Block 6 Heat transport in rivers

Transcription

1 Numerical Hydraulics Block 6 Heat transport in rivers Markus Holzner

2 Contents of the course Block 1 The equations Block 2 Computation of pressure surges Block 3 Open channel flow (flow in rivers) Block 4 Numerical solution of open channel flow Block 5 Transport of solutes in rivers Block 6 Heat transport in rivers 2

3 Why model temperature? Temperature is a universal parameter, which influences all processes There are standards for temperature (or temperature increment) in rivers Rivers therefore have a limited capacity of carrying heat, which has to be managed In lakes the vertical temperature profile controls vertical mixing 3

4 Heat transport equation (1) Extensive (transported) quantity Heat energy in volume Vrc p DT Heat energy per volume rc p DT Unit: J or Ws (old unit cal: 1 cal = 4.2 Joule) Intensive quantity in transport equation Temperature T Temperatur increment DT relative to base temperature T 0 Units C or K Conversion factor rc p = 4.2 x10 6 J/(m 3 K) 4

5 Heat transport equation (2) Transport equation for temperature T T T 1 T H T q u AD Tin T t x A x x rc h A p H(T) is the heat flux (J/m 2 /s = W/m 2 ) Heat exchange through water surface Heat exchange through bed (neglected in the following) The denominator of the heat exchange term represents the heat capacity of the water column of depth h per unit area 5

6 Global energy fluxes 6

7 Heat fluxes through the water surface Long wave radiation from atmosphere HG Incoming short wave radiation (global radiation) = direct + diffuse solar radiation HS H HSW HGW HW HV H K Reflected part Reflected part Long wave radiation Evaporation / Condensation Convection (sensible heat) rshs rghg H w H v H k (1-rG)HG H GW (1-rS)HS H SW (latent heat) 7

8 Global radiation H SW S w r H 1 H SW Heat flux into water surface r S = α albedo (reflection factor), appr w Cloudiness (0 = no clouds, 1 = strongly cloudy) 0.65 takes into account diffuse radiation from clouds H S = clear sky incoming short wave radiation = f(geographic position, date, solar constant) S Top of atmosphere H S = 341 W/m 2 (almost constant) Ground surface on average in Zurich H S =100 W/m 2 (strongly varying: seasons, day-night) You will get a Matlab routine Suncalc which calculates H s 8

9 Atmospheric incoming radiation (1) H GW G H 1 r G H GW Heat flux into water surface r G Reflection factor (appr. 0.03) H G H 0 1 kw 2 G 0 H G Heat flux into water surface cloudless sky k Cloud type factor ( ) w Cloudiness 9

10 Atmospheric back radiation (2) H 0 G 0.069E T ' L Assumption: Black body 0 H G Atmospheric back radiation for cloudless sky Stefan-Boltzmann constant [W/(m 2 K 4 )] E ' E L Vapor pressure of air [mm Hg] T L Air temperature in 2 m height (standard height) [ C] Conversion factor: 1 mmhg = 1.33 hpa 10

11 Long wave radiation from water surface negative sign for outgoing flux T W Water temperature [ C] 0.97 Emissivity, consistent with r G = Stefan-Boltzmann constant [W/(m 2 K 4 )] Depends on water temperature 11

12 Evaporation/Condensation H V v E E T f z L, z S W Condensation if T W EL z E S, v z E L,z Wind speed at height z above water surface Vapor pressure of air at height z (in mm Hg) Depends on water temperature E S f(v z ) saturation pressure (function of water temperature T W ) (in mm Hg) wind forumula, general form: f v c c v c3 z 1 2 z Example: Formula after Trabert (not valid for lakes) c 1 0 W c m mm Hg c 0. 3 Conversion factor: 1 mmhg = 1.33 hpa 14

13 Convection H K f * v T T z L W Depends on water temperature v z Wind speed at height z above the water surface T L Air temperature (in C) T W Water temperature T W (in C) f*(v z ) Wind formula for convection Bowen-Hypothesis f f * c constant 2.03 b 0.49 mmhg K Conversion factor: 1 mmhg = 1.33 hpa 15

14 H GW HSW H tot H K Heat balance of lake Zurich H V H W after Kuhn, from Imboden, Physik aquatischer Systeme 16

15 Heat fluxes through the water surface Which fluxes depend on water temperature? Long wave radiation from atmosphere HG Incoming short wave radiation (global radiation) = direct + diffuse solar radiation HS H HSW HGW HW HV H K Reflected part Reflected part Long wave radiation Evaporation / Condensation Convection (sensible heat) rshs rghg H w H v H k (1-rG)HG H GW (1-rS)HS H SW (latent heat) 17

16 Equilibrium temperature (1) Water temperature T G at which H(T G ) = 0 Depends on the parameters w, v, T,, EL, z z L H S 18

17 Equilibrium temperature (2) Computation by search for zero of function using Newton-method T T G, new G, alt or by interval method HT ( ) G, old H ( T ) G, old if H ( T ) H ( T ) 0 T T L M R M if H ( T ) H ( T ) 0 T T R M L M T L T M T R 19

18 Transport equation for DT (1) Transport equation for temperature T including anthropogenic heat sources W T T 1 T H T q u AD Tin T W t x A x x rc h A Transport equation for natural temperature T n Tn Tn 1 Tn H ( Tn ) q u AD Tin, n T t x A x x rc h A p p n Form difference using DT =T T n and H H ( T ) H ( Tn) ( T Tn) T T n 20

19 Transport equation for DT (2) H DT DT 1 DT T q u AD DT DTin DT W t x A x x rc h A p Steady state case, neglecting dispersion, heat source and/or tributary contained in upstream boundary condition, uniform flow u d T dx H T D T n D rc h p T Solution DT DT e 0 1 H rc hu T p T n x 21

20 Development of temperature increment relative to average equilibrum temperature 22

21 Calculation of equilibrium Temperature if H ( T ) H ( T ) 0 T T L M R M if H ( T ) H ( T ) 0 T T R M L M T L T M T R do until T is close enough to T G Assignment 5: 1) Write function to compute H(T1, Hs, el, tl, w, v) 2) Calculate the equilibrium temperature using the interval method 3) Calculate spatial profile of temperature downstream of a power plant 23

22 Other important processes can be modelled based on the same approach: Transport of Oxygen and BOD-DO model for a stream Transport of Nitrogen Transport of Phosphorus

23 Simple oxygen model (Streeter-Phelps plus) Atmospheric O 2 Reaeration Dissolved oxygen (DO) c C-BOD (L) Bottom sludge Respiration C-BOD = Carbonaceous Biological Oxygen Demand Photosynthesis Chlorophyll A Algae 27

24 Essential processes Gas exchange ( ) Advection/Dispersion ( ) Uptake (-) Photosynthesis (+) Equations: r photo 2 c c c in sat v D 2 c c qin rphoto rbottom rresp rbod, diss k2( c c) t x x 2 L L L in v D ( L L) q 2 in rbod, diss t x x 28

25 Numerical solution Discretize river in N boxes of length Dx Upstream boundary condition: given L 0 and c 0 (containing pollution source) Downstream boundary condition: transmission boundary Use timestep according to Cr=1 (Method of characteristics) L or c (mg/l) L c x (m) 36

26 Improved oxygen model Atmospheric O 2 Reaeration NH 4 + NO 2 - Nitrification Dissolved Oxygen (DO) C-BOD Bottom sludge Sedimentation Nitrification NO 3 - Photosynthesis Respiration/Decaying dead biomass Chlorophyll A Algae Consequence: 7 coupled transport equations, 1 trivial (bottom sludge, u=d=0) 37

27 Advanced oxygen and algae model (QUAL2k) Org. N Reaeration Atmospheric O 2 C-BOD Sedimentation Dissolution from sediment Bact. degradation NH 4 + NO 2 - NO 3 - Nitrification Nitrification Photosynthesis Dissolved oxygen (DO) Respiration Sedimentation Bottom sludge Org. P Bact. degradation Dissolved P Nutrient uptake Die-off of algae Chlorophyll A Algae Nutrient uptake Die-off of algae 38

28 Advanced quality model 10 transport equations, 2 trivial (u=d=0) See One-dimensional river and stream water quality model intended to represent a well-mixed channel both vertically and laterally with steady state hydraulics, non-uniform steady flow, and diel heat budget and water-quality kinetics. 39

29 HEC-RAS

30 Application: Hydro- and thermopeaking