In  Out  Generation or Removal  Accumulation Reactor Concepts Reactor – any device in which an incoming constituent undergoes chemical (or biochemical) transformation, phase transformation, or phase separation Constituent – soluble, colloidal, and particulate substances Starting point for analysis of reactor performance – material balance relationship (input, output, and reaction terms) Net rate of mass transport through the control volume Net rate of mass transformation within the control volume (Vc ) Net rate of mass change within the control volume (Vc ) Net rate of mass input across the CV boundaries Net rate of mass output across the CV boundaries Net rate of reaction within the CV (Vc ) Net rate of accumulation within the CV (Vc ) In  Out  Generation or Removal  Accumulation

Ideal Reactors - CMBRs “Batch”  No input, no output “Completely mixed”  no spatial concentration, density or thermal gradients Useful for determining reaction rates Masses of the target components varying with time only by their reactions MBE  Residence time ( )

Ideal Reactors - CMFRs Instantaneous mixing immediately after fluid elements and constituents entering the reactor  rapid dilution of influent reactants CR = Cout   Advantages – effective and fast mixing of reactants, reduction of loading shock (buffering) MBE 

Ideal Reactors - PFRs Uniform fluid velocity across any cross section normal to the axial flow Fluid elements and constituents intermixed completely throughout the cross-section of the reactor No mixing in the direction of flow axis MBE  point form for dVR At steady state, Residence time ( )

Comparison of Reactor Performance Comparison of residence time based on the same extent of reactions (i.e., reaction rate order, n, and rate coefficient, k)  the required volume of reactor to achieve a given (expected) transformation rate at a given flow rate

Comparison of Reactor Performance Except 0th order reaction, the residence time for CMFR > the residence time for PFR Except 0th order reaction, the reaction efficiency for CMFR < the reaction efficiency for PFR

CMFR in Series A number of CMFRs connected in series  increase of overall process efficiency (reduction of reactor volume, residence time, increase of reaction rate, etc.) With the same flow rate (Q) and reactor volume (VR), the reactor behavior and performance → closed to those of PFR as the number of the CMFR increases.

CMFR in Series If the reaction is the 1st order, Total residence time, If n , by L’Hospital’s theorem,

CMFR with Solids Recycle Increase of residence time for fluids and solids within the reactor for the enhancement of reaction efficiency without increase of reactor volume MBE, Residence time Steady-state per-pass retention time for a CMFR with flow recycle ( , detention time per pass through the CMFR)

CMFR with Solids Recycle Using a Clarifier MBE, If the input solids negligible, CIN = 0, where, R = QR/QIN (recycle ratio) Residence time Mean solid residence time (SRT, ) If rate of solids build-up were small, CR  CMIX  close to steady state

Nonideal Reactors In reality, no assumptions such as “completely (and instantaneously) mixed” and “plug flow” are valid.  CMBR, CMFR, and PFR do not exist. Existence of dispersion condition  behaviors of reactors somewhere between the PFR and the CMFR Due to short-circuiting, recycle, stagnant zones

Nonideal Reactors Flow and mixing characteristics deviated from ideal conditions – determined by experiments  residence time distribution (RTD) analysis using tracers An RTD is obtained by applying stimulus-response analysis.  introduction of reactor responses (readily detectable tracers) to reactors as either pulse or step inputs Pulse (delta) input – instantaneous injection of a fixed mass of tracer to the influent of a reactor Step input – continuous injection of a tracer at a constant concentration to the influent of a reactor Tracers – environmentally acceptable, non-reactive, and readily measureable at low concentrations (e.g., Cl-, Br-, dye, etc.)

C Curve When a pulse or delta input is used, the effluent tracer concentration profile (stimulus-response relationship) is the C curve. C curve  where   C curve shows how fluid elements are distributed in time as they pass through the reactor. Some elements exit in a time shorter than HRT, , while others greater than . Mass balance check! 

E Curve E curve – the exit age or residence time distribution curve Area under the C curve,   E curve is defined as Total area under the E curve,  E curve shows a time-normalized or factional age distribution.    Fraction of fluid elements with age <

E Curve Mean hydraulic residence time,   Mean constituent (dissolved solutes or suspended solids) residence time, = mean value of the residence time of a constituent, which is given by the first moment of the centroid of E curve, Variance of the distribution, which is given by the second moment of the centroid of E curve, Skewness of the distribution, which is given by the third moment of the centroid of E curve,

Class quiz Question: Characterize the RTD for the filters.

Dimensionless E Curve Using dimensionless time, , area under the dimensionless E curve, if   Variance of the distribution,

F Curve When the tracer is introduced as a step-function stimulus (continuous input), the effluent tracer concentration profile (stimulus-response relationship) is the F curve. F curve  where   F(t)  the fraction of tracer substances having an exit age young than t.

F Curve Relationship btw F and E   Mean residence time Variance

RTD Analysis RTD analysis  to compare the performance of real reactors to those of ideal reactors   The RTD patterns of ideal reactors (theoretical) must be analyzed by mathematical description of the responses of ideal reactors (CMFR, PFR, and CMRF in series) to either pulse or step inputs.

RTD Analysis RTDs for ideal CMFR E curve  Mass balance relationship   E curve  Mass balance relationship The initial condition,  This is because the tracer is instantaneously mixed throughout the reactor at time = 0. Integrating the mass balance eqn.,

RTD Analysis E curve Area under the E curve,   Area under the E curve, Dimensionless form by assuming Mean constituent residence time ( ) must be the same as hydraulic retention time ( )

RTD Analysis F curve  Mass balance relationship   Integrating the mass balance eqn., The shaded area above the F curve  Mean constituent residence time Mean constituent residence time ( ) must be the same as hydraulic retention time ( )

RTD Analysis RTDs for ideal PFR Pulse input The response is a spike of infinite height and zero width (a peak with no defined area). Mean constituent residence time must be the same as hydraulic retention time ( ). Step input The response is a instantaneous step increase in the exit concentration from zero to a feed concentration at time . Mean constituent residence time must be the same as hydraulic retention time ( ).    

RTD Analysis RTDs for nonideal PFDR    Starting point for the characterization of real reactors – PFR with dispersion (= plug flow dispersion reactor, PFDR)  advection-dispersion-reaction (ADR) model   For pulse input, One-dimensional (x-axis) form of ADR equation Dimensionless form of ADR equation where, If Nd  0, then the reactor  PFR If Nd  , then the reactor  CMFR

RTD Analysis Boundary conditions   For small amounts of dispersion, the analytical solution to the dimensionless ADR equation Variance,    This is valid only if the amount of dispersion is small and the C curve is reasonably close to symmetrical.

RTD Analysis For large amounts of dispersion,  

RTD Analysis For step input, Boundary conditions   Boundary conditions Analytical solution to the dimensionless ADR equation Variance (for closed vessel),