Important points from previous lecture. What are the major hydrologic fluxes to

Transcription

1 Important points from previous lecture What are the major hydrologic fluxes to and from lakes? How can flow from one gaged river be extrapolated to an ungaged river? How does one convert flow in cm to units of m 3 /sec? What is the definition of water residence time? Of conservative-chemical-residence time? Models & Modeling 1

2 Model Definitions A simplified representation of something; a description or analogy used to help visualize something that cannot be directly observed; a system of postulates, data, and inferences presented as a mathematical description of an entity or state of affairs; Types of Models CONCEPTUAL: a non-quantitative description that illustrates the underlying structures, mechanisms or trends; all models generally begin as conceptual models; system reactivity 2

3 Types of Models: BOX A box model is little more than a conceptual model of flow paths; it may or may not have estimates of transfer rates; Outseepage Atmospheric deposition 16.8 Leaf litter.3 Algae (5) SO 4 2- (1) 53 surface mineralization 4-6 diffusion 45 7 Sediment accumulation 9..4 Groundwater inflow Pools as mmol m -2 ; fluxes as mmol m -2 yr -1 Types of Models: Empirical Empirical models provide a quantitative, mathematical description of data; the mathematical expression in the model may be unrelated to the causal mechanisms. C:S (atomic) 12 L. Sempach L. Baldegg 1 L. Cadagno L. Greifen 8 Trend line C:S = δ 34 S; r 2 =.41; P < δ 34 S 3

4 Concentrations in Rivers Mass flux (kg/d) = Conc. (kg/m 3 ) Flow (m 3 /d) Often, flow is measured continuously, but concentrations are measured only periodically. Because concentration often is related to flow, a Q-C relationship can be used to estimate concentrations that were not measured. DOC m g /l y = 12.38Ln(x) R 2 =.4722 y =.414x R 2 = Flow cfs Mg+Ca meq/l y = -.3x R 2 =.4745 y = -.42x R 2 =.31 1 Mg+Ca Sodium Flow cfs e q/l Na me Bad River 199 Flow (m 3 /s) /1/9 4/11/9 7/2/9 1/28/9 Date Flow x Conc. = Load (m 3 /yr)(g/m 3 ) = g/yr L = QC L = Q(12.3lnQ-39) DOC m g/l y = 12.38Ln(x) R 2 =.4722 y =.414x R 2 = Flow cfs Load (kg) /1/9 4/11/9 7/2/9 1/28/9 Date 4

5 Types of Models: Mechanistic Mechanistic models employ equations based on actual reaction steps or pathways; mechanistic models are quantitative; models may be used in simulation, design or predictive modes. GWi Water column only Organic C IInorganic C Total carbon P R A GWi GWi GWo P R GWo S doc/dt=+gwi - GWo+ P - R - S dic/dt = +GWi - GWo - A -P-K+M+R dtc/dt = GWi - GWo - A - K - S + M K M K S A M GWo Simulation Mode inventory (kmol) 2- SO 4 Year North basin yr mean 4 LRL South basin 2 A k= 2 yr k= 5 yr k= 1 yr -1 2 B with algal uptake C without algal uptake Year 5

6 Water Quality Management Plan Specify Use Establish Criteria Often based on empirical models Determine Cause/Effect Modeling Cost/Benefit Analysis Social/other non-monetary impact analysis Identify Alternatives All models are wrong; some are useful 6

7 Chapra Model c = concentration (M L -3 ) W = loading (M T -1 ) a = assimilation factor (L 3 T -1 ) c = 1 a W Simplest Mass Balance Model Q in, C in Q, C C, V V dc dt = QC QC k CV in in 7

8 Steady State Solution V dc = Q C Q C k C V dt in in dc = dt Q C = Load = W = b C Q + in in kv W W C = = Q+ kv a g Hydrologic Budget River inflows (Q Precipitation (P) in ) Evaporation (E) Outflow (Q) Ground water inflow (G in ) Ground water outflow (G out ) Q in + P + G in (Q + G out + E) = dv/dt = ΔS (= ) 8

9 Hydrologic Budgets Flux (cm) Lake Little Rock Torch Lake Superior Lake Q in 56.3 (41%) 113 (94%) P 8.4 (59%) 82.4 (92%) 77.4 (6%) G in 7.5 (8%) ~ E 39.8 (32%) 5.8 (56%) 55 (4%) Q 85.6 (68%) 116 (96%) G out 39.1 (44%) ~ 9

10 Chapra Model Q in, C in Q, C C, V V dc dt = QC QC k CV in in Chapra model again dc = Steady State t dt Q C = Load = W = C Q + kv C in = in W Q+ kv W = a b g 1

11 Kinetics: Points to learn Rate laws (zero-, first-, second-, pseudo-first-order, Monod, reversible or deficit ); Methods to assess rate law (graphical [integral], differential); Temperature dependence Arrhenius equation θ values Q 1 Little Rock Lake 11

12 SO4 2- (μm) Timestep Effects 1-month 2-month 6-month 12-month Measured values Year LRL Conceptual Model Groundwater Discharge W GWin Atmospheric Deposition W A Lake SO 4 2- Groundwater Discharge W GWout Algal Sedimentation W alg Microbial Sulfate reduction W bac 12

13 Sulfate Reduction 1 8 O 4 2- Conc (μm) 6 4 S Time (hr) Common Rate Laws dc k dt dc = kc dt dc 2 = kc dt dc C = Vm dt K + C = Zero order m First order Second order Michaelis-Menton 13

14 Differential Evaluation 3 n=1.7, k=.4 hr -1 r 2 =.46, P <.5 log(dc/dt) log[so 4 2- ] Second Order Reaction n=2, k=.62 L mole -1 hr -1 r 2 =.65, P <.1 1/[SO 4 2- ] Time (hr) 14

15 Evaluation of rate constant k=.36 hr -1 r 2 =.39, P < o ln n[so 4 ]/[SO4 ] Time (hr) Michaelis-Menton Kinetics Ra ate (μm hr -1 ) V = 11 µm hr -1 m K m = 2 µm r 2 = SO 4 2- Conc (μm) 15

16 Integration revisited n=1, k=.54 hr -1 r 2 =.81, P <.1 ln(c/c o ) Time (hr) Temperature dependence of rates.4 te (nmol cm 3 hr -1 ) Ra Temperature ( o C) 16

17 Temperature Corrections F E A exp H G I K J k = A RT Arrhenius Equation k k 1 2 = Θ θ = exp Q 1 T2 T1 F H G E A RT T 2 1 I KJ kt + = θ = k 1 1 T Free Energy Course of Reaction, ξ E A = Activation Energy ΔG o Temperature dependence of rates 17

18 Process equations: 1. Sulfate reduction (SR, kg yr -1 ) θ T SR = k [ SO ] A k =.37 m yr -1 θ = 1.41 (Q 1 = 1.5) 2. Biological uptake, settling (BU, kg yr -1 ) BU NPP S C algae = F H G I K J algae A SO 4 2- Model for LRL Change in lake SO 2-4 = Sum(Inputs) Sum (Outputs) + Reaction dc 2 SO4 V = Loading-Outflow± SulfateReduction+BiologicalUptake dt dc S V W QC k C A NPP ( ) 2 SO4 T Tr alg ae = θ 2 SO 4 dt C a lg ae 18

19 Nitrogen in Lake Superior Conc. (μg N/L) NO3 NH4 PON DON Year Atmos.Deposition River inflows Total N Denitrification Outflow Sedimentation Porewater diffusion 1 state variable, 6 processes 19

20 Copper Cycle Cu (aq) Biotic uptake Sorption Cu (p) Precipitation Dissolution (leaching) Resuspension Settling 2