Diffusion in PHREEQC TITLE diffusion simulation SOLUTION 1 units mol/kgw Na 0.001 Cl 0.001 SOLUTION 2-50 TRANSPORT -cells 50 -time_step 1000 -diffusion_coefficient 1.0e-4 -flow_direction[r]
(1)Physical Chemistry of Minerals and Solutions 2008/2009 Advection and Diffusion Physical Chemistry of Minerals and Aquous Solutions D.M Sherman, University of Bristol Diffusion-Limited Reactions •Reactions for which the elementary chemical steps are very fast will be diffusion controlled •Diffusion constants of ions in water range from x 10-6 to x 10-4 cm2/s •If diffusion is fast enough, the rate of the reaction will be advection controlled Page ‹#› (2) Physical Chemistry of Minerals and Solutions 2008/2009 Diffusion at the Mineral-water interface Diffusion: Ficks First Law The flux (J, in moles/cm2-s) of a chemical species is given by the concentration gradient: J = "D ! #C #x Diffusion constants of ions in water range from 10-6 to 10-4 cm2/s Page ‹#› (3) Physical Chemistry of Minerals and Solutions 2008/2009 Diffusion: Ficks Second Law The concentration as a function of t and x is found By solving the differential equation (Fick’s 2nd law): "C " 2C =D "t "x With the appropriate boundary conditions to define the problem ! Diffusion: Analytical Solutions For a simple constant-source problem, the boundary condition is C(0,t) = C0 The solution to Fick’s Law is: C(x,t) = x C0 (1" erf ) 2 Dt ! Page ‹#› (4) Physical Chemistry of Minerals and Solutions 2008/2009 Diffusion: A Cheap Solution A general approximation for any geometry is that C(x,t) " C0 when x = Dt ! Numerical Diffusion Simulations Divide system into cells of length l Allow components to diffuse from one cell to another Page ‹#› (5) Physical Chemistry of Minerals and Solutions 2008/2009 Diffusion in PHREEQC TITLE diffusion simulation SOLUTION units mol/kgw Na 0.001 Cl 0.001 SOLUTION 2-50 TRANSPORT -cells 50 -time_step 1000 -diffusion_coefficient 1.0e-4 -flow_direction diffusion_only -shifts 50 -punch_frequency 125 SELECTED_OUTPUT -file nacl.txt -totals Na Cl END Diffusion in PHREEQC Page ‹#› (6) Physical Chemistry of Minerals and Solutions 2008/2009 The Advection-Diffusion Equation: "Ci " 2Ci "C = D #v i "t "x "x ! Isotherm-Based Reactive Transport Si = f (Ci ) ! "Ci " 2Ci "Ci Ri = Di # v "t "x "x Where Ri is the retardation factor: ! Ri = 1+ "b $Si # $Ci ! Page ‹#› (7) Physical Chemistry of Minerals and Solutions 2008/2009 Problems with the Kd approach "Si = Kd "Ci is not constant! Moreover, the retardation factor approach will fail for pptn-dissolution reactions since : "Si "Ci ! is undefined ! Advection Simulations Divide system into cells Transfer solution from cell n-1 to cell n at each step Cell is always full The flow velocity = (cell length)/(time step) Page ‹#› (8) Physical Chemistry of Minerals and Solutions 2008/2009 Advection Rainwater; pH 5.6, pCO2 = 3.5 Buried DU g Schoepite; pCO2 = 3.5 With FeOOH 1g FeOOH; pCO2 = 3.5 1g FeOOH; pCO2 = 3.5 No FeOOH 1g FeOOH; pCO2 = 3.5 1g FeOOH; pCO2 = 3.5 Sorption by FeOOH retards transport The Advection-Diffusion with Kinetics: If a dissolved species is being produced (consumed) at a rate k(Cs-C) where Cs is the saturation concentration, then the concentration C will obey "C " 2C "C = D #v + k(Cs # C) "t "x "x Ignoring diffusion, the concentration will have a steady State profile when k(Cs " C) = v ! #C #x ! Page ‹#› (9) Physical Chemistry of Minerals and Solutions 2008/2009 Calcite Dissolution: Kinetics + Advection TITLE Kinetic dissolution of calcite SOLUTION pH 7.00 temp 25.00 EQUILIBRIUM_PHASES CO2(g) -3.5 1000 SOLUTION 1-20 pH 7.00 temp 25.00 KINETICS 1-20 Calcite -m0 1.0 -parms 0.3 TRANSPORT -shifts 200 in m -cells 20 -lengths in seconds to give v= 1m/300s -time_step 300 -punch_frequency 25 Calcite Dissolution: Kinetics + Advection [Ca] = 0.44 mmol/l at sat’n 0.003 m/s 0.01 m/s 0.1 m/s Resulting steady state profiles Page ‹#› (10)