Applying Bernoulli's equation to problems involving a perfect gas

In summary, applying Bernoulli's equation to problems involving a perfect gas allows for the analysis of fluid flow and pressure changes in gas systems. This approach assumes the gas behaves ideally, enabling the derivation of relationships between velocity, pressure, and elevation. Key considerations include the gas's density, temperature, and the impact of compressibility at varying flow speeds. By integrating these factors, engineers and physicists can predict gas behavior in various applications, such as aerodynamics and HVAC systems.
  • #1
Hak
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I would like to know the opinions of experienced forum users regarding an issue that seems to happen often in problems: namely, applying Bernoulli's equation to perfect gas. Is it permissible to do so, even if only to find reasonable estimates? Two examples I found out might be:

- The problem of finding the internal pressure trend in a leaky space station.

-Estimating the pressure change due to a train passing through a tunnel.
(I just don't see how this can be done without Bernoulli!).

Then I also read in an old article (which I can't find now) that it is possible to prove that the viscosity of a perfect gas is well defined and depends on the root of the temperature, but I haven't found much on the web. Does anyone have any idea how to do this? Could you give me some help in this regard?
 
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  • #2
Hak said:
Then I also read in an old article (which I can't find now) that it is possible to prove that the viscosity of a perfect gas is well defined and depends on the root of the temperature, but I haven't found much on the web. Does anyone have any idea how to do this? Could you give me some help in this regard?
”ideal gas” is used more commonly than “perfect gas” in English. I found several links searching on this.
 
  • #3
See Chapter 1 of Transport Phenomena by Bird, Stewart, and Lightfoot for answer to your question about viscosity of a gas in the ideal gas limit. This same book also has a section on Bernoulli equation customized to ideal gases.
 
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  • #4
Chestermiller said:
See Chapter 1 of Transport Phenomena by Bird, Stewart, and Lightfoot for answer to your question about viscosity of a gas in the ideal gas limit. This same book also has a section on Bernoulli equation customized to ideal gases.
Thanks.
 
  • #5
Hak said:
I would like to know the opinions of experienced forum users regarding an issue that seems to happen often in problems: namely, applying Bernoulli's equation to perfect gas. Is it permissible to do so, even if only to find reasonable estimates? Two examples I found out might be:

- The problem of finding the internal pressure trend in a leaky space station.

-Estimating the pressure change due to a train passing through a tunnel.
(I just don't see how this can be done without Bernoulli!).

Then I also read in an old article (which I can't find now) that it is possible to prove that the viscosity of a perfect gas is well defined and depends on the root of the temperature, but I haven't found much on the web. Does anyone have any idea how to do this? Could you give me some help in this regard?
I believe Bernoulli's is ok for low speed gas flows (Mach 0.3 or lower), so long as the flow is steady. High speed flows, unsteady flows, and or both high speed/unsteady would need adjustment.
 
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  • #6
erobz said:
I believe Bernoulli's is ok for low speed gas flows (Mach 0.3 or lower), so long as the flow is steady. High speed flows, unsteady flows, and or both high speed/unsteady would need adjustment.
Thank you.
 
  • #7
Chestermiller said:
This same book also has a section on Bernoulli equation customized to ideal gases.
Is this section on Chapter 3?
 
  • #8
Hak said:
Is this section on Chapter 3?
Yes, Eqn. 3.5-12
 
  • #9
You might also search for "Sutherland's law" for finding the viscosity of gases like air.
 

FAQ: Applying Bernoulli's equation to problems involving a perfect gas

What is Bernoulli's equation and how is it applied to a perfect gas?

Bernoulli's equation is a principle of fluid dynamics that describes the behavior of a moving fluid. For a perfect gas, it can be expressed as \( P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \), where \( P \) is the pressure, \( \rho \) is the density, \( v \) is the velocity, and \( gh \) represents the gravitational potential energy per unit volume. When applying this to a perfect gas, we often use the ideal gas law \( PV = nRT \) to relate pressure, volume, and temperature.

How do you account for changes in temperature when using Bernoulli's equation for a perfect gas?

Changes in temperature can affect the density of a perfect gas. Using the ideal gas law \( \rho = \frac{P}{RT} \), where \( R \) is the specific gas constant, you can determine the density at different temperatures. This adjusted density is then used in Bernoulli's equation to account for temperature variations.

Can Bernoulli's equation be used for compressible flow of a perfect gas?

Bernoulli's equation is generally derived under the assumption of incompressible flow. For compressible flow, such as that of a perfect gas at high velocities, modifications are needed. The energy equation for compressible flow, which includes changes in internal energy and enthalpy, should be used instead. For low-speed flows where compressibility effects are negligible, Bernoulli's equation can still be applied with reasonable accuracy.

How does the conservation of mass relate to Bernoulli's equation in the context of a perfect gas?

The conservation of mass principle, or continuity equation, states that the mass flow rate must remain constant in a steady flow. For a perfect gas, this can be expressed as \( \rho_1 A_1 v_1 = \rho_2 A_2 v_2 \), where \( A \) is the cross-sectional area and \( v \) is the velocity at different points. This relationship helps in applying Bernoulli's equation by ensuring that changes in velocity and cross-sectional area are consistent with the conservation of mass.

What are common applications of Bernoulli's equation for a perfect gas in engineering?

Bernoulli's equation for a perfect gas is widely used in various engineering applications, such as in the design of airfoils and wings in aerodynamics, where it helps predict lift. It is also used in the analysis of gas pipelines, HVAC systems, and in the study of gas dynamics in nozzles and diffusers. These applications often involve assumptions

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