CHAPTER FOUR Fundamentals of Water Quality

Transcription

1 HAPTER 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 hapter 3. In this chapter, it will be applied to mixing at a node and transport in a pipe. onstituent balance in a storage tank is discussed in hapter 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 ONSERVATION OF A ONSTITUENT MASS As presented in hapter 3, conservation of constituent mass (Eq. 3-18) is: dmc d( V ) in Qin outqout + (4-1) dt dt in flows outflows is the volume concentration [M c /L 3 ], 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, dm c d t [M c /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: m M V L 3 Units of in SI are typically given as g/m 3 or mg/l where l is liters. onversion factors from g to mg and m 3 to l are the same so the concentration value for both units is the same. 1 g/m 3 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/m 3 and ppt (parts per thousand) are identical to g/l. Bacteria are 4-1

2 4- HAPTER FOUR commonly in the range of micrograms/liter (µg/l). Equivalent measures to µg/l are mg/m 3 and ppb (parts per billion). The constituent mass added to water per unit time is the loading rate, m& c 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,, in the volumetric flow rate, Q or: m& c d m d t Q M T 3 L T M L 3 Example 4.1 Problem: The chlorine concentration of water flowing at a rate of 0.3 m 3 /s is 0.03 mg/l. The desired concentration is 0. 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 mg/l. The loading rate is then: 3 m& l Q 0.3 m / s ( ) g / m mg / s 184 mg / hr The mass per month is the loading rate times the duration of injection or: 3 m l m& l t mg / s 3600 s / hr 4 hr / d 30 d / mo mg / mo 13 g / mo 4. TRANSPORT AND MIXING IN THE PIPE NETWORK As discussed in hapter 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

3 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 Mixing at Junctions Simple Junctions At junction nodes within a pipe network, water quality changes due to dilution and injection. onservation 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 dm dt equals zero. Under these conditions, Eq. 4-1 becomes: c d( V ) dmc Q Q + Q dt in out dt in out Q 0 (4-) onsider 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 4-4 HAPTER FOUR Applying conservation of a constituent mass with no storage, we can write Eq. 4- as: 1 Q 1 + Q + 3 Q 3-4 Q 4-5 Q 5 out - out q 0 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, out, or: 1 Q 1 + Q + 3 Q 3 - out out Q4 - out Q5 - out q 0 Since the inflow concentrations are known, this equation can be solved for out or: 1 Q1 + Q + 3 Q3 4 (1.5) + 3 (.5) + 5 () out out 3. 9 out Q + Q + q Eq. 4- can be written for a general node as:

5 WATER QUALITY FUNDAMENTALS 4-5 out l J l Q + l q in (4-3) out q + Q l J out in l in Here, flow enters the node through J in pipes and as an externally supplied flow,. Each of these flows may have a different constituent concentration, in q l and in, respectively. Outgoing flow consists of the nodal withdrawal, q out, and the flow in the set of J out pipes carrying flow from the node. With the complete mixing assumption, all outflows have the same concentration, out. Equation 4-3 shows that the outgoing concentration is a flow weighted average of the incoming concentrations Junctions with Injected onstituent In most models, a constituent can be introduced to the system in five ways. First, for a dynamic simulation, the initial constituent concentrations, 0, 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, dm c dt in Eq. 4-1 is not equal to zero. A common injection approach is municipal chlorinators (Figure 4-) that can be used within the water treatment plant or at booster locations. hlorine 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 Source oncentration 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, in in Eq. 4-3, is described as the source concentration.

6 4-6 HAPTER FOUR Figure 4-: 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 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: inj in m& in dmc inj m& in in Ql (4-4) dt l J in where is the effective concentration resulting from injecting the defined constituent mass. Substituting this term in Eq. 4-1 and solving results in:

7 WATER QUALITY FUNDAMENTALS 4-7 out l Jin l q Q + out l + in q in Q l l Jout + m& in l Jin l Q + l q out in + q in + Q l l Jout inj in l Jin Q l (4-5) Example 4. Problem: A.5 lb (1.1 kg) tablet that is 16% l 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 kg of l (16%*1.1) to the flow. In the main pipeline, the mass rate of kg/hr is dissolved in the flow of 50 lps. Thus, m& c Q ( kg / hr)10 6 ( mg / kg) / 3600 ( s / hr) 50 ( l / s) pipe 0.98 mg / l * pipe 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: m& c Q ( kg / hr) / 3600 ( s / hr) 1.5 ( l / s) feeder 3.6 mg / l * feeder Example 4.3 Problem: A mass loading rate of 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, 4-4. inj in, can be computed from Eq. m& in inj in Ql l Jin inj in / 3600 ( units / s) ( Q1 + Q + Q3 )

8 4-8 HAPTER FOUR inj 3 3 inj in unit L ( ) L / s in / 0.46 units / L The concentration of flow leaving the node is determined by Eq out l J in l q Q + out l + in l J q out in Q l + m& in ( ) ( + 3) units / L 6 3 The injected mass increased the outflow concentration by 0.46 units/l 3 from 3.9 to 4.38 units/l 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: dm dt c inj out out q + Ql l J out (4-6) Substituting this term in Eq. 4-1 and assuming no nodal storage gives: in flows dmc inqin outqout + 0 dt outflows l l Jout Q l + in q in out q out + l l Jout Q + inj out q out + Q l l Jout that can be solved to show: out q l l Jout out Q + + l in Q l l Jout q in + inj out (4-7)

9 WATER QUALITY FUNDAMENTALS 4-9 A flow paced booster adds a user defined fixed concentration of inj out to all outflows. A setpoint booster fixes out and determines the required out to meet the setpoint. Using different logic, both boosters can be modeled in a water quality computer model with Eq 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, available concentration. inj out is set to zero and the out is set to the inj Example 4.4 Problem: Determine the constituent concentration for the node in Figure 4-1 if a flow paced booster injects 0.5 units/l 3 to all flows leaving the node. Solution: The concentration without any constituent injected at the node was determined in the text to be 3.9 units/l 3 using Eq. 4-3 which is equivalent to inj the first term on the left hand side of Eq The injected concentration, out, is given as 0.5 units/l 3. Eq. 4-7 is then applied to determine the outlet concentration, out. out in l Ql + in q l Jout inj 3 + out units / L out q + Ql l Jout 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 4.5 units/l 3. What is the injection concentration and loading rate of constituent injected? Solution: A setpoint booster defines the outlet concentration at the node, out in Eq As noted, the concentration without the addition of injected constituent is 3.9 units/l 3 which is the first LHS term in Eq The required injection inj concentration,, can be determined by substituting the known values in Eq. 4-7 giving: out out 4.5 l J out q l out Q + + l l J out in Q l q in + inj out inj out

10 4-10 HAPTER FOUR inj out 0.33 units / L As defined in Eq. 4-6, the loading rate is the flow rate leaving the node times the injection concentration or: 0.33 m& c inj out out q + Q l l J out 3 3 [ unit / L ]( 1 + ( + 3) )[ L / s] unit / s 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. onsider 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

11 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, L seg, 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: L τ (4-8) V Figure 4-3a: Polyester foam pipeline pigs can be bare foam or coated with a polyurethane material. oated 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/ft 3 ) open cell polyurethane foam with a polyurethane coating. (ourtesy of Girard Industries Incorporated ( See also Pigging Products and Services Association at V Figure 4-3b: Movement of pig with mean flow velocity through pipeline.

12 4-1 HAPTER 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 1.5 ft/s the travel time is 1000 ft/1.5 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 hapter 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 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:

13 WATER QUALITY FUNDAMENTALS 4-13 V t dmc ( Q) in ( Q) out + (4-9) dt 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 inflowoutflow terms are: ( Q) in ( Q) out Q ( ) A x x V V t A x t ( ) x V x (4-10) 1 dm Substituting this term in Eq. 4-9 and defining c r( ) results in the final V dt differential form of conservation of constituent mass for a pipe element: t + V x r() (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 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.0 ft 3 /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 ft 3 /s, the flow velocity is: The travel time is: Q V m/s A τ ( π D 4) ( π ((8 / 1) ) 4) L V s.6 minutes

14 4-14 HAPTER 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 m 3 /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: t inj 3 (g) / 0.1 (g/s) 30 s 0.5 min The concentration of fluoride in the water is: The velocity is: 3 3 m& c / Q 0. 1( g / s) / 0. 3 ( m / s) g / m mg / l Q V m/s A ( π D 4) ( π ((0. 8) ) 4) Since the fluoride is injected over a 30 s period, the length of the pulse is: L t V 30 s m / s seg inj 18 The front of the segment begins at time 0. The travel time through the reach is: τ L V m 167 s. 78minutes where L is the length of the pipe. Thus, the front edge of the segment will reach the end of the pipe at time.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.78 minutes minutes 3.8 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,.78 minutes, and 3.8 minutes.

15 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 +

16 4-16 HAPTER 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.78 min (167 s) The travel time for the pipe is.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 8 m (0.60 m/s * 137 s) (Figure E4.8d). e) Time 3.8 min (197 s) The front of the segment reached the outlet of the pipe at time 167 s. After 30 s ( s), 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... 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: out l J l Q + l q in (4-3) out q + Q l Jin One equation of this form can be written for each node. Pipe transport is described by Eq or: in l in t + V x r() (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

17 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. hapter 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 (hapter 7). Several methods (Eulerian and Lagrangian) for dynamic water quality modeling are discussed in hapter 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 with concentrations of 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., 0) Solution: a) alculate the outflow from node 3 (q 3 ) b) ompute the flow velocities and the travel times for each pipe. c) alculate node 3 s constituent concentration, 3, at times t 0, 1, 30, and 45 minutes. 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: Q + Q q q b) Given the flow rates, the velocities are computed by V Q/A and travel times are found using τ L/V. m 3 / s

18 4-18 HAPTER FOUR Figure E4-9a: Two-pipe system with fluoride injection. 6 oncentration (mg/l) Pipe 1 Pipe Node Time (minutes) Figure E4-9b: oncentration versus time for Example 4-9. oncentrations are shown for flows at the downstream end of pipes 1 and as they enter node 3 (Pipe 1 and Pipe ) and the resulting weighted concentration leaving node 3. Although pipe s concentration is 5 mg/l, the pipe only contributes /9 of the nodal flow. So the node 3 concentration does not increase dramatically when the constituent arrives in that pipe. Pipe 1: 3 Q m /s V1. 48 m/s A π D 4 m π ( 0. 6 ) 4 1 1

19 WATER QUALITY FUNDAMENTALS 4-19 Pipe : L1 340 m τ1 943 s min V. 48 m/s 1 Q V m/s A π D 4 π( 0. 4 ) 4 L 300 τ 010 s min V 159. 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: ( Q + Q ) ((0) (0. 7) + (0) (0. ) ) 1 1 3, t 0 Q The concentration remains equal to zero until the first constituent reaches node 3. This holds until flow from pipe that contains constituent reaches the node. Since the travel time in pipe 1 is 15.7 minutes, node 3 s concentration at time 1 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 ( τ ). 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: ( Q + Q ) (( ) ( 0. 7) + ( 0) ( 0. ) ) 1 1 3, t 30 min mg / l Q Finally, after water containing constituent in pipe 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: ( Q + Q ) (( ) ( 0. 7) + ( 5) ( 0. ) ) 1 1 3, t 45 min. 67 mg / l Q 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. 0

20 4-0 HAPTER FOUR 4..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) 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

21 WATER QUALITY FUNDAMENTALS 4-1 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 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 (hapra, 1997) or: t D m y (4-1) where D m is the molecular diffusion coefficient [L /T], which is on the order of 10-5 cm /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.

22 4- HAPTER FOUR It may become important in low velocity conditions that occur in dead-end pipes under constant or intermittent conditions 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. 4-1 but with a turbulent diffusion coefficient, D t [L /T]: t D t y (4-13) where D t is typically in the range of cm /s 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.

23 WATER QUALITY FUNDAMENTALS 4-3 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. 4-1) or: t D disp y (4-14) where D disp is a dispersion coefficient that is usually in the range of 10 6 cm /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 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:

24 4-4 HAPTER FOUR t + V x ( D m m + Dt ) ( D r + Dt ) x r( ) (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 D t is dropped and dispersion is added so the resulting advective dispersion equation for laminar flow is: t + V x D m ( D D ) m + disp r x r( ) (4-16) Alternative formulations of Eqs and 4-16 are discussed below and solution methods and results are presented in hapter 6 for longitudinal transport and Section for radial transport. Table 4-1: Impact of alternative transport mechanisms in radial and longitudinal directions. Molecular Turbulent diffusion Dispersion diffusion ause Movement of molecules within fluid. Eddy transport in turbulent flow Variation of velocity across a pipe section Longitudinal transport Radial transport Parameter oefficient magnitude (cm /s) Little impact in moving fluid. May be important in static water. Little impact in moving fluid. May be important in static waters No impact since turbulence causes uniform velocity profile ( R > ) * Significant impact causing complete mixing across section Significant in laminar flow ( R < 300 ) * No impact in fully turbulent flow No impact Molecular Eddy diffusivity, D t Dispersion diffusivity, D m coefficient, D disp * The range in which turbulent diffusion and dispersion fully occur is shown in the Table. With 300 < R < mixed flow will occur that is not fully laminar or turbulent. lear 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

25 WATER QUALITY FUNDAMENTALS 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: t + V x m ( D + Ddisp ) x r( ) (4-17) Eq 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 (001) 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, 001). 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 hapter 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: t + V x m ( D + Dt ) r r( ) (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

26 4-6 HAPTER 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 REATION KINETIS As discussed in hapter 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 (onnors, 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-). 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-: Water quality problems associated with water age (from EPA/AWWA white paper ( hemical issues Biological issues Physical issues Disinfection by-product formation Disinfection by-product biodegration Temperature increases Disinfectant decay Nitrification Sediment deposition orrosion control Microbial olor effectiveness regrowth/recovery/shielding Taste and odor Taste and odor onservative (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

27 WATER QUALITY FUNDAMENTALS 4-7 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 socalled 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 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, dm c dt. In developing conservation of a constituent mass for an unsteady system with only advective transport and constituent reactions, dm c dt was defined as V r (). With this assumption the resulting relationship is Eq or: t + V x r() (4-11) A simple reaction relationship is a first order relationship in which the reaction is linearly related to the concentration or: r ( ) k (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. onsider chlorine in a tank that acts as a continuously stirred reactor (STR), i.e., the tank is completely mixed and 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 is zero and assuming chlorine decays following a first order relationship the equation becomes: t r( ) k (4-0)

28 4-8 HAPTER FOUR where k will be negative since chlorine is a decaying substance. Eq. 4-0 is solved by separating variables and integrating: d k dt kt 0 e (4-1) where 0 is the initial concentration in the tank. Thus, the chlorine concentration decreases exponentially from 0 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 -0.15/hr, the concentration after 3 hours will be.06 mg/l ( o e kt 3 e (-0.15*3) 3 * 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) * 1/4 (day/hr) (1/hr). Since k is less than 0.5, this implies that 1.5% 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 r() for decay and growth processes are: r( ) r( ) k ( * ) c k ( * ) c n 1 n 1 (4-) (4-3) respectively, where * is the limiting concentration or non-reactive portion of the constituent, k is the reaction constant, and n c is the reaction order (e.g., n c 1 defines a first order reaction). ommon mathematical forms for alternative reaction types are listed in Table 4-3. In the previous chlorine decay example, the final chlorine concentration, *, was 0 and n c equaled 1 so Eq. 4- became k. 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 lark (1998) and extended by others. Others have (e.g., Lu et al, 1995; Munavalli and Kumar, 004) 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.

29 WATER QUALITY FUNDAMENTALS 4-9 Table 4-3: Reaction types and mathematical forms. Reaction type n c * k Rate units r() Example constituent onservative Fluoride Zero order growth M c /(L 3 T) k Water age First order decay 1 0 <0 1/T k hlorine First order saturation growth 1 * >0 1/T k(* - ) Trialomethanes (THM) Second order 0 <0 L 3 /(M c T) k Initial chlorine decay Second order reactions for dependent constituents reactions 0 L 3 /(M c T) k A B hlorine-thm (where A and B 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 can be solved analytically for different cases of r() for steady conditions. Recall that t in Eq 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 onservative onstituents onservative 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, r() for conservative constituents equals zero since k 0. So Eq becomes: t + V x 0 (4-4) Under steady conditions, t 0, V 0 (4-5) x

30 4-30 HAPTER FOUR onsidering a pipe segment (Figure 4-8), if we separate variables and integrate Eq. 4-5, the left hand side equals zero or: x d 0 dx (4-6) x 1 1 where 1 and are the pipe segments inflow and outflow concentrations. Eq. 4-6 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 Zero Order Decay/Growth Kinetics For zero-order decay, n c 0 and r() is then: r( ) k ( *) n 1 c (0 1) k ( 0) k (4-7) Note the dimensions of k for a zero order reaction are (M c /L 3 )/T. Thus, the constituent mass decreases/increases by k units per unit mass per unit time. Under steady state conditions, Eq reduces to: V k x (4-8) Separating variable and integrating Eq. 4-8 yields: x x1 1 k k τ1 (4-9) V

31 WATER QUALITY FUNDAMENTALS 4-31 The fraction on the right hand side is the travel time, τ 1-, for flow to move from section 1 to. 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., M c 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/L 3 /T] representing an increase of one unit of time per unit time. With this definition in Eq. 4-9, the difference in concentrations (water age) is the travel time. Thus, the difference in water age, 1, in Eq. 4-9 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 First Order Growth/Decay Kinetics As noted earlier, decaying constituents often approximately follow first order reactions with n c equal to 1 and * equal to 0. hlorine and other disinfectants in the distribution network fall in this category. Substituting these values for r() in Eq. 4- gives: r( ) k ( * ) For steady state conditions, V k x n 1 c (1 1) k ( 0) k (4-30) (4-31) where a k value less than zero denotes a decaying constituent. For a pipe, Eq can be solved by separating variables and integrating along the pipe length or: d k After substituting, 1 d x 1 x x dx dx ln( k ) x V k (4-3) x V V