Material balances also called mass balances, is an application of the law of conversation of mass, which states matters is neither created nor destroyed. Material balances are vital for chemical engineers as they are the basis of process design and allow the design of units themselves as they determine the quantities of raw materials required for the quantity of product required.

The material balance can be seen as the accounting for materials that enter and leave a process and accounts for all materials even if they undergo chemical reactions, separation, heating, cooling, mixing, drying or any other operation (excluding nuclear reactions) that occurs in the system.

Material balances are a really useful tool for studying plant; operations, troubleshooting, checking actual plant performance versus the design performance (what the plant works at against what it was designed for), can extend the amount of data from plant instrumentation, checks if instruments are calibrated correctly and can help locate sources of material loss.

Key Definitions

 A list of definitions that must be understood when doing material balances:

• Steady-state – Conditions at all points within the process are constant with time.
• Unsteady state – Conditions within the process vary with time.
• System boundary – the imaginary box is drawn around the system that is being analysed.
• Open system (or flow system) – Materials crosses the system boundary during the process.
• Closed system – no materials cross the system boundary except the beginning and end of the process, and changes take place within the system and heat may be transferred.

System Boundary

The system or region needs to be defined by an imaginary closed box which is called the system boundary, which should always be drawn to avoid making any mistakes (figure 1). A system can be one single process unit, a collection of process units or an entire process.

Figure 1: How to define a system boundary on a system

Material Balance Equation

 As stated previously the material balances obey the law of conversation of mass, thus mass in is equal to mass out. The equation that needs to be remembered by every chemical engineer at all levels is:

Accumulation = (Mass in – Mass out) + Generation – Consumption

• Accumulation – The change in quantities of materials inside the system
• Mass in – Materials entering the system that cross the system boundary
• Mass out – Materials leaving the system that cross the system boundary
• Generation – Materials produced by a chemical reaction that takes place within the system
• Consumption – Materials used by a chemical reaction that takes place within the system

Simplification For The Material Balance Equation

 In certain cases, the material balance equation will be affected:

• No net accumulation – as the process is in steady-state or when you have a batch vessel which starts empty and then ends empty, so no accumulation within the process, the equation becomes:

In + Generation = Out + Consumption

• No reaction – No materials are used up such as when blending, washing or drying. Thus, no generation of a new product of consumption of starting materials, the equation becomes:

Accumulation = In – Out

• No net accumulation or reaction – nothing is consumed, and no new products are created, and no materials remain in the process, the equation becomes:

In = Out

Choosing A Basis

A basis is a reference point that you can choose when wanting to solve a problem where the information gave is limited or purposely missed out such as an exam type question. A good choice of basis makes material balances much easier to solve, and can be a flow rate, a unit of time or an amount.

When choosing your basis, always state your basis clearly to remove any issues that could arise from colleagues not understanding where your values came from, and for exam markers to be able to get the marks given for choosing a basis.

Other things that need to be considered are; the units to be used, the most convenient basis value to be used and what the question is asking. Having this checklist will make creating a basis straightforward and minimise mistakes.

In a scenario where one stream has enough data, then use that stream as it has the most data and if no data then assume a basis for a stream with known components, and when mass fractions are known then choose either the total mass or the mass flow rate as the basis. Similarly, if mole fractions are given then choose the total number of moles or the molar flow rate.

When having to choose a basis for a continuous steady-state process a sensible choice for the basis is based on the amount of material entering or leaving the system within a period i.e. Kg/hr. For a batch or semi-batch process then base the basis on the inlet amount at the feed initially or the output at the end of the process i.e. Kg/batch.

Procedure For Solving A Material Balance

• Carefully analyse the situation now what you are being asked to find or work out.
• Label all quantities on the block flow diagram (BFD) or process flow diagram (PFD) using the units given and doing any conversions to make sure all units are the same.
• Choose your basis, keep this simple by using easy numbers i.e. 100 kg or 1 hour.
• Draw your system boundary.
• State any assumptions made.
• Draw a calculations table to make it easier to see the values you have worked out
• Check your solution to make sure it makes sense!

Material Balance Example

A centrifuge is used to separate particles of a diameter of 0.5 – 50 mm from a liquid.  Yeast cells are recovered from a broth (a liquid mixture containing cells) using the centrifuge. Find out the amount of cell-free discharge if per hour if 1000 L^-1 is the feed stream that contains 500 mg cells L^-1 with a product stream that contains 50 wt% cells. Assume a density of 1 g cm ¹.

Try to do this question without looking at the answer, if you can do this first time without looking at the answer it will be very impressive, but if you cannot don’t worry getting used to material balance questions takes time.

Hint: Use the procedure to solve this question and only use the information you need to be careful of red herrings!

Show Answer

Dr. Adam Zaidi

Dr. Adam Zaidi, PhD, is a researcher at The University of Manchester (UK). His doctoral research focuses on reducing carbon dioxide emissions in hydrogen production processes. Adam’s expertise includes process scale-up and material development.’

The post An In Depth Guide To Material Balances appeared first on Engineeringness.

For reactor design, Fogler’s is a really good start for understanding the basics of reactor design and then books such as by Levenspiel are really useful once the basic knowledge has been acquired, but we will be focusing on Fogler’s algorithm for understanding of problems for reactor design that will be essential for exams and coursework’s.

With reactor design relatively simple theories and models help to give insight into what is going on inside the reactor and if any changes can be made to increase the performance, however, due to this simplicity making alterations to existing designs is a more preferred practice.

The shape of reactors can vary but they will resemble a tank or a tube for the most part and are either a batch reactor or a continuous reactor. Batch reactors have constant changing compositions and are simple to operate and aren’t steady-state, whilst continuous reactors can be two kinds PFR, Plug Flow reactor or CSTR, continuously stirred tank reactors which are used a lot in the industry due to their larger flow rates handling capabilities and control over product quality.

Batch Reactor

Batch reactor schematic diagram (The Essential Chemical Industry – online, 2013)

A batch reactor whilst the reaction is taking place has no flow into or out of the reactor, this is because during the reaction the batch reactor is a closed system. The reaction process in Batch reactors will produce a high conversion of the reactants however the disadvantage is the long reaction time which adds to costs such as labour costs as well as issues that are encountered in the industry such as unreliable batch qualities (Hafeez, 2019).

Plug Flow Reactor (PFR)

PFR schematic diagram (Wikipedia, 2020)

The plug flow reactor model, PFR (sometimes called continuous tubular reactor, CTR) is a cylindrical reactor with a tubular design. The type of flow going through the PFR is called plug flow, which is modelled as infinitely thin coherent plugs (see diagram above), that travels in an axial direction with each ‘plug’ being a different entity and is effectively a small batch reactor per each plug with each plug having a different composition from before or after it. the assumption that is made for a PFR is that the fluid will be perfectly mixed in the radial direction but not mixed at all in the axial direction. The residence time (total time spent in the reactor) is an impulse (a small narrow spike function), and is derived from the position of the fluid in the PFR, and is a key factor when scaling up flow reactors (Vapourtec, 2020).

Continuously Stirred Tank Reactors (CSTR)

CSTR diagram (Wikipedia, 2020)

A continuous stirred tank reactor (CSTR) is a basically batch reactor with an impeller or other mixing device to provide efficient mixing. A CSTR is often referred to an idealised agitated vessel used to model operational variables used to attain specific outputs. Using a single CSTR leads to issue of sever back mixing, extremely poor residence time control which limit the performance of the CSTR and have a negative impact on product yield, selectivity and space-yield. Thus, to combat these issues CSTRs are used in cascades of 3 or 4 to promote better residence time control and reduce back mixing (Vapourtec, 2020).

Reactor Design – Equations

To be able to find out a design parameter for a reactor you require, and the first step required a mole balance (can be called Molecular mole balance) to find out from the established system boundary of the reactor on what mass; enters, leaves, stays in the reactor or is converted into a new species.

A material balance must be developed, which is simply:

Inlet Flow + Generation = Outlet Flow + Consumption + Accumulation

(1.11)

If the composition is uniform (the same at all points) in the reactor then the material balance can be done for the whole reactor, if it is not uniform then it needs to be done on an infinitesimal volume which needs to be integrated, this will become more clear when looking at batch and continuous reactors. Going forward there will be a range of notations used and making yourself familiar with these expressions is extremely important.

N_i – Number of moles of species ‘i’

F_i – Molar flow rate this is used for continuous reactors at a point in the system

X_i – Conversion of species ‘i’ during the chemical reaction

C_i – Concentration of species ‘i’ at a point during the chemical reaction

v – Volumetric flow rate

t – Time

Subscript ‘0’ – donates initial value when t = 0

Stoichiometry

For flow reactors:

(1.12)

(1.13)

 For a batch reactor:

(1.14)

(1.15)

Conversion

Conversion refers to the number of moles of a species that has been changed or converted into a new species, for a batch reactor this will be in terms of moles and for a continuous flow reactor this will be in terms of molar flow rates, this is simply just a division of how much of species A has been used up over the original amount of species A at t = 0.

(1.16)

(1.17)

There is a special case when the densities are constant, or we assume constant density we then assume the volume is constant as (density = mass/volume) for the fluid element and conversion becomes:

(1.18)

Equation 1.18 can be further simplified due to the fact that C_A/C_A0 = 1 so

(1.19)

then this expression for conversion can be written as:

(1.20)

C_AO will be constant and the X_A and C_A will be changing throughout the reaction. For liquids, when the density is constant volumetric flow rates are the same throughout:

(1.21)

Space-Time and Space-Velocity

For a continuous flow reactor, space-time and space-velocity are used instead of reaction time which is used in batch reactors which represent the amount of time the reaction is going on for. Space-time (τ) is the time for the time taken for the one reactor volume of feed to go through the reactor and space-velocity (s) is how many reactor volumes of feed can be treated per unit of time.

(1.22)

Ideal Batch Reactor Mole Balance

Assuming the composition is the same throughout and is well-mixed (this helps in making the mass balance easier by eliminating terms), we can do a balance on the whole reactor volume, thus the mass balance becomes

0 = Consumption + Accumulation

(1.23)

Then stating that the consumption of species let’s call it A within the given volume:

Consumption of A is equal to:

-rAV←consumption is the reaction rate multiplied by the volume.

(1.24)

Thus the Accumulation of A is equal to:

(1.25)

This gives a mole balance of:

(1.26)

The reason for the differential is because we need to find out the conversion of species A over time.

Integrating gives and rearranging for t:

(1.27)

With a constant density and thus a constant volume:

(1.28)

(to make it easier a document will have all the final forms of the equations)

CSTR Mole Balance

CSTR Mass balance: Input = output + consumption

(1.29)

CSTR Performance Equation

 The performance equation relates the reaction rate, volume, feed rate, and conversion of the species and we can rearrange the CSTR mole balance to get the performance equation and be able to work out the space-time or space velocity.

(1.29)

Then we can use the space-time equation seen previously:

PFR Performance Equation

 Assuming constant density,

C_AF – final concentration of species A

If the elementary reaction is given, we can easily form an expression for r_A in terms of conversion and then integrate to find out the space-time.

To be able to solve the integral, Simpson’s rule will need to be used, and depending on your university lecturer you could be given the Simpson’s rule formula, but if not it isn’t too hard to learn and the best way to learn is to practice and try as many variations of questions at different levels of difficulty to get used to answering these equations which are usually heavily weighted in terms of marks.

The equation below is Simpsons 3^rd rule this can usually be used most of the time unless stated otherwise.

(1.35)

Changing Density

When the number of moles of gas, temperature or pressure changes then the density will no longer be constant, and we can then use the ideal gas law and then using the initial conditions and the final conditions inside the batch reactor and rationing them out:

Ideal gas law: pV =nRT

p – Pressure

V – Volume

n- Moles

R – Universal gas constant

T – Temperature

Initial conditions of gas:

Final conditions of gas: pV =nRT

This is simply the initial number of moles plus the change in the total number of moles due to the reaction taking place.

Now we will introduce epsilon,  which is the change in the number of moles of the limiting reactant after the reaction has taken place, divided by the original number of moles.

(1.39)

Then we can say, in a situation where species A is the limiting reagent

(1.40)

Remember, the total number of moles includes the leftover reactants that are in excess and any inert species that are present.

Using the change in moles formula, equation 1.37 into the volume equation 1.37:

For a flow reactor, the volume is replaced with the volume flow rate:

(1.41)

When the density is constant, we can assume that is equal to zero.

Example – limiting reagent

 Air is comprised of roughly 21% oxygen, and the input feed to a reactor is 100 moles and sulphur dioxide is added as well and the input mixture is 28% sulphur dioxide and the remainder is air, the desired product is sulphur trioxide, what is the limiting reagent? and why?

2SO2 + O2 → 2SO3

References

The Essential Chemical Industry – online. (2013, March 18). Chemical reactors. Retrieved from The Essential Chemical Industry – online: https://www.essentialchemicalindustry.org/processes/chemical-reactors.html

Vapourtec. (2020). Continuous Stirred Tank Reactor (CSTR). Retrieved from Vapourtec: https://www.vapourtec.com/flow-chemistry/continuous-stirred-tank-reactor-cstr/

Vapourtec. (2020). Plug flow reactor. Retrieved from Vapourtec: https://www.vapourtec.com/flow-chemistry/plug-flow-reactor/

Wikipedia. (2020). Continuous stirred-tank reactor. Retrieved from Wikipedia: https://en.wikipedia.org/wiki/Continuous_stirred-tank_reactor

Wikipedia. (2020). Plug flow reactor model. Retrieved from Wikipedia: https://en.wikipedia.org/wiki/Plug_flow_reactor_model

Dr. Adam Zaidi

Dr. Adam Zaidi, PhD, is a researcher at The University of Manchester (UK). His doctoral research focuses on reducing carbon dioxide emissions in hydrogen production processes. Adam’s expertise includes process scale-up and material development.’

The post An In-Depth Breakdown | PFR and CSTR Reactor Design appeared first on Engineeringness.