MASS BALANCE EQUATIONS FOR CSTR

MASS BALANCE EQUATIONS FOR CSTR

( DERIVATION AND SOLUTION )

Consider the following bioreactor.

What are the implications of assuming it is a CSTR ?

-348615286385FEED STREAM

Q1

S1

X1

00FEED STREAM

Q1

S1

X1

1784985238760

19278605080 CSTR

( V S X )

0 CSTR

( V S X )

right252730EXIT STREAM

Q1 + Q2

S

X

00EXIT STREAM

Q1 + Q2

S

X

-3200401214755RECYCLE STREAM

Q2

S2

X2

00RECYCLE STREAM

Q2

S2

X2

373380038100013087346146800088963513843000

Kinetic Equations

Assume Monod kinetics: = max S / (Ks + S) (1)

rx = net production rate of biomass = X kd X = ( kd) X (2)

rs = substrate consumption rate = ( X / Y) (3)

Steady-state Biomass Balance over CSTR

Biomass entering CSTR + Biomass generated in CSTR = Biomass leaving CSTR

Q1 X1 + Q2 X2 + V rx = (Q1 + Q2) X (4)

Steady-state Substrate Balance over CSTR

Substrate entering CSTR - Substrate consumed in CSTR = Substrate leaving CSTR

Q1 S1 + Q2 S2 - V rs = (Q1 + Q2) S (5)

Solution of CSTR Model

Model has 5 equations and 16 variables ( not a square system cannot solve model)

Can reduce model size by substitution.

Eliminate by substituting (1) into equations (2) and (3)

Eliminate rx by substituting (2) into equation (4)

Eliminate rs by substituting (3) into equation (5)

Model now has 2 equations and 13 variables (i.e. 3 equations and 3 variables have been eliminated).

Q1 X1 + Q2 X2 + V [max S / (Ks + S) kd] X = (Q1 + Q2) X (6)

Q1 S1 + Q2 S2 V [max S / (Ks + S)] X / Y = (Q1 + Q2) S (7)

Remaining variables are: (Q1 Q2) (S1 S2 S) (X1 X2 X) (max Ks kd Y) V

Need to specify 11 variables gives a square system of 2 equations in two unknowns.Can specify any 11 variables one set of specifications might be:

Q1 = 25 m3 / hrQ2 = 25 m3 /hr

S1 = 400 mg BOD / L S2 = 0

X1 = 0 X2 = 4000 mg VSS / L

max = 0.18 hr-1Ks = 100 mg BOD / L

kd = 0.0025 hr-1Y = 0.6 mg VSS / mg BOD

V = 300 m3

One way to solve equations (6) and (7) for the unknowns S and X is to use Solver in Excel.

S = 5.466 mg BOD / L

X = 2085 mg VSS / L

Do the results seem reasonable ?

Assignment Part .1 instructions

REPORT FORMAT

One of my key teaching objectives in this course is to help you make the transition from university students to graduate engineers. Thus, you need to learn how to present technical information as anengineering report(not simply as the answers to a series of tutorial type questions).

There is no single format for an engineering report but I have given you one possibility below.

MODELLING AN ACTIVATED SLUDGE CIRCUIT SYSTEM

In Part 1 of this assignment, the bio-reactor is treated as a CSTR (= Continuous Stirred Tank Reactor). For those of you who are not familiar with modelling a CSTR, I have put two documents in theSupplementary MaterialModule:

CSTR Mass Balances.docs Word document that describes how to model a single CSTR where the reaction taking place is described by Monod kinetics.

CSTR Mass Balances.xlsx Excel file that shows how to use theSolverfacility in Excel to solve the reactor model (2 equations in 2 unknowns).

I have also included in theSupplementary MaterialModule a Word document that describes modelling the entire activated sludge circuit (not just the CSTR bio-reactor):

Wastewater Circuit Modelling.docsPlease note that when a team submits its Part 1 report, itmustalso send a copy of the Excel spreadsheet that they used to solve the activated sludge circuit (using theSolverfacility) to the course tutor. This spreadsheetmustbe well documented and formatted so that it is easy for us to test and grade.

How to format your assignment report

When you start work as a young engineer, you willnotbe given tutorial type questions (conveniently broken up into bite-size chunks) by a friendly boss who already knows the answers and has them in the top drawer of his/her desk! You or your team will be faced with unsolved problems that people (e.g. your boss, your company, your client) are willing to invest time & money to answer. You or your team will have to provide answers to the problems posed in a form that iseasily understood by all shareholders.

So, for the assignment, I don't want 'tutorial answers' to each of the questions posedI expect anengineering style report(i.e. the sort of thing that your company would be proud to give to a client - and charge them money for).

There is no single correct format but one way to write thePart 1report would be as follows:

PROBLEM DESCRIPTION

Briefly state the problem (i.e. food production plant; variable wastewater; need to get effluent BOD down to a maximum of 5 mg/L).

Lay out what your company has been asked to do (i.e. carry out preliminary sizing calculations for an activated sludge system and look at various alternative aerobic digester options).

SUBSTRATE REMOVAL KINETICS

Where did the growth kinetics come from?

These kinetics can be nicely summarized by including:

(i) a labelled figure showing(S) and

(ii) a table showing rx(S) and the importance of endogenous respiration (one way to do this is as the ratio of the cell death rate to the new biomass creation ratepoint out to the reader that as S gets lower, so endogenous respiration becomes more important).

End this section by stating why you expect these kinetics to be adequate for a preliminary sizing study.

INITIAL MASS BALANCE CALCULATIONS

Probably a good idea to provide a labelled figure showing an activated sludge circuitdon't assume that everyone will know what you are talking about.

Break this section up into a number of sub-sectionswant to make your report easy for the reader to follow.

Modelling Assumptions

Allmodelling requires assumptions what were your key assumptions?why were they made (i.e. how did they affect the modelling)?how realistic would they be for a real activated sludge plant?

Development of Mass Balance Model

Give just enough detail to show how the various equations were derived.

Point out that if additional assumptions are made (e.g. = 1), then it is possible to solve this circuit modelby handgive this solution in your reportpoint out that this is important as it allows your computer solution (given later) to be easily checked.

If a numerical value is set for a variable (e.g. =1), explain why this value was used.

Solution of Mass Balance Model using Excel

Show that your Excel model agrees with the previous hand calculations.

Give enough detail so that the client is comfortable that you know how to useSolverin Excel to solve a set of nonlinear algebraic equations.

Point out that a working, easy-to-understand copy of your Excel spreadsheet has been made available for inspection.

Sensitivity of Circuit to Key operational Parameters

Quantify how sensitive the activated sludge circuit behaviour is to changes in, and .Point out that and aresetvariables (i.e. you can decide on which values to use) but is acalculatedvariableso explain briefly to the client how you handled these two different types of variable.

What circuit variables did you chose to look at in terms of changes and say why these variables were chosen.

It might be best to give any circuit changes relative to a 'base case' set of conditionsremember that in an engineering report, it is always preferable to give results in a Table or a Figure rather than embedded in pages of (hard to follow) text.

CONCLUSIONS

Give a very short summary of what you've done and what you've found out.

Need to connect Part 1 to Part 2so point out that treating the aerobic digester as a single CSTR simplifies the analysis/modelling but gives you a large digester volume as the biomass growth rate is low because the effluent BOD is only 5 mg/Lso investigating options for reducing the digester volume will be one of the key considerations in Part 2.

Of course, there are other ways of formatting your report butDON'Tjust hand in a collection of tutorial answers to Question 1, Question 2 etc - this is not an engineering report

CHNG5005 WASTEWATER ENGINEERING

TUTORIAL 6 (GROUP ASSIGNMENT: PART 1)

INTRODUCTION

A plant producing a wide range of dairy-based products discharges its wastewater to the local sewage system after undergoing on-site primary treatment. As is typical in such plants, the wastewater varies considerably in both flow and quality. The values below show the observed ranges of what is discharged to sewer over a typical week:

Flow:500 - 700 m3/day

BOD:350 - 450 mg/L

However, the regional sewage treatment plant is nearing its maximum capacity and the water authority is insisting that the plants wastewater be further treated on-site before it is discharged to sewer. Thus, the company has recently been advised that any future discharge should have a maximum BOD level of 5 mg/L. Your consulting company has been contracted to carry out preliminary sizing calculations for an on-site secondary treatment plant.

As is often the case in such consulting work, this (two part) project will be carried out in small groups. In any design process, there is rarely a single correct answer. An important aspect of engineering design is the ability to make realistic assumptions, as the assumptions you make will invariably influence the answers you get. Thus, it is essential that for both parts of this project you make it clear what assumptions you have made and why you think that these are reasonable.

PHASE 1: SUBSTRATE REMOVAL KINETICS

An Internet search provides laboratory data for wastewater from a skimmed milk plant that is being successfully treated in an aerobic digester where the volatile suspended solids (VSS) has been measured at around 3000 mg/L.

The biomass growth rate (rx in mg VSS/L.hr) was found to be adequately described by the following equation:

where S is the substrate level (in mg BOD per L).

The yield (Y) was reported to be 0.6 mg VSS/mg BOD.

Question 1

Sketch the behaviour of the specific growth rate ( in hr-1) as a function of S over the likely range of interest (i.e. S up to 500 mg/L).

Question 2

Calculate the biomass growth rate (rx in mg VSS/L.hr) when S = 5, 50 and 250 mg/L for the measured VSS loading. Note that the biomass loading (X in mg/L) can be taken as equivalent to the VSS loading in an aerobic reactor.

Question 3

Comment on the importance of the endogenous respiration term in the published expression for .

PHASE 2: INITIAL MASS BALANCE CALCULATIONS

A basic aerobic digestion system consists of the following units:

A mixer (to combine the fresh feed with recycled activated sludge).

An aerated digester.

A clarifier (producing a treated wastewater overflow and a sludge underflow).

A splitter (producing a recycled sludge stream and a wasted sludge stream).

For preliminary calculations, the following assumptions may be made:

The digester is a single, well-mixed reactor (i.e. essentially a CSTR).

Substrate consumption (and thus biomass generation) only occurs in the digester.

Substrate consumption can be adequately described by Monod kinetics with no rate restrictions imposed by oxygen transfer limitations.

The clarifier operates perfectly (i.e. all biomass in the feed to the clarifier reports to the underflow stream).

Question 4

Discuss how reasonable each of the above assumptions is likely to be for a real plant.

Question 5

Such systems are typically operated so that F:M (i.e. the food to microorganism ratio or the sludge loading rate) falls in the range 0.2 0.5. Explain the consequences of operating outside this accepted range.

Question 6

Using the above assumptions, develop a set of mass balance equations that describe this basic aerobic digestion system. Clearly note any additional assumptions that you make. Also indicate which variables in your model are known and which variables must be calculated by solving your set of model equations.

It is strongly advised that at this point you solve your mass balance model by hand. This will allow you to see whether your model gives reasonable answers (noting that a common mistake in any model development exercise is not checking that the units for all variables are consistent), as well as providing values that can be used to check your computer (i.e. Excel) solution.

Question 7

Develop an Excel spreadsheet that solves your mass balance model using Solver. Your model should give the substrate level (in mg BOD/L) and the biomass level (in mg VSS/L) at all key points around the circuit, as well as the required digester volume (in m3), and the hydraulic and solids retention times (in days).

Your spreadsheet should have the following characteristics:

It should allow the user to readily change any operational variable or circuit parameter.

It should make clear how the mass balance equations are solved, noting that in various scenarios it will be necessary to solve the model for different sets of variables.

In terms of the above considerations, the spreadsheet should be nicely formatted so as to assist external assessment.

Solving your model will require you to determine (or set) values for the circuit parameters , and . The first two determine the recycle activated sludge flowrate or RAS (i.e. Q where Q is the feed flowrate to the circuit) and the wasted activated sludge flowrate or WAS (i.e. Q), while is the fraction of biomass entering the secondary clarifier that reports to the underflow stream). At this stage, you should set = 1 (i.e. assume that you have a perfect clarifier).

Note that an alternative to setting is to provide a value for the biomass concentration in the overflow from the clarifier.

Explain clearly how values were determined for and .

Question 8

Use your Excel spreadsheet model to examine the sensitivity of this aerobic digester circuit to the parameters , and . In looking at such sensitivity issues, it may be useful to define a base case set of conditions around which changes are then made.

THINGS TO CONSIDER (for both Parts 1 and 2)

This assessment task is a group assignment. So one of the key factors in producing two good reports (for Parts 1 and 2) is to work well as a team. In my professional career, a good group has almost never been made up of personal friends. A good group is made up of people who know what is expected of them as a whole, who know what is expected of them individually, who understand (and can stick to) the agreed project timeline, and who take real pride in delivering a quality product. At university, you can sometimes fall short in what your group comes up with and still pass. As a professional engineer in any organisation, merely doing the minimum necessary is totally unacceptable.

In line with the above comments, I would strongly recommend that each group decides very early on how the work is to be divided up and when each part should be completed.

The central part of this assignment is built around a mass balance based model. In developing such a model, there are several distinct stages:

Deriving the set of equations that form the model.

Dividing the model variables up into those that are known (i.e. given in the data or can be assumed) and those that are unknown (i.e. must be calculated by solving the model).

Solving the model (or a simplified version of it) by hand (i.e. not using Excel).

Solving the model equations by computer and checking the answers against your hand calculations.

Using your (hopefully correct) computer model to explore the behaviour of the system.

As a professional engineer, the way in which your work is presented (to your boss or to an external client) is really important. It does not matter how clever you are personally or how brilliant your computer calculations are, if the reader cannot quickly and easily understand what is before them, then that report is a failure, it is a complete waste of everyones time. In this assessment task, we expect to see clear evidence that some real thought has been put into presenting your calculations, ideas and recommendations to the reader so that they are easily understood.

Pages limits have not been given for either Part 1 or Part 2. Remember that as a professional engineer the golden rule is that any report should be just long enough to clearly get your message across to the reader. In keeping each part of this assignment short and to the point, summarising your results in well formatted tables and/or figures is always a far better option than having long slabs of text.

Copies of the marking sheets used by the course tutor for both Part 1 and Part 2 have been placed on Canvas. Each shows how the marks are allocated between the various questions. Note that in Part 1, each group has to submit a copy of the Excel spreadsheet that they used to solve their mass balance model.

The submission dates for Parts 1 and 2 are several weeks apart so that students can obtain useful feedback after the first submission.