Adiabatic Expansion of Pressurized Air in a Piston-Cylinder Setup

In summary, the conversation discusses a task involving a pressurized air cylinder and a closed-bottom cylinder, with a tightly fitting piston and hot air inside. The calculations involve considering air as an ideal gas with specific heat capacity and specific gas constant. The question of the homework asks for the final height of the piston and the temperature of the system after opening the pressurized cylinder. The approach used by the expert involves equations for energy conservation and pressure-volume relationship, but there may be mistakes in the calculations.
  • #1
Peter_parker
3
0
Homework Statement
Uncertain how to take into account the temperature of the hot cylinder?
Relevant Equations
U=m*cv*dt
p*V=m*Rs*T
Task:
A thermally insulated pressurized air cylinder, B, was initially placed inside a closed-bottom, circular hollow cylinder A with an inner diameter of 50 cm. Then a tightly fitting, frictionless sliding piston with a mass of 20 kg was installed. Using the outlet valve, the height of the piston was set to h1 = 100 cm. The pressurized air cylinder B contains 10 liters of hot air with a temperature of 150 °C (TB) and an absolute pressure of 20 bar. The air in container A, that is, outside the pressurized air cylinder B, initially has the ambient temperature.

For your calculations, consider air as an ideal gas with a specific heat capacity at constant volume, cV = 0.718 kJ/(kg·K), and specific gas constant, Rs = 0.287 kJ/(kg·K).

The intrinsic volume of the bottle material should be neglected. The following environmental condition exists: ambient pressure pamb = 1 bar, ambient temperature Tamb = 20 °C.

Question of the homework:
After opening the pressurized air cylinder B, the entire system quickly reaches an equilibrium state. The process is adiabatic. At what height is the piston now located? What temperature does the system reach?

What i did:
Equation1 / Part1:
Condition 1: U1=m_a*cv*T1+m_a*cv*TB= m_both*cv*(T1+TB)
Condition 2: U2=m_both*cv*(T2)
Connection via:U12=Q+Wv with Q =0 and Wv=p*(V1-V2) to m_both*cv*(T2-(T1+TB))=p*(V1-V2)
Equation2 / Part2:
p(V2+VB)=m_both*Rs*T2

By combining Equation 1 and Equation 2, I get T2 / V2 --> then used the area of the cylinder to calculate h2.

However, I'm getting a very unrealistic value for h2. I'm wondering if I made a mistake in the way I included TB in Condition 1. Can anyone spot an mistake in my approach?

Thanks!
 
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  • #2
Let’s see some numbers.

What is the initial pressure in A?
What is the mass of air in A?
What is the mass of air in B?
 
Last edited:
  • #3
Peter_parker said:
Homework Statement: Uncertain how to take into account the temperature of the hot cylinder?
Relevant Equations: U=m*cv*dt
p*V=m*Rs*T

Task:
A thermally insulated pressurized air cylinder, B, was initially placed inside a closed-bottom, circular hollow cylinder A with an inner diameter of 50 cm. Then a tightly fitting, frictionless sliding piston with a mass of 20 kg was installed. Using the outlet valve, the height of the piston was set to h1 = 100 cm. The pressurized air cylinder B contains 10 liters of hot air with a temperature of 150 °C (TB) and an absolute pressure of 20 bar. The air in container A, that is, outside the pressurized air cylinder B, initially has the ambient temperature.

For your calculations, consider air as an ideal gas with a specific heat capacity at constant volume, cV = 0.718 kJ/(kg·K), and specific gas constant, Rs = 0.287 kJ/(kg·K).

The intrinsic volume of the bottle material should be neglected. The following environmental condition exists: ambient pressure pamb = 1 bar, ambient temperature Tamb = 20 °C.

Question of the homework:
After opening the pressurized air cylinder B, the entire system quickly reaches an equilibrium state. The process is adiabatic. At what height is the piston now located? What temperature does the system reach?

What i did:
Equation1 / Part1:
Condition 1: U1=m_a*cv*T1+m_a*cv*TB= m_both*cv*(T1+TB)
This algebra is incorrect.
Peter_parker said:
Condition 2: U2=m_both*cv*(T2)
Connection via:U12=Q+Wv with Q =0 and Wv=p*(V1-V2) to m_both*cv*(T2-(T1+TB))=p*(V1-V2)
Equation2 / Part2:
p(V2+VB)=m_both*Rs*T2

By combining Equation 1 and Equation 2, I get T2 / V2 --> then used the area of the cylinder to calculate h2.

However, I'm getting a very unrealistic value for h2. I'm wondering if I made a mistake in the way I included TB in Condition 1. Can anyone spot an mistake in my approach?

Thanks!
 
  • #4
Chestermiller said:
Let’s see some numbers.

What is the initial pressure in A?
What is the mass of air in A?
What is the mass of air in B?
Pressure in A: pamb + F/A, here Mass of the piston * g, together = F, and the area of cylinder A
Mass of A: mA=(pA * V A ) / (RS / *TA) , same for B with pB etc.
together A and B are mboth
 
  • #5
Chestermiller said:
This algebra is incorrect.
Can you go in detail please?
 
  • #6
Peter_parker said:
Pressure in A: pamb + F/A, here Mass of the piston * g, together = F, and the area of cylinder A
Mass of A: mA=(pA * V A ) / (RS / *TA) , same for B with pB etc.
together A and B are mboth
Let's see some numbers for these in Pa and kg.
 
  • #7
Peter_parker said:
Can you go in detail please?
$$m_ac_vT_1+m_bc_vT_B \neq m_{both}c_v(T_1+T_B)$$
 
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  • #8
Peter_parker said:
Homework Statement: Uncertain how to take into account the temperature of the hot cylinder?
Relevant Equations: U=m*cv*dt
p*V=m*Rs*T

Condition 1: U1=m_a*cv*T1+m_a*cv*TB= m_both*cv*(T1+TB)
Condition 2: U2=m_both*cv*(T2)
Connection via:U12=Q+Wv with Q =0 and Wv=p*(V1-V2) to m_both*cv*(T2-(T1+TB))=p*(V1-V2)
Equation2 / Part2:
p(V2+VB)=m_both*Rs*T2
Learn how to use LaTeX so that we can read your equations. What is the principle that makes Conditions 1 and 2 true? Is it conservation of energy? Because energy is not conserved in this process.

Also, I don't see where you are using the fact that the process is adiabatic.
 
  • #9
Mister T said:
Learn how to use LaTeX so that we can read your equations. What is the principle that makes Conditions 1 and 2 true? Is it conservation of energy? Because energy is not conserved in this process.

Also, I don't see where you are using the fact that the process is adiabatic.
Adiabatic: ##\Delta U=-W##
 

FAQ: Adiabatic Expansion of Pressurized Air in a Piston-Cylinder Setup

What is adiabatic expansion in a piston-cylinder setup?

Adiabatic expansion in a piston-cylinder setup refers to the process where pressurized air expands within the cylinder without exchanging heat with its surroundings. This means that all the work done by the expanding air is at the expense of its internal energy, leading to a drop in temperature.

How does the first law of thermodynamics apply to adiabatic expansion?

The first law of thermodynamics, which states that energy cannot be created or destroyed but only transformed, applies to adiabatic expansion by indicating that the work done by the expanding gas is equal to the decrease in its internal energy. Mathematically, this is expressed as ΔU = -W, where ΔU is the change in internal energy and W is the work done by the gas.

What is the significance of the adiabatic index (γ) in this process?

The adiabatic index, often denoted by γ (gamma), is the ratio of specific heats (Cp/Cv) at constant pressure and volume, respectively. It plays a crucial role in determining the relationship between pressure, volume, and temperature during adiabatic processes. For an ideal gas undergoing adiabatic expansion, the equation PV^γ = constant applies, where P is pressure, V is volume, and γ is the adiabatic index.

How can we calculate the final temperature after adiabatic expansion?

The final temperature after adiabatic expansion can be calculated using the relation T2 = T1 (V1/V2)^(γ-1), where T1 and T2 are the initial and final temperatures, V1 and V2 are the initial and final volumes, and γ is the adiabatic index. This equation is derived from the principles of adiabatic processes for ideal gases.

What are the practical applications of adiabatic expansion in engineering?

Adiabatic expansion has several practical applications in engineering, including in internal combustion engines, refrigeration cycles, and gas turbines. In these systems, the principles of adiabatic expansion are used to improve efficiency, control temperature changes, and optimize performance by managing the energy transformations within the system.

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