2 k c c c c d d v s i dry
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IIT Bombay Slide 5 Advection-Diffusion equation Combined advection-diffusion equation 2 K C C C C = d D v . . s. i. dry 2 t z t z C = f (t,z) D i : Diffusion

  1. IIT Bombay Slide 5 Advection-Diffusion equation • Combined advection-diffusion equation ∂ ∂ ∂ ∂ 2 K C C C C = − − ρ d D v . . s. ∂ ∂ ∂ ∂ i. dry η 2 t z t z C = f (t,z) D i : Diffusion coefficient K d : Distribution coefficient Environmental Geomechanics Lecture No. 24 D N Singh

  2. IIT Bombay Slide 6 Factors deciding type of Contaminant transport mechanism • Grain size • Density • Seepage velocity • Concentration • Viscosity • Hydraulic conductivity Factors affecting the behavior of contaminant • Contaminant • Soil condition • Mechanism Environmental Geomechanics Lecture No. 24 D N Singh

  3. IIT Bombay Slide 7 Concentration, C, of a contaminant in the porous media ( ) = μ, ρ C f D, S, V , T , , g, l, l , t, soil properties μ s f f C : the concentration of contaminant in the pore water (ML -3 ) µ : the dynamic viscosity of the fluid (ML -1 T -1 ) D: the diffusion coefficient (L 2 T -1 ) S : the mass of the adsorbed contaminant/unit volume (ML -3 ) V s : corresponds to the interstitial flow velocity (LT -1 ) T f : the surface tension of the fluid particle interface (MT -2 ) ρ f : the fluid density (ML -3 ) g : the acceleration due to gravity [LT -2 ] l : the characteristic macroscopic length [L] l µ : the characteristic microscopic length (particle size) [L] t: the time [T]. Environmental Geomechanics Lecture No. 24 D N Singh

  4. IIT Bombay Slide 8 Coefficients of Contaminant Transport Mechanisms Dimensionless Number Dimension Evaluation Ensures similarity of C concentrations at homologous Concentration Number ρ points in the model and f prototype Ensures kinematic similarity of V s . t Advection Number motion in the model and l prototype Ensures similarity of diffusion Dt Diffusion Number process in the model and 2 l prototype Ensures similarity of capillary ρ . g . l . l u f Capillary Effects Number effects in the model and T prototype f Ensures similarity of adsorption S Adsorption Number process in the model and ρ f prototype gt 2 Scaling is not done for Dynamic Effects Number contaminant flows. Significant in l the case of dynamic events only. Environmental Geomechanics Lecture No. 24 D N Singh

  5. IIT Bombay Slide 9 Discrepancies ρ It is N times higher in the model. V s . l f u Reynolds Number (R e ) Scaling is not required if R e <1 µ (i.e. for laminar flow) It is N times higher in the model. For V l low velocities dispersion is s . u Peclet Number (P e ) dependent of velocity and can be D modelled accurately (i.e. P e <1) The relation between Pe and Re numbers depends only on the contaminant. P e = µ /( ρ f D ) R e µ : the viscosity of the contaminant (solution) ρ f : the density of the contaminant solution D : the coefficient of diffusion for the contaminant ρ f : the fluid density V s : the seepage velocity l u : the characteristic microscopic length (such as particle size) and is equal to either d 10 (or d 50 ) or the mean particle size of the soil. Environmental Geomechanics Lecture No. 24 D N Singh

  6. IIT Bombay Slide 10 100 10 Peclet number dispersion advection 1 advection-diffusion diffusion 0.1 0.01 0.1 1 10 Reynolds number Sreedeep S., Berton, C., Moronnoz, T. and Singh, D. N., "Centrifuge and Numerical Modeling of Contaminant Transport Through the Unsaturated Silty Soil", ISSMGE International Conference on From Experimental Evidence towards Numerical Modelling of Unsaturated Soils , September 18/19, 2003, Bauhaus-Universität Weimar. 2003. Environmental Geomechanics Lecture No. 24 D N Singh

  7. IIT Bombay Slide 11 Sorption Absorption Adsorption atoms or molecules move atoms or molecules move from into the bulk of a porous the bulk phase (that is, solid, material, liquid, or gas) onto a solid or e.g. the absorption of water liquid surface. by a sponge e.g. purification by adsorption where impurities are filtered from liquids or gases by their adsorption onto the surface of a high-surface-area solid such as activated charcoal Environmental Geomechanics Lecture No. 24 D N Singh

  8. IIT Bombay Slide 12 Terms related to Sorption • Adsorbates - molecules that have been adsorbed onto solid surfaces • Substrate or Adsorbent - the surface to which adsorbates are adsorbed • e.g. in case of adsorbed cations tightly held on surfaces of negatively charged dry clay particles, clay particle is substrate and cations are adorbates. cations Clay particle Environmental Geomechanics Lecture No. 24 D N Singh

  9. IIT Bombay Slide 14 Simple representation of these processes Adsorbed Water cation molecule Clay layers Dry condition (cations strongly The water molecules The cations get fully sorbed on clay wedge into the inter-layer particles) hydrated, which results in after adding water their desorption from clay surface Environmental Geomechanics Lecture No. 24 D N Singh

  10. IIT Bombay ENVIRONMENTAL GEOMECHANICS CE-641 Department of Civil Engineering DR. D. N. SINGH dns@civil.iitb.ac.in www.civil.iitb.ac.in/~dns

  11. IIT Bombay 7.11.2015 Lecture No. 24 Lecture Name: Geomaterial Characterization Sub-topics Thermal Characterization Importance Methodologies Thermal properties Influence of Various soil specific Parameters Centrifuge Modelling Environmental Geomechanics Lecture No. 24 D N Singh

  12. IMPORTANCE IIT Bombay (in REAL LIFE SITUATIONS) HIGH LEVEL RADIOACTIVE WASTE DISPOSAL HIGH VOLTAGE UNDERGROUND POWER CABLES ROADS, PIPELINES, STRUCTURES IN COLD REGIONS AGRI- & AQUA-CULTURE FIELDS/ SOLAR PONDS GROUND IMPROVEMENT TECHNIQUES (SOIL HEATING & FREEZING) ENERGY CONSERVATION SCHEMES TRANSMISSION OF HOT FLUIDS (CHEMICALS/GAS) HEAT LOSS FROM THE BASEMENTS OF BUILDINGS Environmental Geomechanics Lecture No. 24 D N Singh

  13. IIT Bombay THERMAL PROPERTIES THERMAL RESISTIVITY (inverse is Conductivity, k) R T (inverse is Conductivity, k) THERMAL DIFFUSIVITY ( α ) SPECIFIC HEAT (C p ) C p =(R T . ρ . α ) -1 ρ is the density of the media K CAN BE CORRELATED TO HYDRAULIC CONDUCTIVITY Environmental Geomechanics Lecture No. 24 D N Singh

  14. IIT Bombay Factors Influencing Thermal properties of Geomaterials Type of Soil Moisture Content Distribution and Size of the Grains Density of the Soil Temperature and Pressure Presence of Contaminants Method of Measurements C p , R T , and α can be used for geomaterial characterization Environmental Geomechanics Lecture No. 24 D N Singh

  15. IIT Bombay The Transient Method Thermocouple Power supply leads leads (T-type) 95 mm Thermocouple Thermocouple leads Nichrome wire 6mm dia copper tube Stainless steel tube of dia 1.2mm Thermal probe Insulated T-type Thermocouple Grounded junction T-type thermocouple Environmental Geomechanics Lecture No. 24 D N Singh

  16. IIT Bombay Thermal probes and thermocouples Environmental Geomechanics Lecture No. 24 D N Singh

  17. IIT Bombay A.C. Power Supply • Constant Power Supply Unit Set Off on 0 300 600 900 1200 1500 Switch • • • • 0000 000.0 big small S Current 0 Timer Temperature indicator 000.0 000.0 000.0 Temperatures Field Thermal Probe Fine tuning Coarse tuning Environmental Geomechanics Lecture No. 24 D N Singh

  18. IIT Bombay Various Devices used for Thermal Property Determination Laboratory thermal probe Field thermal probe THERMODET DDTHERM (software) Environmental Geomechanics Lecture No. 24 D N Singh

  19. IIT Bombay Transient Method r Governing Equation for Line Heat Source in an Infinite Medium ∂θ  ∂ θ ∂θ  2 1 = α  +    ∂ ∂ ∂ 2   Initial and boundary conditions: t r r r θ = θ 0 , for t = 0, r = ∞ ∂ θ = − π lim 2 . k. r Q ∂ → r r 0 Solution of the Differential Equation: ( ) ( )   − ∞ n n Q 1 u ∑ − = − − − 2 θ θ γ   r ( ) lnu = 0 π u 4 k n.n!   = 1 n α 4 t γ is the Euler’s constant and is equal to 0.5772. Environmental Geomechanics Lecture No. 24 D N Singh

  20. IIT Bombay For r → 0 and t →∞ , the higher order terms of u can be neglected 100 (a) Q t 80 − = θ θ 2 ( ) ln 2 1 π 4 k t 60 s 1 40 − 1   Q = R s. θ ( 0 C) 20   0.1 1 10 100 T  4 π  100 (b) 80 60 40 20 0 5 10 15 20 25 30 35 40 t (min) Environmental Geomechanics Lecture No. 24 D N Singh

  21. Details of the thermal property detector IIT Bombay (THERMODET) Power leads Thermocouple leads Cap of the probe Rubber washer Top cap 25 mm thick Styrofoam 5 mm thick Perspex disk 220 mm long SS tube Compacted soil 140 mm Thermocouple Thermal probe 25 mm thick Perspex disk 20 mm thick Styrofoam Rubber washer Bottom cap 70 mm Environmental Geomechanics Lecture No. 24 D N Singh

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