1. MASS BALANCE REACTOR THEORY
Dr-Ing Marisa Handajani, ST, MT

2. Uses
Essential tp solution of a greater number of environmental engineering and science problems, such as:
Pollutant movement
Monitoring operating efficiency in ‘day to day’ operation of process (process control)
Calculations for design and development of a process i.e. quantities required, sizing equipment, number of items of equipment

3. Law of conservation of mass
“Mass can neither be produced nor destroyed”
The principle of conservation of mass means that if the amount of a chemical somewhere (for example, in a reactor) increase, then that increase canot be the result of some “magic” formation.
The chemical must have either carried into the reactor or produced via chemical or biological reaction from other compounds that were already in the reactor.
If the reaction produced the mass increase of this chemical, they must also have caused a corresponding decrease in the mass of some other compound(s)

4. (mass that entered from t to t + Δ t ) –
Mass Balance
(Mass at time t + Δ t ) =
(mass at time t ) +
(mass that entered from t to t + Δ t ) –
(mass that exited from t to t + Δ t ) + (net mass of chemical produced from other compounds by reaction between t and t + Δ t )
The conservation of mass provides a basis for compiling a budget of mass of chemical.
This budget keeps track of the amounts of chemicals entering the reactor, the amounts of chemicals leaving the reactor, and the amount formed or detryed by chemical reaction.
Thus budget can be balanced over a given time period.
This balance is useful when there is a clear beginning and end to balance period, so the change of mass over the balance period can be determined

5. Mass Flux Balance Rate at which mass enters or leaves a system
Mass which enters or leaves a system in a certain time range Δ t
Dividing the mass balance equation by Δ t
F = mass flux unit (mass/time)
𝒎𝒂𝒔𝒔 𝒂𝒄𝒄𝒖𝒎𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝒓𝒂𝒕𝒆 = 𝒎𝒂𝒔𝒔 𝒇𝒍𝒖𝒙 𝒊𝒏 + 𝒎𝒂𝒔𝒔 𝒇𝒍𝒖𝒙 𝒐𝒖𝒕 + 𝒏𝒆𝒕 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒄𝒉𝒆𝒎𝒊𝒄𝒂𝒍 𝒑𝒓𝒐𝒅𝒖𝒄𝒕𝒊𝒐𝒏
𝑑𝑁 𝑑𝑡 = 𝐹 𝑖𝑛 − 𝐹 𝑜𝑢𝑡 + 𝐹 𝑟𝑥𝑛

6. A mass balance is only meaningful in terms of a specific region of space, which has boundaries across which the term of min and mout are determined
Control Volume has boundaries over which min and mout can be calculated
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7. Mass Balance on Reactive System
In - out + gen - cons = accumulation
A mass balance for the system is
NA is the mass of “A” inside the system.

8. The reaction term can be written in more familiar terms,
GA = rA V
V is volume of the system.
Note that the units for this relation are consistent:
If GA (and hence rA) varies with position in the system volume, we can take this into account by evaluating this term at several locations. Then DGA1 = rA1 DV1,

9. Chemical Reaction (generation) Rate Forms Possibilities
Conservative Compounds
No chemical formation or loss within the control volume
Not affected by chemical or biological reaction
𝒅𝑪 𝒅𝒕 𝒓𝒆𝒂𝒄𝒕𝒊𝒐𝒏 𝒐𝒏𝒍𝒚 = 𝑮 𝑨 =𝟎
Zero Decay
Rate of loss of the compound is constant
𝒅𝑪 𝒅𝒕 𝒓𝒆𝒂𝒄𝒕𝒊𝒐𝒏 𝒐𝒏𝒍𝒚 = −𝒌 𝐚𝐧𝐝 𝑮 𝑨 =−𝑽𝒌
First order Decay
Rate of loss of the compound is directly proportional to its concentration
𝒅𝑪 𝒅𝒕 𝒓𝒆𝒂𝒄𝒕𝒊𝒐𝒏 𝒐𝒏𝒍𝒚 = −𝒌𝑪 𝒂𝒏𝒅 𝑮 𝑨 =−𝑽𝒌C
Production
Rate depends on the concentration of other compounds in the reactor
𝒅𝑪 𝒅𝒕 𝒓𝒆𝒂𝒄𝒕𝒊𝒐𝒏 𝒐𝒏𝒍𝒚 >𝟎

10. Summing the reactions over the entire volume yields:
As (that is, as we decrease the size of these cubes and increase their number)
which gives

11. Generalized Design Equation for Reactors
In - out + gen - cons = accumulation

12. 𝑑𝑁 𝑑𝑡 = 𝑑 𝑉.𝐶 𝑑𝑡 Mass Accumulation rate (dN/dt)
Assume that the volume control is completely mixed
𝑑𝑁 𝑑𝑡 = 𝑑 𝑉.𝐶 𝑑𝑡
So the concentration and hence the mass within the control volume remains constant
In steady state 𝑑𝑁 𝑑𝑡 =0

13. Process Classification
Chemical processes can be classified as batch, continuous or semi-batch and as either transient or steady state
Batch process is one in which the feed is charged into the system at the beginning of the process, and the products are removed all at once some time later
Continuous process is when the inputs and outputs flow continuously across the boundaries throughout the duration of the process.

14. Types of Reactors Batch Fed- Batch Continuous
No flow of material in or out of reactor
Changes with time
Fed- Batch
Either an inflow or an outflow of material but not both
Continuous
Flow in and out of reactor
Continuous Stirred Tank Reactor (CSTR)
Plug Flow Reactor (PFR)
Steady State Operation

15. Batch Reactor or Generalized Design Equation for Reactors
No flow into or out of the reactor, then, FA = FA0 = 0
Good mixing, constant volume or
Batch reactors are often used in the early stage of development due to their ease of operation and analysis

16. If the reaction is describe by first order kinetics
𝑑𝐶 𝑑𝑡 =𝑘𝐶
The reaction time for realizing a desired reactant concentration can be determined by integrated between the limit C0 (initial) and Cd (desired)
𝑡= 1 𝑘 𝑙𝑛 𝐶 0 𝐶 𝑑
Batch reactors are often used in the early stage of development due to their ease of operation and analysis
Very limited use in field-scale biological wastewater treatment process, although it should considered for some operation in small plants and for sludge digestion

17. Fed Batch Reactor Reactor Design Equation No outflow FA = 0
Good Mixing rA dV term out of the integral

18. Convert the mass (NA) to concentration
Convert the mass (NA) to concentration. Applying integration by parts yields
Since
Then
Rearranging Or
Used when there is substrate inhibition and for bioreactors with cells.

19. Continuous Stirred Tank Reactor
Q CA0
Q, CA
V, CA
Assume rate of flow in = rate of flow out
FA = Q CA and FA0 = Q CA0
Q = volumetric flow rate (volume/time)

20. CSTR - continued General Reactor Design Equation Assume Steady State
Well Mixed
So or
If the reaction rate is assumed a first order reaction 𝑟 𝐴 =𝑘 𝐶 𝑒 , the above eq. is rearranged into the form
𝐶 𝑒 𝐶 0 = 1 1+𝑘 𝑉/𝑄

21. The nominal hydraulic retention time in CSTR is defined as
𝑡 𝐶𝑆𝑇𝑅 = 𝑉 𝑄
𝐶 𝑒 𝐶 0 = 1 1+𝑘 𝑡 𝐶𝑆𝑇𝑅
The reaction time required to achieve a desired reactant concentration :
𝑡 𝐶𝑆𝑇𝑅 = 1 𝑘 𝐶 0 𝐶 𝑒 −1

22. CSTR in Series 𝑛𝑡 𝐶𝑆𝑇𝑅 = 𝑛 𝑘 𝐶 0 𝐶 𝑒 𝑛 −1
Q, C0
V,C1
V,C2
Q, C2
𝐶 1 𝐶 0 = 1 1+𝑘 𝑡 𝐶𝑆𝑇𝑅 𝐶 2 𝐶 1 = 1 1+𝑘 𝑡 𝐶𝑆𝑇𝑅
𝐶 2 𝐶 0 = 𝐶 1 𝐶 0 𝐶 2 𝐶 1 = 𝑘 𝑡 𝐶𝑆𝑇𝑅 2
𝐶 𝑛 𝐶 0 = 𝑘 𝑡 𝐶𝑆𝑇𝑅 𝑛
𝑛𝑡 𝐶𝑆𝑇𝑅 = 𝑛 𝑘 𝐶 0 𝐶 𝑒 𝑛 −1

23. Plug Flow Reactor (PFR)
Tubular Reactor
Pipe through which fluid flows and reacts.
Poor mixing
Difficult to control temperature variations.
An advantage is the simplicity of construction.

24. PFR Design Equation Design Equation
Examine a small volume element (DV) with length Dy and the same radius as the entire pipe.
If the element is small, then spatial variations in rA are negligible, and
Flow of A out of Element
Flow of A into Element
Assumption of “good mixing” applies only to the small volume element

25. If volume element is very small, then assume steady state with no changes in the concentration of A.
Simplify design equation to:
rA is a function of position y, down the length of the pipe and reactant concentration
The volume of an element is the product of the length and cross-sectional area,
DV = A Dy
Design Equation becomes:

26. 𝑑 𝐹 𝐴 𝑑𝑉 = 𝑄.𝑑𝐶 𝐴.𝑑𝑦 = 𝑄.𝑑𝐶 𝐴.𝑣.𝑑𝑡 = 𝑄.𝑑𝐶 𝑄.𝑑𝑡 = 𝑑𝐶 𝑑𝑡
take the limit where the size of a volume element becomes infinitesimally small
or because Dy A = V,
𝑑 𝐹 𝐴 𝑑𝑉 = 𝑄.𝑑𝐶 𝐴.𝑑𝑦 = 𝑄.𝑑𝐶 𝐴.𝑣.𝑑𝑡 = 𝑄.𝑑𝐶 𝑄.𝑑𝑡 = 𝑑𝐶 𝑑𝑡
In case of first order reaction
𝑑𝐶 𝑑𝑡 =−𝑘𝐶
𝐶 𝑡 𝐶 0 =𝑒𝑥𝑝 −𝑘𝑡
The time required to obtain a desired reactant concentration in the effluent
𝑡 𝑃𝐹 = 1 𝑘 𝑙𝑛 𝐶 0 𝐶 𝑒
This is the Design Equation for a PFR
Bioapplications - Sometimes hollow fiber reactor analysis is simplified to a PFR

27. Operating Characteristic of Different Reactor Systems
Reactor Type
Variation of Composition with time
Variation of Composition with position in reactor
CMB
Yes No
CSTR PF

28. Homework
Compare the total volume requirement for the following reactor systems:
CMB reactor
Single CST reactor
Two CST reactor connected in series
PF reactor
It is desired that the reactant concentration be reduced from 100 mg/L to 20 mg/L for the flow 50m3 day-1. Assume the first order reaction kinetics are followed and the rate constant has a value of day-1.