Question

The journals listed at the erid of Chapter 1 may be useful for part (b). (a) Example \(1-1\). Consider the mass transfer-limited reaction $$A \longrightarrow 2 B$$ What would your concentration (mole fraction) profile look like? Using the same values for \(D_{\mathrm{A} \mathrm{h}},\) and so on. in Example \(11-1,\) what is the flux of \(\mathrm{A} ?\) (b) Example \(11-2 .\) How would your answers change if the temperature was increased by \(50^{\circ} \mathrm{C}\). the particle diameter was doubled. and fluid velocity was cut in half? Assume properties of water can be used for this system. (c) Example \(11-3 .\) How would your answers change if you had a \(50-50\) mixture of hydrazine and helium? If you increase \(d_{p}\) by a factor of \(5 ?\) (d) Example \(11-4 .\) What if you were asked for representative values for \(\mathrm{Re}\), Sc, Sh, and \(k_{\varepsilon}\) for both liquid-and gas-phase systems for a velocity of 10 \(\mathrm{cm} / \mathrm{s} \text { and a pipe diameter of } 5 \mathrm{cm} \text { (or a packed-bed diameter of } 0.2 \mathrm{cm}) ?\) What numbers would you give? (e) Example \(11-5 .\) How would your answers change if the reaction were carried out in the liquid phase where kinetic viscosity varied as $$v\left(T_{2}\right)=v\left(T_{1}\right) \exp \left[-\frac{4000 K}{T}\right] ?$$ to predict the drug delivery as a function of time. Compare this result with that where the drug in the patch is in a dissolving solid and a hydro-gel and therefore constant with time. Explore this problem using different models and parameter values. Additional information $$\begin{aligned} &H=0.1, D_{\mathrm{AB} 1}=10^{-6} \mathrm{cm}^{2} / \mathrm{s}, D_{\mathrm{AB} 2}=10^{-5} \mathrm{cm}^{2} / \mathrm{s}, A_{\mathrm{p}}=5 \mathrm{cm}^{2}, V=1 \mathrm{cm}^{3}\\\ &\text { and } C_{\mathrm{AP}}=10 \mathrm{mg} / \mathrm{dm}^{3} \end{aligned}$$

Short answer

The concentration profile for the mass transfer-limited reaction \(A \longrightarrow 2 B\) will be a decreasing curve for A and an increasing curve for B. For the given values of the diffusivity, the flux of A can be calculated using Fick's law. The answers in part (b), (c), and (e) depend on the changes in the parameters provided and will require recalculating the mass transfer rates, concentration profile, and flux of A. In part (d), calculate the Reynolds number (Re), Schmidt number (Sc), Sherwood number (Sh), and the turbulent kinetic energy dissipation rate (k_ε) for both liquid and gas-phase systems with the given velocities and pipe diameters.

Step-by-step solution

A reacts to form 2 B, so the reaction is written as \(A \longrightarrow 2 B\). In this problem, mass transfer-limited reaction means that the rate of reaction is limited by the rate of mass transfer. #Step 2: Concentration profile#

In a mass transfer limited reaction, the concentration of A decreases as it moves towards the reaction location, while the concentration of B increases. The concentration profile will be a decreasing curve for A and an increasing curve for B. #Step 3: Determine the flux of A#

We need to use the same values for \(D_{Ah}\) as in Example 11-1. The flux of A can be determined using Fick's law as: \(N_{A}=-D_{A} \frac{dC_{A}}{dx}\) Where \(N_A\) is the flux of A, \(D_A\) is the diffusivity of A, and \(\frac{dC_{A}}{dx}\) is the concentration gradient of A. Use the value for \(D_{Ah}\) from Example 11-1 and calculate the flux of A. (b) #Step 1: Determine the changes in parameters#

The changes given are: temperature increased by \(50^\circ\mathrm{C}\), particle diameter doubled, and fluid velocity cut in half. #Step 2: How these changes affect the answers#

Each of these changes will affect the mass transfer rates and the values of various dimensionless numbers. The concentration profile and the flux of A will be affected by these changes. Use the properties of water to recalculate the concentration profile and the flux of A. (c) #Step 1: Determine the changes in parameters#

The changes given are: a mixture of 50-50 hydrazine and helium, and an increase in \(d_p\) (particle diameter) by a factor of 5. #Step 2: How these changes affect the answers#

These changes in the composition and the particle diameter will affect the mass transfer rates and the values of various dimensionless numbers. The concentration profile and the flux of A will be affected by these changes. Recalculate the concentration profile and the flux of A with these new conditions. (d) #Step 1: Find representative values for Re, Sc, Sh, and k_ε#

Using the given velocity of 10 cm/s and a pipe diameter of 5 cm (or a packed-bed diameter of 0.2 cm), calculate the Reynolds number (Re), Schmidt number (Sc), Sherwood number (Sh), and the turbulent kinetic energy dissipation rate (k_ε) for both liquid and gas-phase systems. (e) #Step 1: Understand the given variation in kinetic viscosity#

Kinetic viscosity varies with temperature as follows: \(v\left(T_{2}\right)=v\left(T_{1}\right) \exp \left[-\frac{4000 K}{T}\right]\) #Step 2: Analyze the effect of varying viscosity on the answers#

This change in kinetic viscosity will affect the mass transfer rates and the values of various dimensionless numbers, which in turn will affect the concentration profile and the flux of A. Recalculate the concentration profile and the flux of A with this new variation in kinetic viscosity.

Key concepts

Chemical Reaction Engineering

Chemical Reaction Engineering (CRE) is a core subject in chemical and biochemical engineering, focusing on the design and operation of chemical reactors to optimally control chemical reactions. Excellent understanding of CRE is pivotal as it combines an understanding of chemistry with mathematics and material science to predict optimum conditions for industrial-scale chemical reactions. For example, consider a mass transfer-limited reaction, which suggests that the rate at which reactants come together in the reactor is the limiting step for the reaction rate rather than the intrinsic kinetics of the chemical reactions itself. In reality, this can be visualized when reactant A is converted into product B, as per our exercise, it is paramount to optimize several factors such as temperature, reactant concentration, and reactor design to control the reaction rate.

Examining how varying conditions such as temperature changes and reactant mixtures influence the reaction showcases the essence of CRE. It involves complex calculations of flow, heat transfer, mass transfer, and chemical kinetics, emphasizing the interdisciplinary nature of CRE.

Concentration Profile

The concentration profile describes how the concentration of a substance, reactant A in our case, varies over a certain distance within a system at a given moment. In the context of a mass transfer-limited reaction, the concentration profile is particularly important as it depicts the distribution of reactants and products over the phase where the reaction occurs. Typically, the concentration of reactant A would decrease from the outer edges towards the reaction zone, resulting in a gradient. This is visualized through a curve that slopes downwards for A, indicating its consumption, while the curve for product B would slope upwards, representing its formation. The sharpness of the concentration gradients directly affects the mass transfer rate and it's crucial for designing efficient reactors to ensure the reactants are adequately converted to products.

The concentration profile can also demonstrate the limitations of a system and aid in identifying areas that may require improved mixing or different reactor geometries to enhance reaction rates.

Fick's Law

Fick's Law of Diffusion is an essential principle in chemical engineering, providing insight into the rate at which components diffuse within a medium. The law stipulates that the flux of a substance from high concentration to low concentration is proportional to the negative gradient of the concentration. Mathematically, this is expressed as:

\(N_A = -D_A \frac{dC_A}{dx}\),

where \(N_A\) is the molar flux of component A, \(D_A\) the diffusion coefficient, and \(\frac{dC_A}{dx}\) the concentration gradient of A. In our exercise, using Fick's Law allows us to calculate the flux of A, which is a measure of the amount of A being transported per unit area per unit time due to diffusion. The relationship highlights the direct proportionality to the diffusivity and the inverse proportionality to distance, emphasizing how critical this law is for reacting flows and reacting systems design.

Dimensionless Numbers

Dimensionless numbers play a pivotal role in the modeling and scale-up of chemical processes. They are used to compare different physical phenomena and extrapolate data from one system to another without the need for dimensionally consistent units. In studies involving fluid flow, heat, and mass transfer, dimensionless numbers offer insight into the behavior of the system by encapsulating complex physical variables into simpler forms.

Reynolds (Re), Schmidt (Sc), and Sherwood (Sh) numbers are primary examples from our problem set. The Reynolds number characterizes the flow regime, indicating whether the flow is laminar or turbulent, which significantly impacts the mass transfer. The Schmidt number is a measure of the relative effectiveness of momentum to mass transfer, affecting the concentration profile for diffusion. The Sherwood number is analogous to the Nusselt number in heat transfer, relating the convective mass transfer to the molecular mass transfer, crucial for understanding the mass transfer at a phase boundary. Collectively, these numbers guide engineers in improving reactor design and predicting the outcomes of varying operating conditions.