Environmental Process Analysis, 2: Dynamic Steady States

Transcription

1 Environmental Process Analysis, 2: Dynamic Steady States T. Torgersen Department o Marine Sciences, University o Connecticut, Groton, CT , Thomas.Torgersen@uconn.edu B. Branco Department o Marine Sciences, University o Connecticut, Groton, CT J. Bean Department o Marine Sciences, University o Connecticut, Groton, CT ABSTRACT Environmental analysis requires an understanding o processes that contribute to a system and the concept o dynamic balances. The Conservation o Mass (Heat) equation or a system deteres whether (e.g.) concentration (heat) in the system will increase, decrease or remain constant. The rate at which change occurs in a system and the magnitude o that change are unctions o the dynamic balance and the rate constants (residence times -1 ) or individual processes. We express this mass (heat) balance concept as a simpliied algebraic expression ( IPOLA ) and use it to evaluate the dynamics o systems. We present a classroom activity that can be accomplished in a short time with imal cost to demonstrate these principles. Our experience suggests that this activity and the IPOLA equation build knowledge by developing a conceptual understanding o systems and their component processes. INTRODUCTION Many people perceive that net change in the environment around them is small over a lietime (e.g., net climate change rom increased CO 2). This perception may lead to the erroneous conclusion that nothing is changing and the environment has an inherent resiliency that resists large change. However, environmental systems are governed by multiple individual processes that interact to produce a dynamic system. It is the balance or imbalance o inputs and production vs. outputs and losses that detere whether change will occur, in what direction and magnitude, and how ast. These concepts are embodied in the equations or (1) Conservation o Mass, (2) Conservation o Energy and (3) Conservation o Momentum (e.g., Bird et al., 1960) that are the basis or most engineering and scientiic analysis. We have ound that a simpliied algebraic expression or the Conservation o Mass or Heat in a system provides a undamental starting point or teaching students to quantiy observations o environmental systems. The conservation equations can be algebraically expressed as: I + P O L = A (1) I j k l where Σ designates the summation; I and O are inputs and outputs respectively across the boundaries o the system; P and L are production and loss respectively inside the boundaries o the system; and A is accumulation or change inside the boundaries o the system (see also Torgersen et al. 2004). The component parts in eqn (1) are expressed in terms o stu [N] (which can be moles, calories, grams, etc.), volume [L 3 ], area [L 2 ] and time [t], e.g. [N t -1 ], [N L -3 t -1 ] and [N L -2 t -1 ]. (We use the common practice o denoting units in square brackets). The observation that nothing is changing is quantiied by A=0, which is the balance between sources and sinks in environmental systems. I i., P j, O k, L l or ΣI, ΣP, ΣO, and ΣL could all be very large quantities and very important processes that coincidentally sum to A=0. Change occurs in the system when A 0. The amount o stu in the system is increasing when A>0 and decreasing when A<0; the larger the magnitude o A, the aster the change. Our experience with environmental science students over the last decade is that (1) the concept o a dynamic steady state and (2) how ast a system will respond to a change in contributing processes, e.g. I i, P j, O k, L l, are poorly understood. We describe here a classroom activity to learn the undamentals o (environmental) process analysis, response times and dynamic steady states. The activity is designed to provide a progression o learning rom the concrete to the abstract that is consistent with scientiic inquiry (AAAS, 1990). The response time o the system under investigation (How long will it take to? How ast will?) is detered rom direct observations o change, and is also calculated rom measurements o the component processes contributing to the change. We apply equation (1) and have ound that the acronym IPOLA can become a valuable mnemonic or students that prompts a methodological approach to complex problems. In its present coniguration, the activity has been used primarily or second year university students although signiicant portions can be translated directly to the high school classroom. Our experience over the last decade with this activity shows that the analysis o real data generates a greater acility with the concepts o dynamic environmental systems than plug-and-chug problems rom the textbook and imparts the physical meaning to numbers. The activity thus oers students an opportunity to do and to talk science (Abrams, 1998) and to engage in scientiic inquiry that supports carry-over o principles to the real world. BACKGROUND For students with limited calculus, the concept o Conservation o Mass (Heat) balance is presented algebraically (equation 1) and students are amiliarized with the concepts o residence times and reaction times (Torgersen et al., 2004). This laboratory activity is then conducted to investigate the individual components o a dynamic steady state (each term in IPOLA) and the time scale upon which a new dynamic steady state can be achieved given a change in the governing conditions. The questions to be addressed are: Torgersen et al. - Environmental Process Analysis 2 331

2 Figure 1 Experimental apparatus arranged or investi - gating dynamic steady states. 1. What is a dynamic steady state? 2. How is a dynamic steady state maintained? 3. What parameters aect the dynamic steady state? 4. How is the rate o change rom one dynamic steady state to another quantiied in terms o the processes that contribute to the system? 5. How does one measure the components o a dynamic system and quantiy their relative importance? Because chemical reactions are hard to control (and typically occur too rapidly), this activity was designed to evaluate a heat balance rather than a mass balance. However, we emphasize that the concepts are identical. The activity is conducted in a well-mixed with an input (F i, [L 3 t -1 ]) and an output (F o, [L 3 t -1 ]) controlled by a peristaltic pump (see Figure 1, Table 1). Production (P) is controlled with a magnetically stirred hot plate and loss (L) is detered by the temperature dierence between the hot and the cooler room air. In this experiment, a Kalorie [Ka] is deined as the relative concentration o heat per unit volume in the (an analog to a well-mixed lake) above a deined zero point equal to a temperature o 0C. Thus, [ Ka] = T ρ H [cal L -3 ] (2) where T is the temperature in the in C, ρ is the density o water (noally 1.0g ml -1 or this experiment), and H is the heat capacity o water (noally 1 cal C -1 g -1 ). For the dynamic steady state, equation (1) can be expressed as (IPOLA): [ Ka] ( F T H ρ) + P ( F T H ) L V A i i o o ρ = = = 0(3) t where F i (F o) is an inlow (outlow) rate, T i (T o)is the temperature o inlow (outlow) water and V is the volume o the. The water balance (equation 1) or the system is expressed as (IPOLA): F i - F o = 0 (4) and thus F i=f o. With an experimental deteration o P and L, the dynamic steady state can be investigated as a unction o F i=f o. For the unsteady state system (A 0) the rate o change to the new dynamic steady state can be investigated and evaluated in terms o the controlling process(es). THE EXPERIMENT Students construct an analog to a well mixed environmental system using a side-arm and a magnetically stirred hot plate (see Table 1, Figure 1 and or a complete experimental description; note especially cautions with regard to good experimental design and saety. We also suggest that the instructor conduct the experiment himsel beore conducting it with students.) The goals o this 1.5hr activity are to make measurements over time within the system in order to quantiy the rate o change or the system and to explore the processes that control change. The experimental design is such that (1) contributions rom each component part o the system heat balance can be investigated, (2) the response time o the system can be observed, and (3) the response time o the system can be evaluated rom knowledge o the components. The principles or the quantiication o the rate constant (λ) or change or the response time (τ =λ 1 ) or change generate knowledge that may then be applied to systems that cannot be explicitly observed (e.g., in naturally occurring environmental systems with large time and space scales). Questions that help students prepare or the activity include: 1. What measurements do I make? 2. How oten should I make a measurement? 3. When can I stop making measurements? 4. How can I interpret my measurements? As students investigate the dynamic system (Figure 1) and conduct speciic experiments to exae the 332 Journal o Geoscience Education, v. 52, n. 4, September, 2004, p

3 Figure 2 Data rom the production (P) o Ka (heat) deteration. P-L, the apparent heating rate is 5C 1 or (rom eqn (5a)) 2875 cal -1. components o IPOLA, they can be guided towards good experimental and analytical techniques. The ollowing discussion ollows the pedagogical development o the activity. TASK 1: Observe and document the apparent rate o Ka production (P-L) in the. Students irst detere the apparent rate o heat production (P-L) by monitoring the temperature in a well-mixed as a unction o time. Most students are aware o the concept o a warm-up time rom kitchen experience and the experimental design should target a constant heating rate. It is also necessary to discuss the amount o water to place in the. To best approximate the dynamic steady state system that will be constructed, the should be illed to the sidearm (hence the volume in a noal 500ml is greater than 500ml and is equal to the volume in the during lowing water conditions). Figure 2 shows the temperature in the as a unction o time or the system. Note that this experimental system is deined as P-L=A; we measure A and calculate P-L. TASK 2: Observe and document the apparent rate o Ka loss (L) rom the. For this experimental system, -L=A and the loss rate is detered rom ~90C to near room temperature. This task takes more time than is typically available during a normal class period. Thereore, the instructor can either obtain the necessary data beore hand or continue taking the measurements ater the class has been dismissed. The accumulation term is observed (a negative quantity) with the thermometer using a ull on an unheated magnetic stir plate (Figure 3a). Students should note that the cooling plot is curved whereas the heating plot o Figure 2 appears linear. TASK 3: Calculate the apparent rate o Ka production (P-L, cal -1 ) in the. How does the apparent rate o production vary with [Ka] (temperature)? I your ield o vision was restricted only to the interior o the with no knowledge o processes ongoing outside o the, could you tell i heat was produced internally or externally? Figure 3a Data rom the loss (L) o Ka (heat) deteration. The cooling data are it to dierent temperature ranges. The slopes indicated or various temperature ranges are C -1 and thereore L (rom eqn (5)) is cal -1. Figure 3b The loss rate (L) as a unction o temperature as calculated rom sequential pairs o data points and plotted at the time midway between the two sequential points. As the dierence between temperature inside the and the room air decreases, the loss o Ka (heat) rom the (L) decreases linearly with temperature inside the as detered by Fick s Law. Figure 2 shows the least squares it to the temperature change rate to be 5.0C -1. The apparent production rate (P-L) is calculated as: T P L t V H [ Ka] = ρ = V = A t (5a) In the example case shown in Figure 2, dt/dt is +5C -1, V =575ml and thus P-L equals 2875 cal -1 rom Equation (5a). Discussion should lead to the observation that the calculated apparent production rate, P-L, applies to all the water inside the, although heat also goes to warg up the itsel. However, by deining the system as only the water inside the, P-L does not need to include the unknown heating o the. This underscores the need to speciically deine your system, what you need to measure and what your measurement speciically quantiies. The discussion can also be guided towards how Ka is produced. I it was not Torgersen et al. - Environmental Process Analysis 2 333

4 known how the water was heated (your ield o vision did not include the hot plate), one could still quantiy the apparent production and hypothesize mechanisms that would produce Ka in the by either internal reactions or by luxes across boundaries rom external sources. TASK 4: Calculate the rate o Ka (heat) loss (L, cal -1 ) in the system (). How does the rate o heat loss vary with temperature? Figure 3a shows the cooling curve or the system (-L=A) and the result is nonlinear. Students should calculate the cooling rate or dierent ranges o water temperature (e.g., Figure 3a). T L = = = t V H [ Ka] ρ A t (5b) Because T/t is negative, A is negative and the L calculated rom equation (5b) will be a positive quantity. Loss rates (L) o 0.9C -1 to 0.19C -1 (517 cal -1 to 109 cal -1 ) characterize the system rom 90C to 35C. The loss rate (L) can also be calculated rom sequential pairs o data points and plotted at the time midway between the two sequential points (Figure 3b) ollowed by additional linear analysis. Students with algebraic backgrounds should be encouraged to discover that L will be dierent or each dynamic steady state temperature achieved under the low conditions imposed during this demonstration. Students with some calculus as well as exposure to Fick s Law (e.g., Bird et al., 1960) should recognize that the primary mechanism o heat loss is diusion through the glass walls o the, and thereore L is proportional to the temperature gradient. It is thereore not surprising that as the dierence between temperature inside the and the room air decreases, the lux o heat rom the (L) will decrease: T L = D x A heat Fick s Law (6a) where D heat is the thermal conductivity o heat in silicate glass and A is the surace area through which the loses heat (likely the sides only because the cork and the bottom are pseudo-insulated). Converting Fick s Law to a inite dierence equation and substituting the temperature dierence that exists across the wall thickness ( x) o glass and the air boundary layer: T T L D A D T i o o = heat ( ) = ( air X X glass lask T room airboundarylayer ) A (6b) where D air is the thermal conductivity o air. Thus, L is a linear unction o T that is conirmed by the plot o Figure 3b. (Continuity requires that the temperature o the inside surace o the is equal to the temperature o the well-mixed water in the, etc.). Similarly to Task 3 above, i your ield o vision were restricted, this calculation o L could be reinterpreted as an output in which the actual outlow was not within your view. TASK 5: Which is more important to the dynamic steady state, P or L? Discuss the validity o your Figure 4 The temperature o water in the system () as a unction o time or two low rates. The symbols indicate the change to low rate #2 which occurred at 17.5 utes. estimate o P. Can you make a better estimate o P? I (P-L)= 2875 cal -1 and L is 109 cal -1 to 517 cal -1, students easily understand that production (P) is the more signiicant process because the magnitude is larger. Within the temperature range o this exercise (40C to 50C), L is expected to be approximately 173 cal -1. Thus, the P is = 3048 cal -1 with an error o cal -1. TASK 6: Create a ull apparatus (system) to achieve a dynamic equilibrium (Figure 1). Calculate the low rate o room-temperature (24C) water that will result in a temperature o approximately 50C. Set the pump to this low rate and test your Ka (heat) balance. What is the residence time o water in the or this low rate? This calculation demonstrates the utility o the simpliied algebraic expression (IPOLA) to answer the question. I you start rom IPOLA, you can identiy all the relevant controlling processes and detere unit conversion actors such that the IPOLA equation is written in consistent and uniorm units. Then, the low rate that results in a dynamic steady state temperature o 50C can be calculated algebraically. It has been our experience that deriving the low rate rom IPOLA generates knowledge o the system, conidence in numbers and their meaning, and understanding o the concepts. Thereore, we advocate this approach or detering low rate rather than giving students a ormula into which numbers are blindly entered ( plug and chug ) and results obtained. To derive a low rate with IPOLA, construct a water balance using equation (4) and a heat balance using equation (3). For the measurements illustrated here, the calculated low rate is 111 ml -1 but the closest we could set the pump was 115 ml -1 and the residence time o water (Torgersen et al, 2004) in the is (575ml/115ml -1 ) 5 utes. TASK 7: With the assembled apparatus to achieve a dynamic steady-state (Figure 1), start the pump (low rate #1) and record temperature as a 334 Journal o Geoscience Education, v. 52, n. 4, September, 2004, p

5 unction o time, and observe the approach to a dynamic steady state. The questions posed under The Experiment will again prepare students or this portion o the activity. Figure 4 shows the data collected as the system moves toward a dynamic steady state that is veriied by our successive temperature readings o 49C over 4 utes. This dynamic steady state was achieved over an elapsed time o 13.5 to 14.5 utes. Recall that a goal o this activity is to observe that no change does not mean that nothing is happening; no change means that sources (I+P) and sinks (O+L) are balanced (A=0). A change in any one process will alter the system. (I=P=O=L=A=0 is the trivial case.) The temperature achieved in this dynamic steady state (49C) in ~14 utes is much less than the >92C achieved in ~14 utes in the heating-only case (Figure 2). The dynamic steady state temperature is less than the no-low, heating-only case because heat is being actively removed by the outlow o water. Given that the residence time o water in the (τ) is 5 and that 3τ is a good estimate or 95% o change to be accomplished (see Torgersen et al, 2004), the comparison between ~14 utes and 3x5 = 15 utes is consistent with theory. TASK 8: Once the dynamic equilibrium temperature has been reached, increase the low rate by ~1.5-2x (low rate #2) and observe the temperature in the as a unction o time. Beore altering the low rate, students should discuss the new dynamic steady state temperature. Will the new state be hotter or colder? Why? The algebraic solution to IPOLA (equation 3) suggests that temperature should be lower given an increased low rate. However, it can also be reasoned that the temperature should decrease because the residence time o water in the decreases. Thus, there is on average, less time available or P to act on the water and the water will reach a lower temperature. The interrelationship o residence time and the dynamic steady state condition should become apparent. Figure 4 also shows temperature as a unction o time ater the change to low rate #2 (180 ml -1 ) at 17.5 utes. The new dynamic steady state is established at 41C (the thermometer has 1C precision). At this point, it is useul to emphasize the signiicance o time as a relative variable. In Figure 4, t=0 corresponds with the initial observation o temperature and the time at which low was initiated. In the analyses o Task 12 (Figure 5), time is reset to t=0 when the pump low rate is changed to low rate #2. The pedagogical advantage o presenting all the data on the same relative time scale (Figure 4) is to demonstrate the change rom one dynamic steady state condition to another by changing the low rate. However, no previous knowledge o the irst transient is required to analyze the second transient. I this type o irst order change (A) in a variable as a unction o time were seen in ield data, a change in IPOLA could be hypothesized and that rate o change could be quantiied (Torgersen et al., 2004). The activity and principles explored in this paper could then be used to evaluate whether change was the result o changes in I, P, O, or L. TASK 9: Write the Ka (heat) balance or the inal condition o low rate #1 and or the inal condition o low rate #2. Since I, P, O have now been deined and/or measured, solve or L (the rate o Ka (heat) loss rom the ) and compare the dynamic steady state calculated values o L to the measured values o L detered in Task 4. Does the system obey theory? Why or why not? The Ka (heat) balance or the dynamic steady state at a low rate o 115 ml -1 is expressed by the IPOLA equation: ( F T H ρ) + P ( F T H ρ) L = A = 0 (3) i i o o At the steady state temperature o 49C with P= 3048 cal -1, we solve or L with low rate #1 (F i=f o=115 ml -1 ): 115ml 24C 1cal 3048cal 115ml 49C 1cal 173cal + = = L C g ml C g ml For a low rate #2 (180 ml -1 ), the value o L is: 180ml 24C 1cal 3048cal 180ml 41C 1cal + = 12cal = L C g ml C g ml The Ka (heat) balance calculations above provide two estimates o L or this dynamic system, +173 cal -1 and -12 cal -1. Estimates o the heat loss rate rom Figure 3a are L= +173 cal -1 or the temperature range 44C 61C and L=+109 cal -1 or the temperature range 38C 47C. From the linear it in Figure 3b, L is +170 cal -1 and +90 cal -1 or temperatures o 49C and 41C respectively. (A 1C error in temperature and the slope detered rom Figure 3b suggests an error o the order 10 cal -1.) While the agreement is very good or low rate #1, low rate #2 requires some discussion o error. Choosing L=90 cal -1 at 41C as true loss, a low rate o 174 ml -1 would put low rate #2 in perect agreement. The dierence between 180 ml -1 and 174 ml -1 represents only a 4% error in low rate measurement. Agreement between calculated L and measured L could also be achieved i the dierence between inlow temperature and outlow temperature were (T o T i) = 16.4C. With the thermometer reading in whole degrees (0.5C), the temperature dierence or dynamic steady state #2 could have been as low as (40.51C-24.49C)=16.02C which we would have recorded as (41C-24C)=17C. Thirdly, the dierence between calculated L and measured L could be achieved i P=2890 cal -1, well within experimental error. Thus, the experiment conirms the theory within observational error. TASK 12: Calculate the rate constant by which temperature increased with low rate #1. Calculate the rate constant by which temperature decreased with low rate #2. To calculate the rate constant that controls how ast the new dynamic steady state is achieved, the data are normalized with respect to the initial and the inal temperature. For the increase to dynamic steady state, the deining equation is (e.g., Torgersen et al., 2004): C C C i C t = e λ (9) Torgersen et al. - Environmental Process Analysis (7) (8)

6 Figure 5 Plots o ln(normalized temperature) vs. time or deteration o the rate constant or change rom one dynamic steady state to another. Both low rates have been it to all points with values o ln(normalized T) > -3. The dashed lines indicate the theoretical slopes that would be predicted IF the residence time as detered by volume/low-rate were valid. For the decrease to dynamic steady state, the deining equation is: C C C i C i t = ( e λ 1 ) (10) where λ (τ -1 ) is the rate constant or the process, C is the measured value (C=(t)), in this case Kalories (relative heat content), C i is the initial value and C is the inal value. The graphical solution or λ is obtained by plotting natural log o normalized temperature (let-hand side o equation (9 or 10) vs. time since initialization o the process. The slope o the line is λ and the residence time is λ -1. Figure 5 illustrates the graphical solution. The error bars on the data are conservative estimates o error. The rate constant or low rate #1 is (τ =4.4 ); the rate constant or low rate #2 is (τ= 2.8 ). Note the step-like patterns or low rate #2 in Figures 4 and 5. The patterns are a result o the thermometer resolution, which is not precise enough to capture temperature changes less than 1C. Students should recognize that the limitations o their instrumentation may give the alse impression that the system has achieved a dynamic steady state (T is constant or successive readings). Thus, measurements should continue long enough to ensure that the dynamic steady state has been achieved. Within an arguable error, the system rate deterations and Ka residence times are equal to the water residence times in the as calculated by dividing the volume by the low rate (5 and 3.2 or low rate #1 and low rate #2 respectively). The dashed lines in Figure 5 represent the expected results i the rate constant or the system was the same as the rate constant or the inlow and outlow. The residence time o water in the deteres the amount o time available to heat water in the system, thus the system must respond on a timescale as detered by the low rate. TASK 13: For the inal temperature o low rate #1 and low rate #2, calculate the residence time o heat with respect to (a) production, (b) loss rom. Conirm that the residence time o water is the best estimate o the residence time o heat in the system. For inal temperature o low rate #1, there are: 25C 1cal 575ml = 14, 375cal C g ml in the above the inlow condition (24C). From Figure 3b, L at 49C is 170 cal -1 and P = 3048 cal -1. Thus, the residence times are calculated to be: τ P τ L τ water 14375cal = = cal 14375cal = = cal 575ml = = ml For inal temperature o low rate #2, there are: 17C 1cal 575ml = 9, 775cal C g ml in the above the inlow conditions (24C). From Figure 3b, L at 41C is 89 cal -1. Thereore: τ P 9775cal = = cal Within the error o the demonstration (including measurements o temperature, low rates, production and time), the residence time o water in the is equal to the residence time or the production o Ka in the. This conirms that the residence time o water in the is critical to the estimate o how high a temperature will be achieved in this dynamic steady state. The calculation o the residence time with respect to heat loss conirms that the heat loss takes ar longer to act and that the heating thereore doates the cooling. CONCLUSIONS The activity described in this paper illustrates the undamental concepts o dynamic steady state, time to dynamic steady state, and allows both advanced and beginning students an insight into the controls on the dynamic systems that are exempliied in environmental systems. The activity provides adequate proo by calculation o why dynamic steady states exist and how ast new dynamic steady states can be achieved. Our experience suggests that this activity and the simpliied 336 Journal o Geoscience Education, v. 52, n. 4, September, 2004, p

7 equation or the Conservation o Mass or Heat, IPOLA, builds knowledge by developing a conceptual understanding o dynamic systems and their component processes. In addition to understanding the speciic system modeled by this activity, the tasks create a learning environment in the sciences that has signiicant opportunities or inquiry, and developing and practicing the language o science. The activity also provides students with hands-on, mentally challenging experiments that serve as a model or real-world processes. Commensurate with research on approaches to improve science education (e.g., Yager, 1996; Roth, 1989), the activity described in this paper emphasizes approaches and applications to common problems and the necessary decision-making skills through the mass balance approach (IPOLA) and attempts to help students develop a conceptual understanding o science and its ways o describing, predicting, explaining, and controlling natural phenomena (National Research Council, 1996). ACKNOWLEDGEMENTS This work was supported in part by NSF grants EAR and EAR We thank the many students who have participated in this activity over the past decade. They have all contributed in some way to the reinement o this activity. REFERENCES Abrams, E., 1998, Talking and doing science: important elements in a teaching-o-understanding approach, In, Mintzes, J.J., Wandersee, J.H. and Novak, J.D. (editors), Teaching Sciences or Understanding: a human constructivist view, San Diego, Caliornia), Academic Press p American Association or the Advancement o Science, 1990, Science or All Americans, New York, Oxord University Press, p Bird, R.B., Stewart, W.E. and Lightoot, E.N., 1960, Transport Phenomena, New York, John Wiley and Sons, 780pp. National Research Council, 1996, The Role o Scientists in the Proessional Development o Science Teachers, Washington, DC, National Academy Press, p. 1-25, Roth, K.J., 1989, Science education: It s not enough to do or relate, American Educator, p , Torgersen, T., Branco, B.F., Bean, J.R., and Sytsma, R., 2004, Environmental Process Analysis, 1: residence time and irst order processes, Journal o Geoscience Education, v. 52, p Yager, R. E., editor, 1996, Science/technology/society as reorm in science education, Albany (New York), University o New York Press, p , Torgersen et al. - Environmental Process Analysis 2 337