Mass Balances

Fundamental Principle of (Dynamic) Mass Balances The rate at which something accumulates in a region of interest (a “control volume”) equals the net rate at which it enters by physical movement plus the net rate at which it is generated inside the control volume by chemical reactions.

Processes for Transport Across the Boundaries of an Aquatic System • Advection: Bulk flow, carrying the substance of interest with it • Molecular Diffusion: Random thermal kinetic motion, leads to net transport down a concentration gradient • Dispersion: Random motion of small packets of fluid, leading to same result as molecular diffusion, but usually faster

The Mass Balance in Words Rate of change of the amount of i stored within the system (rate of accumulation) = Net rate (in  out) at which i enters by advection + Net rate (formation  destruction) at which i is created by chemical reaction

The ‘Storage’ or ‘Accumulation’ Term d Vci  Rate of Accumulation  dt

Special Cases Well-Mixed, Fixed Volume: Steady State:

dci Rate of Accumulation  V dt

Rate of Accumulation  0

The Advective Term Net Advective Inflow 

Q c

in i ,in

inlets



Q

c

out i , out

outlets

Special Case Batch system: Net Advective Flow  0

The Reaction Term Mass or moles of i formed Reaction Rate ri   Volume   Time 

Net Reaction Term  rV i CV ri  fcn  ci , c j , ck ,..., T 

The Reaction Term Special Cases Non-reactive Substance (Conservative Tracer):

ri  0 nth-order Reaction Dependent Only on ci:

ri  k n c

n i

The Overall Mass Balance for Constant-Volume Systems

d  cV i CV    Qinci,in   Qout ci,out  dt inlets outlets



all reactions

rV i CV

Approach • Formulate idealized models for reactor flow patterns • Model behavior of non-reactive tracers in each ideal type of reactor, for pulse and step-change tracer input patterns • Model behavior of reactive species in idealized reactors for steady-state reactors • Model behavior of reactive species in idealized reactors not at steady-state • Consider effects of non-ideal flow patterns (qualitatively)

Idealized Model Reactors • Limiting Case #1. Unidirectional advection with no mixing: A Plug Flow Reactor (PFR). Often used to model rivers, pipe flow, settling basins, disinfection processes. Q, cout

Q, cin x

L

All parcels of fluid have identical residence time: L LA V    vx vx A Q

Concentration

Anticipated Tracer Output for a Pulse Input to a PFR

Output

Input

0

Time



Concentration

Anticipated Tracer Output from Step Input to a PFR

Input

F t  

cs  t  cin ,t 0

0 Time



More Realistic Tracer Profiles after a Pulse Input into a PFR-Like Reactor Dimensionless Conc., c /c o Concentration

12 t / = 0.3

10

0.5 0.7

8

0.9

6 4 2 0 0.0

0.2

0.4

0.6

Dimensionless Distance, x /L

0.8

1.0

Idealized Model Reactors Limiting Case #2. Advection with intense mixing: a Completely Stirred Tank Reactor (CSTR, CMFR, CFSTR, CMR). Often used to model lakes, reservoirs, flocculation basins. Q, cin V Q, cout •

All parcels of fluid have identical chance of exiting in any instant, so they have a wide range of residence times; still avg = V/Q

CSTR Response to a Pulse Input of Tracer, Steady Flow Evaluate from t = 0+ to 

Q, cin

Q, cout

• Constant V, Q • cin = 0 (at t>0+) • ci = ci,out • No reaction

d  ciV   Qin ci ,in  Qout ci ,out  rV i dt dci V  Qin ci,in  Qout ci  ri V dt

dci V  Qci dt

CSTR Response to Pulse Input dci V  Qci dt ct 



 

c 0

t

t

dci 1 Q    dt    dt  0 ci V 0

ln

c t 

c 0







t



 t M  t c  t   c  0  exp     exp      V   

CSTR Response to Pulse Input 1.0 0.9 0.8

 t c  t   c  0 exp    

0.6 0.5 0.4 0.3

1/e

0.2 0.1 0.0 0



2

Time

3

4

5

6

c t  t ln   c 0

0 -1 -2 ln(cp /c o)

cp /c o

0.7

-3

1

-4

1

-5 (b)

-6 -7

0



2

3 Time

4

5

6

Representing Intermediate Degrees of Mixing • PFR with Dispersion: Zero dispersion is PFR; increasing dispersion increases mixing; infinite dispersion is a CSTR • CSTRs in Series: Inserting baffles (keeping V and Q constant) segregates segments of reactor and decreases mixing; as Nbaffles increases, overall mixing decreases, and reactor becomes more PFR-like

Summary of Key Points • Reactor hydraulics can be characterized by the range of residence times of entering water ‘packets’ • Ideal, limiting cases include PFRs (no mixing) and CSTRs (complete, instant mixing) • For both PFRs and CSTRs, the average residence time, , is V/Q. For PFRs, all the fluid spends time  in the reactor; for CSTRs, different packets of fluid spend different amounts of time in the reactor, but the average is 

Summary of Key Points • We can assess whether a reactor behaves like one of the limiting cases by a carrying out a tracer test with a pulse or step input, combined with a mass balance analysis • Intermediate mixing can be modeled as dispersion and/or CSTRs in series

Designing and Evaluating Systems with Chemical Reactions

Extent of Reaction in a Batch Reactor dc VV  Qc Qout coutVrV r Q cininQc dt dc r dt c t 

t

dc c 0 r  0 dt  t  

1st-Order Reaction in a Batch Reactor 

In a disinfection process, bacterial kill follows the first-order reaction expression: rX = (1.38 min1)cX. How long is required for 99% disinfection?

ct 

c t 

c t 

dc dc dc 1 1 c t  t        ln r k1c k1 c 0 c k1 c  0  c 0 c 0

1 t ln  0.01  3.3 min 1 1.38 min

Extent of Reaction in a CSTR at Steady State Q

Q

Cin

Cout

d Vc  dt

 Qcin  Qcout  Vr

V

Cout

cout

V  cin  r  r Q

cout  cin  r

1st-Order Reaction in a CSTR at Steady State  What

average residence time is required for the same, 99% kill of bacteria, if the reactor is a CSTR? If the flow rate is 1.5 m3/min, how large must the reactor be? cout  cin 1  100    71.7 min 1  k1c  1.38 min  1

 m3  3 V  Q  1.5   71.7 min   108 m s  