Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

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

2. MASS TRANSPORT

3. RELEVANCE • 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”

4. RELEVANCE • 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

5. IDEAL STEADY-FLOW CHEMICAL REACTORS • 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)

6. IDEAL STEADY-FLOW CHEMICAL REACTORS Basic chemical reactor types

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

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

9. PFR • Increment in reactor volume Then: • If can be uniquely related to local reagent mass fraction wA, then vessel volume, VPFR, required to reduce reactant composition from feed (wA1) to reactor exit (wA2):

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

11. 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

12. WSR Hence: From previous Figure, when rate increases monotonically with reagent concentration, VWSR > VPFR

13. WSR • 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

14. WSR • 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

15. 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

16. CONVECTION • 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)

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

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

19. 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

20. 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

21. SOLUTE DIFFUSIVITIES • Di,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

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

23. DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES • Sedimentation: • gi,eff gravitational body force • c = ci,s = settling or sedimentation velocity

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

25. DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES • Thermophoresis: • gi,effdue to temperature gradient, proportional to –grad(ln T) • c = ci,T= thermophoretic velocity • rici,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)

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

27. PARTICLE “SLIP”, INERTIAL SEPARATION Stk  Stokes’ number, given by • Inverse Damkohler number governing dynamical nonequilibrium • << 1 => particles follow host fluid closely • > 10-1 => separate momentum equations required for each coexisting phase

28. 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,

29. CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS • ji,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

30. 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

31. 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

32. 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 wi,w << wi,∞ , , and: • Generally < (1/2) hcap since Aw,proj ≤ (1/2) Aw