Reactor Theory: hydraulics TVM 4145 Vannrenseprosesser / unit processes Reactor Theory: hydraulics Prof. TorOve Leiknes torove.leiknes@ntnu.no

Principle of Mass balance: System boundary Reaction Transport in Transport out Accumulation = In - Out + Generation where: In – Out is net transport in the system Generation is net production or consumption Accumulation is net product left over

Mass balance principle applied to reactor theory: Flow-sheet: Equation form: Reactor Qi, Co Qe, Ce V, C Where rc is reaction rate Reaction rate; rc where: k is the rate coefficient C is the concentration in the reactor n is a constant.

Definition of reactor types: PF: Plug flow reactor NO - Stempelstrømning CMB: Completely mixed batch reactor NO - Ideell blanding CMFR: Completely mixed flow NO - Ideell blanding

CMB: Assumptions: closed system reactor volume is constant homogeneous mixing

With a first order reaction and reduction of the concentration: CMB: With a first order reaction and reduction of the concentration:

CMFR: Assumptions: Q, Co Q, C V, C, rc closed system reactor volume is constant homogeneous mixing Q, Co Q, C V, C, rc

CMFR develop theoretical basis for a simplified system consider the system under steady state conditions assume a simple 1.order reaction for removal of a substance i.e. the process kinetics is: Solve the equation with regard to effluent concentration:

What is the goal?

CMFR in series: Assume steady state and 1.order reaction for rc: V1, C1, rc1 V2, C2, rc2 V3, C3, rc3 Q Co Q C1 Q C2 Q C3 Assume steady state and 1.order reaction for rc: Reactors er equal, ki=k, Vi/Q=th:

What happens when V ≠ constant? Change in volume: HRT

Important / useful equations: 1. Calculation of effluent concentration: 2. Average hydraulic retention time (HRT): For one reactor: Total for n reactors: 3. Total retention time necessary for desired reduction:

PF: Assumptions no mixing in the reactor Q constant velocity through the reactor retention time = theoretical retention time no displacement of liquid elements concentration gradient in along length Q v Consider an element in the reactor: x Q, C Q, C + C A

Assume steady-state to analyze the PF reactor: vx x x x = L Assume 1.order reaction and steady-state:

Comparison of reactor types: Average retention time, th: Reaction order 0.order 1.order 2.order n.order (n1) CMFR: PF:

How will the reactors behave? ”Tracer” studies where an inert substance (colour, salt) are used to analyze the hydraulic conditions through the reactor. Time Ideal completely mixed Ideal plug flow C C (gitt volum) Q Reactor Measure C in the outlet

Co = concentration distributed in tank volume t/th C/Co 1.0 Co = concentration distributed in tank volume C = concentration measured in outlet th = theoretical hydraulic retention time (V/Q) t = measured time tmean = center of mass, (1.moment) tmode = average of max concentration tmdian= time for å measure 50% of tracer in outlet Analysis of central tendencies: none: times coincide short-circuits: (tmean -tmedian)/ tmean or (tmean -tmedian)/ tmean dead-zones: th- tmean ; in practice tmean < th Short circuits given by: Distribution given by: t10/t90 or expressed by the variance (s2)

Retention time (th) as a function of recycling: Q rQ Q + rQ V recycling fraction Average retention time is calculated by considering an element(Dq): average retention time:

A fraction of the fraction is recycled: Total retention time for the element: This is a geometric series:

PFD: Dispersion model Fick’s lov: Where: vx x L Dispersion In Bulk flow In Bulk flow Out Dispersion Out  reaksjon Fick’s lov: Where: D is the dispersion coefficient u is average water velocity in x

 - dispersion large, CMFR The equations is made dimensionless by: z = x/L og  = tu/L where Is the dispersion number If 0 - dispersion negligible, PF  - dispersion large, CMFR

Dispersion number is determined by ”Tracer studies” use an inert / stable substance which is easy to measure, but with no reaction introduce substance as a pulse with known concentration measure the outlet concentration of the substance over time plot concentration against time, commonly normalized (C) graphs and plots are available for different reaction orders (i.e. Levenspiel & Bischoff)

Interpretation of dispersion number: Based on tracer study

Example:

Applied in biological systems: Mass balance principle applies Reaction kinetics for biological systems must be used Activated sludge processes: Reactor Sedimentation Q, So, Xo Q(1+r) Qe, Se, Xe Inflow Outflow V, S, X Return sludge Excess sludge Qr, Xr Qw, Xw =Xr

Definitions: Mass balance on cell mass: (steady state) Hydraulic retention time: Average cell mass retention time: Mass balance on cell mass: (steady state)

Solutions/adjustments of equations: From reaction kinetics: Change in cell mass: Change in substrate concentration: