Solve Thermodynamics Problem: Determining Exit Temp & Velocity

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In summary, the conversation discusses the determination of the exit temperature and velocity of a gas flow through a nozzle supplied with a steady gas stream at a temperature of 800K and pressure of 300kN/m^2. The gas expands adiabatically through the nozzle to a pressure of 158kN/m^2 following the law Pv^1.4 = constant. The exit temperature can be found using the equation T2/T1 = (P2/P1)^[(n-1)/n], but the method for finding the exit velocity is still being discussed. The suggestion of using the steady flow energy equation is mentioned but has not yet been successful.
  • #1
bill nye scienceguy!
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A nozzle is supplied with steady gas stream at temperature = 800K and pressure = 300kN/m^2. If the gas expans adiabatically through the nozzle to a pressure of 158kN/m^2 following the law Pv^1.4 = constant, determine the exit temperature and velocity of the gas flow.


I've already found the exit temperature using:

T2/T1 = (P2/P1)^[(n-1)/n]

so how would i go about finding the exit velocity? I tried using the steady flow energy equation but to no avail.
 
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  • #2
bill nye scienceguy! said:
I tried using the steady flow energy equation but to no avail.

Why not? I think that the steady flow energy equation will give you the answer.
 
  • #3


To find the exit velocity, we can use the continuity equation, which states that the mass flow rate at the inlet is equal to the mass flow rate at the exit. This can be expressed as:

m_dot = ρ1 * A1 * V1 = ρ2 * A2 * V2

Where m_dot is the mass flow rate, ρ is the density, A is the cross-sectional area, and V is the velocity. We can rearrange this equation to solve for V2, the exit velocity:

V2 = (ρ1 * A1 * V1) / (ρ2 * A2)

To solve for ρ1 and ρ2, we can use the ideal gas law:

P1 * V1 = m1 * R * T1

P2 * V2 = m2 * R * T2

Where m is the mass of the gas, R is the gas constant, and T is the temperature. We can rearrange these equations to solve for ρ1 and ρ2:

ρ1 = (P1 * V1) / (m1 * R * T1)

ρ2 = (P2 * V2) / (m2 * R * T2)

Substituting these values into the equation for V2, we get:

V2 = [(P1 * V1) / (m1 * R * T1)] * A1 * V1 / [(P2 * V2) / (m2 * R * T2)] * A2

Simplifying and rearranging, we get:

V2 = (P1 * V1 * A1 * V1 * m2 * R * T2) / (m1 * R * T1 * P2 * A2)

Finally, we can substitute the given values into this equation to solve for V2:

V2 = (300kN/m^2 * A1 * 800K * V1 * m2 * R * T2) / (m1 * R * 800K * 158kN/m^2 * A2)

Since we know that the gas is expanding adiabatically, we can assume that there is no heat transfer and therefore the mass flow rate remains constant. This means that m1 = m2, and we can simplify the equation further:

V2 = (300kN/m^2 * A1 * 800K
 

1. What is thermodynamics?

Thermodynamics is the branch of physics that deals with the relationships between heat, work, and energy. It also studies the effects of these relationships on the physical properties of matter.

2. How do you determine the exit temperature and velocity in a thermodynamics problem?

To determine the exit temperature and velocity in a thermodynamics problem, you need to apply the laws of thermodynamics, specifically the conservation of energy and mass. You also need to consider factors such as heat transfer, work done, and changes in internal energy.

3. What are the key equations used in solving thermodynamics problems?

The key equations used in solving thermodynamics problems include the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred or converted into different forms; the second law of thermodynamics, which states that the total entropy of a closed system will never decrease over time; and the ideal gas law, which relates the pressure, volume, temperature, and number of moles of an ideal gas.

4. How do you handle complex thermodynamics problems?

To handle complex thermodynamics problems, it is important to break them down into smaller, more manageable parts. This can be done by identifying the key variables and equations involved and then applying them step by step. It is also helpful to draw diagrams and use mathematical tools such as calculus to simplify the problem.

5. What are some common mistakes to avoid when solving thermodynamics problems?

Some common mistakes to avoid when solving thermodynamics problems include not properly accounting for all forms of energy, not considering all relevant variables, and using incorrect units of measurement. It is also important to carefully check your calculations and ensure that they make physical sense in the context of the problem.

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