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

MASS TRANSPORT 2

 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

 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

 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

6 Basic chemical reactor types IDEAL STEADY-FLOW CHEMICAL REACTORS

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

  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

 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

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

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

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

 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

 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

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

 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

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

 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

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

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

 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

 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

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

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

 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

 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

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

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

 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

 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

 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

 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