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.

• Zeroth Rate Law or Zero Order Reaction
• First Rate Law or First Order Reaction
• Second Rate Law or Second Order Reaction

Each of the above rate laws provide a unique insight into reaction dynamics.

Half Life Definition

The half life which we derive from those integrated rate laws , is the time it takes for the concentration of reactants to fall to half.

The Integrated Zeroth-Order Rate Law

The integrated rate law for a zeroth-order reaction is expressed as:

[A]_t = -kt + [A]₀

• [A] is the concentration of reactant A at time t.
• k is the rate constant for the reaction.
• [A]₀​ is the initial concentration of reactant A at t=0.

Zeroth order reactions are quite unusual as the rate is independent of the concentration of reactants, i.e. Rate = k.

Where Can Zero Order Reactions be Found in Real Life?

Zero order kinetics are usually found where the surface area of a heterogenous catalyst is the main factor controlling the rate.

An example where Zero Order Reactions can be found in real life is within the Haber Process where the decomposition of NH₃ is done in the presence of Tungsten or Molybdenum.

How To Derive Half Life Equations for A Zero Order Reaction

To derive the half life equation for a zero order reaction the following can be done:

Question: Consider a zeroth-order reaction with an initial concentration of a reactant A, [A]=0.50 , and a known rate constant k=0.10 M/s. Calculate the half-life of this reaction.

Answer: The half-life of a zeroth-order reaction can be calculated using the formula: t_1/2=[A]₀ /2k (as shown above)

Given that [A]₀=0.50 M and k=0.10 M/s , we can substitute these values into the formula: t_1/2 =0.50M/(2×0.10 M/s)

to give an answer of t_1/2 = 2.5s

Therefore, the half-life of this zeroth-order reaction is 2.5 seconds

The Integrated First-Order Rate Law

The integrated rate law for first order reactions can be written in two formats :

1. Exponential Form of the Integrated Rate Law: The exponential form is expressed as [A] = [A]₀e^−kt.
   • Here, [A] is the concentration of reactant A at time t.
   • [A]₀​ is the initial concentration of reactant A at t=0.
   • k is the rate constant
   • This equation predicts that the concentration of A will decrease exponentially over time.
2. Logarithmic Form of the Integrated Rate Law: By taking the natural logarithm of the exponential form, we can rearrange it to obtain the logarithmic expression: ln[A]=ln[A]₀ −kt
   • This is another way of expressing the relationship between the concentration of A and time t, where ln denotes the natural logarithm.

Both forms are useful for different purposes: the exponential form can be used for predicting concentrations at any given time, while the logarithmic form is particularly useful for finding the rate constant k and for plotting data to determine reaction order.

Plotting A First Order Rate Reaction

When plotting the natural logarithm of the concentration of reactant A, denoted as ln[A], against time (t), the resulting graph is a straight line. This line has a slope of -k (the negative rate constant) and an intercept equal to ln[A]₀, which is the natural logarithm of the initial concentration of reactant A.

Deriving Half Life Equations of First Order Chemical Reactions

The half life is how long it takes for the concentration of reactants to fall to half, you can derive the half life equation for first order using the integrated rate law as such :

Thus, for a first-order reaction, each successive half-life is the same length of time and is independent of [A] as shown below:

Question on first order chemical reactions :

Question: Determine the half-life of a first-order chemical reaction where the rate no constant is 0.0011 s^-1?

Answer: You simply divide 0.693 by 0.0011 as indicated in the first order half life equation t_1/2 = 0.693/k

The Integrated Second-Order Rate Law

The integrated rate law for a second-order reaction can be expressed as: 1/[A] = kt + 1/[A]₀.

• Here, 1/[A] represents the inverse of the concentration of reactant A at time t.
• k is the rate constant for the reaction.
• 1/[A]₀​ is the inverse of the initial concentration of reactant A at t=0.

This equation predicts that the inverse of the concentration of A increases linearly over time.

Plotting and Interpreting A Second Order Reaction

When plotting the inverse concentration 1/[A] against time t, you will obtain a straight line.

The slope of this line is the rate constant k, providing a direct measurement of the reaction rate.

The intercept on the y-axis is 1/[A]₀, representing the inverse of the initial concentration.

Deriving half life equation for second order rate law :

As concentration decreases, the half life of second order reactions increases.

Question for second order rate law

Question: The reaction 2NOBr(g) → 2NO(g) + Br₂(g) is a second order reaction with
respect to NOBr. k = 0.810 M-1s-1 at 10°C. If initial concentration of NOBr = 7.5 × 10^-3 M, what quantity of NOBr remains after a reaction time of 10 minutes?

Answer: The integral for a second order reaction is 1/[NOBr] = kt + 1/[NOBr]0

1/[NOBr] = 0.810*600 + 1/ 7.5*10^-3

Adrian Michaels

Adrian graduated with a Masters Degree (1st Class Honours) in Chemical Engineering from Chester University along with Harris. His master’s research aimed to develop a standardadised clean water oxygenation transfer procedure to test bubble diffusers that are currently used in the wastewater industry commercial market. He has also undergone placments in both US and China primarely focused within the R&D department and is an associate member of the Institute of Chemical Engineers (IChemE).

The post Mastering Chemical Kinetics | A Comprehensive Guide to Integrated Rate Laws and Reaction Dynamics appeared first on Engineeringness.

– Vibratory motors are essential components in various industries, providing efficient and reliable vibration for a wide range of applications.
– These motors are designed to generate controlled vibrations, which can be used for tasks such as conveying, compacting, screening, and sorting.
– Vibratory motors offer several advantages, including low maintenance requirements, energy efficiency, and the ability to customize vibration intensity and frequency.
– They are commonly used in industries such as mining, construction, food processing, and pharmaceuticals.
– Understanding the working principles and different types of vibratory motors can help in selecting the right motor for specific applications.

Introduction

Vibratory motors play a crucial role in numerous industries, providing controlled vibrations for various applications. These motors are designed to generate mechanical vibrations, which can be utilized for tasks such as conveying, compacting, screening, and sorting. With their ability to deliver efficient and reliable vibration, vibratory motors have become indispensable in industries ranging from mining and construction to food processing and pharmaceuticals. In this article, we will explore the fascinating world of vibratory motors, their working principles, different types, and the advantages they offer.

Working Principles of Vibratory Motors

Vibratory motors operate on the principle of electromagnetic induction. They consist of a stator and a rotor, with the stator being the stationary part and the rotor being the rotating part. The stator contains a coil of wire, which is energized by an alternating current (AC). This alternating current creates a magnetic field that interacts with the rotor, causing it to rotate.

The rotation of the rotor generates centrifugal force, which is responsible for the vibrations produced by the motor. The intensity and frequency of the vibrations can be controlled by adjusting the current supplied to the motor. This allows for precise customization of the vibrations according to the specific requirements of the application.

Types of Vibratory Motors

There are several types of vibratory motors available, each designed for specific applications. Some of the commonly used types include:

1. Unbalanced Motors

These motors consist of a single rotor with an unbalanced weight attached to it. As the rotor rotates, the unbalanced weight creates centrifugal force, resulting in vibrations. Unbalanced motors are widely used in applications such as vibrating screens, feeders, and compactors.

2. Electromagnetic Motors

These motors utilize the principle of electromagnetic induction to generate vibrations. They consist of a coil of wire and a magnetic core. When the coil is energized, it creates a magnetic field that interacts with the magnetic core, causing it to vibrate. Electromagnetic motors are commonly used in applications such as vibrating conveyors and feeders.

3. Eccentric Rotating Mass (ERM) Motors

ERM motors consist of an eccentric mass attached to a rotating shaft. As the shaft rotates, the eccentric mass creates centrifugal force, resulting in vibrations. ERM motors are commonly used in applications such as mobile phones, pagers, and handheld devices.

4. Linear Resonant Actuators (LRAs)

LRAs are compact and lightweight vibratory motors that generate vibrations in a linear motion. They consist of a coil and a magnet, with the coil attached to a movable mass. When an alternating current is passed through the coil, it interacts with the magnet, causing the mass to move back and forth, generating vibrations. LRAs are commonly used in applications such as haptic feedback in smartphones and gaming controllers.

Advantages of Vibratory Motors

Vibratory motors offer several advantages over other types of motors and vibration-generating devices. Some of the key advantages include:

1. Low Maintenance

Vibratory motors have a simple design and require minimal maintenance. They do not have any brushes or commutators, which eliminates the need for regular maintenance tasks such as brush replacement.

2. Energy Efficiency

Vibratory motors are highly energy-efficient, as they convert electrical energy into mechanical vibrations with minimal energy loss. This makes them a cost-effective choice for applications that require continuous or frequent vibration.

3. Customizable Vibration

The intensity and frequency of vibrations generated by vibratory motors can be easily adjusted by controlling the current supplied to the motor. This allows for precise customization of the vibrations according to the specific requirements of the application.

4. Wide Range of Applications

Vibratory motors find applications in various industries, including mining, construction, food processing, pharmaceuticals, and more. They are used for tasks such as conveying bulk materials, compacting soil or concrete, screening and sorting materials, and providing haptic feedback in electronic devices.

Conclusion

Vibratory motors are essential components in numerous industries, providing efficient and reliable vibration for a wide range of applications. These motors operate on the principle of electromagnetic induction and can generate controlled vibrations for tasks such as conveying, compacting, screening, and sorting. With their low maintenance requirements, energy efficiency, and customizable vibration, vibratory motors have become indispensable in industries ranging from mining and construction to food processing and pharmaceuticals. By understanding the working principles and different types of vibratory motors, industries can select the right motor for their specific applications, ensuring optimal performance and productivity.

Adrian Michaels

Adrian graduated with a Masters Degree (1st Class Honours) in Chemical Engineering from Chester University along with Harris. His master’s research aimed to develop a standardadised clean water oxygenation transfer procedure to test bubble diffusers that are currently used in the wastewater industry commercial market. He has also undergone placments in both US and China primarely focused within the R&D department and is an associate member of the Institute of Chemical Engineers (IChemE).

The post Understanding Vibratory Motors: Working Principles, Types, and Advantages appeared first on Engineeringness.

What Is An Energy Balance?

Energy balance is simply the difference in the energy input and output and 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.

Figure 1. Energy balance diagram for an open system

The Energy Balance Equation For Open Systems:

(1.11)

The energy balance equation flow reactors in open systems (1.11), which is valid for non-isothermal reactors that have no phase change, and a reference species A has been used:

Where:

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

Example:

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 An Brief Overview | Non-Isothermal Reactors appeared first on Engineeringness.

Aluminium Chloride and Sodium Hydroxide Example

The best way to show the factors affecting a chemical reaction is to use an example. The chemical reaction we will look at is the double displacement reaction between Aluminium Chloride (AlCl₃) and Sodium Hydroxide (NaOH) that produces Aluminium Hydroxide (Al(OH)₃) and Sodium Chloride (NaCl) (equation 1).

AlCl3(aq) + 3NaOH(aq)  → Al(OH)3(s)  + 3NaCl(aq)

Equation 1: Double displacement reaction between Aluminium Chloride and Sodium Hydroxide

Factors Affecting The Rate Of A Chemical Reaction:

Concentration

The concentration of reactants in a chemical reaction is related to the rate of a chemical reaction. If we increase reactant concentration, then the rate of the reaction will also increase. The rate of the reaction is dependent on the concentration of both the reactants. Because the AlCl₃ solution is a strong electrolyte, it dissociates completely into two ions, Al³⁺ and Cl^–. Therefore, it means that AlCl₃ and Al³⁺ ions are present in aqueous solution. In this reaction, AlCl₃ is the limiting reagent, as the rate of the reaction is dependent on it.

While sodium hydroxide solution is not an electrolyte, but it dissociates into Na⁺ and OH^– ions. Thus, NaOH and OH^–ions are present in aqueous solution. In this reaction, NaOH is the limiting reagent, in terms of concentration. This is because the rate of the reaction is dependent on it. So, this reaction is an equilibrium reaction, as it depends on the product.

The pH of the reaction must be in the range required by the reaction. If it is outside this range, then the reaction will be slow. For instance, in the case of the reaction between AlCl₃ and NaOH, the concentration of both the reactants are equal. However, in this case, the highest concentration of the product is achieved when the reaction is carried out at pH 3, and the highest concentration of the reactant is at pH 9. So, the reaction requires the pH to be in the range of 3-9.

Physical State

The physical state of the reactants affects the rate of a chemical reaction. For instance, in the case of the reaction between AlCl₃ and NaOH, if both the reactants are in the gaseous state, then it will be difficult to achieve the equilibrium. This is because gaseous reactants move away from each other, and the distance between them has a bearing on the diffusion rate. Moreover, in the gaseous state, the molecules have high kinetic energy, which slows down the rate of reaction.

While if both the reactants are in the solid-state, then it is easier to reach the equilibrium, in this case. Solid reactants do not have great kinetic energy. Moreover, in the solid-state, the molecules are close to each other, so it is easier to achieve equilibrium.

If either of the reactants is in a liquid state, then it is easy to achieve the equilibrium in the liquid state also. But, if both the reactants are in the liquid state, then it will be easier to achieve equilibrium again, as the molecules are closer to each other. Thus, the liquid state is always favourable for a reaction, the reason being that it ensures a greater number of collisions. On the other hand, free gas is less reactive as compared to a gas dissolved in a solution.

Catalyst

 The presence of a catalyst has an impact on the rate of the reaction. For instance, in the case of a reaction between hydrogen and oxygen, a catalyst is not required. This is because neither reaction is extremely fast, nor reactive. However, if we want a faster and more productive reaction, we can use the catalyst in this case, as well. The catalyst’s role, in this case, is to speed up the rate of reaction. But this also depends on the type of catalyst used. For instance, different catalysts react differently under different conditions.

Temperature

The temperature of the reaction has a big impact on the rate of the chemical reaction. If we increase the temperature of the reaction, hence the temperature of both the reactants, then the reaction rate will increase. Hence, the rate of chemical reactions increases with temperature.

In the reaction between AlCl₃ and NaOH, if the temperature of the two reactants is doubled, then the rate of reaction, hence production will also increase. For instance, if we carry out the reaction at a temperature of 98°C, then the rate of reaction will increase by a factor of four, as the rate of reaction doubles with every 10° rise in temperature.

Surface Area

 The surface area of both the reactants has a significant impact on the rate of reaction. It is because the surface area of reactants has a direct relation with the number of collisions per unit time. Therefore, if we want to increase the rate of a chemical reaction, then we must increase the surface area of both the reactants.

In the case of a reaction between AlCl₃ and NaOH, there is no significant change in the reaction rate, whether the number of reactants is increased or the reactants change in shape or size. However, if we increase the surface area of a reactant, then the reaction rate also increases. For instance, if AlCl₃ and NaOH are powdered, then the reaction rate will decrease by a factor of two, as only a small number of ions come in contact with each other. So, we can state that the rate of reaction is dependent on the surface area of the reactants. The maximum size of reactant particles in which a chemical reaction takes place at the maximum rate is called the critical size.

Thus, in the case of the reaction between AlCl₃ and NaOH, both these reactants are powdered, hence the change in the rate of reaction is not significant. Therefore, due to the powdered nature of the reactants, the reaction rate is dependent on the size of the reactants. As a result, the reactants keep on colliding with each other, in case of powdered reactants. Hence, the rate of the reaction is higher in this scenario.

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 Factors Affecting The Rate Of A Chemical Reaction 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.

A Batch reactor plot is a graphical representation of the volume of an isothermal system.

General shape of a Batch Reactor (Advanced Energy Materials Processing Laboratory, 2020)

Batch reactor plot (Advanced Energy Materials Processing Laboratory, 2020)

A Levenspiel plot is a representation of the continuous flow reactor; CSTR and PFR design equations as a function of conversion and is used to determine the volume of the reactor.

Shape of CSTR and PFR (Advanced Energy Materials Processing Laboratory, 2020)

PFR and CSTR Levenspiel Plot Comparison

The rate used for the CSTR is evaluated at the exit stream conditions while for the PFR the rate used is integrated over a range of conditions and we can solve this using Simpsons composite rule,

CSTR and PFR Levenspiel plot (Advanced Energy Materials Processing Laboratory, 2020)

• PFR requires a smaller volume than the CSTR for a given conversion
• When the reaction speed increases for a CSTR the Levenspiel plot will curve downwards as the conversion changes and will require a smaller CSTR volume.

Levenspiel Plot For Reactor In A Series Arrangement

PFR in series act as one large PFR and if the density is constant then the residence time is just the space time at the inlet conditions. For a CSTR multiple CSTRs in series require a smaller volume as a CSTR is evaluated at the output conditions and will make a series of CSTR’s smaller than one large CSTR, as when using multiple CSTRs the first tank operates at a lower conversion so the concentration of reactants will be higher so the rate will be greater and the volume required will be smaller.

CSTRs in series get close to the performance of PFR and the smaller the CSTRs the closer they get, but financial costs and available space and other factors make having lots of small CSTRs not practical when one PFR can be used.

CSTR Levenspiel plot in series (MIT, 2007)

Parallel Reactors

Parallel reactors for equal-sized flow reactors, the feed stream is split evenly between the reactors. Parallel reactor arrangement is used for CSTRs as the reactors will be operating at the lowest conversion will be better to operate in series. For PFRs this arrangement behaves as one large PFR and is a common arrangement as used in industry or in Labourites.

PFR in a parallel arrangement (Santofimio, 2020)

PFR With Recycle

Unreacted reactants can be recycled from the PFR exit stream, we define a recycle ratio, R when it is equal to zero (R = 0) then we have standard/normal plug flow and as R increases we develop mixed flow and the PFR starts to resemble the behaviour of a CSTR.

R = Volume of fluid recycledVolume of fluid leaving PFR

(1.11)

We will be adding two new terms, single-pass conversion X_S and overall conversion X_O, the equations are below and have used species ‘A’ to represent the species used.

Single-pass conversion shows the fraction that is converted when it goes through the PFR once and overall conversion is the fraction converted in the final stream from the total inlet flow.

PFR with recycle diagram (Cheggstudy, 2020)

PFR with recycle is a difficult concept to get your head around and has a lot of keywords, that can trip you up if you don’t pay attention them, the best way is to do an example whilst looking at the answers and see what steps to do to solve this type of question in an exam, if you can do this example exam question without looking at the answers it will be extremely impressive!

Example – PFR with recycle (typical exam question)

In a PFR with recycle a reaction that is elementary and, in the liquid phase takes places, with an R = 1 and a conversion of 2/3, what is the conversion if there is no recycle stream?

References

Advanced Energy Materilas Processing Laboratory. (2020). CHE 309: Chemical Reaction Engineering. Retrieved from Advanced Energy Materilas Processing Laboratory: http://aempl.kist.re.kr/wp-content/files/Lecture-5_Ch2.pdf

Cheggstudy. (2020). Question: Problem 3: Recycle Reactor. Retrieved from Cheggstudy: https://www.chegg.com/homework-help/questions-and-answers/problem-3-recycle-reactor-farmer-michael-process-setting-recycle-reactor-farm-reaction-tak-q26913611

MIT. (2007). PFR vs. CSTR: Size and Selectivity. Retrieved from MIT: https://ocw.mit.edu/courses/chemical-engineering/10-37-chemical-and-biological-reaction-engineering-spring-2007/lecture-notes/lec09_03072007_w.pdf

Santofimio, D. S. (2020). Parallel Reactors.docx. Retrieved from Scribd: https://www.scribd.com/document/242835751/Parallel-Reactors-docx

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 Batch And Levenspiel Plots For Parallel And Series Reactors appeared first on Engineeringness.

The reaction rate, r_i (also referred to as the rate of reaction) is a measure of how fast a reaction is and is defined using several ways:

$r_i = \frac{1}{V}\frac{∆N_i}{∆t} = \frac{Moles of species ‘i‘ formed}{Volume of fluid \times time}$

${r_i}^{‘} = \frac{1}{M}\frac{∆N_i}{∆t} = \frac{Moles of species ‘i‘ formed}{Mass of solid \times time}$

${r_i}^{‘‘} = \frac{1}{S}\frac{∆N_i}{∆t} = \frac{Moles of species ‘i‘ formed }{Surface area \times time}$

${r_i}^{‘‘‘} = \frac{1}{V_s}\frac{∆N_i}{∆t} = \frac{Moles of species ‘i‘ formed}{Volume of solid \times time}$

${r_i}^{‘‘‘‘} = \frac{1}{V_r}\frac{∆N_i}{∆t} = \frac{Moles of species ‘i‘ formed}{Volume of reactor \times time}$

One of the benefits of the reaction rate is that if the reactor is scaled up the rate of reaction will be the same, which simplifies the process of reactor scale up, so no long equations are needed every time the reactor volume changes. Thus, the reaction rate is an intensive quantity as its magnitude is independent of the size of the system (i.e. changing reactor size).

For reagents, reaction rates are negative and for products the reaction rates are positive, this is because reagents are used up in reactions, and products are formed. An example of this would be the reaction:

$aA + bB \to \mathrm{cC}$

to write this out in terms of reaction rates, the species (A, B, or C) and the stoichiometric coefficient (a, b and c) are required, and the concept stated above: the reagents being negative and the products being positive.

$–\frac{r_A}{a} = –\frac{r_B}{b}=\frac{r_C}{c}$

(1.0)

Example: To prove that the reactions rates are equal

A reactor with fluid volume 1 m³ has a reaction with the reaction time being 10 seconds, and we are told that 5 moles of B are formed, prove that the rates are equal.

Hint: use the reaction rate definitions, looking closely at parameters you have been given and equation 1.0.

Show Answer

Rate Equations

For any chemical reaction equation, a rate equation can be used which links the forward reaction rate with the concentration or pressure of the reactants. These are more complicated functions of reagents and (sometimes) product concentrations for a non-elementary reaction.

Reactions can be classified under these terms:

• Homogeneous: consists of only one phase
• Heterogeneous: more than one phase needed for the reaction
• Catalytic or noncatalytic
• Exothermic or endothermic
• Elementary or nonelementary
• Single reaction or multiple reactions (and within latter: series or parallel, and combinations)

Writing out a rate equation:

Taking an example reaction:

$A + B \to \mathrm{C}$

The rate equation would simply be:

$–r_A = K_A{C}_A^a{C}_B^b$

And using equation 1.0 we know that the rate equation concerning species B would be the same as that of species A. K is the rate constant which is proportionally constant that indicates the relationship between the molar concentration of reactants and the rate of a chemical reaction (Helmenstine, 2018), and the rate constant concerning each species is:

$\frac{K_A}{a}=\frac{K_B}{b}$

Given a rate equation:

$–r_A = K_A{C}_A^a{C}_B^b$

The reagents a and b are not always the stoichiometric coefficients, we say that the reaction n order concerning A and b^th order concerning B and the overall order is n = a + b.

The molecularity of an elementary reaction is the number of molecules involved in the reaction. The order can be fractional values and molecularity is always a whole number.

Equilibrium Constant

For an equilibrium reaction, the rate of the forward reaction is the same as the rate of the backward reaction and the concentration doesn’t change.

$–r_{A, forwards }= r_{A, backwards}$

(1.2)

For the reaction:

$\mathrm{aA} + \mathrm{bB} \rightleftharpoons \mathrm{cC} + \mathrm{dD}$

The equilibrium constant K_C (also written as K_eq or K) can be defined as:

$K_C = \frac{{\left[C\right]}_{eq}^c{\left[D\right]}_{eq}^d}{{\left[A\right]}_{eq}^a{\left[B\right]}_{eq}^b}$

(1.3)

The position of the equilibrium is determined by the entropy change, the enthalpy change, and the conditions the reaction is under such as temperature and pressure.

Assuming the reaction is elementary (stoichiometric coefficients are usually 1 for all species) the forward and backward reaction can be stated:

$r_{A, forwards }= –k_1{\left[A\right]}^a{\left[B\right]}^b$

$r_{A, backwards} = k_{–1}{\left[C\right]}^c{\left[D\right]}^d$

k₁ represents the rate constant for the backward reaction and k_-1 represents the rate constant for the forward’s reaction

Thus, we can know to prove equilibrium constant K_C equation 1.3, by first using equation 1.2:

$–k_1{\left[A\right]}_{eq}^a{\left[B\right]}_{eq}^b = k_{–1}{\left[C\right]}_{eq}^c{\left[D\right]}_{eq}^d$

when rearranged will give:

$K_C = \frac{{\left[C\right]}_{eq}^c{\left[D\right]}_{eq}^d}{{\left[A\right]}_{eq}^a{\left[B\right]}_{eq}^b}$

References

Helmenstine, A. M. (2018, September 27). What Is the Rate Constant in Chemistry? Retrieved from ThoughtCo: https://www.thoughtco.com/reaction-rate-constant-definition-and-equation-4175922

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 Basic Reaction Kinetics appeared first on Engineeringness.