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Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 3 Lecture 10 Constitutive Laws: Energy & Mass Transfer

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intrinsic viscosity of (non-turbulent) fluid t turbulent contribution; more dependent on local condition of turbulence than on nature of fluid 2 VISCOUS LIQUID SOLUTIONS, TURBULENT VISCOSITY

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Fourier’s Heat Flux Law: For energy diffusion (conduction) in pure isotropic solids Vector equation, equivalent to 3 scalar components in, say, cylindrical polar coordinates: 3 ENERGY DIFFUSION FLUX VS TEMPERATURE GRADIENT

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k local thermal conductivity Non-isotropic materials => k is a tensor (vector operator) 4 ENERGY DIFFUSION FLUX VS TEMPERATURE GRADIENT

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In multi-component systems (e.g., reacting gas mixtures), each diffusing species also transports energy in accordance with its enthalpy, h i Hence, Fourier’s Law must be generalized: 5 SPECIES DIFFUSION CONTRIBUTION TO ENERGY FLUX

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Consistent with requirement of locally positive entropy production for k > 0, irrespective of sign of grad T Radiative energy transport (“action at a distance”) cannot be treated as a diffusion process, must be dealt with separately. 6 SPECIES DIFFUSION CONTRIBUTION TO ENERGY FLUX

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Energy diffusion (conduction) contributes additively to local rate of entropy production: Quadratic in gradient of relevant local field density Positive for any flux direction 7 ENTROPIC ASPECTS

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In the absence of multi-component species diffusion, entropy diffusion flux vector is given by: Entropy flows by diffusion as well as convection! 8 ENTROPIC ASPECTS

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Experimentally obtained by matching results of steady-state or transient heat-diffusion experiments with predictions based on energy conservation laws & constitutive relations (in the absence of convection) Unit of k: W/ (m K) , thermal diffusivity; m 2 /s k has modest temperature dependence 9 THERMAL CONDUCTIVITY COEFFICIENT

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Chapman – Enskog - Herschfelder Expression: viscosity molar specific heat R universal gas constant 10 k FROM KINETIC THEORY OF GASES

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Mixture: cube-root law 11 k FROM KINETIC THEORY OF GASES

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CORRESPONDING STATES CORRELATION FOR THERMAL CONDUCTIVITY OF SIMPLE FLUIDS 12

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THERMAL CONDUCTIVITY OF LIQUID SOLUTIONS, TURBULENT FLUIDS No simple relations for thermal conductivity of liquid solutions Greater dependence on direct experimental data Gases & liquids in turbulent motion display augmented thermal conductivities 13

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THERMAL CONDUCTIVITY OF LIQUID SOLUTIONS, TURBULENT FLUIDS k intrinsic thermal conductivity of quiescent fluid k t turbulent contribution; more dependent on local condition of turbulence than on nature of fluid 14

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EQUIVALENCE OF THERMAL & MOMENTUM DIFFUSIVITIES Due to additional terms chemically reacting mixtures in LTCE also exhibit augmented thermal conductivities. 15

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MASS DIFFUSION FLUX VS COMPOSITION GRADIENT Fick’s diffusion-flux law for chemical species: In pure, isothermal, isotropic materials, species mass diffusion is linearly proportional to local concentration gradient Directed “down” the gradient 16

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where is local mass fraction of species i D i = Fick diffusion coefficient (scalar diffusivity) for species i transport in prevailing mixture Valid for trace constituent i, and When mixture has only two components (N = 2) 17 MASS DIFFUSION FLUX VS COMPOSITION GRADIENT

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OTHER CONTRIBUTIONS TO MULTI- COMPONENT DIFFUSION Other forces, such as pressure & temperature gradients - grad p, - grad (ln T), etc. Interspecies “drag” or “coupling”, i.e., influence on flux of species i due to fluxes (hence, composition gradients) of other species - grad, where j ≠ i 18

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CHEMICAL ELEMENT DIFFUSION FLUXES Example: local diffusional flux of element oxygen in a reacting multi-component gas mixture 19

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ENTROPY PRODUCTION & DIFFUSION ASSOCIATED WITH CHEMICAL SPECIES DIFFUSION General form of driving force for chemical species diffusion: where chemical potential, dependent on mixture composition via “activity” a i : 20

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and grad T,p spatial gradient, holding T & p constant 21 ENTROPY PRODUCTION & DIFFUSION ASSOCIATED WITH CHEMICAL SPECIES DIFFUSION

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General form of Multi-component Diffusion Flux Law where scalar coefficients, directly measurable Reciprocity relation (L Onsager): 22 ENTROPY PRODUCTION & DIFFUSION ASSOCIATED WITH CHEMICAL SPECIES DIFFUSION

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DIFFUSIONAL FLUX OF ENTROPY For the case of multi-component species diffusion in a thermodynamically ideal solution (a i = x i ): Each bracketed quantity =, partial specific entropy of chemical species i, such that 23

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Convective flux of entropy: Mixing entropy contributions ( origin of minimum work required to separate mixtures into their pure constituents 24 DIFFUSIONAL FLUX OF ENTROPY

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SOLUTE DIFFUSIVITIES IN GASES, LIQUIDS, SOLIDS– REAL & EFFECTIVE D i,eff effective mass diffusivity of species i in prevailing medium May be a tensor for solute diffusion in: Anisotropic solids (e.g., single crystals, layered materials) Anisotropic fluids (e.g., turbulent shear flow) 25

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In such cases, diffusion is not “down concentration gradient”, but skewed wrt –grad Can often be treated as single scalar coefficient, valid in any direction 26 SOLUTE DIFFUSIVITIES IN GASES, LIQUIDS, SOLIDS– REAL & EFFECTIVE

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DILUTE SOLUTE DIFFUSION IN LOW- DENSITY GASES and y j mole fraction of species j y i << 1 D i not very temperature-sensitive, varies as T n, n ≥ 3/2, ≈

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BINARY INTERACTION PARAMETERS 28

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MOMENTUM – MASS – ENERGY ANALOGY For mixtures of similar gases, D i is always of same order of magnitude as momentum diffusivity, (kinematic viscosity) and energy diffusivity, Reason: for gases, mechanisms of mass, momentum and energy transfer are identical viz., random molecular motion between adjacent fluid layers 29

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Both dimensionless ratios are near unity for such mixtures. Sc i can be >> 1 for solutes in liquids, aerosols in a gas 30 MOMENTUM – MASS – ENERGY ANALOGY

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DILUTE SOLUTE IN LIQUIDS & DENSE VAPORS D i estimated using a fluid-dynamics approach Each solute molecule viewed as drifting in the host viscous fluid in response to Net force associated with gradient in its partial pressure 31

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DILUTE SOLUTE IN LIQUIDS & DENSE VAPORS Stokes-Einstein Equation: effective molecular diameter of solute molecule i Newtonian viscosity of host solvent Also applies to Brownian diffusion of particles in a gas, when 32

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SOLUTE DIFFUSION IN ORDERED SOLIDS D i calculated from net flux of solute atoms jumping between interstitial sites in the lattice energy barrier encountered in moving an atom of solute i from one interstitial site to another 33

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SOLUTE DIFFUSION THROUGH FLUID IN PORES Interconnected pores of a solid porous structure where solid itself is impervious to solute Solute mfp D i,eff < D i-fluid Reduction depends on pore volume fraction, 34

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SOLUTE DIFFUSION THROUGH FLUID IN PORES Denominator correction for “tortuosity” (variable direction & variable effective dia of pores) usually determined experimentally Can be computed theoretically for model porous materials e.g., for impermeable spheres, = (1- ) 35

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SOLUTE DIFFUSION THROUGH FLUID IN PORES When solute mfp > mean pore diameter: e.g., gas diffusion through microporous solid media at atmospheric pressure Solute rattles down each pore by successive collisions with pore walls For a single straight cylindrical pore (Knudsen, 1909): (pore diameter plays role of solute mfp) 36

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For Knudsen diffusion in a porous solid, Independent of pressure when fluid is an ideal gas Interpolation formula, rigorous for a dilute gaseous species at any mfp/ pore size combo: 37 SOLUTE DIFFUSION THROUGH FLUID IN PORES

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Widely used to describe gas diffusion through porous solids (e.g., catalyst support materials, coal char, natural adsorbents, etc.) 38 SOLUTE DIFFUSION THROUGH FLUID IN PORES

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SOLUTE DIFFUSION IN TURBULENT FLUID FLOW Effective diffusivity, D i,t, unrelated to molecular diffusivity, but closely related to prevailing momentum diffusivity,, in local flow = number near unity (turbulent Schmidt number) e.g., tracer dispersion measurements near centerline of ducts containing a Newtonian fluid in turbulent flow ( > 2,000) reveal that: 39

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SOLUTE DIFFUSION IN TURBULENT FLUID FLOW Pe eff (Re) weak function of Re,

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SOLUTE DIFFUSION THROUGH FIXED BED OF GRANULAR MATERIAL Similar to turbulent flow in a homogeneous medium D eff nearly proportional to product of average interstitial velocity, u i, and particle size, d p In packed cylindrical duct with Re bed > 100: 41

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SOLUTE DIFFUSION THROUGH FIXED BED OF GRANULAR MATERIAL Peclet numbers weakly dependent on bed Reynolds number, near 10 & 2, resp. Time-averaged solute mixing, apparently anisotropic, much more rapid than expected based on molecular motions alone 42

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