12 - Advective Diffusive Flux

This example corresponds to the “Example 12-Advective and Diffusive Flux of Heat and Solutes” from the Phreeqc manual. It can be retrieved from the Phreeqc Website.

The following example demonstrates the capability of PHREEQC to calculate transient transport of heat and solutes in a column or along a 1D flowline. A column is initially filled with a dilute KCl solution at 25 °C in equilibrium with a cation exchanger. A KNO 3 solution then advects into the column and establishes a new temperature of 0 °C. Subsequently, a sodium chloride solution at 24 °C is allowed to diffuse from both ends of the column, assuming no heat is lost through the column walls. At one end, a constant boundary condition is imposed, and at the other end, the final cell is filled with the sodium chloride solution and a closed boundary condition is prescribed. For the column end with a constant boundary condition, an analytical solution is compared with PHREEQC results, for unretarded Cl - (R = 1.0) and retarded Na + and temperature ( R = 3.0). Finally, the second-order accuracy of the numerical method is verified by increasing the number of cells by a factor of three and demonstrating a decrease in the error of the numerical solution by approximately one order of magnitude relative to the analytical solution.

Studies

This project contains 1 Studies.

  • PhreeqcStudy: “study_12 - Advective Diffusive Flux”

Db used: “Phreeqc_dat” database

TITLE Example 12.--Advective and diffusive transport of heat and solutes.
      Two different boundary conditions at column ends.
      After diffusion temperature should equal Na-conc in mmol/l.
SOLUTION 0   24.0 mM KNO3
    units mol/kgw
    temp  0     # Incoming solution 0C
    pH    7.0
    pe   12.0   O2(g) -0.67
    K    24.e-3
    N(5) 24.e-3
SOLUTION 1-60   0.001 mM KCl
    units mol/kgw
    temp 25    # Column is at 25C
    pH   7.0
    pe  12.0   O2(g) -0.67
    K    1e-6
    Cl   1e-6
EXCHANGE_SPECIES
    Na+ + X- = NaX
    log_k       0.0
    -gamma      4.0     0.075
    H+ + X- = HX
    log_k       -99.
    -gamma      9.0     0.0
    K+ + X- = KX
    log_k       0.0
    gamma       3.5     0.015
EXCHANGE 1-60
    KX    0.048
PRINT
   -reset   false
   -selected_output false
   -status false
TRANSPORT           # Make column temperature 0C, displace Cl
   -cells   60
   -shifts  60
   -flow_direction  forward
   -boundary_conditions flux  flux
   -lengths 0.333333
   -dispersivities        0.0     # No dispersion
   -diffusion_coefficient 0.0     # No diffusion
   -thermal_diffusion     1.0     # No retardation for heat
END
SOLUTION 0   Fixed temp 24C, and NaCl conc (first type boundary cond) at inlet
    units  mol/kgw
    temp 24
    pH  7.0
    pe  12.0   O2(g) -0.67
    Na  24.e-3
    Cl  24.e-3
SOLUTION 58-60  Same as soln 0 in cell 20 at closed column end (second type boundary cond)
    units  mol/kgw
    temp 24
    pH  7.0
    pe  12.0   O2(g) -0.67
    Na  24.e-3
    Cl  24.e-3
EXCHANGE 58-60
    NaX  0.048
PRINT
   -selected_output true
TRANSPORT            # Diffuse 24C, NaCl solution from column end
   -shifts 1
   -flow_direction         diffusion
   -boundary_conditions    constant  closed
   -thermal_diffusion      3.0       # heat is retarded equal to Na
   -diffusion_coefficient  0.3e-9    # m^2/s
   -time_step              1.0e+10   # 317 years give 19 mixes
END

/n

Plots

This project contains 1 Plots.

  • Plot 1: “Diffusion of Solutes and Heat”

DiffusionofSolutesandHeat.svg