Adiabatic steady state flow in a nozzle

In summary, the two equations yield different results based on the assumptions made about the system. The first equation is based on the full energy balance, while the second equation works with only Ideal gas assumptions.
  • #1
jason.bourne
82
0
air enters an adaibatic nozzle steadily at 300 kPa, 200 degree celcius and 30m/s and leaves at 100 kPa and 180 m/s.
find the exit temperature?

when m using energy balance equation: h1 + [ (v1)^2 / 2 ] = h2 + [(v2)^2 / 2 ]
i get the answer 184 degree celcius.

but when i use the other relation i.e, pressure temperature relation

[ T1 / T2 ] = [ P1 / p2 ]^ ((k-1)/k) (i used k = 1.4)

i get very different values of temperature. is this equation not valid?
why m i getting different answers?

is this pressure temperature relation only valid for reversible processes?
 
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  • #2
The first equation (with h1 and h2 in it) works for inviscid adiabatic flows (with no work done on the fluid by propellers, etc). It does not require flow to be reversible. As a result, it will work even across the shock wave.
The second equation works only for isentropic flows - i.e. reversible, adiabaric flows.

Thus, these two equations will yield the same answers (assuming p1, p2, h1, h2 , T1 are somehow known or given) only for isentropic flows.
 
  • #3
jason.bourne said:
air enters an adaibatic nozzle steadily at 300 kPa, 200 degree celcius and 30m/s and leaves at 100 kPa and 180 m/s.
find the exit temperature?

when m using energy balance equation: h1 + [ (v1)^2 / 2 ] = h2 + [(v2)^2 / 2 ]
i get the answer 184 degree celcius.

but when i use the other relation i.e, pressure temperature relation

[ T1 / T2 ] = [ P1 / p2 ]^ ((k-1)/k) (i used k = 1.4)

i get very different values of temperature. is this equation not valid?
why m i getting different answers?

is this pressure temperature relation only valid for reversible processes?

The first equation is the based on the full energy balance (with some assumptions that drop out some terms). It always works.

The second is for an Ideal gas, isentropic, and constant specific heat assumption.

Based on your problem description, the system does not meet the criteria for the second relation. If your system was given as isentropic, ideal gas, with constant specific heats, then you could use the second relation you listed.

BTW you have the T1/T2 and P1/P2 inverted (T2 and P2 are the numerators).

CS
 
  • #4
thank you so much
 

1. What is adiabatic steady state flow in a nozzle?

Adiabatic steady state flow in a nozzle is a phenomenon that occurs when a fluid (such as a gas or liquid) flows through a narrowing channel or nozzle. The term "adiabatic" means that there is no heat transfer between the fluid and its surroundings, and "steady state" means that the flow is constant and unchanging over time.

2. How does adiabatic steady state flow occur in a nozzle?

Adiabatic steady state flow in a nozzle is a result of the conservation of energy and mass. As the fluid flows through the nozzle, its velocity increases due to the narrowing of the channel. This increase in velocity causes a decrease in pressure, following the principle of Bernoulli's equation. The decrease in pressure results in a decrease in the fluid's internal energy, causing it to cool down. This decrease in temperature maintains the adiabatic condition of no heat transfer.

3. What are the applications of adiabatic steady state flow in a nozzle?

Adiabatic steady state flow in a nozzle has many practical applications, such as in jet engines, rocket propulsion, and spray nozzles. It is also used in the design of heat exchangers and refrigeration systems. In these applications, the goal is to achieve a high velocity and low pressure of the fluid in the nozzle to maximize the efficiency of the system.

4. What factors affect adiabatic steady state flow in a nozzle?

Several factors can affect adiabatic steady state flow in a nozzle, including the size and shape of the nozzle, the properties of the fluid (such as density and viscosity), and the initial conditions of the fluid (such as temperature and pressure). Additionally, the presence of any obstructions or changes in the flow direction can also impact the flow in the nozzle.

5. How is adiabatic steady state flow in a nozzle calculated or modeled?

The flow in a nozzle can be mathematically modeled using the laws of thermodynamics and fluid mechanics. This involves solving a set of differential equations, such as the continuity equation and the energy equation, to determine the fluid properties at different points along the nozzle. Computational fluid dynamics (CFD) is often used to simulate and analyze adiabatic steady state flow in a nozzle to understand its behavior and optimize the design for various applications.

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