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Sodium bicarbonate is synthesized by reacting sodium carbonate with carbon dioxide and water at \(70^{\circ} \mathrm{C}\) and \(2.0 \mathrm{atm}\) gauge pressure: $$\mathrm{Na}_{2} \mathrm{CO}_{3}+\mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O} \rightarrow 2 \mathrm{NaHCO}_{3}$$ An aqueous solution containing 7.00 wt\% sodium carbonate and a gas stream containing 70.0 mole\% \(\mathrm{CO}_{2}\) and the balance air are fed to the reactor. All of the sodium carbonate and some of the carbon dioxide in the feed react. The gas leaving the reactor, which contains the air and unreacted \(\mathrm{CO}_{2},\) is saturated with water vapor at the reactor conditions. A liquid-solid slurry of sodium bicarbonate crystals in a saturated aqueous solution containing \(2.4 \mathrm{wt} \%\) dissolved sodium bicarbonate and a negligible amount of dissolved \(\mathrm{CO}_{2}\) leaves the reactor and is pumped to a filter. The wet filter cake contains 86 wt\% sodium bicarbonate crystals and the balance saturated solution, and the filtrate also is saturated solution. The production rate of solid crystals is \(500 \mathrm{kg} / \mathrm{h}\).Suggestion: Although the problems to be given can be solved in terms of the product flow rate of \(500 \mathrm{kg} \mathrm{NaHCO}_{3}(\mathrm{s}) / \mathrm{h},\) it might be easier to assume a different basis and then scale the process to the desired production rate of crystals.(a) Calculate the composition (component mole fractions) and volumetric flow rate \(\left(\mathrm{m}^{3} / \mathrm{min}\right)\) of the gas stream leaving the reactor. (b) Calculate the feed rate of gas to the process in standard cubic meters/min \(\left[\mathrm{m}^{3}(\mathrm{STP}) / \mathrm{min}\right]\) (c) Calculate the flow rate \((\mathrm{kg} / \mathrm{h})\) of the liquid feed to the process. What more would you need to know to calculate the volumetric flow rate of this stream? (d) The filtrate was assumed to leave the filter as a saturated solution at \(70^{\circ} \mathrm{C}\). What would be the effect on your calculations if the temperature of the filtrate actually dropped to \(50^{\circ} \mathrm{C}\) as it passed through the filter? (e) The reactor pressure of 2 atm gauge was arrived at in an optimization study. What benefit do you suppose would result from increasing the pressure? What penalty would be associated with this increase? The term "Henry's law" should appear in your explanation. (Hint: The reaction occurs in the liquid phase and the \(\mathrm{CO}_{2}\) enters the reactor as a gas. What step must precede the reaction?)

Step by step solution

01

Calculate the composition and volumetric flow rate of the gas stream leaving the reactor

Given the composition of the feed gas stream is 70 mole% \( CO_{2} \) and the balance air, calculate the moles of \( CO_{2} \) and air according to the stoichiometry of the chemical reaction. Deduct the moles of \( CO_{2} \) that reacted to get the moles of \( CO_{2} \) in the stream leaving the reactor and add this to the initial moles of air. The mole fraction of the gases in the outgoing gas stream can then be calculated by division with the total moles. The volumetric flow rate can be determined using the ideal gas law, \( PV=nRT \) , knowing that the volume is given by \( V=nRT/P \).

02

Calculate the feed rate of gas to the process in standard cubic meters/min

From the stoichiometry of the chemical reaction, every mole of \( CO_{2} \) reacts to form two moles of \( NaHCO_{3} \). From the production rate of \( NaHCO_{3} \) we can get the moles of \( NaHCO_{3} \) formed per minute and hence the moles of \( CO_{2} \) that reacted. We then add this to moles of \( CO_{2} \) already found in the outgoing gas stream from step one and know the total moles of \( CO_{2} \) in the feed. From \(70%\) mole fraction of \( CO_{2} \), we calculate total emerging moles of gas and convert into volume using ideal gas law at standard condition.

03

Calculate the flow rate of the liquid feed to the process.

First, calculate the fed moles of \( Na_{2}CO_{3} \). Knowing each mole of sodium carbonate consumes a mole of carbon dioxide from the gas feed to produce two moles of sodium bicarbonate, we can calculate the mass flow rate of the liquid feed to process. However, we would need additional information, primarily the liquid feed's density, to translate mass flow rate into volumetric flow rate.

04

Analyze effect of temperature drop on calculations.

If the temperature drops from \(70^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\), it would affect the solubility of the sodium bicarbonate in the filtrate. The recirculation of the filtrate has to be increased to compensate for the decreased solubility, which leads to an increase in the total feed to the reactor.

05

Discuss what could potentially benefit from increasing the pressure and the possible penalty associated with it.

An increase in pressure favors the forward reaction according to Le Chatelier's principle. More \( CO_{2} \) would dissolve in the solution and contribute to more \( NaHCO_{3} \) production rate. This is the benefit of increasing the reactor pressure. However, the penalty associated with this is that a high-pressure system is more expensive to design, construct, operate and maintain.

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Key Concepts

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

Understanding Stoichiometry in Chemical Reactions

Stoichiometry is a fundamental concept in chemistry that involves the calculation of the relative quantities of reactants and products in chemical reactions. It is based on the conservation of mass and the principle that atoms are neither created nor destroyed during a chemical reaction.

For example, when sodium carbonate reacts with carbon dioxide and water to form sodium bicarbonate as stated in the chemical equation \(\mathrm{Na}_{2}\mathrm{CO}_{3} + \mathrm{CO}_{2} + \mathrm{H}_{2}\mathrm{O} \rightarrow 2\mathrm{NaHCO}_{3}\), stoichiometry helps us calculate the precise amounts of each reactant needed to produce a certain amount of the product. In a practical sense, if we know the production rate of sodium bicarbonate crystals is \(500 \mathrm{kg}/\mathrm{h}\), stoichiometry allows us to backtrack and figure out how much sodium carbonate and carbon dioxide is necessary to achieve this output.

In the given exercise, the suggestion to assume a different basis for calculation is a useful strategy. By defining a certain amount of product or reactant, known as the 'basis of calculation,' we can use the stoichiometric ratios to find all the unknown quantities. Once all values are found, we can simply scale them to correspond to the actual production rate.

Calculating Volumetric Flow Rate with the Ideal Gas Law

The volumetric flow rate of a gas is a measure of the volume of gas that passes through a particular point in the system per unit time. In a chemical process, knowing the volumetric flow rate is essential for equipment sizing, process control, and safety.

To find the volumetric flow rate of the gas stream leaving the reactor in the exercise, we used the ideal gas law, represented by the equation \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature. By rearranging the ideal gas law to solve for volume \(V\), and understanding that the conditions of the gas release from the reaction are not standard temperature and pressure (STP), we calculate the volumetric flow rate under the given conditions.

Understanding the ideal gas law is crucial as it relates the four variables of a gas. Any time we know three of these variables, we can solve for the fourth. However, it's important to consider that the ideal gas law assumes gases are 'ideal', which means it works best for gases at low pressure and high temperature. Knowing that gases deviate from ideal behavior under certain conditions, one should be mindful of potential inaccuracies when using this law for real-world applications.

The Role of the Ideal Gas Law in Chemical Process Calculations

The ideal gas law \(PV = nRT\) is an equation of state that describes the behavior of ideal gases. It's an invaluable tool in chemical process calculations as it allows engineers to predict how gases will react when subject to changes in pressure, volume, and temperature without the need to experimentally measure these values.

In our exercise, the ideal gas law helps calculate not only the volumetric flow rate of the gas leaving the reactor but also the feed rate of the gas entering the process. This is because the law can be applied at standard conditions (STP), which are defined as a temperature of 0°C (273.15 K) and a pressure of 1 atm, to compute the volume occupied by the gases if they were at these standard conditions. By determining the feed rate at STP, we can compare the utilization efficiency and process capacity with other systems or standard references.

By employing the ideal gas law, the calculations in chemical processes become standardized, which is essential for designing systems, controlling reactions, and scaling production rates. It's the simplicity and predictive nature of the ideal gas law that makes it such an integral part of chemical engineering and process design.