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Chapter 17: Problem 3

A Fischer-Tropsch slurry bubble column reactor is shown in Figure 17.4. Small catalyst particles are suspended in the liquid product (a mix of hydrocarbons and water), while syngas is vigorously bubbled through the liquid. Due to intense mixing, the compositions of the liquid and the gas are uniform within each phase throughout the reactor, i.e., independent of the location in the reactor. The reactants, \(\mathrm{CO}\) and \(\mathrm{H}_{2}\), are only partially converted in such a reactor. A fraction of the unconverted syngas is recycled and mixed with fresh syngas feed. The main issue is the determination of the composition of fresh syngas meeting a target \(\mathrm{H}_{2} / \mathrm{CO}\) ratio for the gas phase. This latter target must be a suitable one for the Fischer-Tropsch reactions in the catalyst particles exposed to the gas. a. Derive the \(\mathrm{H}_{2}\) and \(\mathrm{CO}\) molar balances over the reactor- recycle-mixer system. Each balance specifies a relation between the concentrations in the fresh syngas and the gas inside the reactor. The balance equations will contain the following parameters: The degree of conversion of \(\mathrm{CO}\) in this reactor \((X)\), relative to intake to reactor The recycle ratio \((r)\) of the unconverted syngas The conversion of \(\mathrm{H}_{2}\) relative to the \(\mathrm{CO}\) conversion in reaction stoichiometry \((3-\alpha)\) b. Using these balances, show how the \(\mathrm{H}_{2} / \mathrm{CO}\) molar ratio \(\left(x_{\mathrm{H}_{2} / \mathrm{CO}}\right)\) in the fresh syngas feed can be related to the corresponding ratio in the gas inside the reactor \(\left(\sigma_{\left.\mathrm{H}_{2} / \mathrm{CO}\right)}\right.\). The gas in the reactor is well mixed, so the \(\mathrm{H}_{2} / \mathrm{CO}\) ratio of the syngas exit stream is equal to the ratio inside. c. Compute the molar \(\mathrm{H}_{2} / \mathrm{CO}\) ratio \(\left(x_{\mathrm{H}_{2} / \mathrm{CO}}\right)\) in the fresh syngas meeting a target \(\mathrm{H}_{2} / \mathrm{CO}\) ratio in the gas inside the reactor \(\left(\sigma_{\mathrm{H}_{2} / \mathrm{CO}}\right)\). This target ratio is expressed as a fraction \((\mu)\) of the ideal stoichiometric \(\mathrm{H}_{2} / \mathrm{CO}\) ratio for the Fischer-Tropsch reactions: \(\sigma_{\mathrm{H}_{2} / \mathrm{CO}}=\mu(3-\alpha)\). The following numerical values are applicable: Chain growth probability: \(\alpha=0.9\) Reduction factor: \(\mu=0.8\) Degree of \(\mathrm{CO}\) conversion: \(X=0.6\) Recycle fraction for syngas: \(r=0.8\)

Short Answer

Expert verified
The molar H2/CO ratio in the fresh syngas required to meet the target ratio inside the reactor is approximately 1.2727.

Step by step solution

01

Derive the molar balances for H2 and CO

To derive the molar balances for \(\mathrm{H}_{2}\) and \(\mathrm{CO}\), we'll consider the moles entering and exiting the reactor. For the \(\mathrm{CO}\) balance, let's denote the moles of \(\mathrm{CO}\) entering the reactor and moles converted as \(n_{\mathrm{CO}}^{in}\) and \(n_{\mathrm{CO}}^{conv}\), respectively. Similarly, we can denote the moles of \(\mathrm{H}_{2}\) entering the reactor as \(n_{\mathrm{H}_{2}}^{in}\). The moles of \(\mathrm{CO}\) entering the reactor is the sum of moles in fresh feed and recycled syngas. The moles of unconverted \(\mathrm{CO}\) is given by \((1-X)n_{\mathrm{CO}}^{in}\), so the moles of recycled \(\mathrm{CO}\) will be \(r(1-X)n_{\mathrm{CO}}^{in}\). Thus, the molar balance for \(\mathrm{CO}\) is: \(n_{\mathrm{CO}}^{in}=(1+r(1-X))n_{\mathrm{CO,fresh}}\) Similarly, the moles of \(\mathrm{H}_{2}\) entering the reactor is the sum of moles in fresh feed and recycled syngas. The moles of \(\mathrm{H}_{2}\) in recycled syngas will be \(r((1-X)n_{\mathrm{H}_{2}}^{in})\), so the molar balance for \(\mathrm{H}_{2}\) is: \(n_{\mathrm{H}_{2}}^{in}=(1+r(1-X))n_{\mathrm{H}_{2,fresh}}\)
02

Relating molar ratios in fresh syngas and inside the reactor

To relate the molar ratio in the fresh syngas feed and the gas inside the reactor, we'll use the molar balances obtained in Step 1: \(\frac{n_{\mathrm{H}_{2}}^{in}}{n_{\mathrm{CO}}^{in}}=\frac{(1+r(1-X))n_{\mathrm{H}_{2,fresh}}}{(1+r(1-X))n_{\mathrm{CO,fresh}}}\) Now, we can cancel the common factor \((1+r(1-X))\) and write the molar ratio in terms of fresh syngas: \(x_{\mathrm{H}_{2}/\mathrm{CO}}=\frac{n_{\mathrm{H}_{2,fresh}}}{n_{\mathrm{CO,fresh}}}\)
03

Compute the molar H2/CO ratio in the fresh syngas meeting the target ratio inside the reactor

We are given the target molar ratio inside the reactor: \(\sigma_{\mathrm{H}_{2}/\mathrm{CO}} = \mu(3-\alpha)\). Now, we'll use the relationship obtained in Step 2 to compute the required molar ratio in fresh syngas: \(x_{\mathrm{H}_{2}/\mathrm{CO}}=\frac{\sigma_{\mathrm{H}_{2}/\mathrm{CO}}}{1+(r(1-X))}\) We'll substitute the given numerical values: \(\alpha=0.9\), \(\mu=0.8\), \(X=0.6\), and \(r=0.8\). \(x_{\mathrm{H}_{2}/\mathrm{CO}}=\frac{0.8(3-0.9)}{1+(0.8(1-0.6))}=\frac{1.68}{1+0.32}=\frac{1.68}{1.32}=1.2727\) So, the molar \(\mathrm{H}_{2}/\mathrm{CO}\) ratio in the fresh syngas required to meet the target ratio inside the reactor is approximately \(1.2727\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Syngas Molar Ratio
In the Fischer-Tropsch process, understanding the syngas molar ratio is crucial. Syngas, a combination of hydrogen (\(H_2\)) and carbon monoxide (\(CO\)), must maintain a specific molar ratio to optimize the reaction. This ratio determines how efficiently the reactants convert into hydrocarbons, the main products of the Fischer-Tropsch synthesis.
To manage this ratio, the process involves balancing the amounts of fresh and recycled syngas introduced into the reactor. Significant factors impacting this balance include the degree of \(CO\) conversion, the recycle ratio, and the stoichiometric conversion ratio of \(H_2\) relative to \(CO\). These factors define the necessary adjustments to achieve an optimal \(H_2/CO\) ratio, crucial for effective catalyst performance and product yield.
Slurry Bubble Column Reactor
A slurry bubble column reactor is vital in the Fischer-Tropsch process. Here, catalyst particles are suspended in a liquid while syngas is bubbled through. This arrangement offers several advantages.
  • Uniform mixing: The reactor ensures gas and liquid compositions remain consistent throughout.
  • Heat distribution: The design allows effective dispersal of heat generated during exothermic reactions.
  • Scalability: It is suitable for large-scale operations, given the uniform phase distribution.

Slurry reactors are distinguished by their ability to maximize contact between gas, liquid, and catalyst. This increases the reaction rate and efficiency. In the context of the Fischer-Tropsch process, the slurry bubble column reactor facilitates the conversion of syngas to hydrocarbons efficiently, maintaining high productivity while ensuring energy efficiency.
Catalyst Particles
Catalyst particles are key to the Fischer-Tropsch synthesis. These small particles are immersed in the liquid within the reactor, providing the surface where the chemical reactions occur. The choice of catalyst influences reaction rates and product composition.
The key roles played by catalyst particles include:
  • Enhancement of reaction rates, boosting the overall efficiency of the process.
  • Control over product distribution, impacting the types and sizes of hydrocarbons produced.
  • Stability under reaction conditions, ensuring long-term operation without frequent replacements.

In the Fischer-Tropsch process, iron and cobalt-based catalysts are commonly used due to their effectiveness in converting syngas into longer-chain hydrocarbons, desirable in many fuel and chemical applications. These catalysts’ physical form and size are tailored to optimize interaction with the syngas and the resident liquid hydrocarbons.
Molar Balances
Molar balances are essential tools in chemical engineering, providing insights into the flow and transformation of species within a system. In the context of the Fischer-Tropsch reactor, molar balances for \(H_2\) and \(CO\) help derive relationships between the input and output streams.
  • Moles entering and exiting the system need precise calculation to ensure balance.
  • The degree of conversion, recycle ratio, and stoichiometry are all intrinsic to these calculations.
  • Ensuring correct balances guarantees that the products' formation is tracked and optimized.

Achieving proper molar balances ensures that the reactor operates efficiently and meets desired product specifications. It also aids in adjusting the fresh syngas composition, achieving the targeted \(H_2/CO\) ratio needed for optimal catalytic reaction within the slurry bubble column reactor.

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Most popular questions from this chapter

Why is it currently more attractive to convert methanol to olefins (MTO) than to gasoline (MTG)?

In multitubular FTS reactors, typically particles of 1 or a few \(\mathrm{mm}\) are used. Decreasing the particle size would increase the catalyst effectiveness. For what reason(s) is the particle size not reduced?

The reaction equation for Fischer-Tropsch synthesis (RX. 17.1) shows that for longer hydrocarbons, the Fischer-Tropsch reaction needs \(\mathrm{H}_{2} / \mathrm{CO}\) in a ratio of about 2:1. However, in practice, often, a lower syngas ratio is used. What are the advantages of using a lower syngas ratio? What are the disadvantages?

What are the advantages of liquid transportation fuels based on biomass over other sustainable energy solutions for transportation purposes?

Suppose the Fischer-Tropsch reactions take place in identical spherical catalyst particles suspended in a homogeneous fluid phase. The reactants hydrogen \(\left(\mathrm{H}_{2}\right)\) and carbon monoxide \((\mathrm{CO})\) diffuse from the fluid at the outer surface of a catalyst particle through the pores to the catalytic reactive sites inside the particle, while the reaction products diffuse out of the particle. The main issue of this problem is what would be a suitable molar \(\mathrm{H}_{2} / \mathrm{CO}\) ratio at the outer surface of the catalyst particles to favor the FT reactions throughout the particles. From a stoichiometric point of view, this \(\mathrm{H}_{2} / \mathrm{CO}\) ratio should be close to \(3-\alpha\), where \(\alpha\) is the chain growth probability (see Section 17.2.2). However, \(\alpha\) is sensitive to the amount of hydrogen. The chain growth probability increases with a lower \(\mathrm{H}_{2} / \mathrm{CO}\) ratio by reducing the termination of chain growth by hydrogen. Another complication is the effect of spatially distributed reaction-diffusion in the catalyst particle. \(\mathrm{H}_{2}\) diffuses faster than \(\mathrm{CO}\) and is consumed in larger amounts. Thus, the \(\mathrm{H}_{2} / \mathrm{CO}\) ratio will vary along the radial coordinate of the particle. This is borne out by the radially distributed concentrations of the reactants, which can be obtained by solving the reaction-diffusion equations for a catalyst particle. Solving such equations with realistic kinetics, e.g., Equation (17.1), requires a numerical approach because coupled nonlinear differential equations are involved. Here, we will use a simplified reaction-diffusion equation for which an analytical solution can be obtained to illustrate effects of unequal diffusion. Our gross simplifications (not suitable for real design cases) involve assuming: \- Zero-order kinetics (obtained by ignoring the partial pressure dependencies in Equation (17.1)) \- Constant \(\alpha\) \- Fickian diffusion, which neglects flux interactions between reactants and products inside the catalyst pores The analytical solution to the reaction-diffusion equation is $$ \begin{aligned} &\mathrm{c}_{\mathrm{i}}(z)=\mathrm{c}_{\mathrm{i}}^{(S)}+\frac{s_{i}}{D_{i}} \frac{a}{6}\left(\mathrm{r}^{2}-z^{2}\right) \quad \text { for } 0 \leq z \leq \mathrm{r} \\ &\mathrm{c}_{\mathrm{i}}(\mathrm{r})=\mathrm{c}_{\mathrm{i}}^{(S)} \quad i=\mathrm{CO}, \mathrm{H}_{2} \\ &\mathrm{c}_{\mathrm{i}}(0) \geq 0 \quad=>\quad \text { physical feasibility : } \frac{\left|s_{i}\right| a}{D_{i} 6} \mathrm{r}^{2} \leq \mathrm{c}_{\mathrm{i}}^{(S)} \\ &a \quad: \text { zero-order reaction rate coefficient }\left(\mathrm{kmol} \cdot \mathrm{s}^{-1} \cdot \mathrm{m}_{\mathrm{cat}}^{-3}\right) \\ &\mathrm{c}_{\mathrm{i}}^{(S)} \quad \text { : concentration of component iat surface }\left(\mathrm{kmol} \mathrm{m}^{-3}\right) \\ &D_{i} \quad: \text { Fickian diffusion coefficient of component } i\left(\mathrm{~m}^{2} \mathrm{~s}^{-1}\right) \\ &\mathrm{r} \quad \text { Radius of spherical catalyst particle }(\mathrm{m}) \\\ &s_{i} \quad: \text { stoichiometric coefficient of component } i(-) \\ &\text { Note: } s_{i}<0 \text { for a reactant } \\ &z \quad: \text { radial coordinate }(\text { from center to surface } S)(\mathrm{m}) \end{aligned} $$ Solving two equations with four unknowns \(\left(\mathrm{c}_{\mathrm{i}}^{(S)}, \mathrm{c}_{\mathrm{i}}(0), i=\mathrm{H}_{2}, \mathrm{CO}\right)\) requires two more specifications: 1\. The \(\mathrm{H}_{2} / \mathrm{CO}\) ratio will increase toward the center of the particle due to faster diffusion of hydrogen. For that reason, the perfect stoichiometric ratio is only imposed as an upper bound in the particle center: $$ z=0: \frac{\mathrm{c}_{\mathrm{H}}(z)}{\mathrm{c}_{\mathrm{CO}}(z)}=3-\alpha $$ 2\. At the outer surface, the \(\mathrm{CO}\) concentration is given, while the \(\mathrm{H}_{2}\) concentration is a degree of freedom to match the previous center condition: $$ z=\mathrm{r}: \quad \mathrm{c}_{\mathrm{CO}}(z)=\mathrm{c}_{\mathrm{CO}}^{(S)}=0.2\left(\mathrm{kmol} \cdot \mathrm{m}^{-3}\right) $$ Typical numerical values (right order of magnitude) for the model parameters (at \(500 \mathrm{~K}\) ) are: $$ \begin{aligned} \alpha &=0.9 ; a=6.0 \times 10^{-3}\left(\mathrm{kmol} \cdot \mathrm{s}^{-1} \cdot \mathrm{m}_{\text {cat }}^{-3}\right) \\ D_{C O} &=5.2 \times 10^{-9}\left(\mathrm{~m}^{2} \cdot \mathrm{s}^{-1}\right) ; D_{H_{2}}=2.7 D_{C O}\left(\mathrm{~m}^{2} \cdot \mathrm{s}^{-1}\right) \\ \mathrm{r} &=0.5 \times 10^{-3}(\mathrm{~m}) ; s_{\mathrm{CO}}=-1 ; s_{H_{2}}=-(3-\alpha) \end{aligned} $$ a. Given the concentration distribution equations with boundary conditions, derive and report the expressions for the \(\mathrm{H}_{2}\) concentration at the outer surface \(\left(\mathrm{c}_{\mathrm{H}_{2}}^{(S)}\right)\) and the expressions for the \(\mathrm{H}_{2}\) and \(\mathrm{CO}\) concentrations in the center. b. Use these expressions to compute the concentrations in the center and the \(\mathrm{H}_{2} / \mathrm{CO}\) ratio at the outer surface \(\left(\sigma_{\mathrm{H}_{2} / \mathrm{CO}}\right)\). Check by which fraction \((\mu)\) this ratio is smaller than the ideal stoichiometric ratio of \(3-\alpha\), where \(\mu=\sigma_{\mathrm{H}_{2} / \mathrm{CO} /(3-\alpha) \text {. }}\)
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