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MTP - Coggle Diagram

- - - - Set up force balans for position
        
        Drag force
        
        6PievR
        
        Net collisional force solvent molecules
      - mulitply by x and rearrange
      - Take the average for many particles, X=0
      - Fill in KbT for mv^2
      - Substitute Z to solve DE
      - Assume a short time response to get rid of constant
      - solve for x^2 and set equal to root(2Dt) to find D
      - Possibly neglect inertia term
    - - D = KbT/6PietaR
  - - - Will eventually follow a gausian distribution
      - Outcome Ldiff = l^2 * n
        
        n = number of steps
        
        l = step size
      - Particle can go to the left or right after each step
      - D = l^2 / tau
        
        l and tau have been derived from KGT
  - - - Tangent with concentration profile
- - - - v = qE/6pietaR
      - Electrophoretic mobility isolated ion = q/6pietaR
      - Is usually a rough estimation
  - - - Derivation
        
        Use MS diffusion model for ions
        
        Assume dilute soltution
        
        Write in flux form
        
        Write in charge density flux Npe = eNa*sum(ziNi)
        
        Fill into constitutive equation
        
        Assume Di = D and state that Conductivity is constant
      - Two forms
        
        Flux
        
        Pe flux
        
        dpe/dt + d/dz(-Ddpe/dz+sigmaEz) = 0
        
        Contains diffusive and electro migration term
        
        Los verder op met gauss law + poisson equation
    - - Nernst planck approach
        
        Apply boundary condtions
        
        Ez = 0 for z --> infinity
        
        Phi = Phi0 for z = 0
        
        Assume ss = no pe flux because flux at boundary = 0
        
        Use the Pe flux form of the nernst planck equation
      - Thermodynamc approach
        
        Define chemical potential for + and - ions
        
        Assume to be in equilibrium d/dz=0
        
        Los op voor de concentratie gegeven de BCs
        
        Assume to have a 1:1 salt
        
        Vul in voor de vergelijking van charge density
        
        Schrijf pe in phi en schrijf rechter term als sinh = e^x-e^-x/2
        
        Geeft hetzelfde resultaat als de potentiaal is < 100 mV
      - Debye huckel double layer
        
        Appears around charged particle/surface
        
        Charesteristics
        
        Charged area: buildup of counterions
        
        Stern layer
        
        Layer of counterions attached to surface
        
        Debye length
        
        Ld = root(D*ep/sigma)
        
        Gives information about the dependence of the concentration, charge density, electric field and potential on position
  - - - Charge density
        
        Gauss law
        
        pe = ep*dEx/dx
        
        pe = eNaCsum(zixi)
        
        Occurs by buildup one respective ion
        
        Poisson equation
        
        pe = -e*d2phi/dx2
      - Conductivity electrolyte
        
        s = eNaFC/RT * sum(Dizi2xi)
        
        Ability to carry current
    - - Electric field
        
        Externally applied or by ionic distribution or surface charge particle
        
        Potential is higher at + electrode, so use E = -dv/dx
        
        Direction electric field (Ex, Ey, Ez) = movement + charged particle
      - Surfaces of dielectric materials aquire a surface charge in water, for example glass
      - For solving
        
        Use Kb = R/Na and D = kbt/6pietaR
        
        Only zeta-potential can be measured
  - - - Use Navier stokes
        
        No pressure gradient
        
        No gravity
        
        1D unidirectional flow in ss
        
        With addtion of PeE
      - vx = system constant * uEOF
        
        Entirely flat except in the debye huckel layer
        
        uEOF = phiepEz/eta
    - - Two electric fields
        
        One external parralel to electric double layer
        
        One caused by the surface potential of tube
      - Tube must be small to have significant flow
  - - - Ld>>R
        
        Muion = q/6PietaR
      - R>>Ld
        
        Phi*ep/eta
        
        Particle moves as consequence of the force of the electroosmotic flow on the particle
- - - - Incompressible
      - Newtonian fluid
      - Geen inertia
      - Kan dus voor unidirectional laminar flow of Re<1
      - Steady state
    - - Electrostatic is een body force (EVPe)
  - - - Number of IC: number of time derivatives
      - Number of BC: number of space derivatives
- - - - Lambda/Vrms
    - - 2mvx momentum transfer between particles and wall
      - Integraal van de dichtheid(N), snelheiddis dx, volume (Avxdt), momentum
      - P = N kb T, zelfde als ideale gaswet
    - - Depends on the particle density and diameter of the molecules
    - - Is symmetric around zero and thus cannot be used to calculate the number of collisions
      - Velocity is a vector (x,y,z)
      - Follows maxwells velocity distribution
    - - Speed is a magnitude
      - Is the product of the three velocity distributions *4V^2dv
      - Is no longer symmetric
      - Formulas
        
        Vrms
        
        Vmp
        
        Vavg
- - - - CO2 drags N2 against gradient
    - - Behoud van massa gaat tegen gradient in
  - - - Opposing friction forces
        
        Frx2(u1-u2) of Fr((N1x2-N2x1)/C)
        
        Friction constant between species x molefraction other specie x difference in diffusive velocity
        
        Friction coefficient = RT/D
      - Driving forces for diffusion
        
        Has units of Newton/mole
        
        Gradient in chemical potential
        
        Assuming an ideal mixture (only mixing entropy)
        
        -dmu/dx=-d/dx(RTln(xi)) = (-RT/xi)*dx/dx
        
        Coulomb forces for ions
        
        F = -FaziE
        
        E=dv/dz
        
        Fa = faraday constant
    - - Simplyfing assumtions
        
        Difference form of MS
        
        Average velocities
        
        Average mole fractions and concentrations
        
        Let op het teken van dx, als er tegengestelde gradienten zijn
        
        Let op concentratie is in moles/m^3
        
        molfractie gas:ideale gaswet of vapour pressure
        
        1D diffusion
        
        Simple film model
        
        gradient is assumed linear in film
        
        Diffusion only occurs in small film next to interface
        
        Bulk is well mixed by eddies
      - Non-ionic mixture
        
        Velocity form
        
        Flux form
        
        Do not forget that flux has direction
      - electrolyte mixture
    - - Bootstraps
        
        Problem specific information
        
        Examples
        
        Trace stagnant
        
        N1=0
        
        Reaction stoichiometry
        
        N1=-rN2
        
        Use alhpa with 2 reactions (alhpa is fraction towards that reaction)
        
        Bulk stagnant
        
        N2=0
        
        Equimolar exchange
        
        N1=-N2
        
        Dilute electrolyte solution
        
        No current N1=N2
        
        Account only friction with solvent
        
        Electroneutrality sum of xizi=0
        
        Must mean that diffusivities of ions must be equal
        
        One ion will drag the counterion
        
        Cause small potential difference
        
        xsol=1
        
        Nsol=0
        
        Om de current te bepalen doe je sumziNi omdat een postieve stroom de andere kant op heeft hetzelfde effect als een negatieve stroom
        
        Constant pressure
        
        N1+N2=0
        
        No accumulation N2=0
        
        Needed to decrease number of unknown variablels
      - Only J-1 equations are independent
        
        Solving for variable: specific adding of the equations
        
        Because all molfractions add up to 1
        
        Can be solved when number of unknown variabels = J-1
- - - - Teveel variatie van eigenschappen
      - 74 nm< voor gassen
      - 8 nm< voor vloeistoffen