Firstly, we will be taking a look into energy balances on reactors to give us a better understanding of the equations and assumptions used in the Non-Isothermal and Non-Ideal Flow reactors. The First 2 terms that need to be understood are, Isothermal and Adiabatic.

What Does Isothermal Mean?

Isothermal which refers to a system at a constant temperature,

What Does Adiabatic Mean?

Adiabatic involves no heat entering or leaving the system.

What Is An Open System?

Open systems refers to mass being able to enter and leave the system. Moreover, Temperature can also be lost to or gained from the surrounding.

What Is An Energy Balance?

An Energy balance is simply the difference in the energy input and output. An energy balance equation can be used on open systems and can be relatively easy to do or challenging when more components and different phases are present.

The energy balance equation for an open system is:

The enthalpy of reaction at a certain temperature (T) is worked out using the enthalpy of products and reactants:

Non-Ideal Flow Reactors

In reality reactors can never be ideal and it cannot be assumed. The flow will always deviate and for flow reactors, non-ideal flow patterns can cause issues.

CSTR (Continuous stirred-tank reactor):

• There will be stagnant or dead regions where no mixing of fluids and no flow. This causes the fluid to sit in place and reduce the volume of the CSTR for reactions.

• In the event where the fluids flow from the inlet to the outlet, this is called short-circuiting and the fluid won’t mix or spend enough time in the reactor.

 PFR (Plug Flow Reactor):

• The ideal plug flow isn’t possible in reality as fluids will mix along the length of the PFR.

• Fluids closer to the walls will travel at a slower rate than fluid closer to the centre of the reactor and this will cause mixing along the axis of PFR.

• Turbulent mixing and molecular diffusion will lead to mixing along the length of the PFR.

What Is Residence Time Distribution and How To Calculate Residence Time Distribution

Residence time distribution (RTD) is the probability distribution of the time that a solid or fluid spends inside the reactor and is the main method that can be measured to understand the types of flow encountered in non-ideal flow reactors. TRD is denoted with the symbol, E and will tell us the amount of time (age) the material spends in the exit stream of the reactor.

t – age. It is time spent in reactor, you will find that age young and older as used to denote time, as long as you remember that age is time spent in reactor. The terms younger and older are the same. They are seen as as less time spent in the reactor (younger) and More time spent in the reactor (older).

the RTD is normalised to unity (integrated between zero and infinity):

How To Measure The Residence Time Distribution

The RTD is measured using the pulse experiment, which involves injecting a pulse dye or tracer dye into the fluid before entering the reactor, then measuring the dye concentration over a period of time and the results when graphed will be normalised. The concentration is proportional to the RTD and gives an area under the graph of unity (1).

You will need to divide the measured concentration curve, C_pulse by the area under the curve. The area under the graph can be found by integration or by the expression M/ (mass of tracer/volumetric flow rate) assuming that C_pulse is mass per unit volume.

Example Residence Time Distribution

Step experiment and F curve:

Here we have a steady yellow tracer flow at t = 0. The tracer concentration at the outlet will increase as time increases, C_step will equal the inlet concentration. This concentration divided by the final concentration is called the F curve. The final concentration of the tracer is mass flow rate divided by volumetric flow rate.

We can now say that at any time greater than t = 0, that the yellow tracer in the exit stream is younger than age t, and the fraction of yellow tracer at the outlet is equal to the fraction of the exit stream younger than time t, which is written as:

State Of Mixing and RTD:

To find out the conversion of the reactants the state of mixing as well as the RTD need to be known. There are two terms we describe the state of mixing:  

The State of Mixing | Macrofluid and Microfluid:

• Macrofluid: Globules of fluids act as their own batch reactors and will spend different amounts of time in the reactor and have a different conversions than other globules. Also, o mixing between globules of fluid that aren’t the same age, such as very viscous fluids or solids.
• Microfluid: Individual globules can move anywhere in the reactor such as an ideal CSTR, examples of these include gases and not very viscous liquids.

For batch reactors, to work out the concentration of a species let’s call it A for an element of age t and average using the RTD the equation would be:

Hassan Ahmed

Hassan graduated with a Master’s degree in Chemical Engineering from the University of Chester (UK). He currently works as a design engineering consultant for one of the largest engineering firms in the world along with being an associate member of the Institute of Chemical Engineers (IChemE).

The post Non Isothermal and Non Ideal Flow Reactors | A Breakdown Of Reactor Design 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.’

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