Characteristics
Introduction
Chemical, 30
Reactor biological and physical processes in nature and in engineered systems usually take place in what we call "reactors." Reactors are defined by a real or imaginary boundary that physically confines the processes. Lakes, segments of a river, and settling tanks in treatment plants are examples of reactors. Most, but not all, reactors experience continuous flow (in and out). Some reactors, experience flow (input and output) only once. These are called "batch" reactors. It is important to know the mixing level and residence time in reactors, since they both affect the degree of process reaction that occurs while the fluid (usually water) and its components (often pollutants) pass through the reactor. Tracer studies can be used to determine the hydraulic characteristics of a reactor such as the disinfection contact tanks at water treatment plants. The results from tracer studies are used to obtain accurate estimates of the effective contact time.
Reactor Classifications
Mixing levels give rise to three categories of reactors; completely mixed flow (CMF), plug flow (PF) and flow with dispersion (FD). The plug flow reactor is an idealized extreme not attainable in practice. All real reactors fall under the category of FD or CMF.
Reactor Modeling
Both the CMF and the PF reactors are limiting cases of the FD reactor. Therefore the FD reactor model will be developed first. Equation 3.1 is the governing differential equation for a conservative (i.e., non-reactive) substance in a reactor that has advective transport (i.e., flow) and some mixing in the direction of flow (x dimension). C C 2 C = -U + Dd 2 t x x C = concentration of a conservative substance U = average fluid velocity in the x direction Dd = longitudinal dispersion coefficient t = time The dispersion coefficient is a measure of the mixing in a system. 3.1
Flow with Dispersion One of the easiest methods to determine the mixing (dispersion) characteristics of a reactor is to add a spike input of a conservative material and then monitor the concentration of the material in the reactor effluent. Assuming complete mixing in y-z plane then transport occurs only in the x direction and the concentration of tracer for any x and t (after t=0) the solution to equation 3.1 gives:
CEE 453: Laboratory Research in Environmental Engineering
Spring 2001
31 x 2 exp A 4 Dd t 4 Dd t M
C(x,t) =
3.2
where M = mass of conservative material in the spike, Dd = axial dispersion coefficient [L2 /T], x' = x - Ut, U = longitudinal advective velocity in the reactor, and A is the cross-sectional area of the reactor. A measure of dispersion can be obtained directly from equation 3.2. From this equation we expect a maximum value of C at t = M x/U. At this time C(x,t) = . If the mass of the tracer input (M) and reactor A 4 Dd t cross-sectional area (A) are known, then Dd can be estimated. The form of equation 3.3 is exactly like the normal distribution curve:
x2 CA 1 = exp 2 M x 2 4 x
where
x2 = 2 Dd t
3.3
3.4
The variance in concentration over space ( x2 ) is the variance in concentrations taken from many different positions in the reactor at some single moment in time, t. The variance in x ( x2 ) has dimensions of length squared. The variance of tracer concentration versus time ( t2 , with dimensions of time squared) can be measured by sampling at a single point in the reactor at many different times and can be computed using the following equations.
=
2 t
C (t )( t t ) dt t
2 0
2
C(t )dt t 2 3.5
=
0
C (t )dt
0
C (t )dt
0
where
t =
t C( t)dt
0
3.6
C (t)dt
0
For discrete data points:
=
2 t
t
i= 0 n
n
2 i
C i t
i
C t
i =0
t 2
3.7
and Reactor Characteristics
32
t =
t
i =0 n
n
i
Ci t 3.8
i
C t
i =0
Inlet and outlet boundary conditions affect the response obtained from a reactor. Closed reactors have little dispersion across their inlet and outlet boundaries whereas "open" reactors can have significant dispersion across their inlet and outlet boundaries. Typically open systems have no physical boundaries in the direction of flow. An example of an open system would be a river segment. Closed systems have small inlets and outlets that minimize dispersion across the inlet and outlet regions. An example of a closed system is a tank (or a lake) with a small inlet and outlet. The reactors used in the lab are closed. The t in equation 3.8 is the measured average residence time for the tracer in the reactor. For ideal closed reactors the measured residence time, t , is equal to the theoretical hydraulic residence time ( = reactor volume/flow rate). The above equations suggest that from the reactor response to a spike input we can compute the dispersion coefficient for the reactor. We have two options for measuring reactor response: 1) synoptic measurements: at a fixed time sampling many points along the axis of the reactor will yield a Gaussian curve of concentration vs. distance. In practice synoptic measurements are difficult because it requires sampling devices that are time-coordinated. By combining equations 3.4, 3.7, and 3.8 it is possible to estimate the dispersion coefficient from synoptic measurements. 2) single point sampling: measure the concentration at a fixed position along the x axis of the reactor for many times. If the reactor length is fixed at L and measurements are made at the effluent of the reactor (observe the concentration of a tracer at x = L as a function of time) then x is no longer a variable and C(x,t) becomes C(t) only. The response curve obtained through single point sampling is skewed. The curve spread changes during the sampling period and the response curve is skewed. Peclet Number The dimensionless parameter Pe (Peclet number) is used to characterize the level of dispersion in a reactor. The Peclet number is the ratio of advective to dispersive transport.
Pe = U Dd L
3.9
In the limiting cases when Pe = 0 (very high dispersion) we have a complete mix regime (CMFR) and when PE = 8 (D d = 0, no dispersion) we have a plug flow reactor (PF). For single point sampling of the effluent response curve, skew increases as the dispersion level in the reactor increases. The degree of skew depends on the
CEE 453: Laboratory Research in Environmental Engineering
Spring 2001
33 dispersion coefficient, the velocity in the x-direction, and the length of the reactor. Peclet values in the range 100<Pe<8 result in a symmetric response curve. Response curve skew makes the assumption of a symmetrical normal distribution curve inappropriate and a new relationship between the variance and the dispersion coefficient (or Pe) has to be determined. Boundary conditions affect the determination of the dispersion coefficient. The relationship between the Peclet number and variance for open systems is given by:
8 2 2 2 t = + 2 Pe Pe For closed systems the relationship is: 2 Pe 2 2 2 t = 2 1 e Pe Pe
3.10
(
)
3.11
2 in equations 3.10 and 3.11 is dominant for Peclet numbers much Pe greater than 10 as is shown in Figure 1. The additional terms in equations 3.10 and 3.11 are corrections for skewedness in the response curve. These skewedness corrections are not very significant for Peclet numbers greater than 10. Thus for Peclet numbers greater than 10 the Peclet number can be determined using equation 3.12 for both open and closed systems.
The term
Pe =
2 2 t2
3.12
Flow through Porous Media Flow through porous media (such as groundwater through soil) is a type of flow with dispersion. The above equations can be applied by recognizing that the relevant water velocity is the pore water velocity. 10000 Q 2/Pe The pore water velocity is U = where 1000 A open A is the cross sectional area of the porous closed 100 media and (volume of voids/total 2 10 volume) is the porosity of the porous 1 media. 2 Completely Mixed Flow Reactor 0.01 In the case of CMF reactors, there is not 0.001 an analytical solution to the advective 0.001 0.01 0.1 1 10 100 1000 10000 dispersion equation so we revert to a Pe simple mass balance. For a completely mixed reactor a mass balance on a Figure 1. Relationship between equations conservative tracer yields the following 3.10 through 3.12. differential equation: Reactor Characteristics
0.1
34
dC = ( Cin C ) Q 3.13 dt where Q is the volumetric flow rate and V is the volume of the reactor. Equation 3.13 can be used to predict a variety of effluent responses to tracer inputs such as the pulse input used in this experiment. If a mass of tracer is discharged directly into a reactor so that the initial concentration of tracer in the reactor is C = 0 M and the input concentration is zero (Cin = 0) the solution to the differential V equation is: V
-t 3.14 C = C0 exp t If a reactor has a complete mix flow regime its response (C/C0 vs. time) to a pulse input should plot as a straight line on a semi-logarithmic plot. The slope of this plot is the negative inverse of the average hydraulic residence time, t , of the reactor. Complete mix flow regimes can be approximated quite closely in practice.
Plug Flow Reactor Plug flow regimes are impossible to attain because mass transport must be by advection alone. There can be no differential displacement of tracer relative to the average advective velocity. In practice some mixing will occur due to molecular diffusion, turbulent dispersion, and/or fluid shear. For the case of the plug flow reactor the advective dispersion equation 3.1 reduces to:
C C = U t x The velocity U serves to transform the directional concentration gradient into a temporal concentration gradient. C In other words, a conservative substance moves with the Co advective flow of the fluid. The solutions to this differential equation for a pulse input and for a step input are shown graphically in Figure 2.
3.15
C C o U U
X p ul se i np ut st ep i np ut
X
Figure 2. Pulse and step input in a plug flow When a pulse of conservative reactor. tracer is added to a continuous flow reactor, all of the tracer is expected to leave the reactor eventually. The mass of a substance that has left the reactor is given in equation 3.16.
Mass Conservation
CEE 453: Laboratory Research in Environmental Engineering
Spring 2001
35
M = QCi ti
i =0 n
3.16
where Q is the flow rate and M i the mass of any substance whose concentration is s given by C. If Q and ?t are constant, then equation 3.16 can be rewritten as
M = Q t C i
i =0 n
3.17
Equation 3.17 can be used to determine if all of the tracer was measured in the reactor effluent.
Conductivity Measurements
We will use a tracer containing salt (NaCl) and red dye # 40 (for visualization). The concentration of NaCl will be monitored using a conductivity probe. Conductivity is the measure of a material's ability to conduct electric current. Conductivity is measured by passing an electrical current between two electrodes and then measuring the voltage. The electrodes can be made of platinum, titanium, gold-plated nickel, or graphite. Conductivity is defined as:
I 3.18 E where G is conductivity, I is the current, and E is the measured voltage. If the current is held constant, as the conductivity of the solution increases the voltage between the electrodes will decrease. For a given current, the measured voltage will increase as the size of the electrodes decreases and as the distance between the electrodes increases. We are interested, however, in measuring properties of the solution, not properties of the conductivity probe! Specific conductivity, C, is a property of the solution. G= L 3.19 A where L is the distance between the electrodes and A is the area of the electrodes. The L term is a property of the conductivity cell and is called the cell constant. In A practice, the cell constant is determined during calibration by measuring the conductivity (G) of a solution with known specific conductivity (C). The units of specific conductivity are Siemens/cm where Siemens are the inverse of Ohms. For a solution to be conductive, it must have ions that can transport the charge + between the electrodes. In pure water, the only ions available are H and OH-. Adding species that disassociate into charged ions increases both the concentration of ions and the specific conductivity. At low concentrations, specific conductivity increases linearly with the concentration of ions. At very high concentrations ion-ion interactions become significant and the relationship is no longer linear. The specific conductivity of several common solutions is given in Table 1. C =G
Reactor Characteristics
36
Conductivity measurements are Table 1. Conductivity of some common temperature dependent. The solutions. conductivity of a solution will Solution Specific Conductivity increase as the temperature increases. pure water 0.055 S/cm The Accumet meter that you will distilled water 0.5 S/cm use in laboratory this compensates for deionized water 0.1-10 S/cm this effect by also measuring the typical drinking water 0.5-1.0 mS/cm temperature and reporting the wastewater 0.9-9.0 mS/cm maximum drinking water 1.5 mS/cm solution specific conductivity at ocean water 53 mS/cm 25C. 10% NaOH 355 mS/cm In this lab sodium chloride will 495 mg/L NaCl 1 mS/cm increase the specific conductivity of the water in the reactors. The concentration of sodium chloride will be low enough so that specific conductivity will be linearly related to the concentration of sodium chloride.
Procedures
A conservative tracer will be used to characterize each of the reactors. A conservative tracer with 20 g NaCl/L and 4 g red dye # 40/L will be used. The salt will increase the conductivity of the water and conductivity will be measured to monitor the salt concentration. The red dye was added to the tracer to make it possible to see the tracer. A common problem when using tracers is that the tracer may have a different density than the fluid that is in the reactors. In this case the salt and dye add significantly to the density of the tracer. The tracer would tend to sink to the bottom of the reactors. To compensate for this problem the density of the water being pumped into some reactors will be increased by using a glucose solution (37 g glucose/L). Glucose is nonionic and thus will not increase the conductivity of the solution. Calibrate Conductivity probe Calibrate the conductivity probe by placing it in a 495 mg NaCl/L standard. Press the conductivity button on the Accumet meter if it is not already in the conductivity mode. Press standardize and enter 1000 S/cm. Press enter and the meter will calibrate and return to the normal display mode. Measure Conductivity of tracer Prepare a calibration curve for conductivity vs. concentration of the tracer (expressed as mg/L of NaCl). The tracer has 20 g/L of NaCl. Measure the conductivity of tracer diluted with distilled water so that the final concentrations of NaCl are 500, 200, and 100 mg/L. As a zero point measure the conductivity of distilled water.
CEE 453: Laboratory Research in Environmental Engineering
Spring 2001
37 Measure Reactor Response to Pulse Input For each reactor add a pulse input of sodium, measure conductivity vs. time in the reactor effluent and use the Compumet software to monitor the conductivity vs. time (see discussion below). Save the collected data for later analysis using a spreadsheet program. The experimental setup is shown in Figure 3. Specific instructions for each type of reactor are detailed below.
Injection port
Porous Media The porous media column is 2.5 Feed Flow with cm in diameter, 60 cm long and solution dispersion (glucose contains 60 cm of glass beads. The solution) Peristaltic or overall porosity including pump headspace and underdrains is Completely mixed approximately 0.4. Use this reactor information to estimate the volume stirrer of water in the reactor. The conductivity probe should be or plumbed into the effluent line. Plug flow reactor 1) Verify that the flow rate is set to 10 mL/min. 2) Inject 10 mg NaCl (0.5 mL of Figure 3. Reactor schematic. Only one reactor tracer) into the influent line. at a time will be connected to the peristaltic 3) Select Set Method from the pump. Compumet control palette. Use automatic data transmission with a timed interval of 1 second. Set channel A to Conductivity and channel B to Off. 4) Select Monitor Sample from the control palette. 5) Start the pump and press the enter key on the keyboard to begin data acquisition. 6) Measure the actual flow rate by collecting a timed sample from the effluent. To get an accurate flow rate you should collect a sample for several minutes. 7) Estimate the width of the tracer pulse when the pulse nears the top of t e reactor h and record the corresponding time. This information will be used to obtain an estimate of the dispersion coefficient. 8) Turn off the pump when the conductivity returns to the baseline conductivity. 9) Stop data acquisition by clicking on the Stop Sampling button. 10) Save the data to \\Enviro\enviro\Courses\453\reactors\netid_porousmedia by selecting Save data from the control palette. The data will be saved in a file (tab delimited format) that can be opened by any spreadsheet program. Completely Mix Flow Reactor (CMFR) 1) Verify that the flow rate is set to 300 mL/min.
Reactor Characteristics
38 2) Fill the CMFR with distilled water to within about 2 mm of the overflow drain . 3) Measure the conductivity of the distilled water. 4) Set the stirrer to the highest setting that doesn't cause splashing (setting 8) and place the conductivity probe near the stir bar. 5) Add 800 mg NaCl (40 mL tracer) directly to the CMFR. 6) Select Set Method from the Compumet control palette. Use automatic data transmission with a timed interval of 10 second. Set channel A to Conductivity and channel B to Off. 7) Select Monitor Sample from the control palette. 8) Start the pump and press the enter key on the keyboard to begin data acquisition. 9) Record the time when water begins flowing out the overflow (this is your actual time zero!) 10) Measure the flow rate by collecting a timed sample from the effluent. To get an accurate flow rate you should collect a sample for several minutes. 11) Turn off the pump after 2 residence times. 12) Stop data acquisition by clicking on the Stop Sampling button. 11) Save the data to \\Enviro\enviro\Courses\453\reactors\netid_CMFR by selecting Save data from the control palette. The data will be saved in a file (tab delimited format) that can be opened by any spreadsheet program. 13) Determine the volume of water in the CMFR. Baffled Tank Reactor The baffled tank reactor is a simple attempt to reduce mixing and short-circuiting. The channels are approximately 4.5 cm wide, 4.8 cm deep and have a total length of 80 cm. 1) Verify that the flow rate is set to 300 mL/min. 2) Determine the volume of water in the baffled tank. 3) Fill the baffled tank with glucose water. 4) Measure the conductivity of the glucose water. 5) Select Set Method from the Compumet control palette. Use automatic data transmission with a timed interval of 10 second. Set channel A to Conductivity and channel B to Off. 6) Select Monitor Sample from the control palette. 7) Inject 200 mg NaCl (10 mL of tracer) into the influent line with a syringe. 8) Start the pump and press the enter key on the keyboard to begin data acquisition. 9) During data acquisition, it is important to gently move the conductivity probe up and down to continually bring the probe into contact with the changing solution. While moving the probe up and down, do not lift the probe so high that the platinum contacts leave the solution
CEE 453: Laboratory Research in Environmental Engineering
Spring 2001
39 10) Measure the actual flow rate by collecting a timed sample from the effluent. To get an accurate flow rate you should collect a sample for several minutes. 11) Turn off the pump when the conductivity returns to the baseline conductivity. 12) Stop data acquisition by clicking on the Stop Sampling button. 13) Measure the average conductivity of the remaining solution in the baffled tank. 12) Save the data to \\Enviro\enviro\Courses\453\reactors\netid_baffled by selecting Save data from the control palette. The data will be saved in a file (tab delimited format) that can be opened by any spreadsheet program. Pipe Flow The pipe flow setup consists of 15.24 m of 6 mm ID tubing. 1) Verify that the flow rate is set to 50 mL/min. 2) Fill the pipe with glucose water. 3) Select Set Method from the Compumet control palette. Use automatic data transmission with a timed interval of 1 second. Set channel A to Conductivity and channel B to Off....