s or 2.48 l/s. Thus the concentration in flow leaving the feeder is:
lmgC
CslhrshrhrkgCQm
feeder
feederc
/8.17
*)/(48.2)/(3600/5.1/)/(238.0
=∆⇒
∆==∆=&
3. Without the additional loading the outflow concentration is computed assuming complete and instantaneous mixing occurs at the node. Thus, the concentrations in pipes 3 and 4 and the nodal withdrawal are the same and C can be computed from Eq. 4-3:
out
mg/lmgC q Q Q
Q C QCC outoutout 79.1/79.1
1.5 2.5 3 (1.5) 3 (2.0) 4 3
43
2211 ==⇒++
+=
+++
=
4-72 CHAPTER FOUR
Based on the rate of mass injected (50 kg/hr), the effective injection concentration, , is the incremental concentration increase or from Eq. 4-4: inj
inC
lmgormgCsmmgC
QQChrshrgQCm
injin
injin
injin
Jll
injinin
in
//98.1/)0.30.4(/
)()/(3600//50000
333
21
=⇒+=
+=== ∑∈
&
Thus the outflow concentrations will be increased by 1.98 mg/l. The concentration of flow leaving the node can be confirmed by Eq. 4-5:
lmgmg
Q q
mqCQC
C
in
in
Jll
out
inin
inJl
ll
out
/77.3/77.37
9.135.12
)5.20.3(5.13600
500000)5.130.24(
3 ==+
=
++
++⋅+⋅=
+
++
=∑
∑
∈
∈
&
Thus, the injected mass increased the outflow concentration by 1.98 mg/l from 1.79 to 3.77 g/m3(=mg/l). 4. A flow paced booster increases the concentration by a defined increment,
(0.25 mg/l in this example). In the previous problem the concentration without any constituent injection at the node was determined to be 1.79 g/m
injoutC
3. The outlet concentration, Cout, is then (Eq. 4-7):
lmgmgCQ q
qCQC
C injout
Jll
out
inin
Jlll
out
in
in /04.2/04.225.079.1 3 ==+=++
+
=∑
∑
∈
∈
The mass injection rate is:
( )
hrkgsg
smmgQ qCminJl
loutinj
outin
/3.6/75.1
/5.235.1/25.0 33
==
++=⎟⎟⎠
⎞⎜⎜⎝
⎛+= ∑
∈
&
Recall that a flow paced booster injects a constant injection rate and mass. So if the inflow concentrations at this node were changed, the outflow concentration would be changed but the injection increment would remain constant.
WATER QUALITY FUNDAMENTALS 4-73
5. The setpoint booster sets the concentration for all flow leaving the node to 1.94 mg/l. The concentration without the addition of injected constituent is 1.79 g/m3 (1.79 mg/l) so the additional constituent is added at a rate of 0.15 mg/l or by Eq. 4-7:
lmgmgC
CCQ q
qCQC
C
injout
injout
injout
Jll
out
inin
Jlll
out
in
in
/15.0/15.0
79.194.1
3 ==⇒
+=++
+
==∑
∑
∈
∈
The mass loading rate is then:
[ ] ( )( ) [ ] sgsmmgQqCminJl
loutinj
outc /05.1/5.235.1/15.0 33 =++=⎟⎟⎠
⎞⎜⎜⎝
⎛+= ∑
∈
&
With a setpoint booster if the flow rate or the inflow concentrations change the injected concentration and mass rate would change to maintain the desired outflow concentration. 6. By continuity, the flow velocity is:
( )( ) m/s..π.
Dπ.
AQV 832
4150050
4050
22 ====
which results in a travel time of:
minutess.V
L 77.1106832
300====τ
7. The flow velocity and travel time are:
( )( ) m/s..π.
Dπ.
AQV 431
4250070
4070
22 ====
and
minutessVL 91.2175
43.1250
====τ
8. At an injection rate of 3 lbs/day, the 0.015 lb is supplied to the pipe over an injection period of:
4-74 CHAPTER FOUR
tinj = 0.015 lb / 3.0 lb/day = 0.005 day = 7.2 min = 432 sec
The fluoride concentration in the water is:
ppm.waterlb/Fluoridelb.ft/lb.
day/s*s/ft./day/lb.Q/mC cinj
370107310312
)8640051(03735
3
=×=×=
==−−
&
The flow velocity is:
s/ft.
π
.Dπ
.AQV 304
4128
514
5122 =
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
===
The travel time through the reach then is:
minutess.V
L 69.9581304
2500====τ
where L is the length of the pipe. Since the fluoride is injected over a 432 s. period the length of the pulse is:
fts/ft.sVtL injseg 185834432 =⋅==
The front edge of the segment will reach the end of the pipe 9.69 minutes after the injection begins. The 1858 ft segment takes 432 s (7.2 min) to fully develop and the back edge of the segment leaves the outlet at time 7.2 minutes. Since the pipe travel time is 9.69 minutes, the back end of the segment will reach the outlet at time 9.69 minutes + 7.2 minutes = 16.89 minutes.
9. The solution is presented for each time step below and on Figure P4-9.
a) Time 0 s. – The fluoride has just begun to be injected so the water has a zero concentration for the entire length of the pipe (Not shown).
b) Time 7.2 min. – After injecting fluoride for 7.2 minutes, the complete segment has developed in the pipe. As computed in the previous problem, the segment front has traveled 1858 ft (Figure P4-9a).
c) Time 8 min. – In the next 0.8 minutes, the segment has traveled an additional 206.4 ft. = 4.30 ft/s * 0.8 min. * 60 s/min. Since the segment
WATER QUALITY FUNDAMENTALS 4-75
does not distort, the front of the segment has traveled to location 1858 + 206.4 = 2064.4 ft from the injector. The back of the segment moved a distance 206.4 ft from the injector. The water in the pipe from 0 to 206.4 ft does not contain any fluoride since the injector was turned off (Figure P4-9b).
d) Time 9.69 min. – The travel time for the pipe is 9.69 minutes so the water passing the injector at time 0 has just reached the pipe outlet. This water contains fluoride and is at the front of the segment. The back of the segment remains 1858 ft to the left of the front. The back of the segment is now 642 ft (=2500 ft – 1858 ft) from the injector. (Figure P4-9c).
e) Time 12 min. – A portion of the segment has now left the pipe. The length of segment that has passed the outlet is 596 ft (= (12-9.69) min * 60 s/min * 4.3 ft/s). The length of the segment with fluoride is now 1269 ft (=1858 ft – 596 ft). The back of the segment is now 1231 ft from the injector (= 2500 ft – 1269 ft) (Figure P4-9d).
Figure P4-9 (a-d): Time history of segment affected by fluoride injection at the left pipe inlet beginning at time 0 s.
4-76 CHAPTER FOUR
At time 16.89 minutes the back of the segment has just passed the pipe outlet. Thus, the pipe contains no fluoride (not shown).
10. Under steady state conditions, the concentrations are time invariant for any given location but vary along the pipe. The time needed for water to move 600 m is 0.37 hours (= L/V = 600 m/(0.45 m/s * 3600 s/hr)). The concentration for location 600 m is computed for each reaction relationship and results for the entire pipe are summarized in Table P4-10 and plotted in Figure P4-10b. a) Conservative constituent: Since k=0, the concentration does not change as it moves through the pipe or Ct = C0 by Eq. 4-26 for all times. Thus, Ct = 0.37 hr = C0 = 12. b) Zero-order growth model: Water age is the time spent in the pipe after entering at the left inlet and is computed using Eq. 4-29 with k = 1 and assuming that the age of water entering the pipe is zero. At 600 meters from the pipe entrance, τ = L/V = 0.37 hrs so:
hrCC mLmL 37.037.006000600 =+=+= == τ
c) First-order decay: The first-order decay follows the exponential relationship (Eq. 4-34). With k = -0.5/hr, the general relationship is:
τττ
5.00 12 −
= == eeCC kt
At 600 feet with τ = L/V = 0.37 hrs, the concentration is 9.97.
Table P4-10: Results for Problem 4-10 for various decay/growth relationships.
Distance (m)
Time (hrs)
Conservative constituent
Zero order
growth
First order decay
First order
growth
Second order decay
Michaelis-Menton decay
0 0 12 0.00 12 0.00 12.00 12 600 0.37 12 0.37 9.97 2.03 3.73 11.35
1200 0.74 12 0.74 8.29 3.71 2.21 10.83 1800 1.11 12 1.11 6.89 5.11 1.57 10.38 2400 1.48 12 1.48 5.73 6.27 1.21 9.99 3000 1.85 12 1.85 4.76 7.24 0.99 9.64 3600 2.22 12 2.22 3.95 8.05 0.84 9.31 4200 2.59 12 2.59 3.29 8.71 0.73 9.02 4800 2.96 12 2.96 2.73 9.27 0.64 8.75 5400 3.33 12 3.33 2.27 9.73 0.57 8.5 6000 3.70 12 3.70 1.89 10.11 0.52 8.27
WATER QUALITY FUNDAMENTALS 4-77
0
2
4
6
8
10
12
14
0 1000 2000 3000 4000 5000 6000
Distance along pipe (m)
Con
cent
ratio
n (m
g/l o
r tim
e)
Conservative Water age First order decay
First order growth limited Second order decay Michaelis-Menton
Figure P4-10b: Plot of constituent concentrations versus pipe distance for different reaction kinetics for Problem 4-10.
d) First-order saturation growth: First-order saturation growth is described by Eq. 4-36. The initial concentration, C0 , is 0. With k = 0.5/hr and C* = 12, the concentration at a location 600 meter from the source or 0.37 hours travel time in the pipeline is:
( ) ( ) lmgeeCCC /03.297.912012120** )37.0(5.05.0 =−=−−=−−= −− ττ
e) Second-order decay: Applying Eq. 4-42 with τ = 0.37 hr, k = -0.5/hr, C* = 0 and C0 = 12 , second-order reaction relationship is:
( ) ( )lmg
CCkCC
CCL /73.3)37.0()012(5.01
0120*)(1
**
0
0600 =
−+−
+=−+
−+== τ
f) Michaelis-Menton decay: The Michaelis-Menton reaction is described by Eq. 4-47. Substituting known values (CM
=15, C0=12, k=-0.5) at τ = L/V = 0.37 hrs gives:
0925)370(50)121215(15
)(
370370
00370370
...lnCCln
kCClnCCClnC
hr.hr.
Mhr.hr.
M
=−+−=−⇒
=−−−
==
==
ττ
ττ τ
4-78 CHAPTER FOUR
Solving by trial and error or using a solver routine in a spreadsheet or calculator gives Cτ=0.37 hrs = 11.35. Results for all locations are listed in Table P4-10 and plotted in Figure P4-10b. 11. The injected mass begins entering the pipe at time zero. After one hour the material that was injected at time zero has moved 1620 m (= 0.45 m/s * 3600s). The back end of the segment was injected at time 1000s. The segment length is 450 m (=0.45 m/s*1000 s). After 1 hour, the back of the segment is at location 1170 m (= 0.45 m/s* 2600 s).
At time 2 hrs, the segment has traveled another 1620 m and the front is now 3240 m from the inlet. The segment length does not change and remains 450 m. The back of the segment is at location 2790 m (=3240 – 450 or 1170 + 1620).
Finally, after 3.7 hrs the front of the segment has reached the pipe outlet. The back of the segment is 450 m from the outlet or at 5550 m (=6000 m – 450 m). The segment location for all three times is shown in Figure P4-11.
Conservative constituent - Times 1, 2, and 3.7 hours – Since the constituent is conservative, the concentration for the entire segment is 12 mg/l for all times. Outside of the segment the concentrations are equal to zero since no constituent is injected and advection is the only transport mechanism.
Water age – Time 1 hour - After one hour the front of the parcel has moved from the entrance to a distance 1620 meter into the pipe. This water had a water age of zero at time 0 s. Thus, its water age at 1 hour is 3600s (= 0s + 3600s). Since the back of the segment entered the pipe at time 1000s, its water age was zero at that time. After 3600 s, it had been in the pipe for 2600s (= 3600s – 1000s).
Time 2 and 3.7 hrs – After the segment is fully within the pipe, it ages with the clock time. So the water ages at the segment front at 2 and 3.7 hours are 7200s and 13320s, respectively. The water ages at the back of the segment are 6200s and 12320s, respectively, since they did not enter the pipe until time 1000s. Water age varies linearly in the segment with the oldest water at the front and youngest at the back of the segment.
First order decaying constituent – Time 1 hour – First order decay is a function of time and the decay constant. Thus, at time 1 hour, the water at the front is 1 hour old and its concentration is computed by:
lmgeeCC kt /3.7)61.0(1212 )1(5.00 ==== −−
The water at the back of the segment is 2600 s old so its concentration is:
lmgeeCC kt /4.8)70.0(1212 )3600/2600(5.00 ==== −−
WATER QUALITY FUNDAMENTALS 4-79
Times 2 and 3.7 hours – At these times, the water age is substituted in the first order decay relationship. At the segment front, the values of the constituent concentration are 4.4 and 1.9 mg/l for hours 2 and 3.7, respectively. At the back of the segment the concentrations are slightly higher due to the lag time in entering the pipe and are 5.1 and 2.2 mg/l for times 2 and 3.7 hrs, respectively. The concentration decreases exponentially from the front to the back of the segment.
Figure P4-11: Location of fluid segment containing constituent for Problem 4-11 at three times.
Second order decaying constituent – Time 1 hour – Second order decay is
represented by Eq. 4-42 with τ equal to the water ages. At the segment front at time 1 hour the travel time is one hour, the initial concentration is 12 units and the limiting concentration is 0. Substituting those values gives:
C1 hr (front) ( )
τ*)(1*
*0
0
CCkCC
C−+
−+= ( )
1)012(5010120−+
−+=
.= 1.71 mg/l
The back of the segment have spent τ = 2600/3600 = 0.72 hrs in the pipe so the resulting concentration is:
C1 hr (back) ( )
τ*)(1*
*0
0
CCkCC
C−+
−+=
( )720)012(501
0120.. −+
−+= = 2.26 mg/l
4-80 CHAPTER FOUR
Times 2 and 3.7 hours – The concentrations are determined using the above equations with these water ages. The concentrations at times 2 and 3.7 hrs for the front are 0.92 and 0.52 mg/l, respectively. The segment back concentrations are 1.16 and 0.56 mg/l for times 2 and 3.7 hours, respectively. These values are identical to those in Table P4-10 for the same locations. For example for second order decay at the end of the pipe Table P4-10 and the value here are both 0.52 mg/l. 12. a) The flow leaving node 3 equals the inflow from pipes 1 and 2 or:
Q1 + Q2 = q3 ⇒ q3 = 20.0 + 9.0 = 29.0 ft3/s
b) The initial constituent concentration, C3, t=0, is equal to zero since it has just been injected at the upstream nodes. c) To determine the concentration at t = 15 minutes, we must determine the travel times for each pipe. For pipe 1, with a flow rate of 20 cfs, the velocity is:
( ) ft/s.πDπA
QV 376412
2420
420
2211
11 ====
and the travel time is:
min.s.V
Lt 6191178376
7500
1
11 ====
Similarly for pipe 2, the velocity and travel time are 5.09 ft/s and 21.3 minutes, respectively. Since t = 15 minutes is less than t1 and t2, the constituent has not reached the node and the concentration at the outlet is still equals to zero. d) At 20 minutes, constituent is supplied from pipe 1 but the contribution from pipe 2 has not yet reached the node.
The outflow concentration at 20 minutes is then:
( ) ( ) l/mg..q
QCQCCtt
t 31029
9020450
3
220
2120
1203 =
⋅+⋅=
+=
===
e) After 30 minutes, both pipes are contributing constituent at the pipe outlet since it occurs after both pipes’ travel times. So, the outflow concentration is:
( ) ( ) l/mg...q
QCQCCtt
t 39029
925020450
3
220
2120
1203 =
⋅+⋅=
+=
===
WATER QUALITY FUNDAMENTALS 4-81
f) A decaying constituent will have the same travel times as the conservative one. So the outflow concentrations for the decaying and conservative constituents are zero until 19.6 minutes. Between times 19.6 and 21.3 minutes, only pipe 1 contributes constituent.
The concentration at pipe’s 1 outlet is:
l/mg.e.eCC */.ktt,in
.t,out 430450 ))6024(619(30
1619
1 === −−==
The resulting concentration at node 3 is computed from Eq. 4-3:
( ) ( ) l/mg..q
QCQCCtt
t 30029
9020430
3
220
2120
1203 =
⋅+⋅=
+=
===
At time 30 minutes, water in pipe 2 that contains constituent reaches node 3. The concentration at the end of pipe 2 at that time is:
l/mg.e.eCC */.ktt,in
.t,out 2390250 ))6024(321(30
2321
2 === −−==
and the resulting node 3 concentration is:
( ) ( ) l/mg...q
QCQCCtt
t 37029
9239020430
3
220
2120
1203 =
⋅+⋅=
+=
===
This problem combines reaction relationships, travel times, and nodal mixing. Although only conservative and first order decay reaction relationships were used the other relationships presented in this chapter can also be applied (Problem 4-10). 13. Plotting the data suggests that the constituent exponentially decays to zero (Figure P4-13a) or:
tkClnClneCC bkt +=⇒= 00
For constituents that follow this reaction type, the concentrations will plot
as a straight line on a semi-log plot (Figure P4-13b). The line’s slope is kb and can be determined by a linear regression analysis in most spreadsheets. Here, the resulting kb is -0.163 hr-1 and the intercept (C0) is 12.25 or:
(rteC 0163.025.12 −= 2 = 0.9518)
4-82 CHAPTER FOUR
where t is in hours. Note that this data was generated from with randomly introduced errors.
teC 015.012 −=
y = 12.234e-0.0163x
R2 = 0.9517
0
2
4
6
8
10
12
14
0 50 100 150 200
Time (hrs)
Mea
sure
d co
ncen
trat
ion
(mg/
l)
Figure P4-13a: Plot of concentration versus time.
0.1
1
10
100
0 50 100 150 200
Time (hrs)
Mea
sure
d co
ncen
trat
ion
(mg/
l)
Figure P4-13b: Semi-log plot of jar concentration versus time. 14. The concentration data is plotted versus time in Figure P4-14a. Based on this result, it is assumed that the concentration goes to zero over time (i.e., C* =
WATER QUALITY FUNDAMENTALS 4-83
0). The differential method requires estimating the derivatives of the concentration decay curve to determine the reaction order and decay coefficient. They are computed numerically by -dC/dt = (Ci+1-Ci-1)/(ti+1-ti-1) and are listed in Table P4-14b. No smoothing is performed on the numerical derivatives. Figure P4-14b is a plot of log(-dC/dt) vs log C. Since no smoothing was applied, the slope points scatter around the curve. Fitting a line to the data gives:
log (-dC/dt) = log k + n log C = -0.845 + 2.059 log C
Thus, the relationship essentially follows a second order decay relationship (n = 2.059) and the decay constant is 0.143/day (= 10-0.845). The reaction equation for this relationship is then:
21430)0(01430)()( C.CC.C*CCkCr −=−−=−=
The data for this problem were developed assuming second order decay with k equals to 0.15/day. Minor deviations were introduced representing measurement errors. The impact of those deviations is small and the decay curve (Figure P4-14a) is relatively smooth. However, they became important at low concentrations and significantly affected the slope estimates resulting in the scattered plotting points.
0.01.02.03.04.05.06.07.0
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
Time (days)
Con
cent
ratio
n
Figure P4-14a: Plot of concentration versus time for constituent in Problem 4-14.
Recall that C* was assumed to be zero after examining Figure P4-14a. If it
is not equal to zero, C* must be subtracted from all concentrations prior to
4-84 CHAPTER FOUR
applying the differential method. Thus, the x-axis in Figure P4-14b becomes log (C – C*) rather than log C. This approach can be verified by adding one unit to all concentrations in Table P4-14a. This linear transformation does not affect n and k but if Figure P4-14b is produced without subtracting C* (= 1) the result is a nonlinear curve and the best fit values for n and k are incorrect.
A rigorous method for simultaneously determining the value of C* is not presented here. If they are to be estimated, it would be necessary to perform a regression analysis to simultaneously determine the best fit for the three parameters, n, k and C*.
Table P4-14b: Calculations for slope of concentration versus time relationship. Column 3 is plotted versus column 5 in Figure P4-14b.
Time (days) Concentration, C log C -dC/dt log (-dC/dt)
0.00 6.01 0.25 4.89 0.69 3.73 0.57 0.50 4.14 0.62 2.58 0.41 0.75 3.60 0.56 1.99 0.30 1.00 3.15 0.50 1.56 0.19 1.25 2.82 0.45 1.14 0.06 1.50 2.58 0.41 0.94 -0.03 1.75 2.35 0.37 0.84 -0.08 2.00 2.16 0.33 0.76 -0.12 2.25 1.97 0.30 0.58 -0.24 2.50 1.87 0.27 0.51 -0.29 2.75 1.72 0.24 0.54 -0.27 3.00 1.60 0.20 0.34 -0.47 3.25 1.55 0.19 0.26 -0.58 3.50 1.47 0.17 0.35 -0.46 3.75 1.38 0.14 0.33 -0.48 4.00 1.30 0.11 0.27 -0.58 4.25 1.24 0.09 0.27 -0.57 4.50 1.17 0.07 0.17 -0.76 4.75 1.16 0.06 0.19 -0.71 5.00 1.07 0.03 0.23 -0.64 5.25 1.04 0.02 0.09 -1.04 5.50 1.02 0.01 0.12 -0.92
WATER QUALITY FUNDAMENTALS 4-85
y = 2.0589x - 0.8452R2 = 0.9636
-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.80
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70
log C
log
(-dC
/dt)
Figure P4-14b: Plot of log C versus log (-dC/dt) for the differential method for determining the parameters k and n for the reaction relationship given the data in Problem 4-14.
15. The total decay coefficient is equal to the bulk and wall decay coefficients or:
wallb kkk +=
where kb is given as -0.30/day. kwall must be computed for the pipe’s flow condition using Eq. 4-70:
))((2
fw
fwwall kkabsR
kkk
+=
R and kw are given at 7 inches (0.58 ft) and -0.25/day, respectively. The mass transfer coefficient, kf, is (Eq. 4-67):
⎟⎠
⎞⎜⎝
⎛=D
DShk m
f
To determine the Sherwood number, we must know the flow regime that is described by the Reynolds’ number or:
4-86 CHAPTER FOUR
⇒=×
==ℜ − 963010091
)1214()090(5.
/.DVν
flow is mixed to turbulent
where V = Q/A = (0.1 cfs)/((π(14/12)2/4) ft2) = 0.09 ft/s. The Sherwood number for turbulent conditions is estimated by Eq. 4-69 with the Schmidt number equal to 839 (ν/Dm = (0.94 ft2/day)/(0.00112 ft2/day)). Note that the kinematic viscosity is converted to ft2/day to be consistent with the diffusivity and Sc is dimensionless. The dimensionless Sherwood number is then:
450)839()9630(0149.00149.0 3/188.03188.0 ==ℜ= ScSh
and the mass transfer coefficient is (Eq. 4-67):
43.0)12/14(
00112.0450 =⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎠
⎞⎜⎝
⎛=D
DShk m
f
Finally, kwall is:
( ) day/...abs/
..kkR
kkk
fw
fwwall 540
430)250()127()430()250(2
)(2
−=+−
−=
+=
and the total decay constant for this pipe is:
daykkk wallb /84.0)54.0()30.0( −=−+−=+=