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Sulfur dioxide is oxidized to sulfur trioxide in a small pilot-plant reactor. SO \(_{2}\) and \(100 \%\) excess air are fed to the reactor at \(450^{\circ} \mathrm{C}\). The reaction proceeds to a \(65 \% \mathrm{SO}_{2}\) conversion, and the products emerge from the reactor at \(550^{\circ} \mathrm{C}\). The production rate of \(\mathrm{SO}_{3}\) is \(1.00 \times 10^{2} \mathrm{kg} / \mathrm{min}\). The reactor is surrounded by a water jacket into which water at \(25^{\circ} \mathrm{C}\) is fed. (a) Calculate the feed rates (standard cubic meters per second) of the \(\mathrm{SO}_{2}\) and air feed streams and the extent of reaction, \(\xi\) (b) Calculate the standard heat of the SO_ oxidation reaction, \(\Delta H_{\mathrm{t}}^{\mathrm{r}}(\mathrm{kJ}) .\) Then, taking molecular species at \(25^{\circ} \mathrm{C}\) as references, prepare and fill in an inlet-outlet enthalpy table and write an energy balance to calculate the necessary rate of heat transfer ( \(\mathrm{kW}\) ) from the reactor to the cooling water. (c) Calculate the minimum flow rate of the cooling water if its temperature rise is to be kept below \(15^{\circ} \mathrm{C}\) (d) Briefly state what would have been different in your calculations and results if you had taken elemental species as references in Part (b).

Step by step solution

01

Determine the feed rates

Given production rate of \(SO_3\) is \(1.00 \times 10^2 kmol/min\) which means in each minute, \(1.00 \times 10^2 kmol\) of \(SO_2\) are getting converted to \(SO_3\). Since the feed contains 100% excess air, conversion of \(SO_2\) is less than expected. Since conversion rate is 65%, actual feed rate of \(SO_2\) is \(1.00 \times 10^2 kmol/min / 0.65 = 153.85 kmol/min\). Therefore, the feed rates of \(SO_2\) and air are \(153.85 kmol/min\) and \(307.7 kmol/min\) respectively.

02

Calculate the heat of reaction

Heat of reaction \(\Delta H_{t}^r\) for the reaction \(SO_2 + 0.5O_2 → SO_3\) can be obtained from the standard heat of formation of the reactants and products. \(\Delta H_{t}^ {r} = Heat of formation of products – Heat of formation of reactants\). The standard heat of reaction is now used to determine the energy flow from the reactor to the cooling water by setting up an enthalpy table for the inlet and outlet streams and writing the energy balance around the reactor.

03

Calculate cooling water flow rate

The rate of heat transfer from the reactor to the cooling system can be calculated using \(q=mc\Delta t\), where \(m\) is the cooling water flow rate, \(c\) is the specific heat of water, and \(\Delta t\) is the temperature difference. This equation gives the minimum flow rate of cooling water needed to keep the temperature rise below \(15^{\circ}C\).

04

Analyses of different reference states

If the calculation was using elemental species as reference, the energy of heat transfer would have been calculated based on the heat of formation values which are zero for the elements in their standard states at the reference temperature. This modification would have changed the energy balance equation and therefore the calculated rate of heat transfer. The conversion rate and feed rates would still be same, but the heat of reaction, and therefore the rate of heat transfer and the cooling water flow rate would have been different.

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Key Concepts

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

Sulfur Dioxide Oxidation

Sulfur dioxide oxidation is a crucial reaction in chemical engineering, particularly in the production of sulfuric acid. This reaction involves the conversion of sulfur dioxide (SO\(_2\)) to sulfur trioxide (SO\(_3\)) using oxygen available in excess air. In the described pilot-plant reactor, the conversion efficiency of SO\(_2\) is 65%. This means that 65% of the sulfur dioxide introduced to the reactor is transformed into sulfur trioxide. Oxidation reactions like this one are significant because they determine the effectiveness and efficiency of industrial processes.
One important aspect of this reaction to notice is the excess use of air. This implies that there is more oxygen available than what is necessary to react with the sulfur dioxide. This ensures the reaction goes to completion as much as possible, minimizing leftover reactants.
Conversion percentage is a key parameter that affects the design and operation of chemical reactors, determining the required feed rates and the overall economic feasibility of the process. Thus, understanding and optimizing this aspect is fundamental to chemical reaction engineering.

Heat of Reaction

The heat of reaction is the heat energy absorbed or released during a chemical reaction. For the sulfur dioxide oxidation reaction, it is important to determine this energy change as it directly affects the reactor operation and its thermal management.
To calculate the heat of reaction (\(\Delta H_{rxn}\),for the oxidation of SO\(_2\) to SO\(_3\), start by considering the standard heat of formation for the reactants and products. These values are typically obtained from thermodynamic tables. The formula used is:\[\Delta H_{rxn} = \text{heat of formation of products} - \text{heat of formation of reactants}\]Due to its impact on energy balance calculations, understanding the heat of reaction is crucial. It dictates whether the process is exothermic (releasing heat) or endothermic (absorbing heat). This energy change helps in designing systems to manage the thermal effects generated during the reaction, often requiring the implementation of cooling systems.

Energy Balance

Energy balance is a fundamental concept in chemical reaction engineering, ensuring that all energy inputs and outputs within a process are accounted for. In this context, it deals with achieving a balance between the energy supplied to the system and the energy absorbed or released during the chemical reaction.
Creating an enthalpy table for the inlet and outlet streams is the first step to writing an energy balance for the reactor. This table lists all the thermal energies of each component entering and leaving the system, based on their specific heat capacities and temperatures.
From there, the energy balance equation can be formulated. This includes the sum of all energies entering the system equated to the sum of the energies leaving the system, along with the energy either absorbed or released by the reaction. This helps in calculating the necessary rate of heat transfer to manage the reactor temperature, which is vital for safe and efficient operation.

Cooling Water Flow Rate

The cooling water flow rate is a key operational parameter in chemical reactors, especially those involving exothermic reactions like sulfur dioxide oxidation. The primary function of cooling water is to remove excess heat from the reactor, preventing overheating and ensuring optimal reaction conditions.
This involves applying the principle of heat transfer, where the equation is expressed as:\[q = mc\Delta T\]Here, \(q\) represents the heat removal rate, \(m\) is the flow rate of the cooling water, \(c\) is the specific heat capacity of water, and \(\Delta T\) is the permissible temperature rise of the water.
In this scenario, the goal is to keep the cooling water's temperature change below 15°C. By substituting known values into the equation, one can solve for the minimum cooling water flow rate necessary. Proper adjustment of this flow rate is essential in maintaining the reactor's stability and preventing any negative impacts on the reaction performance or safety.