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Conservation of Mass, Momentum, and Energy

In this section I am going to discuss the conservation of mass, momentum, and energy.  I will show how to derive these equations and how to use them for different for different control volumes.

Conservation of Mass

The conservation of mass states that mass cannot be created or destroyed.  As a result for a system the conservation of mass principle states that a system is a collection of unchanging contents.  Finally, through use of the Reynolds transport theorem to coincide the system to a control volume, the continuity equation can be derived.

(Eq 1) $ρ_1A_1v_1 = ρ_2A_2v_2$

$ρ$ = fluid density

$A$ = cross-sectional area

$v$ = fluid velocity

In the next article, I go through great detail on how the continuity equation is derived.

Conservation of Momentum

In addition to the conservation of mass, I also discuss the conservation of momentum.  Recall that momentum is a mass times a velocity.  To further this, Newton’s second law of motion will need to be applied.  In turn, this will result in the following linear momentum equation for a fixed, non-deforming control volume.

(Eq 2) $\frac{∂}{∂t}\int{_{CV}}~vρV +\int{_{CS}}~vρv·\hat{n}dA = \sum{F_{CV}}$

$V$ =  volume

$\sum{F_{CV}}$ = sum of the forces on a control volume

To further understand how this equation was derived refer to this article.

Conservation of Energy

Finally, the last thing that I am going to talk about is the conservation of energy.  To do this both the 1st and the 2nd laws of Thermodynamics will need to be applied. By using the 1st and 2nd laws of thermodynamics in addition to the Reynolds transport theorem, the energy equation can be derived.  Below is the energy equation for an incompressible, one-dimensional, steady state flow.

(Eq3) $\frac{∂}{∂t}~\int{_{CV}}~eρdV + \int{_{CS}}~\left(\check{u} + \frac{p}{ρ}+\frac{v^2}{2}+gz\right)ρv·\hat{n}dA = \dot{Q}_{net~in} + \dot{W}_{shaft~net~in}$

$e$ = total stored energy

$\check{u}$ = internal energy per unit mass

$P$ = pressure

$g$ = gravitational constant

$z$ = fluid height

$\dot{Q}$ = heat transfer be transferred across the control surface

$\dot{W}$ = work be transferred across the control surface

The following article will further explain on the energy equation was determined.  In addition, the energy equation can be used to derive Bernoulli equation.  To see how this is done go to the following article.

 

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