Mass Continuity Equation Problem

In summary, the question asks to show that the equation dA/A + dv/v + dρ/ρ = 0 applies to a one-dimensional steady flow. The given equation, ρ*v*A = constant, is the differential equation for this type of flow. The solution involves differentiating ρ*v*A and dividing it by ρ*v*A.
  • #1
tophat22
4
0

Homework Statement



Question Details:

The question reads:

Show that the equation:

dA/A + dv/v + dρ/ρ = 0

applies to a one-dimensional steady flow. (Here 'one dimensional' means that both the density ρ and seed v = - v . n (vectors) are constant across any cross-sectional area A cutting the flow.)

Please help!

Homework Equations




d/dt ∫V ρ dV + ∫A -ρv⋅n dA = 0

The Attempt at a Solution



I know that the equation they gave us above is the differential equation for ρ*v*A = constant, I just don't know where to go from there.
 
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  • #2
Hi tophat22! :smile:

In steady flow, mass in = mass out of any surface, so ρ*v*A = constant.

So differentiate ρ*v*A, and divide by ρ*v*A. :wink:
 
  • #3
I actually ended up getting it before that... thank you though!
 

Related to Mass Continuity Equation Problem

1. What is the Mass Continuity Equation Problem?

The Mass Continuity Equation Problem is a fundamental equation in fluid mechanics that describes the conservation of mass in a fluid system. It states that the rate of change of mass in a control volume is equal to the net mass flow rate into the control volume.

2. Why is the Mass Continuity Equation important?

The Mass Continuity Equation is important because it helps us understand and analyze the behavior of fluids in various systems. It is used to study fluid flow in pipes, pumps, turbines, and other devices, and is essential for designing and optimizing these systems.

3. How do you solve a Mass Continuity Equation Problem?

To solve a Mass Continuity Equation Problem, you first need to define a control volume and identify the inflow and outflow rates of mass. Then, you can use the equation to calculate the rate of change of mass within the control volume. Finally, you can use this information to analyze the behavior of the fluid within the system.

4. What are the assumptions made in the Mass Continuity Equation?

The Mass Continuity Equation assumes that the fluid is incompressible, meaning that its density does not change with pressure. It also assumes that the flow is steady, meaning that the velocity and other properties of the fluid do not change over time. Additionally, the equation assumes that the flow is one-dimensional and that there are no sources or sinks of mass within the control volume.

5. Can the Mass Continuity Equation be applied to all fluid systems?

The Mass Continuity Equation can be applied to most fluid systems, but it may not be applicable in certain cases. For example, it cannot be used for compressible fluids or for systems with unsteady flow. In these cases, other equations, such as the compressible continuity equation or the unsteady continuity equation, must be used.

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