{"id":92203,"date":"2026-04-18T18:00:52","date_gmt":"2026-04-18T18:00:52","guid":{"rendered":"https:\/\/gaeatech.com\/knowledge-center\/?p=92203"},"modified":"2026-04-13T23:35:31","modified_gmt":"2026-04-13T23:35:31","slug":"pollutev10-example-9-phenol-diffusion-freundlich-sorption","status":"publish","type":"post","link":"https:\/\/gaeatech.com\/knowledge-center\/pollutev10-example-9-phenol-diffusion-freundlich-sorption\/","title":{"rendered":"POLLUTEv10 Example 9: Diffusion with Freundlich Non-Linear Sorption (Phenol in Clay)"},"content":{"rendered":"\n<p>In <strong>POLLUTEv10 Example 9<\/strong>, the model advances beyond linear sorption by incorporating <strong>Freundlich non-linear sorption<\/strong> to simulate the diffusion of <strong>phenol<\/strong> through a clay specimen. This example reflects more realistic contaminant behavior, particularly for <strong>organic compounds<\/strong> that do not follow simple linear partitioning.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-problem-overview\">Problem Overview<\/h2>\n\n\n\n<p>This example simulates a <strong>laboratory diffusion test<\/strong> with the following conditions:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Contaminant:<\/strong> Phenol<\/li>\n\n\n\n<li><strong>Soil:<\/strong> Clay (7 cm thick)<\/li>\n\n\n\n<li><strong>Transport mechanism:<\/strong> Diffusion only (no advection)<\/li>\n\n\n\n<li><strong>Sorption model:<\/strong> Freundlich non-linear isotherm<\/li>\n\n\n\n<li><strong>Bottom boundary:<\/strong> Impermeable (zero flux)<\/li>\n\n\n\n<li><strong>Source type:<\/strong> Finite mass<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-source-conditions\">Source Conditions<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Initial concentration (<strong>co<\/strong>): 50 mg\/L<\/li>\n\n\n\n<li>Leachate head (<strong>Hr<\/strong>): 6.5 cm<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-times-of-interest\">Times of Interest<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>200 hr<\/li>\n\n\n\n<li>400 hr<\/li>\n\n\n\n<li>600 hr<\/li>\n\n\n\n<li>800 hr<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-conceptual-model\">Conceptual Model<\/h2>\n\n\n\n<p>The system consists of:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A <strong>finite mass source<\/strong> of phenol at the top<\/li>\n\n\n\n<li>A <strong>clay layer<\/strong> where diffusion and sorption occur<\/li>\n\n\n\n<li>An <strong>impermeable base<\/strong> preventing downward flux<\/li>\n<\/ul>\n\n\n\n<p>Unlike constant concentration sources, the finite source means:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Concentration at the top <strong>decreases over time<\/strong><\/li>\n\n\n\n<li>Transport is influenced by both <strong>diffusion and depletion<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-input-parameters\">Input Parameters<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Property<\/th><th>Symbol<\/th><th>Value<\/th><th>Units<\/th><\/tr><\/thead><tbody><tr><td>Darcy Velocity<\/td><td>va<\/td><td>0.0<\/td><td>cm\/hr<\/td><\/tr><tr><td>Diffusion Coefficient<\/td><td>D<\/td><td>0.019<\/td><td>cm\u00b2\/hr<\/td><\/tr><tr><td>Freundlich Coefficient<\/td><td>Kf<\/td><td>2.0<\/td><td>cm\u00b3\/g<\/td><\/tr><tr><td>Sorption Exponent<\/td><td>\u2014<\/td><td>0.628<\/td><td>&#8211;<\/td><\/tr><tr><td>Soil Porosity<\/td><td>n<\/td><td>0.46<\/td><td>&#8211;<\/td><\/tr><tr><td>Dry Density<\/td><td>\u2014<\/td><td>1.47<\/td><td>g\/cm\u00b3<\/td><\/tr><tr><td>Soil Layer Thickness<\/td><td>H<\/td><td>7.0<\/td><td>cm<\/td><\/tr><tr><td>Number of Sub-layers<\/td><td>\u2014<\/td><td>14<\/td><td>&#8211;<\/td><\/tr><tr><td>Source Concentration<\/td><td>co<\/td><td>50.0<\/td><td>mg\/L<\/td><\/tr><tr><td>Leachate Height<\/td><td>Hr<\/td><td>6.5<\/td><td>cm<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-understanding-freundlich-non-linear-sorption\">Understanding Freundlich Non-Linear Sorption<\/h2>\n\n\n\n<p>The <strong>Freundlich isotherm<\/strong> describes sorption as:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>S<\/mi><mo>=<\/mo><msub><mi>K<\/mi><mi>f<\/mi><\/msub><mtext>\u2009<\/mtext><msup><mi>C<\/mi><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">S = K_f \\, C^n<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>S<\/strong> = sorbed concentration<\/li>\n\n\n\n<li><strong>C<\/strong> = \u056c\u0578\u0582\u056e dissolved concentration<\/li>\n\n\n\n<li><strong>K<\/strong><sub><strong>f<\/strong> <\/sub>= sorption capacity<\/li>\n\n\n\n<li><strong>n<\/strong> = non-linearity exponent<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-key-implications\">Key Implications<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sorption is <strong>not constant<\/strong> (unlike linear Kd models)<\/li>\n\n\n\n<li>Retardation varies with concentration<\/li>\n\n\n\n<li>Transport becomes <strong>concentration-dependent<\/strong><\/li>\n<\/ul>\n\n\n\n<p>With <strong>n = 0.628 (&lt; 1)<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sorption is <strong>stronger at lower concentrations<\/strong><\/li>\n\n\n\n<li>This causes <strong>tailing effects<\/strong> in concentration profiles<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-key-processes-simulated\">Key Processes Simulated<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-1-diffusion\">1. Diffusion<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Governed by concentration gradients<\/li>\n\n\n\n<li>Slower compared to Example 8 due to lower diffusion coefficient<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-2-non-linear-sorption\">2. Non-Linear Sorption<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Phenol interacts with clay via <strong>Freundlich behavior<\/strong><\/li>\n\n\n\n<li>Retardation is <strong>dynamic<\/strong>, not constant<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-3-finite-mass-source\">3. Finite Mass Source<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Source concentration decreases over time<\/li>\n\n\n\n<li>Results in <strong>attenuated diffusion fronts<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-4-boundary-condition\">4. Boundary Condition<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Zero flux at base \u2192 contaminant accumulates within the domain<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-importance-of-layer-discretization\">Importance of Layer Discretization<\/h2>\n\n\n\n<p>This example highlights a critical modeling requirement:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\"><strong>Accuracy depends strongly on the number of sub-layers when using non-linear sorption.<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p>Why?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Non-linear equations require <strong>finer resolution<\/strong><\/li>\n\n\n\n<li>Concentration-dependent retardation must be captured precisely<\/li>\n\n\n\n<li>Coarse discretization can lead to <strong>numerical errors<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-best-practice\">Best Practice<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Use <strong>\u2265 14 sub-layers<\/strong> (as in this example)<\/li>\n\n\n\n<li>Increase layers further for higher accuracy or steeper gradients<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-graphical-output-depth-vs-concentration\">Graphical Output: Depth vs Concentration<\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1007\" height=\"910\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-20.jpg\" alt=\"\" class=\"wp-image-92204\" srcset=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-20.jpg 1007w, https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-20-300x271.jpg 300w, https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-20-768x694.jpg 768w\" sizes=\"auto, (max-width: 1007px) 100vw, 1007px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-pdf-report\">PDF Report<\/h2>\n\n\n<div class=\"ead-preview\"><div class=\"ead-document\" style=\"position: relative;padding-top: 90%\"><div class=\"ead-iframe-wrapper\"><iframe src=\"\/\/docs.google.com\/viewer?url=https%3A%2F%2Fgaeatech.com%2Fknowledge-center%2Fwp-content%2Fuploads%2F2026%2F04%2Fpollute-example9.pdf&amp;embedded=true&amp;hl=en\" title=\"Embedded Document\" class=\"ead-iframe\" style=\"width: 100%;height: 100%;border: none;position: absolute;left: 0;top: 0;visibility: hidden;\"><\/iframe><\/div>\t\t\t<div class=\"ead-document-loading\" style=\"width:100%;height:100%;position:absolute;left:0;top:0;z-index:10\">\n\t\t\t\t<div class=\"ead-loading-wrap\">\n\t\t\t\t\t<div class=\"ead-loading-main\">\n\t\t\t\t\t\t<div class=\"ead-loading\">\n\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/plugins\/embed-any-document\/images\/loading.svg\" width=\"55\" height=\"55\" alt=\"Loader\">\n\t\t\t\t\t\t\t<span>Loading&#8230;<\/span>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"ead-loading-foot\">\n\t\t\t\t\t\t<div class=\"ead-loading-foot-title\">\n\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/plugins\/embed-any-document\/images\/EAD-logo.svg\" alt=\"EAD Logo\" width=\"36\" height=\"23\"\/>\n\t\t\t\t\t\t\t<span>Taking too long?<\/span>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<p>\n\t\t\t\t\t\t\t<div class=\"ead-document-btn ead-reload-btn\" role=\"button\">\n\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/plugins\/embed-any-document\/images\/reload.svg\" alt=\"Reload\" width=\"12\" height=\"12\"\/> Reload document\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<span>|<\/span>\n\t\t\t\t\t\t\t<a href=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/pollute-example9.pdf\" class=\"ead-document-btn\" target=\"_blank\">\n\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/plugins\/embed-any-document\/images\/open.svg\" alt=\"Open\" width=\"12\" height=\"12\"\/> Open in new tab\t\t\t\t\t\t\t<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div><\/div>\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-concentration-profiles\">Concentration Profiles<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Profiles evolve over time (200 \u2192 800 hr)<\/li>\n\n\n\n<li>Slower migration compared to conservative solutes<\/li>\n\n\n\n<li>Gradual flattening due to source depletion<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-non-linear-effects\">Non-Linear Effects<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Stronger sorption at low concentrations causes <strong>extended tails<\/strong><\/li>\n\n\n\n<li>Profiles are <strong>non-symmetric<\/strong> and more complex<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-finite-source-behavior\">Finite Source Behavior<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Peak concentrations decrease over time<\/li>\n\n\n\n<li>Diffusion front weakens as source mass is exhausted<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-practical-applications\">Practical Applications<\/h2>\n\n\n\n<p>This example is especially relevant for:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Organic contaminant transport (e.g., phenol, hydrocarbons)<\/strong><\/li>\n\n\n\n<li><strong>Landfill leachate assessments<\/strong><\/li>\n\n\n\n<li><strong>Clay liner performance evaluation<\/strong><\/li>\n\n\n\n<li><strong>Risk assessment modeling<\/strong><\/li>\n<\/ul>\n\n\n\n<p>It is particularly important when:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Contaminants exhibit <strong>non-linear adsorption<\/strong><\/li>\n\n\n\n<li>Long-term predictions are required<\/li>\n\n\n\n<li>Laboratory calibration data is available<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-best-practices-for-pollutev10-users\">Best Practices for POLLUTEv10 Users<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Always verify <strong>Freundlich parameters (Kf and n)<\/strong> from lab data<\/li>\n\n\n\n<li>Use <strong>fine discretization<\/strong> for non-linear problems<\/li>\n\n\n\n<li>Compare results at multiple time steps<\/li>\n\n\n\n<li>Be cautious when interpreting <strong>retardation (not constant!)<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-conclusion\">Conclusion<\/h2>\n\n\n\n<p>POLLUTEv10 Example 9 demonstrates how incorporating <strong>Freundlich non-linear sorption<\/strong> significantly enhances the realism of contaminant transport modeling.<\/p>\n\n\n\n<p>Key takeaways:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Non-linear sorption leads to <strong>concentration-dependent transport<\/strong><\/li>\n\n\n\n<li>Finite sources introduce <strong>time-varying boundary conditions<\/strong><\/li>\n\n\n\n<li>Numerical accuracy depends heavily on <strong>layer discretization<\/strong><\/li>\n<\/ul>\n\n\n\n<p>This example is essential for modeling <strong>organic contaminants in clay systems<\/strong>, where linear assumptions are often insufficient.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In POLLUTEv10 Example 9, the model advances beyond linear sorption by incorporating Freundlich non-linear sorption to simulate the diffusion of phenol through a clay specimen. This example reflects more realistic contaminant behavior, particularly for organic compounds that do not follow simple linear partitioning. Problem Overview This example simulates a laboratory diffusion test with the following [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":92206,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[858,1632],"tags":[1663,501,1657,24,1662,1661,1660,1664,1659,1627],"class_list":["post-92203","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-contaminant-transport-modeling","category-pollute-examples","tag-clay-soil","tag-contaminant-transport","tag-diffusion-modeling","tag-environmental-engineering","tag-freundlich-sorption","tag-geotechnical-modeling","tag-nonlinear-transport","tag-organic-contaminants","tag-phenol","tag-pollutev10"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.4 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Phenol Diffusion with Freundlich Sorption Explained - 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