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Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 6 Lecture 24 1 Mass Transport: Ideal Reactors & Transport Mechanisms

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MASS TRANSPORT 2

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Transport-controlled situations: In applications where reagents are initially separated, mass-transfer determines reactor behavior e.g., in combustors, fuel & oxidant must “find each other” Observed reaction rate is “transport controlled” RELEVANCE 3

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Kinetically-limited situations: Kinetics play important role, but local reactant concentrations also do e.g., premixed fuel/ oxidizer systems “law of mass action” => mass transport still plays significant role Especially true for “nonideal” reactors Gradients in concentration & temperature RELEVANCE 4

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Overall reactor volume required to convert reactants to products depends on: Apparent chemical kinetics, and Reagent-product contacting pattern in reactor Two limiting ideal cases: Plug-flow reactor (PFR) Well-stirred reactor (WSR/ CSTR) IDEAL STEADY-FLOW CHEMICAL REACTORS 5

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6 Basic chemical reactor types IDEAL STEADY-FLOW CHEMICAL REACTORS

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PFR Reacting fluid mixture moves through vessel (e.g., long tube) in one predominant ( ) direction Negligible recirculation, backmixing, streamwise diffusion Governing equations: Quasi 1D Species A mass balance: (ODE) 7

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A cross-section area-averaged reactant mass fraction j” A,w wall diffusion flux of species A P( ) local wetted perimeter A( ) local flow area Lumping heterogeneous term into an effective (pseudo-) homogeneous term: PFR 8

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Increment in reactor volume Then: If can be uniquely related to local reagent mass fraction A, then vessel volume, V PFR, required to reduce reactant composition from feed ( A1 ) to reactor exit ( A2 ): PFR 9

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10 Reciprocal reaction rate vs reagent composition plot to determine ideal reactor volume required (per unit mass flow rate of feed)

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WSR Intense backmixing even in steady flow All intra-vessel composition nonuniformities rendered negligible Reaction takes place at single composition, nearly equal to exit value Mass balance equations simplify to:, and 11

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Hence: From previous Figure, when rate increases monotonically with reagent concentration, V WSR > V PFR WSR 12

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Physical appearance can be deceiving: At low gas pressures, a short straight tube with axial flow behaves like a WSR (due to molecular backmixing) At high pressures, turbulent stirring action produced by reactant jet injection can make a tubular reactor behave like a WSR (e.g., aircraft gas turbine combustor) Radial-flow, thick annualar bed can perform like a PFR WSR 13

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Real reactor can deviate considerably from both PFR & WSR Momentum, energy & mss transport laws must be applied together for design & scale-up Reaction rate laws can involve transport factors (especially for multiphase reactors) Can often be represented as a network of interconnected ideal reactors WSR 14

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MASS-TRANSPORT MECHANISMS & ASSOCIATED TRANSPORT PROPERTIES Three mechanisms of mass transport: Convection Diffusion Free-molecular flight Analogous to energy transport First two collaborate in continuum (Kn << 1) regime 15

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Two types: Due to motion of host fluid Due to solute drift through host fluid (as a result of net forces applied directly to solute) CONVECTION 16

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Host-fluid convection: For fluid mixture in Eulerian CV, local convective mass flux Local chemical species convective mass flux CONVECTION 17

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Solute convection: “phoresis” In response to local applied force– gravitational, electrostatic, thermal, etc. Quasi-steady drift velocity, c i Relative to local mixture velocity v Contributes mass flux vector CONVECTION 18

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CONCENTRATION DIFFUSION Random-walk contributes net drift of species: Fick diffusion flux For suspended particles: Brownian flux In local turbulent flow, time-averaged mass flux: eddy diffusion flux Actually, result of time-averaging species convective flux Total mass flux of species 19

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FREE-MOLECULAR FLIGHT Travel in the absence of collisions with other molecules Net flux: algebraic sum of fluxes in different directions and at different speeds Mechanism analogous to energy transport by photons 20

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D i,eff effective mass diffusivity of species I in prevailing medium Not always a scalar, but frequently treated as such Can be a tensor defined by 9 (6 independent) local numbers When diffusion is easier in some directions, e.g.: Anisotropic solids (single crystals) Anisotropic fluids (turbulent shear flow) Diffusion not always “down the concentration gradient”, but skewed SOLUTE DIFFUSIVITIES 21

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g i force acting on unit mass of species i m i particle mass c i quasi-steady drift speed, given by: where f i friction coefficient (inverse of mobility) Relates drag force to local slip (drift) velocity Example: Stokes coeff.= 3 i,eff for solutes much larger than local solvent mean free path DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES 22

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Sedimentation: g i,eff gravitational body force c = c i,s = settling or sedimentation velocity DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES 23

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DRIFT VELOCITIES DUE TO SOLUTE-APPLIED FORCES Electrophoresis: g i,eff due to presence of electrostatic field, and either a net charge on species i, or an induced dipole on a neutral species c = c i,e = electrophoretic velocity Determines motion of ions & charged particles in fields (e.g., fly-ash removal in electrostatic precipitators) 24

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Thermophoresis: g i,eff due to temperature gradient, proportional to – grad(ln T) c = c i,T = thermophoretic velocity i c i,T = thermophoretic flux Important in some gaseous systems (e.g., high MW disparity, steep temperature gradients), and in Most aerosol systems (e.g., soot and ash transport in combustion gases, deposition on cooled surfaces) DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES 25

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Single-phase flow: Only if particles or heavy molecules follow host fluid closely Criterion: stopping time, t p Compared to characteristic flow time, t flow (L/U) t p time required for velocity to drop by factor e -1 in prevailing viscous fluid where p particle mass density d p diameter 26 PARTICLE “SLIP”, INERTIAL SEPARATION

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Stk Stokes’ number, given by Inverse Damkohler number governing dynamical nonequilibrium particles follow host fluid closely > => separate momentum equations required for each coexisting phase 27

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CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS mass fraction field for chemical species i (or particle class i) Measurable: local fluxes,, at important boundary surfaces e.g., local rate of naphthalene sublimation into a gas stream or, average flux for entire surface, 28

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j i,w ” diffusional contribution to mass flux Reference values: Dimensionless Nusselt number for mass transport Widely used for quiescent systems, and for forced & natural convection systems 29 CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

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Dimensionless Stanton number for mass transport Used only for forced convection systems area-weighted average of normal component of diffusion flux, i.e., Rarely measured in this manner 30 CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

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Dimensional coefficients: Mass flux per unit driving force May be based on, or on, or on (for gases) System of units becomes important in the definition 31 CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

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Capture Fraction, : When species i is contained in mainstream feed Ratio of actual collection rate ( ) to rate of flow of species through projected area of target, i.e.: When i,w << i,∞,, and: Generally < (1/2) cap since A w,proj ≤ (1/2) A w 32 CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

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