4- Introduction Phs. 645: Environmental Phsics Phsics Department Yarmouk Universit hapter 4 Transport of Pollutants - e cannot avoid the production of pollutants. hat can we do? - Transform pollutants into harmless matter. - Dilute pollutants to harmless concentrations - Forbid pollutants to enter the environment b storing them (example nuclear waste) http://ctaps.u.edu.jo/phsics/ourses/phs645/hapter4 Dr. Nidal M. Ershaidat Phsics All phenomena of transport can be categorized as follows: ) Transport of momenta in flowing media (fluid dnamics) ) Transport of energ (hapter 3) 3) Transport of matter. All these processes are governed b similar equations. For us the transform to non-linear differential equations. 3 The Navier-Stokes equations The N-S equations describe the mass outflow in space-time. To understand transport of matter one should solve these equations. (Numerical analsis is needed) 4 Main idea The tool is to consider a grid at each point of which one should compute some derivative (relative to space and/or time).this implies: - The grid should be adapted to the geometr of each problem - If the dimensions involved are smaller than a possible grid then approximations should be used. Major processes in pollutants dispersion - Turbulent diffusion One should stud turbulences - A simplified approach is the gaussian plume model. 5 4- Diffusion Diffusion is the process during which a concentration of pollutants move through a fluid. The concentration varies thus with time. The origin of such a process is collisions between the atoms/molecules of pollutants and those of the carrier fluid. 6
Molecular Diffusion In this diffusion the size of the suspended molecules and that of the host molecules involved is comparable. Examples: Molecular diffusion of highl radioactive materials in clas or in ground water at rest. If not, this is called Brownian motion and this process is studied in the kinetic theor. Still, dispersion through flows of liquids (rivers, oceans) or gases (air) and turbulence phenomena associated to it are the most important processes. 7 The Diffusion Equation The concentration (x,,z), of a diffusing substance is defined b: M ( x,, z ) ( r ) 4- V here M is the corresponding mass and V is the volume in which the substance is dispersed. V is assumed to be large enough compared to a 3, where a is the mean free path between the diffusing molecules or particles. is assumed so small that V does not change during diffusion 8 Fick s Law The flux oriented in the direction of flow of the particles as a function of the concentration is given b Fick s law: F D F is the mass of diffusing particles crossing a unit area in a unit of time: D Diffusivit or diffusion coefficient e have seen the same form exactl as Fick s Law: The heat diffusion equation (proportionalit between heat flow q and the temperature gradient). Heat (energ) diffused). 9 4- Diffusivit D depends on temperature and the molecular weight of the diffusing particles and can be derived from the kinetic theor of gases. Pollutant in fluid D (m s - )* O in air O in water H S in water * At T 5 and P atm 6.4-6.6-9.36-9 The equation of continuit For a time-dependent concentration ( (r,t)), The equation of continuit, or conservation of mass of the dispersing substance gives : t +. F 4-3 Dispersion within a laminar flow For dispersion within a laminar flow with velocit u independent of the position r, the flux has a component in the direction of u, we have F u D 4-4 The first term u gives simpl the flux of a concentration which has a velocit u. And the equation of continuit becomes: t + u D 4-5
The equation of continuit + u t D The exact differential d/dt can be written as: d + u 4-7 d t t This equation is valid for an quantit varing with time and space (and u could be also a function of time and space) The equation of continuit can thus be written as follows: d D 4-8 d t 3 4-6 -D equation of continuit onsider that we are in a -D space: If we ignore diffusion Media in rest u, Moving Media u Then the local increase in at point (x,t) is: ( x,t + dt ) ( x, t ) ( x dx The concentration at (x,t+dt) originates from the point (x-dx,t) where the velocit is u dx/dt., t 4 ) ( x,t ) 4-9 -D equation of continuit Talor expansion gives: t ( x, t u x + u t x ) dx ( x x, t This is a special case of the eq. of continuit ) 5 4- Diffusion Equation The previous equation was obtained assuming One can define an operator: d d t u d + u d t t And the previous equation remains valid for an kind of media (D is assumed to be constant) D Diffusion Equation 6 Media at rest and moving media The diffusion equation + u t D 8 4-3 Examples of Diffusion can be rewritten using : d d t D is valid for (D is constant): d + u as: d t t a medium at rest (u ) a moving medium with u 4-3
Application Instantaneous Plane source in 3-D For a homogenous medium at rest and Dconstant and in -D, we have for (x): D 4- t x The Gaussian function (Q is an integration constant): x ( 4 D t ) ( x, t ) e π D t is a solution of this st order diff. eq. Q 9 4-3. Instantaneous Plane source /Instantaneous Point source in -D The factor π D t is chosen such that: ( x, t ) Q δ( x ) This can be interpreted in manners: ) At t an amount Q (kg m - ) is released in the plane x and diffuses in the x direction (this is an instantaneous plane source) ) At (point) x, t an amount Q (kg m - ) is released and diffuses in the t direction (this is an instantaneous point source) 4-4 Time dependence in terms of σ If we define the nd moment of the gaussian function (which represents the standard deviation from the center of the gaussian), i.e.: ( t ) σ x ( x, t ) dx D t 4-5 Q Then σ(t) represents the mean square distance to which the particles have diffused at instant t. In other words, at instant t, the particles having diffused in the interval [- σ, σ] represent 68% of Q. Homework: heck the integral. (Diffusion of) A finite Size loud onsider a cloud released at t in the horizontal region b b, i.e., at t, (x,) is defined b (x,) (x,) for elsewhere b b < x < Finding how this cloud diffuses means solving the diffusion equation with the given initial conditions. 4-6 Technique The cloud diffuses in a plane (,z) perpendicular to x. e can consider the cloud as a superposition of a big number of delta functions (laminar concentrations): (x,t) δ(x-x ) dx For each laminar concentration, i.e. for each delta function δ(x-x ), the solution of the diffusion equation is: dx ( x x ) ( ( )) ( ) σ t x, t e 4-8 σ π 3 4-7 Solution The complete solution, b summation on all sub clouds is: ( x x ) σ ( t ) ( x, t ) e σ π b b ( ) dx 4 4-9 4
Back to the error function The solution is a linear combination of error functions (see the Sudden exchange of heat in chapter 3) b + x b x ( x ) F + F 4- σ σ F is the error function F β π β ( ) e x dx 5 4-6 Homework: Stud the following examples: Instantaneous Line and Point Sources in 3-D ontinuous Point Source in 3-D Figure 4-4-4 Flow in Rivers 4-4 Flow in Rivers How is a concentration of pollutants that enter a river dispersed b the flow? (Another example of diffusion in a moving medium with a velocit distribution). u flow x 8 z Figure 4- Flow in Rivers: A one dimensional approach Reducing the problem to a -D problem! Assume that the pollutant is inserted over the total depth and width of the flow, thus dispersion takes place in the x direction. 9 3 Velocit vs. position in a river In a river, the velocit depends on the position : u() will be the largest near the middle of the stream and this complicates the problem a little! Technique: As alwas in phsics, we shall tr to reduce the problem to a one we know. (e saw how to deal with a constant velocit) Figure 4-3 5
h() Variation of depth with Figure 4-4 3 Second hpothesis Introducing averages over the width and weighted with depth h. No change with the vertical coordinate z e define: the weighted average velocit u u( ) h( ) d A And the weighted average concentration ( x, ) h( ) d A A is the cross-sectional surface area of the river, u volume of water per time unit 3 4-3 4-4 Using relative frames If u is the velocit in a frame of reference attached to the flow then: u ( ) u + u ( ) And the concentration (x,,t) is ( x, t ) + ( x,, ) ( x,, t ) t e shall use a Galilean transformation where the average velocit disappears, i.e.: ξ x u t, ' and t' t 33 4-5 4-6 4-7 The diffusion equation The total derivative (exact differential) d/dt can be written in the new frame as: d dt ( x u ) ξ + + t t t ξ t t ( u ) + t u + u ξ t Thus the diffusion equation can be written as: ( + ) ( + ) ( + ) t + u ξ D ξ ξ ξ + 34 4-8 4-9 The diffusion equation The diffusion equation can be reduced to: u D with ξ or )The change of velocit in the direction will dominate and the diffusion in the x direction can be neglected, i.e.: << 4-3 ξ )Mathematics (due to Talor) allow to discard the other terms leaving the one mentioned above with the initial conditions 35 4-3 Phsical arguments e shall work here with the flow ) The change of u and h with x will also be neglected and diffusion is onl determined b the derivative: F with u D 36 4-3 and make the following hpotheses: ) The advective flow (uh) is dominated b the mean concentration (u h) 4-33 6
Hpothesis +d ξ D h u h D h ξ + d u h ξ + dξ Figure 4-5 ξ + dξ The advective flow (u h) is dominated b the mean concentration (u h) 37 Phsical arguments For an rectangle in the moving frame, the total amount of pollutants remains the same. - The continuit equation + F t e found earlier (using Talor expansion) that t u ξ 38 And we can write F u D ξ D h ( ) u 4-34 39 Perpendicular Flow Integrating twice the nd order differential equation: 4-35 u h D h ξ Defining the function f( ) as: e have: ( ) ( ) ( ) f u h d f d ξ D h ( ) ( ) ( ) ( ) 4 4-36 4-37 The J Flow The flow of pollutants through a unit area perpendicular to the x direction becomes: J A ( ξ) u ( ) h( )d onsidering that J ( ξ) ξ A D h u ( ) h( ) u ξ A D h ( ) ( ) f ( ) d u ( ) h( )d ( ) ( ) ( ) h( ) d d d then: 4 4-38 4-39 The longitudinal dispersion coefficient J ( ξ) can be rewritten as follows: J ( ξ) K ξ here K is given b: K u A ( ) h( ) D h f ( ) d d ( ) K is called the longitudinal dispersion coefficient 4 4-4 4-4 7
K Diffusivit in the moving frame The conservation of mass gives: (see equation of continuit) J K t ξ ξ And if K is independent of x then: t K ξ ξ hich has the form of a diffusion equation. K being the diffusivit (the diffusion is in the moving frame) 43 4-4 4-43 -D dispersion equation If we return back from the moving frame (x,) to the frame (x,) then we can prove that: t + u x K x This is the -D dispersion (diffusion) equation we have been looking for. here K plas the role of D (Reminder: e obtained this result with the assumption of a dispersion after a certain time t) 44 4-44 K vs u K u A D h ( ) h( ) f ( ) d d 4-4 ( ) K is inversel proportional to D and depends strongl on how u varies with the width onsider the special case u K and J in the moving frame, which means that there is no transversal mixing 45 The case K In the case where D then K The phsics result is that the left hand term of the equation: K 4-45 t ξ to be finite one should have, ξ which is consistent with the equation: u D with ξ or 46 4-46 4-47 Application Back to point source diffusion The molecular diffusion, quantified b D, is ver small ( x, t ) x ( σ e ) 4-48 Q σ π onsider that an amount Q (kg/m ) of salt is dispersed in still water (u, D.3-9 m s - ), 9 then: σ t D t. 6 ( ) t For a dispersion of (x m) far from the source with a root mean square dispersion (σ), salt will need a time t given b: t σ. ears D. 6 4 9 47 4-5 The influence of turbulence 8
The influence of turbulence Turbulence: Definition A phenomenon due to a flow in motion (water in a river, air in a wind weather) instabilities. As molecular diffusion, turbulence is, from the point of view of statistical phsics, the result of collisions between the molecules, constituents of the fluid, D (diffusion constant) is replaced b a certain turbulent diffusion e) Turbulences are observed in a complex geometr like rivers as well as in uniform pipes 49 Mathematics of turbulences A stationar (no acceleration) fluid (width, depth d) in motion under gravit and friction force in equilibrium For simplicit we ll consider that: - and d are constant. - the flow has a slope such that it causes a flow with constant velocit. 5 The Renolds number hen studing viscosit, we introduce the ratio ui ρ Magnitude of the inertia force t Re 4-5 Magnitude of the viscous force uk µ z Then to estimate this ratio, one takes some characteristic velocit U and a length L where a time is extracted. And using dimensional analsis: ui ρ t ρ U / ( L U ) ρ U L Re U L uk µ U L µ ν µ z ν µ/ρ is the kinematic viscosit See textbook chapter 3 page 54 for more details 5 Limit turbulence-viscosit R e Experience shows that for R e <, viscosit dominates and for values for R e larger than viscosit is no longer capable of slowing the flow motion and turbulence becomes the dominant phenomenon. For the near atmosphere the tpical values of U and ν are respectivel: m s - and -5 m s -. For a vertical length L m, R e 9 For a Horizontal length L km, R e These values indicate a clear predominance of turbulence 5 The tangential stress onsider also that the cross-section is perpendicular. Two forces act on the fluid: - Gravit (m g sin α) - The friction force f f µ m g cosα f f d g α α m g sinα 53 A stationar fluid Instead of the normal forces we shall consider the equivalent forces per length unit. The frictional force per unit area perpendicular to the fluid is called the tangential stress (τ ) (cosα is thus included in the definition of τ) The so-called friction velocit u * is related to τ and ρ b a simple relation: τ u* 4-5 ρ 54 Figure 4-6 9
Friction velocit Putting Ssinα and since there is no acceleration then: τ ρ d g S τ ρ d g S S could also be the driving force, the force which causes the flow to run such as a depression (pressure difference) (Meteorolog) The boundar laer at the bottom of the flow can be described using the above mentioned friction velocit. u * g d S heck the dimensions See discussion and comparison with a pipe 55 4-5 Turbulence as a flux in 3-D If we replace D b the turbulent mixing ε x, ε, ε z coefficients in 3-D in the expression of the flux F : F u D 4-53 e obtain the flux for a real river F u ε x ê x ε ê ε x ε x, ε, ε z are called edd viscosities. z ê z Turbulence appears to be related to the behaviour of the fluid in the boundar laer, and it looks acceptable that u * be included in the definition of edd viscosities. z 56 4-54 Friction velocit and edd viscosities Turbulence appears to be related to the behaviour of the fluid in the boundar laer, and it looks acceptable that u * be included in the definition of edd viscosities. Using dimensional analsis (again), one has to multipl u * b a length to obtain ε. (m s - ). e multipl b the average depth of the river d. For the vertical value ε z, the value ε z.67 u* d is usuall taken.. Because the depth (d) is much smaller than the width rapid mixing (in the z direction) is assumed and ε z can be neglected.. 57 edd viscosities For the transversal edd viscosit, ε, its value depends on the kind of flow (estimations) : ε.57 u * d For straight channel ε.4 u * d irrigating canal ε.6 u * d Meandering steams For longitudinal ε x, the turbulence can be neglected in the x direction and thus transverse turbulence dominates. 58 Turbulence dominates! All the precedent arguments and the discussion in the previous paragraph lead to consider that the diffusion, where turbulence dominates can be described b the equation, replacing D b the transverse ε. + u t x K x K u h( ) u h A ε h ( ) ( ) d d d 59 4-55 with K defined as (Eq. 4-4): (D is replaced b ε ) Applications: Please stud: The Rhine example (pages 3-6) Paragraph 4-3 The Groundwater flow (Dissolution of Pollutants in soil) Darc equation: q - k grad φ q is the discharge vector (m s - ) φ is the hdraulic potential k hdraulic conductivit Hdrolog Stud of liquids (incompressible) using Pressure as the main parameter 6
Application: Dilution of pollutants 6 Application: Dilution of pollutants 6 onsider an industr, source of pollutants discharging 7 litres/da of material with a concentration of ppm of pollutants in a slowl (u m s -, u *. m s - ) meandering wide river ( >> h (assumed to be m)). q 3 3 m 3 s 4 36 ε.6..6 m s - alculate the width of the plume to two standard deviations ( σ) and the maximum concentration at 3 m downstream. max d q 4 π ε x u max.5 ppm Next Lecture hapter 5 Noise Dr. Nidal M. Ershaidat