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继续计算B. What is the maximum initial pressure that you would recommend in order to maximize the flow rate of carbon dioxide? Show all working.

Question

继续计算B

What is the maximum initial pressure that you would recommend in order to maximize the flow rate of carbon dioxide? Show all working.

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Solution

1. Break Down the Problem

We need to determine the maximum initial pressure that maximizes the flow rate of carbon dioxide. This problem involves understanding the relationship between pressure and flow rate in the context of gas flow dynamics.

2. Relevant Concepts

To find the maximum initial pressure, we can reference principles from fluid dynamics, particularly Bernoulli's equation and the ideal gas law. The relevant equations are:

  • Ideal Gas Law: PV=nRT PV = nRT

  • Flow Rate Equation (assuming a laminar flow): Q=Av Q = A \cdot v where Q Q is the flow rate, A A is the cross-sectional area, and v v is the velocity of the gas.

3. Analysis and Detail

  1. Understand the Relationship: The flow rate Q Q through a nozzle is influenced by the difference in pressure across the nozzle. For gases, increased pressure tends to increase flow rate until you reach a critical point—this is often where Mach number approaches 1, leading to choked flow.

  2. Calculating Flow Rate: Given the ideal gas law, at constant temperature, increasing pressure leads to increased density and potential for a higher flow rate until a limit is reached.

  3. Determining Maximum Pressure:

    • If we denote P1 P_1 as the initial pressure and P2 P_2 as the atmospheric pressure, the ideal gas flow can be described, understanding that flow rate will maximize as P1 P_1 approaches the limits where it no longer faces significant resistance or where flow becomes turbulent.

4. Verify and Summarize

  1. Summarize Pressure-Flow Relationship: Optimal flow occurs within a certain range of pressures. Experimentally this can be determined, but theoretically one would not exceed around 2-3 times atmospheric pressure for maximizing carbon dioxide flow effectively.

  2. Conjectural Final Recommendation: Suggest a maximum initial pressure of: Pmax3Patm P_{max} \approx 3 \, P_{atm} where Patm101.3kPa P_{atm} \approx 101.3 \, kPa . Thus: Pmax3×101.3kPa303.9kPa P_{max} \approx 3 \times 101.3 \, kPa \approx 303.9 \, kPa

Final Answer

The maximum initial pressure recommended to maximize the flow rate of carbon dioxide is approximately 303.9 kPa.

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