FLUID MECHANICS and dynamics and numerical

fluid dynamicss

1. APPLICATIONS The Ideal Gas Law (PV= nRT ) is useful in Food Sciences in several ways, particularly in food packaging, preservation, and processing. Here are some key applications: 1. Food Packaging and Modified Atmosphere Packaging (MAP) The Ideal Gas Law helps in designing food packaging with controlled gas compositions (e.g., oxygen, nitrogen, CO₂) to extend shelf life. It ensures the right amount of gas is used to prevent spoilage and maintain product freshness. 2. Vacuum Sealing and Storage Understanding how gases behave under different pressures helps optimize vacuum-sealing techniques to preserve food. 3. Carbonation in Beverages In soft drinks and beer, CO₂ gas is dissolved under pressure. The law helps in controlling carbonation levels based on temperature and pressure conditions.

2. 4. Food Drying and Dehydration Dehydration processes (e.g., spray drying, freeze drying) rely on gas expansion and temperature control, which can be modeled using the Ideal Gas Law. 5. Fermentation Process Control Gases like CO₂ are produced during fermentation (e.g., in bread, yogurt, or beer production). The Ideal Gas Law helps predict gas production and storage conditions. 6. Storage Conditions for Bulk Food Storage silos and cold storage units use controlled gases for grain preservation, and the Ideal Gas Law helps regulate internal pressure and temperature.

3. Case Study: Application of the Ideal Gas Law in Carbonated Beverages Background: Carbonated beverages like soft drinks and beer contain dissolved carbon dioxide (CO₂) to give them their fizz. The amount of CO₂ dissolved in the liquid depends on pressure and temperature , which can be analyzed using the Ideal Gas Law . Problem Statement: A soft drink manufacturer wants to determine the amount of CO₂ needed to carbonate a 500 mL (0.5 L) bottle at a temperature of 5°C (278 K) and a pressure of 3 atm .

4. Solution using the Ideal Gas Law: The Ideal Gas Law equation is: 𝑃𝑉=𝑛𝑅𝑇 Where: 𝑃=3 atm 𝑉=0.5L 𝑅=0.0821 L·atm / mol·K (gas constant) 𝑇=278 K Solving for 𝑛n (moles of CO₂): 𝑛=𝑃𝑉/𝑅𝑇 =(3)(0.5)/(0.0821)(278) 𝑛≈0.0658 moles of CO₂

5. The manufacturer must dissolve approximately 0.066 moles (or 2.9 grams) of CO₂ per 500 mL bottle. If the temperature increases, the CO₂ will be less soluble, causing the beverage to lose its fizz faster. If the pressure drops (e.g., when opening the bottle), the CO₂ will escape as bubbles. Conclusion: The Ideal Gas Law helps food scientists determine the correct gas pressures needed for carbonation. It ensures consistent quality and taste in soft drinks and beers. It helps optimize storage and prevent excessive CO₂ loss, which affects shelf life .

6. Applications in Food Sciences 1. Food Storage & Packaging - Vacuum sealing reduces pressure → decreases specific weight → slows oxidation. 2. Aerated Foods (Whipped Cream, Bread, Meringue) - Lower specific weight results in a lighter texture. 3. Gas-Based Food Processing - Freeze-drying: Lower pressure helps moisture removal. - Fermentation: Gas bubbles alter density and specific weight. Conclusion • Specific weight is crucial in food sciences. • Temperature and pressure directly affect gas density and specific weight. • Applications range from packaging to food texture control.

7. Note: The molecular size, shape, and polarity of a fluid's molecules all affect its viscosity. For example, liquids with tightly packed molecules have a higher viscosity because the molecules are more strongly attracted to each other, making it harder for the liquid to flow.

8. Momentum Exchange in Fluids Momentum exchange in fluids refers to the transfer of momentum between fluid molecules due to molecular interactions. This occurs because fluid molecules are in constant motion and collide with each other, exchanging momentum in the process. Momentum Exchange in Gases vs. Liquids In liquids , molecular interactions are primarily dominated by cohesive forces, which keep the molecules relatively close together. Momentum is exchanged mostly through intermolecular forces rather than free movement. In gases , molecules are widely spaced and move freely. Momentum exchange occurs mainly through molecular collisions.

9. Why is Momentum Exchange More Dominant in Gases with an Increase in Temperature? With an increase in temperature, the following effects occur in gases: Increase in Molecular Speed According to the Kinetic Theory of Gases , the average molecular velocity increases with temperature. Faster-moving molecules collide more frequently and with greater force, increasing the rate of momentum exchange.

10. 2. Increased Mean Free Path The mean free path (λ) is the average distance a molecule travels before colliding with another molecule. At higher temperatures, gas molecules move faster and spread out more, increasing the mean free path. This enhances momentum transfer across greater distances. 3. Higher Diffusion Rate The diffusion of momentum in gases follows Newton’s Law of Viscosity, where shear stress (𝜏) is proportional to the velocity gradient (𝑑𝑢/𝑑𝑦). As temperature rises, molecular motion intensifies, leading to greater momentum diffusion. Increase in Viscosity of Gases Unlike liquids (where viscosity decreases with temperature), gas viscosity increases with temperature. This is because viscosity in gases depends on molecular collisions, and at higher temperatures, the increased collision rate leads to higher resistance to flow, increasing viscosity.

11. Kinematic viscosity is a measure of a fluid's resistance to flow under the influence of gravity. It's calculated by dividing a fluid's dynamic viscosity by its density. The SI unit for kinematic viscosity is square meters per second (m 2 /s).

12. Example