Epc_fa2011_lecture_4 - Kinetics in Environmental Engineering Kinetics is an important environmental engineering element as it provides information

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Unformatted text preview: Kinetics in Environmental Engineering -- Kinetics is an important environmental engineering element as it provides information on how long it takes for a transformation processes to occur. I.e., kinetics tells us how long it takes for a system to reach an equilibrium. -- Engineers are interested in kinetics where a significant change occurs during certain time period in which we are interested, i.e., when and what. -- Also tells about the fate of contaminants as they move through the natural environment. 1 Classes of Kinetic Reactions Heat generation in the reaction - Exothermic = Heat liberated in the reaction - Endothermic = Heat absorbed during the reaction Phases involved in the reaction (gas, liquid, solid) - Homogeneous = Elements in the reaction are all in the same phase gas-gas reaction - Heterogeneous = Elements from different phases are in the reaction gas-liquid reaction Acceleration/Deceleration in reaction - Catalytic = Catalyst is used in the reaction to modify and increase the rate of a reaction - Noncatalytic = Reverse of the catalytic reaction 2 Direction of Reactions a) Irreversible reaction A + B b) Reversible reaction A + B c) Sequential reaction A B d) Parallel reactions A A C B C C C Example) Sequential reaction / Bacterial nitrification process of Ammonia NH 3 ⎯Nitrosomonas → NO 2 ⎯Nitrobacter → NO 3 ⎯⎯⎯⎯ ⎯⎯⎯ ⎯ 3 Rate of Reactions General irreversible reaction aA + bB → cC + dD Rate = r = k [ A]α [ B] β Where A, B C, D [ ] a, b, c, d k α, β Reactant species Product species Molar concentration (M), not in mg/L Stoichiometric coefficients of species A, B, C, D Rate constant (units are dependent on a and b) Empirical exponents (reaction order = α + β) 4 Order of Reactions -- 1 There are different order of reactions commonly used and applied in environmental engineering. Among them, zero-order reaction first-order reaction second-order reaction are most frequently used. 5 Order of Reactions -- 2 Zero-order reactions -- The rate of reaction is independent of the concentration of the reactant. A → Products rA = − CA t CA 0 d CA d[A] =− = k 1CA dt dt 0 ∫ d C A = −k 0 ∫ d t C A − C A 0 = −k 0 t C A = C A0 − k 0 t CA CA0 Slope = ‐k0 Time (t) -- Many biologically induced reactions such as oxidation of ammonia to nitrite, and the oxidation of glucose by aerobic bacteria, i.e., linear loss of material with time. 6 Order of Reactions -- 3 First-order reactions -- The rate of reaction is directly proportional to the concentration of the reactant. A → Products CA ∫ CA 0 rA = − d CA d[A] =− = k 1C A dt dt t d CA = −k 1 ∫ d t CA 0 ln C A − ln C A0 = −k 1 t ln CA ln CA0 ln C A = ln C A 0 − k 1 t C A = C A 0 e − k 1t Slope = ‐k1 Time (t) -- First-order reaction is a most frequently used reaction type in environmental engineering. 7 Order of Reactions -- 4 Second-order reactions -- The rate of reaction is proportional to the second power of reactant(s) being converted to product(s). A + B → Products or A + A → Products r=− d CA d CB d[A] d[B] =− =− =− = k 2CACB dt dt dt dt rA = − d CA d[A] =− = k 2CA 2 dt dt 1 1 = + k2t CA CA0 1 CA Slope = k2 1 CA0 Time (t) -- An important use of a second-order reaction is in describing cometabolic biotransformation of halogenated organic compounds or biological transformations in general where the concentration of the compound transformed is very low (< 1 mg/L). 8 Kinetics of DO-BOD reaction -- 1 -- Dissolved oxygen (DO) is one of the most important water quality parameters. The saturated value of DO in water is a function of the water temperature, and typically in 6 to 10 mg/L range in average quality water bodies. -- Minimum requirements for fish are 3 mg/L (catfish, carp, etc.) to 8 mg/L (trout and gamefish), depending on the species. -- Many types of organic pollutants exert an oxygen demand if discharged into a water body water. These pollutants can cause a lowering of the DO level to the point where fishkills and other harmful effects can occur. (these oxygen demanding wastes are not necessarily harmful in themselves) -- Frequently, the potential [of pollutants] to cause a reduction in DO concentration of the waterbody [such as stream, lake, reservoir, estuary, etc.] is the primary water quality concern. 9 Kinetics of DO-BOD reaction -- 2 -- Once the pollutant gets into water body, usually it is too late, and subsequently it takes a lot longer and expensive to recover/restore back to the prior condition. (via water quality modeling and management) -- Biochemical Oxygen Demand (BOD) is the amount of oxygen required by organisms to biologically digest the pollutants (usually oxidizable organic matters) discharged into the receiving water. -- Usually, BOD is expressed in units of mg/L of O2 -- We need to be able to estimate/predict this oxygen demand prior to discharge so that we know how much oxidizable pollutants can be accepted by the water without causing harmful effects. This is called WLA (Waste Load Allocation). 10 Kinetics of DO-BOD reaction -- 3 Aerobic environments: organic matter (=OM) + O2 + microbes CO2 + H2O + new microbes (i.e., growth) + stable products - stable products may include SO42- (Sulfate), NO3- (Nitrate) and PO43- (Phosphate) Anaerobic environments: organic matter + microbes CO2 + CH4 (Methane) + new microbes + unstable products - unstable products may include H2S (Hydrosulfuric acid) and NH3 (Ammonia) 11 Kinetics of BOD reaction -- 1 Modeling BOD as a First-order reaction -- Assume that the rate at which the microorganisms digest organic matters/pollutants is proportional to the amount of organic matters remaining to be digested (i.e., a first order reaction). d[BOD remaining ] dt where = −k[BOD remaining ] k = first-order BOD reaction rate constant BODremaining = amount of BOD remaining to be digested at time t 12 Kinetics of BOD reaction -- 2 -- Solution form of this rate expression becomes [BODremaining]= [BODultimate] exp(-kt) (A) -- Ultimate BOD, BODU is the total amount of oxygen required by the microbes to digest the organic matters, or simply, the initial BOD when the oxidation (=decay) process begins. -- By applying a simple mass balance; [BODultimate] =[BODt]+[BODremaining] (B) -- Combining the equations (A) and (B), BOD exerted at time t, BODt becomes [BODt ] = [BODultimate](l - exp(-kt)) 13 Kinetics of BOD reaction -- 3 [BODremaining]= [BODultimate] exp(-kt) [BODultimate] =[BODt]+[BODremaining] [BODt ] = [BODultimate](l - exp(-kt)) 14 Quick Recap 1) First-order reaction is a most frequently used reaction type in environmental engineering. C A = C A 0 exp(−kt) 2) Dissolved oxygen (DO) is one of the most important water quality parameters. 3) Biochemical Oxygen Demand (BOD) is the amount of oxygen required by organisms to biologically digest the pollutants in the water. 4) DO = f(BOD), and BOD is expressed with a First-order reaction. 15 Flow Models/Reactors -- 1 -- Flow models are simplified conceptual transport models, and they are used to describe the movement of pollutants through both man-made and natural systems. -- Both reaction kinetics and flow characteristic of a system can be expressed simultaneously using flow models. Great for engineering problem solving and design! Example) A toxic pollutant is discharged into a lake, and that the pollutant decays to form non-toxic products according to a first-order reaction with a rate constant. Flow models can be used to predict the concentration of this pollutant in the lake over a period of time, if we know about the mixing conditions in the lake as well as the rate of flow of water into 16 and out of the lake. Flow Models/Reactors -- 2 -- Basic Flow Models 17 Flow Models/Reactors – Batch Reactor -- A batch reactor is a well-mixed system with no flow in and out. It is the most simplistic flow model. [Accumulation] = [In] - [Out] + [Generation] = - [Decrease due to reaction] Since there’s no inflow into or outflow from the reactor, d CA V = −rV dt d CA = −r = − k C A dt Integrate it to obtain a usual first-order rate expression, C A = C A 0 ⋅ exp( − kt ) 18 Flow Models/Reactors – CSTR -- 1 -- Continuous Stirred Tank Reactor. Also referred to as CMFR, Completely Mixed Tank Reactor -- A CSTR is well-mixed like a batch reactor, but the same amount of flow passes through the system. -- A CSTR is the most frequently applied environmental reactor model due to its simplicity and applicability. [Accumulation] = [In] - [Out] + [Generation] = [In] - [Out] - [Decrease due to reaction] d CA V = C A0 Q in − C A Q out − k C A V dt 19 Flow Models/Reactors – CSTR -- 2 CASE #1) No flow in/out Batch Reactor CASE #2) No Kinetic reaction (k=0) in the system d CA Q = (C A0 − C A ) dt V d CA V = C A0 Q in − C A Q out − k C A V dt Integrating with CA = 0 at t = 0 as initial condition, CA dC Q t A ∫ (C − C ) = V ∫ d t A0 A 0 0 C A = C A0 (1 − e − ⎛ C − CA − ln ⎜ A0 ⎜ C −0 A0 ⎝ Q t V )=C A0 (1 − e − ⎞Q ⎟= t ⎟V ⎠ t θH ) θH= td = V/Q = Retention time 20 Flow Models/Reactors – CSTR -- 3 CASE #3) With a first-order Kinetic reaction at steady-state d CA V = C A0 Q in − C A Q out − k C A V dt CA ∫ 0 ⎛ ⎞ ⎛ V⎞ ⎜ C A0 − ⎜ 1 + k ⎟ C A ⎟ ⎜ ⎟ Q⎠ ⎜ ⎟Q 1 ⎝ − ln ⎜ ⎛ V⎞ ⎟= t V ⎟ ⎜ − CA ⎜1 − k ⎟ C A0 ⎜ ⎟V 1− k Q⎠ ⎝ ⎜ ⎟ Q ⎝ ⎠ ⎛ ⎛⎛ V ⎞ ⎞⎞ ⎜ ⎜ ⎜ 1 + k ⎟ t ⎟⎟ ⎜ Q ⎟ ⎟⎟ ⎜⎜ ⎝ ⎠ C A0 ⎜ 1 − exp⎜ − ⎟⎟ V ⎜ ⎟ ⎟ C ⎛ 1 − exp⎛ − (1 + kθ H ) t ⎞ ⎞ ⎜ ⎜ ⎜ ⎟⎟ A0 ⎜ ⎜ ⎟⎟ ⎟⎟ ⎜ ⎜ Q θH ⎝ ⎠⎠ ⎠⎠ ⎝ ⎝ ⎝ = CA = V 1 + kθ H 1+ k Q d CA C A0 = CA ≅ Qt ∫ dt V0 C A0 C A0 = V 1 + kθ H 1+ k Q at steady-state 21 Flow Models/Reactors – PFR -- 1 -- Plug-Flow Reactor -- No mixing occurs in the direction of flow, but the fluid is wellmixed perpendicular to the flow direction. Example) Flow through a narrow pipe is an example for PFR -- It is like a series of batch reactors moving on a conveyor belt. Each batch reactor acts like a fluid parcel of a homogeneous concentration, and adjacent parcels' concentration may or may not be the same. [Accumulation] = [In] - [Out] + [Generation] = [In] - [Out] - [Decrease due to reaction] ∂ CA ΔV = QC A - Q(C A - ΔC A ) - ΔVkC A = QΔ C A − ΔVk C A ∂t 22 Flow Models/Reactors – PFR -- 2 ∂ CA ΔV = QC A - Q(C A - ΔC A ) - ΔVkC A = QΔ C A − ΔVk C A ∂t (ΔV=AΔx) ∂ CA ∂ CA Q ∂ CA − k CA =− kC A = −U A ∂x ∂t ∂x At S.S. d CA ∂ CA = 0 = −U − k CA dt ∂x → k CA ∂ CA =− U ∂x CA d CA kx =− ∫ dx ∫C U0 A C A0 C A = C A0 ⎛k⎞ ⎜− x⎟ e⎝ U ⎠ = C A0 e (− kθ H ) 23 Quick Recap 24 Mass Flux - Transport Mechanism for pollutants -- Environmental engineering is primarily concerned with the mass flux of pollutants/contaminants at a specific problem domain. -- Flux is defined as flow per unit area per unit time - it gives not just a point value of the concentration change but also provides corresponding extent and spatial magnitude that the concentration change takes place. -- Flux of pollutants/contaminants can be combined into transport models to represent a wide range of both natural and engineered environmental systems. 25 Mass Flux in Environment 26 Common Mass Flux Types 1) Advective Flux 2) Diffusive Flux (=Molecular diffusion Flux) 3) Dispersive Flux (=Turbulent diffusion Flux) 27 Mass Flux Advective Flux (=Advection) -- Vehicle -- Advection is the flux of materials (dissolved and suspended solutes) due to bulk flow of fluid. -- Advective flux is expressed as a function of flow rate, concentration and area. J advection = QC c A = UC c 28 Mass Flux Diffusive Flux (=Molecular Diffusion) -- Driver with attitudes -- Diffusion is due to the migration of a solute in response to a concentration gradient. -- If a concentration gradient exists, then the random motion of molecules leads to a net flux from higher to lower concentration regime until an equilibrium is reached. J M diffusion = −Dm dC c dx 29 Mass Flux Dipersive Flux (=Turbulent Diffusion) -- Driver driving the Vehicle with attitudes -- Dispersion is due to the turbulent diffusion (also called hydrodynamic dispersion or simply, dispersion) -- Turbulence in fluids is caused by the motion of small parcels of fluid or eddies, which in turn caused by various elements such as wind, channel shape and channel bottom topography, etc. J dispersion = −E x dC c dx 30 Mass Flux Total Flux -- A combined form of Advection/Diffusion/Dispersion flux is the foundation for many contaminant transport models used in natural and man-made settings such as surface water quality modeling, subsurface contaminant migration, atmospheric plume models, water quality in water distribution network, etc. Put it all together • J = Jadvection + Jdiffusion + JDispersion • dC dC J = UC c − Dm c − Ex c dx dx 31 Quick Recap Mass Flux 1) The Vehicle -- Advection 2) The Driver with attitudes -- Diffusion 3) The Driver driving the Vehicle with attitudes -- Dispersion 4) Transport models = f(Advection/Diffusion/Dispersion flux) 32 ...
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This note was uploaded on 10/19/2011 for the course CEE 350 taught by Professor Jaewanyoon during the Fall '10 term at Old Dominion.

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