Conservation of Mass for a Control Volume

And this module 5, we are starting with control volume analysis, okay? And the first thing that I'm gonna do in this is I'm gonna look at the conservation of mass, okay? And after that, I'm gonna do conservation of momentum, module six, and conservation of energy, module seven. This whole module is based on the Reynolds transport theorem, okay? And I'm gonna read it. I'm gonna rewrite this from the previous segment from module number four, but this is what it is. This is the DB Dt will be equal to del Del t of triple integral over the control volume Rho b d volume, plus over control surface Rho b V dotted with n dA. So this was my Reynolds transport theorem from the previous segment. So now what I'm gonna do is, I'm gonna actually go ahead and read the title of the module, mass. Some people refer to the conservation of mass as continuity equation. That's quite all right. I would like you to be aware of that. But at the end, I'm gonna do this, capital B is equal to mass, okay? Let’s go ahead and write this. Capital B is equal to mass. What do you think I'm gonna do for the conservation of momentum? I'm gonna say capital B is momentum. What do you think I'm gonna say for conservation of energy? I'm gonna say capital B is energy. If you remember, this capital B was the extensive property, right? And there's a corresponding intensive property that is defined by B divided by mass. In this particular case, by chance, it seems that hey, this is mass over mass. So what do you think this is? Well, one, right? Mass over mass is one, or unity, okay? So then I'm going to insert here one. I'm gonna insert here one. I’m gonna insert here one. So basically, I'm gonna drop those two out. And I'm gonna go ahead and insert in here mass. So then you're gonna get Dm Dt will be equal to del Del t of triple integral control volume Rho times d volume, plus over control surface Rho V dotted with n dA. Now I have to do rethink one last step before I finalize this equation and box it up and call it a day is, remember that this was for Lagrangian, and this is a material derivative, right, capital D. And also, if you think about it, this will be for the system or control mass, right? And right-hand side was for the Eulerian reference frame. And this was for control volume. As you can clearly see, control volume is here, and this is the boundary of the control volume. So now let's reassess left-hand side over here, this is a control mass. If you look at the previous segment from module 4, when I derived Reynolds transport theorem, what I did was, I was moving with the flow. If you remember, I was not adding any mass, or was not leaving any mass from my system. It was a constant number. So in that particular case, for a control mass, the definition was, no mass leaves or enters. Then what this term is gonna be is, it's not going to change over time. My mass is constant, okay? So this term will be zero. Okay, so let's write it over here. Dm Dt zero as m is constant for the system or control mass. I mean, this is zero on the left-hand side. So I'm gonna rewrite this 0 on the right-hand side because, I want think about it, you know summation, you don't say 0 is equal to summation of the forces, right? You say summation of the forces is equal to 0. From the same convenience, I'm gonna write it this way. Del Del t and we have this control volume, Rho d volume, plus over the control surface, Rho V dotted with n dA will be equal to zero. Now I can go out and celebrate this. We have special names. So this very first term that I'm seeing over here is basically, if you look think about it, what's actually indicated here? What is Rho times V? That's mass. This is how the mass changes over time for my control volume. Well, it's fairly self-explanatory, right? So it's time rate of change. Time rate of change of mass for the CV. Okay, and this second term, if you remember, I had it in the B dot. I called that flux, right? So this is the mass flux entering or leaving the control volume. You see, this is a volume, and I have a line across the V symbol, and this is a velocity. Velocity is far more common than volume. You will find out, okay? So I would like you to be careful about differentiating these two. And also, if I continue this approach, what I can say is, do you remember that what was, let's actually write this because I don't think we all remember. So V dot n from the previous segment was, plus V for exit, minus V for inlet, and zero otherwise. Okay, we may want to revisit the previous segment to double-check you know this clearly. So basically, this is Rho times, let's say this is an exit, Rho V A. Rho V A or rho minus V A or Rho zero. Well, that's gone, right? So basically, it is either plus Rho V A or minus Rho V A. Oh yeah, I got the point that hey, there's a double integral in here. But hey, that's just a mathematical operator at the end of the day. It's not gonna change my units. It's not gonna change the way that things behave, okay? So the integral actually, if you remember, is simply a summation sign. So I just take small strips and I add them up under the curve that I have. So this, the last term that I have over there, Rho V dotted with n dA, can be written as m dot exit minus m dot inlet. And m dot, this is a very common terminology that I use. m dot is called the mass flow rate, okay? So I use this a lot. So, this is, you know, one thing, when students see this is they start to celebrate, because, you know, okay, good, I got rid of the integral. I just see what the m dot and subtract from one another. And to calculate the m dot, you need to do the integral. So basically, my m dot exit will be double integral over the exits, Rho V dA. Okay, so this is the net mass flux exiting. I will have a similar for the inlet, obviously. And let's not forget the negative sign, so that's going to be over the inlet. Then I will get Rho V dA.