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Chapter 14: Problem 32

A platinum catalytic reactor in an automobile is used to convert carbon monoxide to carbon dioxide in an oxidation reaction of the form \(2 \mathrm{CO}+\mathrm{O}_{2} \rightarrow 2 \mathrm{CO}_{2}\). Species transfer between the catalytic surface and the exhaust gases may be assumed to occur by diffusion in a film of thickness \(L=10 \mathrm{~mm}\). Consider an exhaust gas that has a pressure of \(1.2\) bars, a temperature of \(500^{\circ} \mathrm{C}\), and a \(\mathrm{CO}\) mole fraction of \(0.0012\). If the reaction rate constant of the catalyst is \(k_{1}^{\prime \prime}=0.005 \mathrm{~m} / \mathrm{s}\) and the diffusion coefficient of \(\mathrm{CO}\) in the mixture is \(10^{-4} \mathrm{~m}^{2} / \mathrm{s}\), what is the molar concentration of \(\mathrm{CO}\) at the catalytic surface? What is the rate of removal of \(\mathrm{CO}\) per unit area of the catalyst? What is the removal rate if \(k_{1}^{\prime \prime}\) is adjusted to render the process diffusion limited?

Short Answer

Expert verified
The molar concentration of CO at the catalytic surface is calculated using the Ideal Gas Law and the given mole fraction, pressure, and temperature values, and is found to be \(C_{\mathrm{CO}} = \frac{0.0012 * P}{R T}\). The rate of removal of CO per unit area of the catalyst is determined using the given reaction rate constant and the molar concentration of CO, which results in \(-k_{1}^{\prime\prime} * C_{\mathrm{CO}}\). To adjust the reaction rate constant for a diffusion limited process, we equate the rate of removal per unit area to the rate limited, given as \(\frac{D_{\mathrm{CO}} * C_{\mathrm{CO}}}{L}\), and solve for the adjusted reaction rate constant, resulting in \(k_{1\,adjusted}^{\prime\prime} * C_{\mathrm{CO}} = \frac{D_{\mathrm{CO}} * C_{\mathrm{CO}}}{L}\).

Step by step solution

01

Calculate the molar concentration of CO

First, let's calculate the molar concentration of CO at the catalytic surface. Since we know the mole fraction of CO (0.0012) and the pressure and temperature of the gas (1.2 bars and 500 掳C), we can use the ideal gas law to find the concentration. The ideal gas law states that \(P V=n R T\), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant and T is the temperature. First, convert the given units to S.I. units: Pressure: \(P=1.2\,bars=1.2*10^{5}\,Pa\) Temperature: \(T=500^{\circ} C = 773.15\: K\) Solve for the molar concentration of the exhaust gas: \[n/V = \frac{P}{RT}\] Using the given value for the CO mole fraction (0.0012) and the calculated exhaust gas concentration, we can now find the molar concentration of CO: \[C_{\mathrm{CO}} = \frac{0.0012 * P}{R T}\]
02

Calculate the rate of removal of CO per unit area of the catalyst

Now let's calculate the rate of removal of CO per unit area of the catalyst, using the given reaction rate constant (\(k_{1}^{\prime \prime}\)) and the molar concentration of CO obtained in step 1: Using the relation, \[rate\, of\, removal\, per\, unit\, area = -k_{1}^{\prime\prime} * C_{\mathrm{CO}}\]
03

Adjust the reaction rate constant for the diffusion limited process

Finally, let's adjust the reaction rate constant to make the process diffusion limited. The rate of removal is determined by the diffusion of CO in this case. We can use the diffusion coefficient of CO (\(10^{-4} \mathrm{~m}^{2} / \mathrm{s}\)) and the film thickness (\(L=10 \mathrm{~mm}\)): The relation for the diffusion limited rate is given as follows: \[rate\,limited = \frac{D_{\mathrm{CO}} * C_{\mathrm{CO}}}{L}\] By setting the rate of removal per unit area equal to the rate limited, we can solve for the adjusted reaction rate constant (\(k_{1\,adjusted}^{\prime\prime}\)), as follows: \[k_{1\,adjusted}^{\prime\prime} * C_{\mathrm{CO}} = \frac{D_{\mathrm{CO}} * C_{\mathrm{CO}}}{L}\] Now we can solve for the adjusted reaction rate constant (\(k_{1\,adjusted}^{\prime\prime}\)) and provide the results for the molar concentration of CO, the rate of removal of CO per unit area of the catalyst, and the removal rate for the diffusion limited process.

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Key Concepts

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

Diffusion in Catalytic Reactors
Understanding how substances like carbon monoxide (CO) move in catalytic reactors is vital for enhancing reaction efficiency. This movement, known as diffusion, is the spontaneous spread of particles from regions of higher concentration to areas of lower concentration. In an automobile's platinum catalytic reactor, diffusion becomes critical as exhaust gases interact with the catalyst surface.

In our example, the diffusion of CO in a film of given thickness is what facilitates the reaction to convert it to carbon dioxide (CO鈧�). The rate of diffusion is influenced by factors such as the diffusion coefficient, which represents how easily CO molecules move through the exhaust gas mixture, and the film thickness, which is the space through which these molecules must diffuse to reach the catalytic surface.

To optimize reactions within catalytic reactors, it's important to balance the reaction rate with the rate of diffusion. When the process is 'diffusion limited', it means that the speed at which CO molecules reach the surface controls the overall rate of the reaction, not the intrinsic speed of the reaction itself. This scenario is particularly crucial in exhaust gas treatment where quick and efficient conversion of toxic gases is required.
Ideal Gas Law Calculations
The ideal gas law is a cornerstone of chemical engineering, allowing us to relate variables such as pressure, temperature, volume, and the number of moles of a gas. The law is typically expressed as the equation, \( PV=nRT \) where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of substance (moles), \( R \) is the ideal gas constant, and \( T \) is temperature.

In our problem, we calculate the molar concentration of CO in the gas phase using the ideal gas law. To do this, we first ensure our temperature and pressure are in the correct units (kelvin and pascals). The CO mole fraction then helps us find the specific concentration of CO at a given pressure and temperature. Calculations like this are essential for engineers to design reactors and control the conditions under which chemical reactions occur efficiently.
Reaction Rate Constant
A reaction rate constant, in the context of a catalytic reactor, is a quantifiable measure of how quickly a reaction proceeds. It factors into the equation describing the rate at which a reactant is consumed or a product is formed over time. In our example, \( k_{1}^{\prime \prime} \) represents this constant for the conversion of CO to CO鈧� on the catalyst surface.

It's crucial to understand that the rate constant is influenced by various conditions, such as temperature and the presence of a catalyst. When we consider that the process may be diffusion limited, we adjust the rate constant to reflect the slower, diffusion-controlled rate. This means that even if the catalyst is highly effective, the overall rate at which CO is removed from the exhaust gas could be limited by how fast CO molecules can diffuse to the catalytic surface.

The rate constant is a key component in the design and optimization of reactors, ensuring that they operate efficiently under the given conditions. Without accurate knowledge of these constants, predicting and controlling the outcome of a chemical reaction would be much more challenging.

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Most popular questions from this chapter

Consider the DVD of Problem 14.49, except now the reacting polymer is blended uniformly with the polycarbonate to reduce manufacturing costs. Assume that a first-order homogeneous chemical reaction takes place between the polymer and oxygen; the reaction rate is proportional to the oxygen molar concentration. (a) Write the governing equation, boundary conditions, and initial condition for the oxygen molar concentration after the DVD is removed from the oxygen- proof pouch, for a DVD of thickness \(2 L\). Do not solve. (b) The DVD will gradually become more opaque over time as the reaction proceeds. The ability to read the DVD will depend on how well the laser light can penetrate through the thickness of the DVD. Therefore, it is important to know the volume-averaged molar concentration of product, \(\bar{C}_{\text {prod }}\), as a function of time. Write an expression for \(\bar{C}_{\text {prod }}\) in terms of the oxygen molar concentration, assuming that every mole of oxygen that reacts with the polymer results in \(p\) moles of product.

Hydrogen gas is used in a process to manufacture a sheet material of \(6-\mathrm{mm}\) thickness. At the end of the process, \(\mathrm{H}_{2}\) remains in solution in the material with a uniform concentration of \(320 \mathrm{kmol} / \mathrm{m}^{3}\). To remove \(\mathrm{H}_{2}\) from the material, both surfaces of the sheet are exposed to an airstream at \(500 \mathrm{~K}\) and a total pressure of \(3 \mathrm{~atm}\). Due to contamination, the hydrogen partial pressure is \(0.1 \mathrm{~atm}\) in the airstream, which provides a convection mass transfer coefficient of \(1.5 \mathrm{~m} / \mathrm{h}\). The mass diffusivity and solubility of hydrogen (A) in the sheet material (B) are \(D_{\mathrm{AB}}=2.6 \times 10^{-8} \mathrm{~m}^{2} / \mathrm{s}\) and \(S_{\mathrm{AB}}=160 \mathrm{kmol} / \mathrm{m}^{3} \cdot\) atm, respectively. (a) If the sheet material is left exposed to the airstream for a long time, determine the final content of hydrogen in the material \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\). (b) Identify and evaluate the parameter that can be used to determine whether the transient mass diffusion process in the sheet can be assumed to be characterized by a uniform concentration at any time during the process. Hint: This situation is analogous to that used to determine the validity of the lumped-capacitance method for a transient heat transfer analysis. (c) Determine the time required to reduce the hydrogen mass density at the center of the sheet to twice the limiting value calculated in part (a).

Nitric oxide (NO) emissions from automobile exhaust can be reduced by using a catalytic converter, and the following reaction occurs at the catalytic surface: $$ \mathrm{NO}+\mathrm{CO} \rightarrow \frac{1}{2} \mathrm{~N}_{2}+\mathrm{CO}_{2} $$ The concentration of NO is reduced by passing the exhaust gases over the surface, and the rate of reduction at the catalyst is governed by a first- order reaction of the form given by Equation 14.66. As a first approximation it may be assumed that NO reaches the surface by one-dimensional diffusion through a thin gas film of thickness \(L\) that adjoins the surface. Referring to Figure 14.7, consider a situation for which the exhaust gas is at \(500^{\circ} \mathrm{C}\) and \(1.2\) bars and the mole fraction of \(\mathrm{NO}\) is \(x_{\mathrm{A}, L}=0.15\). If \(D_{\mathrm{AB}}=10^{-4} \mathrm{~m}^{2} / \mathrm{s}\), \(k_{1}^{\prime \prime}=0.05 \mathrm{~m} / \mathrm{s}\), and the film thickness is \(L=1 \mathrm{~mm}\), what is the mole fraction of \(\mathrm{NO}\) at the catalytic surface and what is the NO removal rate for a surface of area \(A=200 \mathrm{~cm}^{2}\) ?

A thin plastic membrane is used to separate helium from a gas stream. Under steady-state conditions the concentration of helium in the membrane is known to be \(0.02\) and \(0.005 \mathrm{kmol} / \mathrm{m}^{3}\) at the inner and outer surfaces, respectively. If the membrane is \(1 \mathrm{~mm}\) thick and the binary diffusion coefficient of helium with respect to the plastic is \(10^{-9} \mathrm{~m}^{2} / \mathrm{s}\), what is the diffusive flux?

A 100-mm-long, hollow iron cylinder is exposed to a \(1000^{\circ} \mathrm{C}\) carburizing gas (a mixture of \(\mathrm{CO}\) and \(\mathrm{CO}_{2}\) ) at its inner and outer surfaces of radii \(4.30\) and \(5.70 \mathrm{~mm}\), respectively. Consider steady-state conditions for which carbon diffuses from the inner surface of the iron wall to the outer surface and the total transport amounts to \(3.6 \times 10^{-3} \mathrm{~kg}\) of carbon over \(100 \mathrm{~h}\). The variation of the carbon composition (weight \(\%\) carbon) with radius is tabulated for selected radii. $$ \begin{array}{lllllllll} r(\mathrm{~mm}) & 4.49 & 4.66 & 4.79 & 4.91 & 5.16 & 5.27 & 5.40 & 5.53 \\ \text { Wt.C }(\%) & 1.42 & 1.32 & 1.20 & 1.09 & 0.82 & 0.65 & 0.46 & 0.28 \end{array} $$ (a) Beginning with Fick's law and the assumption of a constant diffusion coefficient, \(D_{\mathrm{C}-\mathrm{Fe}}\), show that \(d \rho_{\mathrm{C}} / d(\ln r)\) is a constant. Sketch the carbon mass density, \(\rho_{\mathrm{C}}(r)\), as a function of \(\ln r\) for such a diffusion process. (b) The foregoing table corresponds to measured distributions of the carbon mass density. Is \(D_{\mathrm{C}-\mathrm{Fe}}\) constant for this diffusion process? If not, does \(D_{\mathrm{C}-\mathrm{Fe}}\) increase or decrease with an increasing carbon concentration? (c) Using the experimental data, calculate and tabulate \(D_{\mathrm{C}-\mathrm{Fe}}\) for selected carbon compositions.
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