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I will use the below mentioned form of Reynolds Transport theorem(usually derived in Fluid Mechanics context) to give a relation between a Control mass system(no mass in or out) and Control Volume(randomly chosen time varying region of interest). It gives the relation between a chosen region of interest called control volume with continously changing control mass systems under consideration that they pass through the chosen control volume at any time $t$ (respectively). I will also use $\delta$ to represent differential elements in space as contrasting from $d$ for differntials in time.

$$\dfrac{d}{dt}\int_{sys(t)} \rho \;\eta \;\ \delta V=\dfrac{d}{dt}\int_{CV(t)} \rho \;\eta \;\ \delta V+\int_{\partial CV(t)}\rho \;\eta\; (\vec{V}_{sys}-\vec{V}_{CV}).\delta\vec{A}_{out}$$


First, I let $\eta=1$ and I get the conservation of mass.

$$\dfrac{d}{dt}(M_{sys})=\dfrac{d}{dt}(M_{CV})+\int_{\partial CV(t)}\delta(\dot{m}_{rel})$$

Using $\dfrac{d}{dt}(M_{sys})=0$, I get simply

$$\dfrac{d}{dt}(M_{CV})=(\dot{M}_{rel})_{in}-(\dot{M}_{rel})_{out}$$

And I believe this expression is correct and make intuitive sense with the notion of mass conservation.


Now I let $\eta = e = u + V^{2}/2 + gz$ , it's the energy content defined locally with respect to an inertial frame of reference(and so is everything else to follow). Where each component is obviously both a function of space and time.

$$\dfrac{d}{dt}\int_{sys(t)} \rho \;e \;\ \delta V=\dfrac{d}{dt}\int_{CV(t)} \rho \;e \;\ \delta V+\int_{\partial CV(t)}\rho \;e (\vec{V}_{sys}-\vec{V}_{CV}).\delta\vec{A}_{out}$$

Which can be written as,

$$\dfrac{d}{dt}(E_{sys})=\dfrac{d}{dt}(E_{CV})+\int_{\partial CV(t)}\rho \;e (\vec{V}_{sys}-\vec{V}_{CV}).\delta\vec{A}_{out}$$

Now using the first law of thermodynamics, I can write for the system that.

$$\dfrac{d}{dt}(E_{sys})=(\dot{Q}_{sys})_{net-in}-(\dot{W}_{sys})_{net-out}$$

Using this, I get.

$$\dfrac{d}{dt}(E_{CV})=(\dot{Q}_{sys})_{net-in}-(\dot{W}_{sys})_{net-out}-\int_{\partial CV(t)}\;e \;\delta(\dot{m}_{rel})$$

I now attempt the following reasoning work to recover the flow work term from the general work term. We can think that the system in our case is continuously being applied upon a pressure $P$ on its boundary $\partial sys(t)$ by the surrounding fluid. Therefore, a boundary work is being done on the system by the surrounding fluid by virtue of pressure. Assuming finite time and space. We can approximate with finite $\Delta$ for a small time and say for the time being some small area $\delta A$. Then the work done on the system by this pressure $P$ is. (Note that we are assuming that all deviatoric stresses are zero, not true in general).

$$\delta(\Delta W_{flow}) = P \;\delta A . \Delta x_{n}= -P \;\delta \vec{A}_{out} . \Delta \vec{x}=-P \;\delta \vec{A}_{out} . \vec{V}_{sys} \Delta t$$

Under the limit of $\Delta t \to 0$, I will get $\delta(\dot{W}_{flow})=-P \;\delta \vec{A}_{out} . \vec{V}_{sys}$

Integrating over the system boundary, I get. $(\dot{W}_{flow})_{in}=- \int_{\partial sys(t)=\partial CV(t)}P \;\delta \vec{A}_{out} . \vec{V}_{sys}=-\int_{\partial CV(t)}(Pv)\; \rho \;\delta \vec{A}_{out} . \vec{V}_{sys}$ $=-\int_{\partial CV(t)}(Pv)\; \rho \;\delta \vec{A}_{out} . (\vec{V}_{sys}-\vec{V}_{CV})-\int_{\partial CV(t)}(Pv)\; \rho \;\delta \vec{A}_{out} . (\vec{V}_{CV})$ $=-\int_{\partial CV(t)}(Pv)\; \delta(\dot{m}_{rel})-\int_{\partial CV(t)}(Pv)\; \rho \;\delta \vec{A}_{out} . (\vec{V}_{CV})$

Is there any error in my reasoning, if not, what is the meaning of the additional term given next in the energy equation after enthalpy term?

$$\dfrac{d}{dt}(E_{CV})=(\dot{Q}_{sys})_{net-in}-(\dot{W}_{nf-sys})_{net-out}-\int_{\partial CV(t)}\;\theta \;\delta(\dot{m}_{rel})-\int_{\partial CV(t)}(Pv)\; \rho \;\delta \vec{A}_{out} . (\vec{V}_{CV})$$ $$=(\dot{Q}_{sys})_{net-in}-(\dot{W}_{nf-sys})_{net-out}-\int_{\partial CV(t)}\;e \;\delta(\dot{m}_{rel})-\int_{\partial CV(t)}(Pv)\; \rho \;\delta \vec{A}_{out} . (\vec{V}_{sys})$$ $$=(\dot{Q}_{sys})_{net-in}-(\dot{W}_{nf-sys})_{net-out}-\int_{\partial CV(t)}\;e \;\delta(\dot{m}_{rel})-\int_{\partial CV(t)}(Pv)\; \delta(\dot{m}_{sys})$$

Or is it the case that the term carries no physical meaning and it's better to write it in just terms of $e$ and some flow work?


I have also done the same thing for entropy balance and I think I have derived the correct result. Can someone verify?

Now I let $\eta = s$ , it's the entropy defined locally.

$$\dfrac{d}{dt}\int_{sys(t)} \rho \;s \;\ \delta V=\dfrac{d}{dt}\int_{CV(t)} \rho \;s \;\ \delta V+\int_{\partial CV(t)}\rho \;s (\vec{V}_{sys}-\vec{V}_{CV}).\delta\vec{A}_{out}$$

Which can be written as,

$$\dfrac{d}{dt}(S_{sys})=\dfrac{d}{dt}(S_{CV})+\int_{\partial CV(t)}\rho \;s (\vec{V}_{sys}-\vec{V}_{CV}).\delta\vec{A}_{out}$$

Now using the second law of thermodynamics and Clausius inequality(augmented with $\dot{S}_{gen}$ term), I can write for the system that.

$$\dfrac{d}{dt}(S_{sys})=\int_{\partial sys(t)=\partial CV(t)}\dfrac{\delta(\dot{Q}_{sys})_{in}}{T}+ \dot{S}_{gen-sys}$$

Using this, I get.

$$\dfrac{d}{dt}(S_{CV})=\int_{\partial CV(t)}\dfrac{\delta(\dot{Q}_{sys})_{in}}{T}+ \dot{S}_{gen-sys}-\int_{\partial CV(t)}\;s \;\delta(\dot{m}_{rel})$$

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  • $\begingroup$ Forgive my ignorance, but what is RTT? Is it any of the common usages on Wikipedia? $\endgroup$ – Michael Seifert Mar 19 at 17:29
  • $\begingroup$ Oh, I am really sorry. I will add this to the question. $\endgroup$ – Mann Mar 19 at 17:31
  • $\begingroup$ Can the votes to close be please justified? How is the question unclear and how it is off-topic? $\endgroup$ – Mann Mar 20 at 14:16
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Your original transport equation is based on a closed system. In this system the velocity of the system boundary and the velocity of the material at the system are always identical.

When you transition to the Eulerian/open system perspective via the RTT, you need to worry about the fact that there are (in general) two velocities at the system boundary: the velocity at which the boundary is moving, and the velocity at which the local material is moving. One can also construct a third velocity by subtracting - this gives the velocity of the local material relative to the boundary. The key thing to remember is that the velocity (or displacement) that appears in the closed-system work equation now corresponds to the velocity of the local material, which is no longer the same as the velocity of the boundary because the definition of the system boundary has changed (closed to open). One could just leave the work term expressed in terms of the material velocity $\vec{v}_\text{material}$, but it is often more convenient to say

$$ \vec{v}_\text{material} = \underbrace{(\vec{v}_\text{material}-v_\text{boundary})}_{\displaystyle\vec{v}_\text{rel}} + v_\text{boundary}. $$

The expansion/compression work term $\vec{F}\cdot\vec{v}_\text{boundary}$ then gets split into a "flow work" term (associated with the motion of the fluid relative to the boundary at $\vec{v}_\text{rel}$) and a "boundary work" term (associated with the motion of the boundary at $\vec{v}_\text{boundary}$), which are often easy to evaluate individually in simple thermo systems. Neither "work" provides a complete picture of the energy transfer by work, but their sum represents the total work associated with expanding/contracting the material.

Note that for a closed system $\vec{v}_\text{rel} = 0$ and the total expansion work done/accepted by the material reduces to the "boundary" work. The real physical work is still associated with the expansion/contraction of the material, but in the closed case this motion happens to coincide with the motion of the system boundary. While textbooks sometimes encourage us to think of boundary work as a the "root" quantity (the thing that first principles say must appear the first law), first principles instead say that it is the work associated with expanding/contracting the material that must appear in the first law.

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  • $\begingroup$ Is my reasoning then correct? Also, I still don't make sense of the term flow work on a control volume, because that physical quantity has no meaning, I suppose. Flow work is done explicitly and only on the system and hence we should be considering only systems velocity right? I mean it's only better to write is simply as $P . v_{sys}$ $\endgroup$ – Mann Apr 3 at 8:00
  • $\begingroup$ Anyway that would still be a matter of choice. $\endgroup$ – Mann Apr 3 at 8:06
  • $\begingroup$ I believe that after "(Note that we are assuming that all deviatoric stresses are zero, not true in general)" the $\Delta x$ that you introduce should be the displacement of the material and you have conflated it with the displacement of the boundary. I am not sure what you mean that flow work is done "explicitly." It is not a term that comes directly from thinking "what kinds of work are being done here?" - it is a mathematical trick that comes from chopping up the real, physical work into two easier-to-handle components. $\endgroup$ – user1476176 Apr 3 at 8:08
  • $\begingroup$ Oh sorry, what I mean is: my "sys" correspond to "material" and CV correspond to a "boundary". So, when I am saying my system's velocity I only worry about velocity of the material $\endgroup$ – Mann Apr 3 at 8:17
  • $\begingroup$ Whereas the CV means any arbitrary region through which some material is going in or out. But I agree with your point in general System's boundary and material boundary and arbitrary control volume boundary velocities are different $\endgroup$ – Mann Apr 3 at 8:18
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In the open system (control volume) version of the first law of thermodynamics, the rate of doing work is split into two separate parts: 1. Work associated with pushing fluid into and out of the control volume by fluid behind at the inlet and fluid ahead at the outlet acting like pistons and 2. All other work, called “shaft work” because it oftn involves a rotating shaft like a turbine. The work in category 1. Is lumped together with the internal energy of the inlet and outlet streams mathematically to obtain the enthalpy of these streams.

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  • $\begingroup$ Hi, I have updated the answer. Do you think my reasoning is correct? $\endgroup$ – Mann Mar 19 at 19:11
  • $\begingroup$ I don't see exactly what you did, but whatever it was, it looks OK. $\endgroup$ – Chet Miller Mar 19 at 20:38
  • $\begingroup$ I am having one additional term after the Enthalpy inlet-outlet term in the final equation for a general Control Volume(maybe moving or deforming). I am unable to understand the meaning of last term, or whether it's just an error. $\endgroup$ – Mann Mar 19 at 20:55
  • $\begingroup$ Sorry, I don't follow. Maybe it is because you are allowing for a moving/deforming control volume. For a fixed control volume, there shouldn't be an additional term. $\endgroup$ – Chet Miller Mar 19 at 21:17
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The flow work you are talking about is "reversible flow work", $\dot{V}\frac{\Delta P}{L}$, and it is not a standalone energy that would arise in the RTT. We can, however, do some analysis to see what an energy equation with flow work would look like.

Consider flow through a pipe of perimeter $P$ in the axial direction $x$, with bulk (averaged over the cross section) enthalpy $h(x)$, bulk kinetic energy $\frac{\dot{m}}{2}\bar{u^2}(x)$, input heat flux $q$, and input shaft work $\dot{W}$.

enter image description here

An infinitesimal energy balance (i.e., RTT for energy) on a differential control volume of length $dx$ in a pipe yields the bulk flow energy equation: $$ \dot{m}\frac{dh}{dx} + \frac{\dot{m}}{2}\frac{d\bar{u^2}}{dx} = qP + \dot{W}$$

Notice that flow work is not a quantity that arose in this energy balance. We can do some thermodynamic analysis, however, to see where the flow work is and how it goes into this energy balance.

From thermodynamics,

$$ dh = Tds + vdP$$

Solving for entropy and applying the bulk flow model,

$$ \dot{m}T\frac{ds}{dx} = \dot{m}\frac{dh}{dx}-\dot{V}\frac{dP}{dx}$$

Substituting our expression for bulk enthalpy,

$$ \dot{m}\frac{dh}{dx} = qP + \dot{W} - \frac{\dot{m}}{2}\frac{d\bar{u^2}}{dx} -\dot{V}\frac{dP}{dx}$$

$$ \dot{m}\frac{dh}{dx} = qP + q_d$$

where $q_d = \dot{W} - \frac{\dot{m}}{2}\frac{d\bar{u^2}}{dx} -\dot{V}\frac{dP}{dx}$ is the heat dissipation per unit pipe length. This is the mechanical energy dissipation that increases the entropy, whereas $qP$ is the thermal energy input that raises the entropy. We can use this definition of heat dissipation with the first equation to obtain

$$ \dot{m}\frac{dh}{dx} = qP + q_d + \dot{V}\frac{dP}{dx}$$

So you can see that we can easily modify an energy balance to include flow work, but it's not a standalone quantity that would appear in an energy balance without manipulating the final result.

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  • $\begingroup$ How can you say for a deformable and moving control volume that while doing a simple RTT balance instead of writing $e$ in the boundary term we write $h$? $\endgroup$ – Mann Mar 20 at 17:14

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