User Interface Tutorial — 1D Equilibrium Sorption with Solute Decay

This tutorial describes the 1D Equilibrium Sorption with Solute Decay applet, which computes analytical solutions for one-dimensional advective-dispersive solute transport with equilibrium linear sorption and optional first-order decay. Five solution types are available, ranging from an instantaneous pulse in an infinite domain to a finite-duration input with source decay.

For the mathematical background, see the Solution Tutorial.

1 Interface Overview

The applet window has three regions: a model selector bar at the top, a left parameter panel, a center plot canvas, and a right data panel.

Transport Parameters

Disp. Coeff. (D)

☐ Retardation (R)

Pore Water Vel. (v)

Concentrations / Pulse

C(0,t) input / Mass (M)

C(x,0) initial / Area (A)

Solution Type

● Fixed pos. ○ Fixed time

Scaling & Options

● Auto ○ Manual

☐ Fill Curves ☐ Multiple

Compute

Clear Curves

Show Data Table

UI Tutorial

Plot canvas — C vs x or C vs t curve(s)

Credits bar — UIUC logo and attribution

Cursor Data

Color x/t Conc.

Move cursor over plot

User Data Points

↑ Load File

≡ Paste Data

x/t: [___]

Conc: [___]

+ Add

Model selector bar (top): Drop-down to choose one of the five solution types. Changing it updates the visible parameter fields.
Left panel: All input parameters, solution type, scaling options, and action buttons.
Center canvas: Plots concentration C as a function of distance x (Fixed Time) or time t (Fixed Position). Multiple curves can be overlaid.
Right panel: Live cursor readout of C values on the plot, and user data point import for comparing measured data to the computed solution.

2 Model Selection

The drop-down at the top selects the analytical solution. Each model corresponds to different boundary and initial conditions. Changing the model updates which parameter fields are visible in the left panel.

Model	Name	Description
Model 0	Instantaneous Pulse, Infinite Domain	A slug of solute mass M is released at x = 0, t = 0 in an infinite column. No inlet/outlet boundaries.
Model 1	Continuous Input, Semi-Infinite — First-Type BC	Constant source concentration C₀ applied at x = 0 for all t > 0 (Dirichlet / first-type boundary condition).
Model 2	Finite Duration Input, Semi-Infinite — First-Type BC	Source concentration C₀ is applied from t = 0 to t = t_c, then shut off. First-type inlet BC.
Model 3	Continuous Input, Semi-Infinite — Third-Type BC	Constant source applied via a flux (Cauchy / third-type) boundary condition at x = 0.
Model 4	Finite Input, Semi-Infinite — Third-Type BC with Decay	Finite-duration flux input with first-order decay of both the source (α) and dissolved/sorbed solute (λ).

Note: First-type (Dirichlet) and third-type (Cauchy/flux) boundary conditions give different results, especially at early times and high Péclet numbers. The third-type BC is more physically correct for most column experiments.

3 Parameters

Transport Parameters (all models)

Parameter	Symbol	Default	Description
Disp. Coeff.	D	0.5	Hydrodynamic dispersion coefficient [L²/T]. Combines mechanical dispersion and molecular diffusion.
Retardation	R	1.00	Linear equilibrium retardation factor R ≥ 1. Enable with the checkbox; when unchecked, R = 1 (no sorption). R = 1 + (ρ_b/n)K_d.
Pore Water Vel.	v	1.0	Average seepage (pore water) velocity [L/T].

Pulse Parameters (Model 0 only)

Parameter	Symbol	Default	Description
Mass of Solute	M	1.0	Total mass of solute released at t = 0 [M].
X-sect. Area	A	1.0	Cross-sectional area of the column [L²].
Porosity	n	1.0	Effective porosity of the porous medium [dimensionless].

Concentration Parameters (Models 1–4)

Parameter	Symbol	Default	Description
C(0,t) input	C₀	1.0	Source concentration applied at the inlet x = 0 [M/L³].
C(x,0) initial	C_i	0.0	Initial concentration throughout the column at t = 0 [M/L³].
Pulse Duration	t_c	10.0	Duration of the finite-duration source input (Model 2 only) [T].

Decay Parameters (Model 4 only)

Parameter	Symbol	Default	Description
Input Decay	α	0.0	First-order decay rate of the source concentration: C_source(t) = C₀ e^−αt [1/T].
1st-order Decay	λ	0.0	First-order decay rate of solute in the aqueous and sorbed phases [1/T].

Tip: Setting α = λ = 0 in Model 4 reduces it to a continuous third-type input without decay, equivalent to Model 3.

4 Solution Type

Choose what the x-axis of the plot represents:

Option	Plot	Enter value for	Result
Fixed Position	C vs Time	Distance x [L]	Breakthrough curve — concentration over time at a fixed observation point downstream.
Fixed Time	C vs Distance	Time t [T]	Concentration profile snapshot — concentration along the column at a fixed instant in time.

Enter the fixed position distance or fixed time value in the adjacent numeric field. The Points field controls the number of sample points along the curve (default 150).

5 Scaling

Mode	Description
Auto	The x-axis range is determined automatically from the computed solution. The concentration (y) axis always auto-scales.
Manual	Enter explicit min and max values for the x-axis (distance or time). Useful for comparing curves computed with different parameters over a consistent range.

6 Compute & Export

Compute and Clear

Button	Action
Compute	Evaluates the analytical solution with the current parameters and draws the curve on the canvas.
Clear Curves	Removes all curves from the plot and resets the canvas.

Multiple Curves Mode

Check Multiple Curves in the Options section before clicking Compute. Each subsequent Compute adds a new curve in a different color without erasing the previous ones. An Active Curves list appears in the left panel showing each curve's color and parameters, with a × button to remove individual curves.

Tip: Use Multiple Curves mode to compare the effect of varying a single parameter — for example, increasing D while holding v and R constant — without clearing previous results.

Fill Curves

Check Fill Curves to shade the area under each curve. Useful for visualizing integrated solute mass passing a point or present in the column.

Show Data Table

Click Show Data Table to open a modal with the full numerical dataset for all active curves. Use the tabs at the top to switch between curves. Export options:

Button	Output
Download CSV	Comma-separated file with x/t and concentration columns for the selected curve.
Download Excel	Excel-compatible file (.xls) with the same data.

7 Cursor Data

Move the cursor over the plot canvas. The Cursor Data table in the right panel updates live, showing one row per active curve with:

Column	Description
Color swatch	Matches the curve color on the plot.
x / t	The x-axis coordinate at the cursor position (distance or time depending on solution type).
Conc.	Interpolated concentration value on that curve at the cursor position.

8 User Data Points

Measured field or laboratory data can be overlaid on the computed curves for visual comparison. Three ways to add points:

Method	Description
Load File	Upload a CSV, TSV, or plain text file with two columns: x/t and concentration. Header rows are detected and skipped automatically.
Paste Data	Paste tab-, comma-, or space-separated data directly from a spreadsheet into the text area. Accepts the same two-column format.
Manual Entry	Type x/t and concentration values into the input fields and click + Add to add one point at a time.

Added points appear as filled circles on the plot. Use − Remove Last to remove the most recently added point, or Clear All Points to remove all user data at once. Individual points can also be deleted using the × button in the user points table in the right panel.