CHEN30001 Fluid mechanics module 8 notes (Lecture) PDF

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Module 8 fluid mechanics lecture notes for 2021 calender...

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CHEN3001 Reactor Engineering

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Module 8: Solid Catalysed Fluid Phase Reactions Levenspiel’s Chemical Reaction Engineering References: • Chapter 17: pages 369-375 • Chapter 18: pages 376-426 • Chapter 19: pages 427-446

Module Overview • Structure of Porous Catalysts • Regimes of Heterogeneous Catalysis • Diffusion & Reaction Control

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Mixtures of Catalysts Temperature Effects Arbitrary Kinetics Modelling a Packed Bed and an MFR Containing Heterogeneous Catalysts.

Intended Learning Outcomes • Develop an understanding of the structure and behaviour of solid catalysts for fluidphase reactions, and in particular the meaning of diffusional resistance and reaction resistance. • Understand how to calculate the rate of reactions in the presence of porous solid catalysts, and the meaning of the Thiele modulus and the effectiveness factor. • Model a packed bed and mixed flow reactor containing porous solid catalysts for a fluid-phase reaction. 8.1 Introduction to Solid-Catalysed Fluid-Phase Reactions (Slides 1-8)

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In this module, we will be analysing solid catalysts for fluid-phase reactions. As you well know, a catalyst is a substance which does not participate in the reaction, but instead changes the reaction rate. At the most basic level, it does this by reducing the activation energy barrier for both the forward and reverse reactions. This increases the rate of both the forward and reverse reaction (c.f. Arrhenius’ Law), which increases the rate of approach to equilibrium without changing the equilibrium state. To this point in your chemical engineering career, you may have only modelled fluid-phase catalysts, which are dissolved in a fluid in order to increase the rate of reactions which occur inside the same fluid. For example, when the enzyme carbonic anhydrase is dissolved in water, it catalyses the reaction between dissolved CO2 and H2 O, increasing its rate by up to 4 orders of magnitude. The catalytic effect of carbonic anhydrase is critical to the efficient operation of your lungs; similar catalysts are used in the separation and capture of CO2 from industrial gas streams. A major problem with fluid-phase catalysts is that, once reactants have turned into products and the catalyst’s job is done, the catalyst may be very difficult to recover from the product stream. In general, catalysts are used in very low concentrations, and, as you know from 8-1

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thermodynamics (or by Googling a Sherwood Plot (Sherwood, 1959; House et al., 2011; Dahmus et al., 2007)) it is very difficult to separate out trace species from a fluid. Catalysts are also often expensive, and so constantly supplying new catalyst, which enters with the reactants and leaves as a contaminant in the product stream, doesn’t often makes sense. For this reason, in many parts of the chemical and process industry, we prefer to use solid catalysts for fluid-phase reactions. When solid catalysts are used, the problem of separating a dissolved catalyst from the fluid-phase product stream is avoided. We cite two common examples: •

In the synthesis of ammonia from hydrogen and nitrogen via the Haber-Bosch process, various metal catalysts may be used (Leigh, 2004). The most common is an iron-based catalyst, though uranium, osmium and ruthenium-based catalysts have also been used. In each case, the rate of the gas-phase reaction, N2 + 3H2 ↔ 2NH3

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is increased when the gas is exposed to large surface areas of a solid metal catalyst. In the back of a car, a catalytic converter (see Figure 1) is used to convert carbon monoxide and oxygen into carbon dioxide by flowing the exhaust gas over a solid platinum catalyst coated onto a monolith support. Once again, the rate of a thermodynamically favourable reaction is increased by exposing the fluid to a highsurface area solid.

Figure 1: Catalytic converters are used to oxidise the toxic by-products of incomplete combustion. By increasing the surface area of a solid catalyst, we may increase the rate of reaction. The simplest way of increasing the solid surface area would be to grind the catalyst into very fine particles. However, such particles may be hard to recover from the product fluid stream, and, if they were used in a packed bed, the pressure drop required to force the fluid through the particles may be too great. Instead, it is more common for catalysts to be incorporated into high surface area, porous materials. These porous solids may be extruded into cm-sized pellets or monoliths (Figures 2 and 3). These materials have extremely large 8-2

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internal surface areas – one gram of such a catalyst material may contain over a football field of surface area on the tortuous pores within, and a large fraction of this area may be composed of active catalyst sites (see Figure 4). The manufacture of these materials is a complicated process, and there are many different approaches. In many cases, most of the porous catalyst pellet is composed of binding and filling agents (such as clay materials) which provide structural integrity to the pellet, and the active catalyst only makes up a small fraction of the total mass of the pellet. It is also possible to directly coat an active catalyst onto the surface of a monolith through which a fluid flows, but even in these cases the surface is often treated beforehand to be rough and irregular, so that on the microscopic level it has as great a surface area as possible.

Figure 2: Pelletised porous catalysts.

Figure 3: Catalysts shaped into monoliths.

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Figure 4: Diagram of macroscopic and microscopic structure of porous solid catalyst. (Bird, Stewart and Lightfoot, 1960) Solid catalysts for fluid phase reactions are ubiquitous within the processing and chemicals industry. The development of new catalysts - which are faster, more robust, or which can operate under a wider range of temperatures – is ongoing in almost every field. Oftentimes, the design of an entire process is determined by the properties of the available solid catalysts. For instance, in the water-gas shift reaction, multiple reactors, at various temperatures and using various catalysts, are often used in series. Process designs are strongly dependent on the performance of the various catalysts at different temperatures, and also their tolerance to contaminants such as sulphur (Newsome, 1980). Many examples of industrially-relevant heterogeneous catalysts may be found in the 8 volumes of the Handbook of Heterogeneous Catalysis (Kiennemann et al. 2008). Incidentally, this book is edited by Gerhard Ertl, who won the Nobel prize in chemistry in 2007 for describing the molecular mechanisms behind both of the examples in the dot points above.

8.2 Mechanisms of Heterogeneous Catalysis It may seem surprising that the presence of a nearby solid could increase the rate of reaction inside a gas or a liquid. Solid surfaces are approximately two dimensional, while a fluid exists in three dimensional space. In any solid-catalysed system, even those with very large surface area, only a tiny fraction of the molecules in a fluid will be close to the solid surface at any given time. Even if the reaction at the surface occurs extremely quickly (or even ‘instantaneously’), this will not lead to a fast reaction unless reactant molecules may be quickly and continuously transported to the surface, and product molecules quickly and continuously transported away. 10

In order for the fluid-phase reaction 𝐴 → 𝐵 to be catalysed by a solid catalyst, a number of processes must occur in series. •

Film Diffusion) The reactant 𝐴 must first be transported from the bulk of the fluid to the outer surface of the porous catalyst. This transport process may occur via convection (if the bulk fluid is in motion) or via diffusion. We typically model this 8-4

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resistance as a thin stagnant ‘film’ through which reactants must diffuse in order to move from the bulk fluid to the surface of the porous catalyst. Of course, such a stagnant film is a mathematical idealisation which may not exist in practice (though in some cases there is a physical justification – see Prandtl’s Boundary Layer Theory (Anderson, 2005)) but we can use correlations (based on the geometry, the Reynolds number, the Sherwood number, etc.) to calculate the fluid-phase mass transfer coefficient, 𝑘𝑔 or 𝑘𝑙 , for this film, in order to quantify the film mass transfer resistance. Pore Diffusion) In a porous catalyst particle, the catalyst is coated on the surface of tortuous, thin pores which tunnel throughout whole volume of the particle. For such a catalyst, diffusion of the reactant, A, to the macroscopic surface of the porous catalyst is only the first step – reactants must then diffuse through the the stagnant fluid in the pores inside the particle itself, until they find a free catalytic site on the internal, microscopic surface of the pores on which they can react. We refer to transport within the porous pellet itself as pore diffusion. Surface Kinetics) Once the reactants have reached the catalyst surface, they must react. The rate of reaction depends on many factors – the catalyst, the local concentration of reactants, the temperature, etc.

The three processes described above – diffusion to the catalyst particle, diffusion through the pores of the catalyst particle, and reaction on the internal catalyst surface - must occur in series for the reactant, A, to be converted to product, and these three process will dominate the kinetics of most catalysts we will consider in this course. Because they occur in series, if any one process is much slower than the others, it will determine the rate of the overall process, and we say it is rate controlling. When one process is rate controlling, we say that we are in a particular kinetic regime (e.g. if the surface reaction is so slow that it is rate controlling, we say we are in the reaction-controlled regime.) Other processes which may also occur in real systems include: •

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Surface Diffusion) Reactant molecules adsorbed on the solid surface may diffuse on the surface, from regions of high concentration to low concentration. In some systems, this surface diffusion may be significant and must be accounted for. Product Diffusion) Once the products have been formed, they must diffuse back to the bulk of the fluid. The rate of product diffusion is often irrelevant for irreversible reactions, but may be important for reversible reactions. Temperature Gradients) In some systems, heat transfer into or out of the particles may strongly influence the rate of reaction.

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For a porous catalyst catalysing a gas-phase reaction, surface kinetics are often rate limiting for slow reactions, while diffusion through the pores may be rate limiting for fast reactions. For a liquid-phase reaction, pore diffusion and film diffusion are often more significant, as diffusivities in a fluid are typically several orders of magnitude smaller than in a gas.

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As shown in Figure 5, different factors influence the rate of reaction within catalyst particles. For a catalyst-coated surface (e.g. a catalyst-coated monolith) mass transfer is 8-5

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governed by the rate of reaction on the solid surface (surface reaction), and the rate of transfer from the bulk of the fluid to the surface. For a porous catalyst particle, mass transfer is typically governed by surface reaction and also mass transfer through the internal pores of the particle, as the reactants diffuse through the pellet to the catalyst surface within (pore diffusion). For the combustion of a liquid or solid particle (which is not a catalytic process, and is largely outside the scope of this course) temperature differences induced by exothermic reactions on the surface can play a strong role in governing the kinetics of the system.

Figure 5: Factors which Influence the Rate of Reaction of Particles 13

On the catalyst surface itself, the kinetics are governed by the following steps: 1. The reactant must be adsorbed to the surface and attached to an active site. 2. The reactant must react, either with another adsorbed molecule, a molecule in the fluid phase, or by decomposing on its own. 3. The product must desorb from the surface. 8.3 Mathematical Theory of Heterogeneous Catalysts NOTE: In what follows, we have only provided an overview of the mathematical theory of heterogeneous catalysts. A more thorough derivation for one particularly common case is provided in the Appendix 8.A at the end of these notes.

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We now model diffusion and surface reaction inside a single, cylindrical pore whose walls are coated with a solid catalyst. While this may seem idealised compared to a real porous catalyst pellet (which has irregular, tortuous pores of various sizes) it turns out that the exact same model may be used in a real porous catalyst provided the molecular diffusivity is replaced by an ‘effective diffusivity’, which accounts for the tortuosity of the pores and the voidage within the catalyst (see Appendix A). Consider the long cylindrical pore shown in Figure 6 below. The fluid inside the pore contains some species, 𝐴, which may react to form products; this reaction is catalysed by a solid catalyst coated on the walls of the pore.

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Figure 6: Concentration profile within long cylindrical pore in which gas may react on the surface. At the open end of the pore (𝑥 = 0) the concentration is high, and as we move through the pore the concentrations gets smaller and smaller, as more and more 𝐴 reacts away. The decreasing concentration profile shown in the figure above is the steady-state profile, and represents the balance between the reaction of 𝐴 on the pore walls (which removes 𝐴 from the system) and the diffusion of fresh 𝐴 from the surface at 𝑥 = 0 (which adds 𝐴 to the system.) In terms of the terminology introduced in the previous section, the rate of reaction depends on two processes which occur in series: pore diffusion and surface reaction. Imagine what would happen if we suddenly made the diffusivity of 𝐴 in the fluid very, very large in the figure above. Then the reaction would not be fast enough to consume much of the 𝐴 diffusing in from the bulk, and the concentration profile would increase until it became approximately flat, taking a constant value of 𝐶𝐴 ≈ 𝐶𝐴𝑠 all the way from 𝑥 = 0 to 𝑥 = 𝐿 (see the Reaction Controlled case in Figure 7). Because the rate of reaction is much slower than the rate of diffusion, we say that the reaction is rate controlling and that this system is reaction controlled. In such a system, accessing the catalyst surface is easy (even the surface all the way at the end of the pores near 𝑥 = 𝐿 , far from the pore inlet) but the reaction on the catalyst surface is slow, and that’s what’s holding the process up. Under these circumstances, if the reaction is first order, the reaction rate in the pore per unit catalyst surface area would be: ′′ 𝑟A,no diffusional resistance

=

1 𝑑𝑁𝐴 = 𝑘𝐴′′ 𝐶𝐴𝑠 𝑆 𝑑𝑡

Note that 𝑆 is the surface area of the solid catalyst coating the pore, and 𝑑𝑁𝐴 /𝑑𝑡 is the rate of reaction of A (in mol/s). Hence, 𝑟𝐴′′ refers to the reaction rate per unit area of solid catalyst, and 𝑘𝐴′′ is a rate constant referring, again, to the rate of reaction per unit of catalyst surface area available. In the above equation, the concentration is 𝐶𝐴𝑠 because we are

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considering a system with very large diffusivity, and so the concentration is everywhere equal to the surface concentration.

Figure 7: Concentration profiles inside a reaction controlled pore and a diffusion controlled pore. Having the concentration in the pores at the constant and large value of 𝐶𝐴𝑠 will lead to a greater reaction rate than will occur in the real system with only finite diffusivity, for which 𝐶𝐴 < 𝐶𝐴𝑠 inside the pores (see Figure 6, and also the Diffusion Controlled case in Figure 7). How great will this discrepancy be? We traditionally quantify this discrepancy by defining a fudge-factor, 𝜀 , known as the effectiveness factor of the catalyst, so that the actual average reaction rate per unit area inside the pore is: ′′ 𝑟A′′ = 𝜀𝑘𝐴′′ 𝐶𝐴𝑠 = 𝜀𝑟A,no diffusional resistance

The effectiveness factor, 𝜀 , relates the actual reaction rate to the reaction rate if there were no diffusional resistance, in which case the diffusivity is considered infinite and 𝐶𝐴 = 𝐶𝐴𝑠 everywhere inside the pore. For all real systems, 𝜀 ≤ 1 . If the pores are long and narrow, and the reaction at the surface is fast, so that the concentration inside the pores quickly drops and most of the catalyst surface is not well used, then 𝜀 ≪ 1 and we say the catalyst is strongly diffusion controlled (see the Diffusion Controlled case in Figure 7). Under these circumstances, chopping the catalyst into finer pieces will increase the rate of reaction, as this will allow more of the internal pore surface to be utilised. On the other hand, if diffusion is fast and/or the reaction is slow, so that 𝐶𝐴 ≈ 𝐶𝐴𝑠 throughout the whole catalyst, then 𝜀 ≈ 1. Under these circumstances, we say the pellet is reaction controlled, and changing the size of the pellets will have no effect on the reaction rate. Another way of thinking of the effectiveness factor is that it is the actual overall rate of reaction, divided by the hypothetical reaction rate if the catalyst pellets were cut into infinitely small pieces, so that all of the internal catalyst surface was exposed to the concentration of reactant in the bulk of the fluid.

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By conducting a shell balance on the system shown in Figure 6 and integrating the resulting differential equations, it is possible (see Levenspiel (1998) and Appendix A) to find an expression for 𝜀 . In particular, it may be shown that, for a linear pore, 8-8

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𝜀=

tanh 𝑀𝑇 𝑀𝑇

where 𝑀𝑇 is the Thiele Modulus of the catalyst. This, in turn, is defined as 𝑀𝑇 = 𝐿 √

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𝑘 𝐷

where 𝐷 is the diffusivity, 𝐿 is the length or the pore, and 𝑘 is the reaction rate constant. While the single pore shown in Figure 6 is an idealised system, the same mathematics may also be used to model a real slab of porous catalyst, in which the reactant, A, diffuses through many microscopic, tortuous pores and reacts on the internal catalytic surfaces. For a porous catalyst, the appropriate diffusivity to use when calculating the Thiele Modulus is the effective diffusivity, 𝐷𝑒 . This is the diffusivity of the whole porous material as if it were treated as a homogenous medium, and it depends on the pore size, tortuosity and the molecular diffusivity within the fluid. The Thiele modulus is a dimensionless number, and so the reaction rate constant, 𝑘, must have dimension of 1/time. The reaction rate constant 𝑘𝐴′′′ (note the 3 dashes this time) is defined by: 𝑟𝐴′′′ =

1 𝑑𝑁𝐴 = 𝑘𝐴′′′ 𝐶𝐴 𝑉𝑠 𝑑𝑡

where 𝑉𝑠 is the volume of the catalyst. Hence 𝑘𝐴′′′ refers to the rate of reaction per unit volume of catalyst, while 𝑘𝐴′′ , which was used above, referred to the rate of reaction per unit area of catalyst. 𝑘𝐴′′′ is the appropriate reaction rate constant to use when calculating the Thiele modulus (see Appendix A for a more rigorous derivation). Hence, for a real porous catalyst the Thiele Modulus defined as: 𝑀𝑇 = 𝐿√

𝑘𝐴′′′

𝐷𝐴eff

The relationship between the effectiveness factor and the Thiele modulus is shown in Figure 8 below. The Thiele modulus is a dime...