Material Balance (Nazaroff & Alvarez-Cohen, Section 1.C.2 + lots more here)

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1 Material Balance (Nazaroff & Alvarez-ohen, Section lots more here) In the environment, we are not dealg only with local processes, such as chemical reactions; we are mostly dealg with terconnected systems, implyg that the system under consideration is most often connected to other systems through imports and exports. We must therefore reckon with material balances the presence of open systems. Inflow Exchange with the atmosphere Outflow Water example: Inside: hemical reactions Biological processes Physical settlg Step 1: hoose a control volume. This is an art rather than a science. Rule 1: Rule 2: Boundaries must be defed clearly; you need to know whether somethg is side or side. The budget must yield practical formation. Step 2: Select the substance for which the budget is to be made. Rule: Be specific (For example: Budget for S or SO 2?) Step 3: onsider all imports and exports. Add sources, subtract sks. 1

2 Examples of practical control volumes plume section urban airshed entire lake Distction (Nazaroff & Alvarez-ohen, Section 5.A) ompletely Mixed Flow Reactors (MFRs) are control volumes for which spatially uniform properties may be assumed. Examples: A room a buildg, a small pond, or an urban airshed. Plug-Flow Reactors (PFRs) are systems along which properties vary. They need to be split to a series of sequential control volumes. Examples: A river, an estuary, or a smokestack plume progressg wd. The analysis on the next slide applies only to MFRs or to a short segment of a PFR. 2

3 Mass balance for a MFR, Accumulation = imports exports + sources sks amount at ( t ) amount at ( t) Accumulation duration ( t ) ( t) d( ) mass volume of carryg fluid volume time mass volume of carryg fluid volume time imports exports sks (Decay constant) (Amount present) K sources S (to be specified accordg to application) S K, where = volume of control volume ( L) = concentration of substance ( g/l) = volumetric flux of fluid ( L/s) S = sum of emissions ( g/s) K = decay constant ( 1/s) Particular cases 1. Steady state: oncentration remag unchanged over time 0 S K 0 S K 2. onservative substance: No source (S = 0) and no decay (K = 0) 3. Isolated system: No import and no export (all s = 0) S K S K S K Kt ( t) itial e (if S is constant over time) 3

4 Example of Material Balance A lake contas = 2 x 10 5 m 3 of water and is fed by a river dischargg upstream = 9 x 10 4 m 3 /year. Evaporation across the surface takes away evaporation = 1 x 10 4 m 3 /year, so that only stream = 8 x 10 4 m 3 /year exits the lake the stream stretch of the river. The upstream river is polluted, with concentration = 6.0 mg/l. Inside the lake, this pollutant decays with rate K = 0.12/year. upstream evaporation K Take volume of the entire lake as the control volume. Assume steady state (= situation unchangg over time). Budget is: Budget reduces to: Solution is: upup evapevap 0 ( K ) up up up stream K ( ) 4 3 up (910 m /yr)(6.0 mg/l) 5.19 mg/l K (810 m /yr) (0.12/yr)(2 10 m ) ariation on the precedg example Un-aided (natural) remediation Suppose that the source of pollution the upstream river has now been elimated. The enterg concentration the lake has thus fallen to zero. Slowly, the concentration of the pollutant decays the lake because of its chemical decay (K term) and flushg ( term). The equation becomes: up up evap K evap K The solution to this equation is: ( t) itial exp K t (5.19 mg/l) exp( 0.52 t) 4

5 From this solution, we can fd that: It takes 1.33 years for the concentration to drop by 50%. If the acceptable concentration is 0.10 mg/l, it takes 7.6 years. uestion: If 7.6 years is too long, what can be done? heck time scales of the problem: Residency time = / = (2 x 10 5 m 3 )/(8 x 10 4 m 3 /yr) = 2.50 years Decay time = 1/K = 1 / (0.12/yr) = 8.33 years onclusion: Decay is slow and flushg comparatively fast. Flushg is primarily responsible for the natural cleang of the lake. Addg a chemical to speed up decay would only brg cremental change, while creasg the flushg rate would have greater impact. 5

6 Mass balance for a PFR (Nazaroff & Alvarez-ohn, Section 5.A.3) Most often when the Plug-Flow Reactor model is used, the actual system under consideration (river, scrubber, electrostatic precipitator) exhibits no perceptible variation over time and contas no ternal sources. The only thg that is happeng is a transformation along the movement of the substance beg traced. The formulation below holds the case of a gradual removal of the substance, modeled as a first-order reaction. The material budget is most easily established if one takes the control volume as a little piece d of the movg material (thus contag d amount of the substance). Followg this little piece of material, there is no addition or removal from it; it is a mi closed system, with the balance for the embedded substance reducg to: Kt d K( d ) The solution to this equation is: ( t) (0) e onnectg the end concentration = (t=) to the entrance concentration = (t=0), we write: e K which = / is the total transit time through the system also called the residence time the system. omparison between MFR and PFR (Mihelcic & Zimmerman, pages ) At equal system volume and throughflow, the plug-flow reactor (PFR) is more efficient than the completely mixed flow reactor (MFR). Indeed: MFR PFR e K K K e 1 1 K either way with So, why don t we always use PFRs stead of MFRs? 6

7 Residence time (Nazaroff & Alvarez-ohen, page 212) The concept of residence time (alternatively called detention time or retention time ) exists to answer the question: How long does the stuff stay there? Defition: Residence time = ontag volume divided by flow rate: Notation:, theta, the Greek letter for th. The dimension is time (T) because / has dimensions of L 3 /(L 3 /T) = T. While the standard unit is the second, typical residence times the environment are measured days, weeks, months or even years, dependg on the size of the contag volume (ex. small pond to large lake). Example of residence time (Mihelcic & Zimmerman, Section 4.1.6, page 128) Age of water at Niagara Falls? Lake olume (x 10 9 m 3 ) Outflow (x 10 9 m 3 /yr) Residence time (years) Superior 12, Huron 3, Erie We are now Subtractg 203 years brgs us back to Total: What was happeng the world 1815? 7

8 6 February 1815 New Jersey grants the first American railroad charter to John Stevens. The Great September Gale of 1815 New England A hurricane struck New England on September 1815 with 135 mph (215 km/h) sustaed wds, creatg serious damage Providence, Rhode Island. At least 38 people died. The event was so traumatic that it became memorialized an engravg now possession of the Rhode Island Historical Society and a plaque on the wall of the Old Market House. 8