Chapter 4. Hydraulic Characteristics of Water Treatment Reactors and Their Effects on Treatment Efficiency

4.2



CHAPTER four



INTRODUCTION

The treatment processes at the heart of the production of safe drinking water almost always

occur in continuous flow reactors. How water flows through those reactors can have a

remarkable influence on the treatment efficiency that is achieved. This fact is well understood in the case of disinfection, where it is embedded in the regulations of the Safe Drinking

Water Act (SDWA) in the United States. However, the same principles apply for essentially

all of the treatment processes in a drinking water treatment plant. In this chapter, we first

explain the common descriptions of the hydraulic characteristics of reactors, beginning

with ideal flow reactors and proceeding to non-ideal flow reactors; this hydraulic characterization primarily consists of determining the residence time distribution. Subsequently, the

ramifications of the different flow characteristics (or different residence time distributions)

for some types of chemical reactions are explained, with reference to specific common

water treatment objectives.



CONTINUOUS FLOW REACTORS: IDEAL AND

NON-IDEAL FLOW

A typical water treatment plant for a surface water pumps water through a pipe from the

source (lake, reservoir, river, or stream) to a series of reactors, usually consisting of a rapid

mix tank (where chemicals are added to the water for a variety of purposes), flocculation

tank(s), sedimentation or flotation tanks, either granular media or membrane filters, and a

clearwell that serves both as a disinfection reactor and a flow equalization basin. In most of

these cases, all of the water flows into the reactor in a single stream and likewise flows out

of the reactor in a single stream. An exception is the sedimentation or flotation unit, which

has a single influent stream but two effluent streams, one for the clarified water (which

contains most of the water and few particles) and one for the residuals (which contains a

small fraction of the water and most of the particles). Because the residuals stream in such

reactors is such a small fraction of the water flow, it can often be ignored in the characterization of the overall water flow. Granular media or (dead-end) membrane filters have a

periodic backwash that creates a residuals stream, but while treating water, they have only

a single effluent. Therefore, for most of the analysis that follows, we consider all reactors

to have a single influent and single effluent stream.

Some reactors are designed to promote mixing while others attempt to prevent mixing

as much as possible. In this context, mixing is taken to mean the extent to which the water

(and any constituents in the water) that comes into the reactor in some small increment

of time (say 1 s or 0.001 s) mingles with water that comes into the reactor at any other

time (before or after the increment of focus). At one extreme, absolutely no mixing of one

entering packet with any other packet occurs, and every packet travels from the influent

to the effluent without any exchange of water molecules with any other packet. A reactor

with this type of flow pattern is known as a segregated flow reactor; a plug flow reactor

is a special type of segregated flow reactor in which all packets have the same residence

time within the reactor; there is no mixing up of packets in the direction of flow. A plug

flow reactor is often thought of as a conveyor belt. Just like boxes on a conveyor belt do

not pass one another or exchange material between one another, the packets of water that

enter the reactor sequentially stay in the same order and do not exchange water molecules

with one another.

At the other extreme, we can imagine complete and instantaneous mixing of every new

packet of water entering a reactor with all of the water already in the reactor; this type of







Hydraulic Characteristics and Treatment Efficiency



4.3



reactor is known as a continuous flow stirred tank reactor (CFSTR) or as a continuous flow

completely mixed reactor (CFCMR); although the latter term is perhaps more precise, we

follow tradition and refer to such a reactor throughout this book as a CFSTR. Because water

is not only entering but also constantly being removed from the reactor, some molecules

go essentially directly from the influent to the effluent and spend very little time (and, in

the limit, zero time) in the reactor. On the other hand, some molecules of water continually

get mixed with new incoming water and escape being in the packets of water leaving the

reactor for a very long time, far longer than the theoretical detention time (τ) of the reactor,

defined as





τ=



V



Q



(4-1)



Here V is the volume of water in the reactor and Q is the volumetric flow rate of water

entering and leaving the reactor. Although the distribution of residence times of different

molecules of water in this second type of ideal reactor flow is (probably) not intuitively

obvious to the reader, quite clearly it is very different from the plug flow reactor where

every water molecule stays in the reactor for the same amount of time. The residence time

distribution of a CFSTR is developed subsequently.

Between the two ideal extremes of the plug flow reactor (zero mixing in the direction

of flow) and the continuous flow stirred tank reactor (complete and instantaneous mixing

of the entire contents of the reactor) is a broad spectrum with some, but incomplete or noninstantaneous, mixing of the influent with the water in the reactor. This spectrum constitutes non-ideal flow. Although the ideal extremes are difficult to accomplish in reality, they

are useful concepts and can be approached quite closely in real reactors. Throughout the

chapter, the behavior in ideal reactors is used to frame that in non-ideal reactors.



TRACER STUDIES

The primary hydraulic characteristic of interest in most cases is the residence time distribution, that is, a description of how long different molecules of water stay in the reactor. If one

could somehow label all of the molecules of water that enter the tank over some small increment of time and then observe the time that each labeled molecule leaves the reactor, the

resulting information would yield the desired residence time distribution. One could say, for

example, that of all the molecules that came into the reactor in some tiny period of time (say,

1 s in a reactor with a theoretical detention time of 2 h), 5 percent came out of the reactor in

less than 6 min, an additional 5 percent came out in the next 8 min, and another 7 percent

came out in the following 4 min. The same information could be expressed cumulatively by

saying that 5 percent stayed in the reactor less than 6 min, 10 percent stayed in the tank less

than 14 min, and 17 percent left the reactor within 18 min. If this description were carried

out until 100 percent of the labeled water molecules had been accounted for, we would have

the complete residence time distribution.

Because it is not easy to label some water molecules in a way that differentiates them

from other water molecules, we add a tracer to the influent water; the tracer must be a measurable and nonreactive soluble constituent that travels through the reactor in exactly the

way that water does. By measuring the effluent concentration of this tracer as a function of

time, we can learn how water molecules travel through the reactor; in essence, the tracer

molecules act like the set of labeled water molecules.

Tracer studies can be performed in two primary ways. In a pulse input tracer test, a known

mass of tracer (M) is added to the influent of the reactor all at once; the instant of the addition



4.4



CHAPTER four



is defined as time zero (t = 0), and the concentration of the tracer in the effluent (Cp (t)) is

monitored for a long time thereafter, until all of the tracer is accounted for. Alternatively, in a

(standard) step input tracer test, a steady concentration of tracer (Cin) is added to the influent

beginning at time zero and continuously thereafter, theoretically forever but practically until

the effluent concentration matches the influent concentration. The concentration of tracer in

the effluent (CS (t)) is monitored throughout the time period of tracer addition.

In both types of tracer tests, the chosen tracer is usually a constituent that is not naturally

in the water, so that the baseline concentration of the effluent is zero; however, this condition is not required. A nonzero (but steady) baseline concentration could be subtracted from

all of the effluent concentration measurements to obtain the concentration that is caused

by the tracer. This idea can be extended to obtain the desired residence time information

from the depletion of tracer concentration after a step input tracer test is ended, as explained

subsequently. First, it is necessary to present how the tracer information is mathematically transformed to yield residence time distribution information. Although tracer tests

in real systems always involve discrete data points (concentrations at discrete times), the

explanations are given first as if the data were continuous functions; how discrete data are

manipulated to yield the same information is explained subsequently.

The Exit Age and Cumulative Age Distributions

Consider first a pulse input tracer test on some arbitrary (i.e., non-ideal) reactor; the mass

M of tracer is put into the influent of the reactor instantaneously at time zero, and the

tracer concentration in the effluent is monitored over time. At and before time zero, the

concentration in the effluent is zero. After sufficient time, all of the tracer will have exited

the reactor and the effluent concentration will again be zero; in the meanwhile, the effluent concentration would be nonzero. Most real reactors yield results for Cp(t) that show a

relatively smooth rise followed by a relatively smooth and continuous fall in the concentration, but the shape of the Cp(t) curve could be almost anything depending on the specific

design of the reactor. To emphasize the arbitrariness of the possible response of the reactor,

a double-humped curve for Cp(t) is shown in Fig. 4-1a. To keep the number of data points

relatively small in this example, data are shown for only every 2.5 min, but more frequent

data points, especially in the first 25 min or so where the effluent concentration is changing

dramatically, would yield better information.

If another pulse input tracer test were performed on the same reactor at the same flow

rate but twice the mass of tracer were put into the reactor, we would expect to see the effluent concentration at every time to be twice that of the original test. Since our interest is in

the distribution of times that water, not tracer, stays in the reactor, it is useful to eliminate

the effect of the mass of tracer used in the test. To do so, we multiply the effluent concentration values by the flow rate Q and divide by the mass M of tracer input to the reactor.

The product of the effluent concentration (mass/volume) and the flow rate (volume/time)

is the instantaneous rate (at time t) at which the mass of tracer is leaving the reactor (i.e.,

dimensions of mass/time). The subsequent division by the total input mass then yields the

instantaneous fractional rate at which the input mass is leaving the reactor, with dimensions

of inverse time. This normalization is so useful that we give it its own symbol, E(t); that is,





E (t ) =



Q

C p (t )

M



(4-2)



The set of all values of E(t) is called the exit age distribution. If the value of Q/M for the

experiment under consideration is 0.002 l/mg-min = 0.002 m 3 /g-min , the resulting E(t)

curve is shown in Fig. 4.1b. (The origin of this value of Q/M is explained subsequently.)



4.5



Hydraulic Characteristics and Treatment Efficiency







30

(a)



Cp (t) (mg/L)



25

20

15

10

5

0

0.06



(b)



E(t) (min–1)



0.05

0.04

0.03

0.02

0.01

0



1

(c)



F(t) (–)



0.8

0.6

0.4

0.2

0



0



10



20

30

Time (min)



40



50



Figure 4-1  Pulse input tracer test: (a) effluent concentration profile; (b) exit age

distribution; (c) cumulative age distribution.



4.6



CHAPTER four



The exit age distribution describes the distribution of times that water stays in the reactor;

that is, the fraction of water from any aliquot of influent that stays in the reactor between

any time t and t + dt is E (t ) dt. For example, the value of E(t) in Fig. 4-1b at 15 min is

0.03 min–1, and if we assume that that value is constant for the 1 min from 14.5 to 15.5 min

(i.e., dt ≈ D t = 1 min), then E (t ) dt ≈ E (t ) D t = (0.03 min -1 ) (1 min) = 0.03. That is, 3 percent

of the water that enters the reactor at one instant will stay in the reactor between 14.5 and

15.5 min. Likewise, if we take a sample of the effluent at some instant, we can say that

3 percent of the water in that sample stayed in the tank between 14.5 and 15.5 min, or in

other words, that 3 percent of the water in that sample came into the reactor between 14.5 and

15.5 min earlier. The exit age distribution describes the distribution of residence times that

different molecules of water stay in the reactor.

Often, it is more useful to know what fraction of the water stays in the reactor less than

a certain accumulated time (say 15 min) than to know how much stays in for some incrementally small time. That information is contained in the dimensionless cumulative age

distribution F(t), which is found as the integration of E(t) from time zero to time t; that is,

t







F (t ) = ∫ E (t ) dt



(4-3)



0



For the experiment under consideration, the values of F(t) are shown in Fig. 4-1c. As

can be seen from the example in the figure, F(t) has the characteristics that it starts at zero

and rises continuously toward the ultimate value of 1. At t = 15 min, the value of F(t) can

be seen to be just less than 0.5 (the precise value is 0.483), which means that 48.3 percent

of the water that enters this reactor stays in the reactor for less than 15 min.

All of the tracer put into the reactor at time zero in a pulse tracer test eventually exits

the reactor; that is, 100 percent of the tracer stays in the reactor for less than infinite time.

Mathematically, this fact means that the ultimate value of F(t) is 1 (unity) and that the area

under the entire E(t) distribution is also 1. That is,

∞







∫ E (t ) = F (∞) = 1



(4-4)



0



In real systems, of course, a practical definition of time infinity might be 4 to 15 times

the detention time, with the smaller part of this range being acceptable for reactors that are

nearly plug flow. The criterion for stopping the monitoring in a pulse input test is that no

appreciable mass of tracer remains in the reactor.

A step input tracer test ultimately yields the same information for the exit age distribution E(t) and the cumulative age distribution F(t), but the raw data, and therefore the analysis, are different from the pulse input case. For the standard step input tracer test (in which

the baseline concentration of the tracer in the influent water and throughout the reactor is

zero), the effluent concentration (Cs(t)) always increases with time because water that came

into the reactor prior to the initiation of the test (with no tracer) is replaced with water that

entered after the start of the test (with concentration Cin). An example of the effluent concentration is shown in Fig. 4-2a using an influent concentration of tracer of 10 mg/L; the

reactor is the same as used in the example for the pulse input.

A unit volume of effluent with concentration CS (t) is a mixture of a fraction of water

(i.e., F(t)) that came into the reactor less than time t ago with concentration Cin and the rest

(1 - F(t)) that came in earlier with concentration zero. That is, the cumulative age distribution F(t) is found by normalizing the effluent concentration by the influent concentration.





F (t ) =



CS (t )



Cin



(4-5)



4.7



Hydraulic Characteristics and Treatment Efficiency







10

(a)

Cs (t) (mg/L)



8

6

4

2

0



1

(b)



F(t) (–)



0.8

0.6

0.4

0.2

0



0.06

(c)



E(t) (min–1)



0.05

0.04

0.03

0.02

0.01

0



0



10



20

30

Time (min)



40



50



Figure 4-2  Step input tracer test: (a) effluent concentration profile; (b) cumulative

age distribution; (c) exit age distribution.



4.8



CHAPTER four



The results of this normalization are shown in Fig. 4-2b. As before, F(t) has the characteristics of being zero at time zero, rising continuously over time, and eventually reaching

a value of 1.

When the baseline concentration in the influent and throughout the reactor prior to time zero

is not zero, a sample of effluent with concentration CS (t) is a mixture of a fraction (F(t)) that

came in less than time t ago with concentration Cin and the rest (1 - F(t)) that came in with

concentration Cbaseline . The result is that the baseline concentration must be subtracted from both

the numerator and denominator to obtain the cumulative age distribution. That is,





F (t ) =



CS (t ) - Cbaseline



Cin - Cbaseline



(4-6)



Equation 4-6 is completely general and can be used for any (properly performed) step

input test, including the possibility that Cbaseline > Cin . Equation 4-5 is a simplification of

Eq. 4-6, where Cbaseline = 0. At the end of a step tracer test with Cbaseline = 0, the reactor has

the concentration equal to Cin throughout the reactor, so it is possible to perform a second

test by continuing the effluent monitoring after the influent concentration has been returned

to zero (at a time designated as a new time zero). In that case, the new baseline is the old Cin

and the new Cin is zero. Regardless of how a step input tracer test is performed, the criterion

for being able to stop the monitoring is that the difference between the influent and effluent

concentrations is minimal or immeasurable.

As noted in Eq. 4-3, the cumulative age distribution is the integral of the exit age distribution; conversely, the exit age distribution is the derivative of the cumulative age distribution at any time t. That is,





E (t ) =



dF (t )



dt



(4-7)



The data for F(t) for Fig. 4-2b were analyzed according to (a discretized version of)

Eq. 4-7 to find the E(t) values for this reactor, and the resulting values are plotted in

Fig. 4-2c. As noted above, this step input tracer study was performed on the same reactor as

the pulse input test shown earlier; as a result, the exit age distributions and cumulative age

distributions for the two tests shown in Figs. 4-1 and 4-2 are identical. Both types of tests

ultimately yield the identical information, and therefore, the type test that is performed is

a matter of choice (perhaps influenced by issues of measurement and precision, discussed

subsequently).

Discrete Data in Tracer Tests

The conceptual understanding of the exit age and cumulative age distributions presented

above presupposes that these distributions are continuous functions and that the tracer data

are likewise continuous. However, the data in real tracer tests are always obtained at discrete

times, and the figures given above indicate that samples were taken at discrete times. In this

section, we explain how to analyze discrete data from both pulse and step input tracer tests

to obtain the desired age distributions. The data for a pulse test used to generate Fig. 4-1 are

shown as a spreadsheet in Fig. 4-3. The raw data from the test—the times that samples were

taken and the corresponding values of Cp(t)—are given in columns A and B, respectively.

In column C, the average concentration in the time interval between adjacent samples is

shown in the row corresponding to the end of the time period and is calculated as





Cave , i =



Ci-1 + Ci



2



(4-8)



4.9



Hydraulic Characteristics and Treatment Efficiency







1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23



A

Time

(min)

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

25.0

27.5

30.0

32.5

35.0

37.5

40.0

42.5

45.0



B

Cp(t )

(mg/L)



C

Cave

(mg/L)



0

5

14

23

26

21

15

15

17

21

14

10

8

5

3

2

1

0

0



D

Cave* ∆t

(mg-min/L)



2.5

9.5

18.5

24.5

23.5

18.0

15.0

16.0

19.0

17.5

12.0

9.0

6.5

4.0

2.5

1.5

0.5

0.0



6.25

23.75

46.25

61.25

58.75

45.00

37.50

40.00

47.50

43.75

30.00

22.50

16.25

10.00

6.25

3.75

1.25

0.00



sum(Cave*∆t)



E

E(t )

(1/min)

0.000

0.010

0.028

0.046

0.052

0.042

0.030

0.030

0.034

0.042

0.028

0.020

0.016

0.010

0.006

0.004

0.002

0.000

0.000



F

Eave

(1/min)

0.005

0.019

0.037

0.049

0.047

0.036

0.030

0.032

0.038

0.035

0.024

0.018

0.013

0.008

0.005

0.003

0.001

0.000



G

F(t )

(–)

0.000

0.013

0.060

0.153

0.275

0.393

0.483

0.558

0.638

0.733

0.820

0.880

0.925

0.958

0.978

0.990

0.998

1.000

1.000



500



Figure 4-3  Spreadsheet for handling pulse input tracer test data.



So, for example, the value of 9.5 mg/L shown in cell C5 is the average of 5 and 14 mg/L

shown in cells B4 and B5, respectively. In the subsequent column (col. D), the average concentrations are multiplied by the associated time interval ( D t = ti - ti -1 ) ; in this example, all

of the time intervals are 2.5 min, although the time intervals are often varied to obtain more

data when the changes are rapid. The value of 23.75 mg-min/L in cell D5 is obtained as

9.5 mg/L × 2.5 min. The sum of all of the values in column D from the entire test is given

in cell D23. This value is found because, when multiplied by the volumetric flow rate, the

product is the mass of tracer accounted for in the effluent throughout the tracer test; that is,

Mout = Q∑ Cave ,i D ti







(4-9)



all i



This calculation constitutes a quality control on the entire tracer test, because the resulting value can be checked against the known value of the mass input to the reactor. If Min

and Mout are not in close agreement, the cause should be determined before using the tracer

data to calculate the age distributions and ultimately to make treatment decisions. In some

cases, one might decide that using Mout rather than Min in normalizing from Cp(t) to E(t) is

a more accurate approach. In the generation of the exit age distribution in this spreadsheet

and in Fig. 4-1, it was assumed that M in = Mout, so that





l

Q

Q

1

1

= 0.002

=

=

=



mg-min

M Q∑ C p , ave , i D ti ∑ C p , ave , i D ti 500 mg-min /l

all i



all i



The values for E(t) shown in column E in Fig. 4-3 are the product of this value and the

values of C p (t ) in column B; the resulting units for E(t) are min–1.



4.10



CHAPTER four



The values for Eave, i shown in column F are calculated from the values in column E

in the same way that the Cave ,i values in column C are calculated from column B. Finally,

after recognizing that the value of F (0) = 0, we calculate the subsequent values of F(t) in

column G as

Fi = Fi-1 + ( Eave , i )(ti - ti-1 )







(4-10)



So, for example, the value in cell G5 is found as 0.013 + (0.019)(2.5) = 0.060; the value of

0.013 shown in cell G4 is rounded from the precise 0.0125, and the spreadsheet uses this

more precise value.

The analysis of discrete data for a step input tracer test is shown in Fig. 4-4, with the raw

data of time and measured effluent concentration shown in columns A and B. As indicated

by Eq. 4-5 (or Eq. 4-6 with Cbaseline = 0), the values of F(t) in column C are calculated by

dividing the values of CS (t) in column B by Cin, which is 10 mg/L in this case.

Next, the estimate of the derivative of F(t) over a time interval is calculated in column D

as follows:

 dF  ≈  D F  = Fi - Fi -1

 dt  i  D t 



 i ti - ti -1







(4-11)



So, for example, the value of 0.019 in cell D5 is calculated as (0.060 - 0.013) /(5.0 - 2.5) min–1.

In reality, the values in column D are average values of E(t) over the interval from ti-1 to ti

and are shown in the spreadsheet associated with the end of the interval. After recognizing

that the initial value of E(t) equals zero, the remaining values of E(t) for the specific values

of t are calculated in column E as

E (ti ) = 2 Eave , i - Ei-1







1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21



A

Time

(min)

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

25.0

27.5

30.0

32.5

35.0

37.5

40.0

42.5

45.0



B

Cs (t )

(mg/L)

0.00

0.13

0.60

1.53

2.75

3.93

4.83

5.58

6.38

7.33

8.20

8.80

9.25

9.58

9.78

9.90

9.98

10.00

10.00



C

F(t )

(– )

0.000

0.013

0.060

0.153

0.275

0.393

0.483

0.558

0.638

0.733

0.820

0.880

0.925

0.958

0.978

0.990

0.998

1.000

1.000



(4-12)



D

dF /dt

(=Eave)

0.005

0.019

0.037

0.049

0.047

0.036

0.030

0.032

0.038

0.035

0.024

0.018

0.013

0.008

0.005

0.003

0.001

0.000



E

E(t )

(1/min)

0.000

0.010

0.028

0.046

0.052

0.042

0.030

0.030

0.034

0.042

0.028

0.020

0.016

0.010

0.006

0.004

0.002

0.000

0.000



Figure 4-4  Spreadsheet for handling step input tracer test data.







Hydraulic Characteristics and Treatment Efficiency



4.11



For example, the value of 0.028 in cell E5 is calculated from the values in cells D5 and E4 as

2(0.019) - 0.010. The values in columns B, C, and E for CS (t), F(t), and E(t), respectively,

were plotted against the time values in column A to make the plots shown in Fig. 4-2.



Choosing a Step or Pulse Input  Tracer  Test

As shown above, both types of tracer tests ultimately yield the same information, so the

question of which type of test to perform is a matter of choice. The monitoring of the effluent

is essentially the same in both cases, although a step input test has the advantage that it

usually requires measurements over a much narrower range of concentrations. The ability

to put tracer into the influent either as a pulse or as a step often dictates the choice. For

a pulse test, a substantial mass of tracer must be entered in a very short period of time

(theoretically, instantly) into the influent in the same way that water enters the reactor; the

design of a reactor and its influent structure often makes this impossible. For a step test,

the tracer is added continuously to the influent in a small flow rate (relative to the influent

flow); it is often easier to accomplish this type of addition with either the existing facility

or a slight modification. Many tracer tests in water treatment plants in the United States are

undertaken in response to current disinfection regulations that require determining the time

(usually designated as T10) at which F(t) = 0.10; although this information is available from

either type of test, it is far simpler in terms of data handling to find it from a step input test.

In some cases, it is convenient to simply turn off the feed of a chemical, such as fluoride

which is added continually and is nonreactive,1 to perform a test in which the concentration

decreases from the normal (post addition) value to the natural baseline value; when normal

operation is resumed, the effluent concentration will rise and can continue to be monitored

as a second tracer test. For all of these reasons, step input tracer tests are performed more

often than pulse input tests, although site specific considerations might dictate the opposite

choice in some cases. Teefy (1996) delineated many of the issues involved in choosing both

the tracer material and the type of tracer test to perform.

Ideal Plug Flow

From the description of a plug flow reactor above, the results of tracer studies on these ideal

reactors can be predicted intuitively. In a pulse test, all of the mass that is put into the reactor

at time zero travels together from one end to the other, and it all leaves together at the time

equal to the theoretical detention time τ. No tracer comes out of the reactor prior to t = t,

nor does any come out afterward; the entire spike of tracer exits the reactor at exactly t = t.

Because M in is not mixed with surrounding fluid, it is difficult to convert to concentration

units, but the fact that all of the water that comes in at one time comes out of the reactor at

a time τ later leads directly to the exit age distribution shown in Fig. 4-5 (with an assumed

theoretical detention time τ of 20 min). The function shown is known as the Dirac delta

function and has the following properties: a value of zero everywhere except one time, the

value of infinity at that time, and an area under the curve (from time zero to infinity, and

therefore under the spike) of 1. In terms of the water age distribution, we can interpret this

function in three ways. First, all of the water that enters at one time exits exactly τ later.

Second, all of the water that is in a sample of the effluent came into the reactor τ earlier.



1

Fluoride is reactive with aluminum, so plants that use aluminum-based coagulants cannot use fluoride as a tracer

until after filtration. Other commonly considered choices for the tracer include lithium, sodium, and chloride.



4.12



CHAPTER four



Exit Age Distribution, E(t) (min–1)



To infinity



0



10



20



30



40



50



60



Time (min)

Figure 4-5  Exit age distribution for a plug flow reactor with theoretical

detention time of 20 min.



Cumulative Age Distribution, F(t) (–)



Third, the probability that a molecule of water will stay in the tank for a time equal to τ is 1,

and the probability that it will stay in the reactor for a time less than or greater than τ is 0.

Translating the exit age distribution to the cumulative age distribution means integrating

E(t) from 0 to time t, as indicated in Eq. 4-3. For the plug flow reactor under consideration,

the value of F(t) is 0 for all t < τ and 1 for all t ≥ τ. Using the same 20-min detention time

indicated in Fig. 4-5, the cumulative age distribution F(t) for the plug flow reactor is shown

in Fig. 4-6.

The interpretations of the cumulative age distribution are similar to those of the exit

age distribution. First, all of the molecules of water in a sample of the influent will stay in



1.0

To Infinity

0.8

0.6

0.4

0.2

0.0



0



10



20



30

40

Time (min)



50



60



Figure 4-6  Cumulative age distribution for a plug flow reactor with theoretical detention time of 20 min. (The function jumps from 0 to 1 at t = τ; the vertical

dashed line is simply to aid the visual effect, but the function does not take on any

value between zero and one.)