Calculating ideal gas compressor work and final temperature

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SUMMARY

The discussion focuses on calculating the minimum work required for an adiabatic compression of three moles of ideal gas from an initial state of 5 bar and 150°C to a final pressure of 50 bar. The relevant equations include the steady state energy balance and the relationship between temperature and pressure in adiabatic processes, specifically T2/T1=(P2/P1)^((γ-1)/γ). The constant pressure heat capacity (Cp) is given as 29 J/(mol*K), which is essential for determining the change in enthalpy (ΔH) and subsequently the work (W) done by the compressor.

PREREQUISITES
  • Understanding of adiabatic processes in thermodynamics
  • Familiarity with the ideal gas law and its applications
  • Knowledge of heat capacities (Cp and Cv) and their relationships
  • Ability to apply the steady state energy balance equation
NEXT STEPS
  • Learn how to derive T2 using the adiabatic process equation T2/T1=(P2/P1)^((γ-1)/γ)
  • Study the implications of constant pressure heat capacity (Cp) in thermodynamic calculations
  • Explore the concept of work done in adiabatic processes and its relation to internal energy (ΔU)
  • Investigate the application of the ideal gas law in various thermodynamic scenarios
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Students and professionals in mechanical engineering, chemical engineering, and thermodynamics who are involved in the design and analysis of compressors and other thermodynamic systems.

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A compressor is being used to adiabatically pressurize three moles of ideal gas from 5 bar and 150 C to 50 bar. Determine the minimum work that must be supplied to the compressor and the outlet temperature of the ideal gas. Assume the ideal gas has a constant pressure heat capacity of 29 J/(mol*K).

Relevant equations: steady state energy balance: 0=Mk(deltaH)+Q+W
Cp=deltaH/deltaT

Attempt at solution: The only problem I'm having is determining how to find T2. With T2 I can solve for delta H and plug it into the energy balance to find W, but I have no clue how to get T2. I don't think I can do anything with the ideal gas law either so I'm not sure what to do. I was thinking maybe P1/T1=P2/T2 but I don't believe the conditions are quite right for that, pllus we've never used that equation in class. Anyone have any ideas or advice?

Thank you for your time.
 
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Cp=Cv+R

γ=Cp/Cv

For adiabatic Processes:
T2/T1=(P2/P1)^((γ-1)/γ)

Solve for T2

Since adiabatic Q=0 therefore work=ΔU
 

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