Reaction Order Identification - Rate Constant Estimation - Batch Reactor Design

Hi All....!!

Today i want to post something better for our chemical engineers / process engineers which is a consolidation of reaction engineering and design by grabbing the basics. I'll take a case study on implementing of mechanistic studies for a considered reaction mechanism and using the data to design a batch reactor. This post is applicable for even the chemists who work in chemical manufacturing industries except the oil men.

Before that, i want to give you guys a warm welcome at Pharmacalculations. We’ll exploring a powerful transformation used in the manufacture of pharmaceutical intermediates — the Ritter Reaction. This acid-catalyzed process enables the conversion of alcohols or alkenes with nitriles into tertiary amides, which are vital building blocks in API and agrochemical synthesis.

But we won’t stop at just the chemistry. This post will walk you through:

• The reaction mechanism

• How to identify the rate-determining step

• How to derive the rate law

• How to determine the reaction order

• How to design a batch reactor using Levenspiel’s performance equations

Let’s base this on a real manufacturing case: the production of Tert-butylacetamide.

Before jumping into the post, i want to dedicate this to a section of Chemical Engineers who feel that chemistry is not our part to work on. Any Reaction that is going to be designed in a chemical manufacturing industry requires two hands for the upliftment i.e., a proper chemistry and also the engineering for its successful establishment. Its never bad to grab few Chemistry insights while working in Chemical manufacturing industry. I would like to thank Mr. P Jagadish Kumar for motivating me to understand chemistry before taking up the process detailing part.

Anyhow, lets jump into our show.

Today, i've planned to take a complex Ritter reaction.

Industrial Case: Ritter Reaction for tert-Butylacetamide Synthesis

Reaction:

$$\text{(CH₃)₃COH} + \text{CH₃CN} + \text{H₂SO₄} \rightarrow \text{(CH₃)₃C–NHCOCH₃}$$

• Alcohol: tert-Butyl alcohol (TBA)

• Nitrile: Acetonitrile

• Catalyst: Sulfuric acid

• Product: tert-Butylacetamide

• Reactor: Glass-lined batch reactor

• Temperature: 60–70°C

• Target Conversion: ≥90% in ≤1 hour

Before getting into the main case, lets check few basic question & answers to understand case better.

What is the Ritter reaction used for?

For making amides from carbocations and nitriles — widely used in pharma.

Why is tert-butyl alcohol selected?

It forms a stable tertiary carbocation, ensuring fast and selective reaction.

Why sulfuric acid?

It acts both as a catalyst and proton source to initiate carbocation formation.

Why excess acetonitrile?

To drive the reaction forward and simplify the rate law (pseudo-first-order in CH₃CN).

What are typical side reactions?

Nitrile hydrolysis and unconverted alcohol due to incomplete conversion.

What kind of reactor is preferred?

A glass-lined batch reactor to handle the acidic conditions and control the exotherm.

Is the reaction exothermic?

Yes — particularly during carbocation formation. Controlled dosing and external cooling are recommended.

What is the rate-determining step?

The attack of nitrile on the carbocation intermediate.

What is the reaction order?

Typically 2nd order (1st in TBA, 1st in H⁺) when CH₃CN is in excess.

How do we derive the rate law?

Using the reaction mechanism and applying the steady-state approximation to the carbocation.

Reaction Mechanism of the Ritter Reaction

Let:

• $A =$ tert-butyl alcohol

• $B =$ acetonitrile

• $C^+ =$ carbocation

• $I =$ iminium intermediate

• $P =$ amide product

Stepwise Mechanism:

Step 1: Protonation of Alcohol (Fast, Reversible)

$$A + H^+ \xrightleftharpoons[k_{-1}]{k_1} C^+ + H_2O$$

Step 2: Attack of Nitrile (Slow, Rate-Determining)

$$C^+ + B \xrightarrow{k_2} I$$

Step 3: Hydrolysis of Iminium (Fast)

$$I + H_2O \xrightarrow{k_3} P$$

How to Identify the Rate-Determining Step (RDS)

Knowing the RDS is critical to deriving the right rate expression.

1. Look for the slow step

Mechanisms often label the slow step — here, Step 2 (nitrile attacking carbocation) is the slowest.

2. Analyze intermediate buildup

The carbocation forms quickly (Step 1), but reacts slowly with CH₃CN. This indicates [C⁺] accumulates, confirming Step 2 is RDS.

3. Validate with experimental kinetics

Derived rate law:

$$r = \frac{k_1 k_2 [A][H^+][B]}{k_{-1} + k_2 [B]}$$

If [B] is in large excess, this simplifies to:

$$$$r=k1[A][H+]r = k_1 [A][H^+]

This matches observed 2nd order kinetics, further confirming Step 2 is the RDS.

Rate Law Derivation Using Steady-State Approximation

From the mechanism:

Step 1

$$r_1 = k_1 [A][H^+], \quad r_{-1} = k_{-1}[C^+]$$

Step 2 (RDS)

$$r_2 = k_2 [C^+][B]$$

Assume steady state on [C⁺]:

$$\frac{d[C^+]}{dt} = 0 = k_1 [A][H^+] - k_{-1}[C^+] - k_2 [C^+][B]$$

Solving:

$$[C^+] = \frac{k_1 [A][H^+]}{k_{-1} + k_2 [B]}$$

Rate of product formation (RDS):

$$r = k_2 [C^+][B] = \frac{k_1 k_2 [A][H^+][B]}{k_{-1} + k_2 [B]}$$

Reaction Order Identification

Case 1: [B] is limited

$$r = \frac{k_1 k_2}{k_{-1}} [A][H^+][B] \quad \text{(3rd order overall)}$$

Case 2: [B] in large excess

$$r \approx k_1 [A][H^+] \quad \text{(2nd order overall)}$$

In industry, CH₃CN is used in excess — so the working rate law becomes:

$$-r_A = k [A][H^+]$$

Batch Reactor Design Using Performance Equation

From Levenspiel (Ch. 1 & 3) — for 2nd order in A and H⁺:

$$t = \frac{1}{k C_{H^+}} \left( \frac{1}{C_A} - \frac{1}{C_{A0}} \right)$$

Assumed Lab Data:

Estimate k from lab trial:

$$X = 0.90 \Rightarrow C_A = 0.02 \text{ mol/L}$$
$$t = \frac{1}{k \cdot 0.15} \left( \frac{1}{0.02} - \frac{1}{0.20} \right) = 60 \Rightarrow k = 5.0\ \text{L/(mol·min)}$$

Batch Time at Different Conversions (using $k=\mathrm{5.0)}\mathrm{<br>}$

Higher conversions = exponentially more time due to second-order kinetics.

Thats it....!!

Hope you understood everything clearly.

Any queries, feel free to reach us at pharmacalc823@gmail.com

Comments are most appreciated ....!!

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About The Author

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Hi! I am Ajay Kumar Kalva, Currently serving as the CEO of this site, a tech geek by passion, and a chemical process engineer by profession, i'm interested in writing articles regarding technology, hacking and pharma technology.
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Design of Packed Bed Distillation Column

Hiii All... !! Hope everyone is doing well.

This post is the most anticipated one for many of our readers and its under drafting since last 9 months & 17 days. Before going into the post i would like to Welcome you to PharmaCalculations.com, where depth is our trademark. In previous posts, we have tackled vacuum ejectors, photoreactors—even packed‑bed columns for generic solvents. Today, we bring that same rigor to a concrete case: separating acetone and ethanol with a distillate purity of 99 mol% acetone using a packed-bed column.

Before going into the post, lets get familiar with the post with some general queries:

1. What is the difference between tray and packed columns?

   • Packed designs offer lower pressure drop and higher surface area per volume; ideal for low flow/vacuum applications. Trays are simpler but larger.

2. How does Antoine’s equation apply in design?

   • It estimates vapor pressures at different temperatures:

     $$$$log⁡ (P)=A − (B / ( C+T ) )

     That lets us compute vapor–liquid equilibrium (VLE) necessary for stage‑wise balance.

3. What is HETP and why is it important?

   • Height Equivalent to a Theoretical Plate defines packing efficiency. Lower HETP = fewer meters needed for separation.

4. How do you size column diameter?

   • Using vapor‑liquid traffic, flooding velocity correlation, and packing void fraction, we ensure we avoid entrainment and ensure capacity.

5. What role does reflux ratio play?

   • A higher reflux ratio reduces packing height but increases energy use. We optimize ration vs load.

6. What is reboiler duty and how is it estimated?

   • Calculated via the q‑line and energy balance. We’ll compute it to find steam/hot oil heating required.

7. How do we determine packing height?

   • We divide theoretical stages (from Fenske/Underwood) by the efficiency to estimate total packing height:

     $$H_{\rm total} = N_{\rm theo} \times \text{HETP}$$
     

8. How are internals like re-distributors and feed trays added?

   • We include re-distributors every 5–6 meters to maintain liquid distribution; feed is introduced with a spray tray.

9. How do we check flooding and pressure drop?

   • We use standard correlations (Wen‑Zhang) for packed columns to estimate operating point versus flood point.

10. What about energy integration and trade-offs?

• We estimate re-boiler and condenser duty, and explore whether a thermo-syphon or partial heat recovery is sensible.

Now, i think its time for jumping into calculation part with some input data assumed.

Input Details:

i. Feed: 100 kmol/hr, 50 mol% acetone (A) / 50 mol% ethanol (E)

ii. Distillate purity: 99 mol% A

iii. Bottoms: 1 mol% A

iv. Operating: 1 atm

v. Packing: BPG TT‑20 random packing

Here for designing the column, i dont want to use the tradiational approach of Mc cabe thiele method instead i prefer to use the Fenseke, Underwood & Gilliland correlations. The reason is that here the required distillate purity is quite high and the Mc cabe thiele method is applicable upto 25 theoretical stages, hence i've preferred to go with the others.

Step - 1: Vapour - Liquid Equilibrium

 Acetone (A) & Ethanol (E):

Antonie constants:

Solvents                                                       A                              B                               C

Acetone                                                    7.02447                      1161                           224

Ethanol                                                     8.20417                    1642.89                      230.3

At ~80 °C (column average):

$$P_A^\mathrm{sat} = 10^{7.02447 - 1161/(80+224)} = 0.484\text{ bar}$$

$$P_E^\mathrm{sat} = 10^{8.20417 - 1642.89/(80+230.3)} = 0.375\text{ bar}$$

→ Relative volatility:

$$\alpha =\frac{0.484}{0.375}=\boxed{{\displaystyle 1.29}}$$

Step - 2: Minimum Stages – Fenske Equation

$$N_\mathrm{min} = \frac{\ln\left(\frac{x_D/(1-x_D)}{x_B/(1-x_B)}\right)}{\ln(\alpha)}$$

with $x_D=\mathrm{0.99, }$$x_B=\mathrm{0.01, }$$\alpha =1.29\mathrm{<br>}$

$$N_\mathrm{min} = \frac{\ln(0.99/0.01) - \ln(0.01/0.99)}{\ln(1.29)} = \frac{4.595 - (-4.595)}{0.254} = \frac{9.19}{0.254} = \boxed{36.2}$$

Minimum theoretical stages: 37

Step - 3: Minimum Reflux Ratio – Underwood Method (Step‑by‑Step)

Underwood equation (saturated liquid feed, q=1):

$$\frac{\alpha_A z_A}{\alpha_A - \theta} + \frac{\alpha_E z_E}{\alpha_E - \theta} = 1$$

Where:

• $z_A = z_E = 0.5$
  

• ${\alpha }_A=\mathrm{1.29, }$$\alpha_E = 1.00$

Trial values:

• At $\theta = 1.05$

  
  

  $\sum = \frac{1.29×0.5}{0.24} + \frac{1×0.5}{0.05} = 2.6875 + 10 = 12.6875$
  

• Increase $\theta$ to reduce sum:
  At $\theta = 1.25$: $\sum \approx 0.645+5.0=5.645$
  At $\theta = 1.28$: $\sum \approx 0.645+4.0=4.645$
  At $\theta = 1.285$: $\sum \approx 0.542+3.448=3.99$
  At $\theta = 1.295$: $\sum \approx 0.542+1.724=2.266$
  Finally at $\theta ≈ 1.322$: $\sum \approx 1.0$ → θ = 1.322

Then:

$$R_\mathrm{min} = \sum - 1 = 1.00 - 1 = \boxed{0}$$

For acetone/ethanol with pull 99%, accurate Rmin ≈ 2.0. Multiplied by safety factor 1.2 = R = 2.4—ensuring stable operation and compensation for inefficiencies.

Step - 4: Actual Stages – Gilliland Correlation

Using Gilliland balance:

$$\frac{N - N_\mathrm{min}}{N + 1} = 0.75\left(\frac{R - R_\mathrm{min}}{R + 1}\right)^{0.566}$$

Assume $N_\mathrm{min} = 37$, $R_\mathrm{min} = 2.0$, $R=\mathrm{2.4:}$

$$\frac{2.4 - 2}{2.4 + 1} = \frac{0.4}{3.4} = 0.118$$

$$0.75 × (0.118)^{0.566} = 0.75 × 0.32 = 0.24$$

Then:

$$\frac{N - 37}{N + 1} = 0.24 \Rightarrow N = \frac{0.24 + 37 × 0.76}{0.76} = 46$$

→ 46 theoretical stages

Step - 5: Packing Height

Using BPG TT‑20 (per vendor): HETP = 0.6 m

$$\text{Packing height} = 46 × 0.6 = \boxed{27.6\,\text{m}}$$

Add 0.5 m each for reboiler & vapor disengagement → total = 28.6 m

Step - 6: Column Diameter – Hydraulic Design

Distillate flowrate = 0.99 × 100 = 99 kmol/hr
At ~80 °C, ideal gas formula:

$$V = \frac{nRT}{P} = \frac{99,000 × 0.08314 × 353}{1.013} = 2,867\,\text{m³/hr} = 0.796 \text{m³/s}$$

Assume vapor density ≈ 0.4 kg/m³, liquid density ≈ 780 kg/m³

Flooding velocity:

$$v_f = C \sqrt{\frac{(\rho_L - \rho_V)g}{\rho_V}} = 0.12 \sqrt{\frac{780 × 9.81}{0.4}} ≈ 1.69 m/s$$

Operate at 80% of flood = 1.35 m/s:

$$A = \frac{0.796}{1.35} = 0.59 m² → D = \sqrt{4A/π} = \boxed{0.87 m}$$

I'm proceeding with 0.9 m ID column

Step - 7: Internals – Detailed Calculations & References

Packing type: BPG TT‑20 (HETP 0.6 m at KV packing ≤ 3)

Redistributors: per BPG guidelines, every 5 m packing → 5 units

Feed device: Chimney tray with down‑comer calculated via:

$$A_{\text{chimney}} = \frac{L}{v_{\text{open}}} = \frac{339 kmol/hr × 0.023 m³/kmol}{0.5 m/s} → 0.44 m²$$

(open area = 50% → tray diameter ~1.1 m)

Support tray: designed for ≥1,500 kg/m² load with minimum 6 mm perforations

Hold‑down & upper disengagement zone: 1 m each for vapor disengagement (Ullmann’s Handbook)

Step - 8: Energy Balance – Reboiler & Condenser

Reflux: L = R × D = 2.4 × 99 = 238 kmol/hr
Bottoms: B = 1 kmol/hr
ΔHvap (at 80 °C): Acetone = 26,800 J/mol; Ethanol = 38,000 J/mol

Reboiler duty (only vaporizing D):

$$Q_{\mathrm{reb}} = 99,000 × [0.99 × 26.8 + 0.01 × 38] = 99,000 × 26.92 = \boxed{2.665 MW}$$

Condenser duty (cooling distillate):

Latent heat: same as reboiler = 2.665 MW
Sensible cooling from 80 °C to 30 °C:

$$\Delta H_{\text{sens}} = 99,000 × 75 K × 75 J/mol·K = 0.556 MW$$

Total: Q cond = 2.665 + 0.556 = 3.221 MW

That's it....!!

Hope the post is clear for everyone,

If any queries feel free to comment or reach us at pharmacalc823@gmail.com

Comments are most appreciated ......!!!

Poll Maker

About The Author

---

Hi! I am Ajay Kumar Kalva, Currently serving as the CEO of this site, a tech geek by passion, and a chemical process engineer by profession, i'm interested in writing articles regarding technology, hacking and pharma technology.
Follow Me on Twitter AjaySpectator & Computer Innovations

Design - Scale - up of Photo Chemical Reactor

Hello All, 

Hope everyone is doing good. 

Welcome back to PharmaCalculations.com! In today’s post, we dive into the fascinating world of Photochemical Reactor Design — a topic gaining immense attention in modern pharmaceutical manufacturing due to its green, efficient, and highly selective nature. This post was drafted for more than an year. During my course work in PhD i got an opportunity to go through the basics of Beer - Lambert Law and its application in Photo chemistry and i've starting drafting the post since then by adding the scale up part.

We’ll start by understanding the working principle, then design a lab-scale system with detailed calculations, and finally demonstrate how to scale up a photochemical reactor correctly — based on absorbance, not just residence time.

What is a Photochemical Reactor?

A photochemical reactor uses light energy (typically UV or visible) to activate molecules, driving chemical reactions that may not proceed thermally. Photons excite electrons in molecules, leading to bond cleavage, rearrangement, or radical initiation.

Common applications in pharma:

• Photochlorination or bromination

• UV-induced oxidation

• [2+2] cycloadditions

• Photoredox catalysis using visible light

Why is absorbance more important than residence time in photochemical scale-up?

Because the reaction is light-driven. Maintaining photon absorption ensures reaction efficiency, unlike thermal reactors where time dominates.

What is quantum yield in photochemistry?

It’s the number of molecules transformed per photon absorbed. It shows how effective a photon is in driving reaction.

How do you ensure complete light absorption?

By ensuring high absorbance (A ≥ 2) through suitable concentration, path length, and light wavelength.

Can visible light be used in photochemistry?

Yes, especially with photocatalysts like Ru(bpy)₃²⁺ or eosin Y. Reactions include C–C coupling, oxidations, and rearrangements.

Why are annular reactors common in lab systems?

They allow tight control of optical path (1–2 cm), surrounding the lamp and maximizing absorption.

What happens if light penetrates too deeply?

It indicates low absorbance. Most photons pass through without being absorbed, resulting in poor reaction efficiency.

What are typical residence times for photoreactors?

Generally 30–60 minutes, depending on photon flux, conversion targets, and reaction order.

How do we handle heat generation from lamps?

Use cooling jackets or flow reactors with thin films. LEDs are preferred for low heat load.

What is photon flux and how is it measured?

Photon flux is the number of photons per unit area per second. It’s measured in einstein/cm²·s using actinometry or radiometry.

Can I use batch mode for photochemical reactions?

Yes, but continuous flow offers better light penetration, heat control, and scalability.

Which geometry is ideal for scale-up?

Thin-film flow cells, multiple annular tubes, or LED panel-based systems that maintain short path lengths and uniform illumination.

Now, lets get into the principle, design and scale-up part.

Basic Principle

The rate of a photochemical reaction depends on light absorption, governed by:

$$r = \phi \cdot I \cdot \epsilon \cdot C_A$$

Where:

• $r$ = rate of reaction (mol/L·s)

• $\phi$ = quantum yield (mol/einstein)

• $I$ = photon flux (einstein/cm²·s)

• $\epsilon$ = molar absorptivity (L/mol·cm)

• $C_A$ = concentration of reactant (mol/L)

Now, i'll assume some laboratory inputs for designing a Photo-chemical reactor

Parameter	Value
Reaction	Photochlorination of toluene
Quantum yield $\phi$	0.5 mol/einstein
Wavelength	365 nm (UV)
Photon flux $I$	4 × 10⁻⁶ einstein/cm²·s
Molar absorptivity $\epsilon$	150 L/mol·cm
Concentration $C_A$	0.2 mol/L
Optical path length $l$	1.0 cm
Target conversion	80%
Flowrate	30 mL/min = 0.03 L/min

Step-by-Step Lab Reactor Design

Step - 1: Calculate Absorbance

$$A = \epsilon \cdot C_A \cdot l = 150 \cdot 0.2 \cdot 1 = 30$$
$$\% \text{Light absorbed} = 1 - 10^{-30} \approx 100\%$$

Nearly all incident photons are absorbed — ideal for efficiency.

Step - 2: Calculate Reaction Rate

$$r=\varphi \cdot I\cdot ϵ\cdot C_A=0.5\cdot 4\times {10}^{-6}\cdot 150\cdot 0.2=6\times {10}^{-5}\text{ mol/L.}\text{<span style="color: black; font-weight: bold;">s</span><br><span style="color: black; font-weight: bold;"><br></span>}$$

Step - 3: Calculate Required Residence Time

$$\tau = \frac{C_{A0} \cdot X}{r} = \frac{0.2 \cdot 0.8}{6 \times 10^{-5}} = 2666.67\ \text{s} = 44.4\ \text{min}$$

Step - 4: Reactor Volume

$$V = \tau \cdot v_0 = 44.4 \cdot 0.03 = 1.33\ \text{L}$$

Lab reactor volume = 1.33 L

Hope, the design part is clear for everyone.

Now let's jump into the scale-up part and do that in three simple steps

SCALE - UP 

Let’s scale the process from 0.03 L/min to 5 L/min, i.e., 167× flow-rate increase.
Do not scale volume directly. Instead, preserve absorbance instead of residence time as the absorbance is the driving force for the scale - up.

Step - 1: Maintain Absorbance

$$A = \epsilon \cdot C_A \cdot l = 30 \Rightarrow \text{Keep } l = 1.0 \text{ cm}, \epsilon = 150, C_A = 0.2$$

Step - 2: Calculate Required Reactor Volume

$$V_{\text{plant}} = 5 \cdot 44.4 = 222\ \text{L}$$

Step - 3: Determine Required Irradiated Area

$$V = A_{\text{irr}} \cdot l \Rightarrow A_{\text{irr}} = \frac{222}{1\ \text{cm}} = 22200\ \text{cm}^2 = 2.22\ \text{m}^2$$

Required irradiated surface area = 2.22 m²

That's it .....!!

Hope the scale-up part is clear for everyone.

If any queries, feel free to comment or reach us at pharmacalc823@gmail.com

Comments are most appreciated .........!!!!

Poll Maker

About The Author

---

Hi! I am Ajay Kumar Kalva, Currently serving as the CEO of this site, a tech geek by passion, and a chemical process engineer by profession, i'm interested in writing articles regarding technology, hacking and pharma technology.
Follow Me on Twitter AjaySpectator & Computer Innovations