 In this segment what we're going to be doing, we're going to be taking a look at the Reynolds number that is defined for condensation on vertical or inclined surfaces and we'll be using that as we develop some of the correlations in the next lecture. Now the thing about condensation when we're looking at heat transfer with condensation there are many many different Reynolds numbers that we're going to be looking at depending upon how you define it but this is one definition of the Reynolds number. So V is the average velocity of the condensate flowing down the surface, rho is the density of the liquid, mu is the dynamic viscosity of the liquid, dH, I'm going to write it up here, this is the hydraulic diameter and it is defined as being 4 times the cross-sectional area AC divided by the perimeter and that would be the cross-sectional area perpendicular to the direction of which the film is moving so the velocity and that's going to depend upon the nature of the shape. There are two areas that we're going to use in condensation so be very very careful not to confuse them. We have wetted area which is the one that we typically use whenever we're using Newton's Law of Cooling and that will be A and then we have this AC and that is essentially the cross-sectional area perpendicular to the wall or perpendicular to the velocity. So if we look at a surface where we have a film growing like that, this here if we look at that that is typically delta and let's say this plate is of width B so I'll say the width is B in this direction. Okay so there's our film coming down the film is moving in that direction. AC is essentially this cross-sectional area here AC which is going to be it will be the perimeter times delta where perimeter it would be for a unit length that would be B. So just to make a comment about that AC is equal to the perimeter times delta and delta is the thickness of the film and the perimeter for example if we're looking at a vertical plate it would be equal to B as we just put in this drawing. If we're looking at a vertical cylinder however it's going to be different. Now with the vertical cylinder the radius of the cylinder so that would be there. R needs to be much larger than the thickness of the film of delta but what will happen is you get condensation forming around and here the area is going to be related to the perimeter or the cross-sectional area so the perimeter as we've defined is equal to pi times D and that would be for a vertical cylinder. Okay so just be aware that this cross-sectional area AC and that is different from A which is wedded area that we will see in a moment as we look at Newton's law of cooling when we start coming up with the convective heat transfer coefficient and so you can imagine here this cross-sectional area for a cylinder will be a little bit more complex than it would be just for the plate but anyways there are two different areas we're going to use don't get them confused hopefully I haven't confused you with all of those stuff here. Alright so what we're going to do we're now going to take a look at a generic plate a vertical plate and and this is one that is often used when looking at condensation flowing on a surface and here is our plate and depending upon the book that you're using sometimes TS will be the wall temperature sometimes TW will be the wall temperature in any event we're assuming that the gravity vector is going in that direction and our coordinate system we will draw y normal to the plate and x in the direction of the plate and what happens is we begin with a film that might grow something like that and let me clean that up a little okay so it's coming down and then it goes into a wavy regime and then eventually it becomes turbulent and and so depending upon the Reynolds number or actually in different regimes here and so we will be noting those in a moment before I do that however I want to put in a few other variables that are within this on the outside of the film we will call that the saturation temperature and then further out here we have the vapor and and so you could have a scenario where it's superheated vapor and and it's above the saturation temperature or you could have it at the saturation temperature usually we assume it to be at the saturation temperature but in the event that it's higher I will denote it with TV freestream and so here what is happening is we have mass flow rate of condensate moving down and as the film gets thicker and thicker the mass flow rate is going to increase because we keep adding more and more fluid as we have more fluid condensing and so m dot is going to be a function of x and the other thing that we have here we typically define delta as being the thickness of our layer and consequently we know that delta is a function of x and the other thing that we did and I define this in the Reynolds number that I just presented a moment ago with the Reynolds number I said that there is some average or bulk velocity in reality there's going to be a velocity profile here but for right now what we're going to do is assume that we have some bulk velocity of the condensate moving down due to the body force or gravity so if we're to look at this what we have up at the top this would be where we have Reynolds number approximately equal to zero and in this region here we refer to it as being laminar and it's also called wave three then when you get to a Reynolds number of approximately 30 what happens is you get a laminar but it is wavy and so those waves would be instability waves developing within the fluid the film itself and instability waves grow and it's the process of transition from laminar to turbulent but it is characterized by waves that we would see on the surface and that will exist until we get to a Reynolds number of about 1800 and once we get to a Reynolds number of 1800 then our film becomes turbulent and just like with when we look at the flat plate boundary layer or a lot of the other things we've been doing in this course whenever you transition from one regime to another your correlations are going to change and so we will see that when we look at the experimental correlations that have been collected for the condensation on a vertical surface or plate like we're looking at here so I think that's the majority of the things that I wanted to write it is okay so now what we do I want to define a Reynolds number and we looked at Reynolds number a moment ago but I'm gonna basically massage that Reynolds number we're gonna come up with different Reynolds numbers which you'll find is quite common when you're looking at condensation and we're going to begin by looking at a definition for the mass flow rate and so we can see the mass flow rate is going to be equal to the density of the liquid in the film and it's going to be equal to it will be rho av and so the a here is going to be remember the cross-sectional area that is going to be for example if you have a vertical plate that's going to be delta times b where b would be the width of the plate and that will also be multiplied by capital v which is the bulk velocity of the fluid going through so we're going to take that mass flow rate and we're going to use it in a definition of the Reynolds number because if you look back when we had Reynolds number here we have some of those components there we have rho and we have v and the cross-sectional area is embedded within the hydraulic diameter which we can pull in so it's going to enable us to be able to recast the Reynolds number in terms of the mass flow rate so let's take a look at that now and when i have rho here and mu i should denote that that is for the liquid phase so what i've done here i've recast the Reynolds number with the mass flow rate and i've also rewritten it with this capital gamma and that is the mass flow rate in the film which is function of x divided by the perimeter and so the perimeter in the case of a vertical flat plate would just be b in the case of a cylinder vertical cylinder would be pi times diameter which we saw earlier so that is one form of the Reynolds number now there's another form of the Reynolds number that we can come up with and for that one what we do is we take this and we relate it to the convective heat transfer coefficient and i'll show you how to do that in a moment here so what we can do we can come up with yet another Reynolds number and for this one what i'm going to do i'm going to use Newton's law of cooling so we have some average convective heat transfer over our surface multiplied by a this is a different a from the ac this is the wetted area so if you have a vertical plate like this a would be that area there so the that and if you're looking at a cylinder it would be the external area around the entire perimeter it's not the cross-sectional area of the film it is the wetted area but that will be multiplied by the saturation temperature minus the surface or the wall temperature and that is equal to m dot so the mass flow rate times the heat of vaporization so going from a vapor to a liquid or a liquid to a vapor and you get that out of steam tables and one thing that i will say be careful because the tables often are in kilojoules per kilogram you need to pull that out of that table in joules per kilogram or you're going to make mistakes so don't make that mistake when you're pulling it out of the table make sure that you get it in joules per kilogram an h bar here is average convection convective heat transfer and then finally a is going to be the length the vertical length and so usually what we'll do is we will denote our vertical length like this so it will be l times the perimeter and remember i said the perimeter could be b in the case of a plate or it'd be pi times d in the case of a cylinder where d is our diameter okay so we have that now what are we going to do with it well i want to recast and write a new form of the Reynolds numbers so what we're going to do we're going to take the m dot from here and basically replace it with the m dot that is up here in that Reynolds numbers so let's go ahead and do that so this here what i'm writing is one of the important equations that you're going to be using is going to be heat transferred divided by the heat of vaporization and we said heat transfer through Newton's law we know that it's this divided by hfg and when we plug in the mass flow rate we get a new Reynolds number so that becomes another form of the Reynolds number that we can use so what we've looked at here three different Reynolds numbers that you can use if you're dealing with convective condensation heat transfer i should say where was the other one the first one was right here okay so those are three different Reynolds numbers and then we look at correlations we're going to look at yet another one there are many many different Reynolds numbers that you can have when you're looking at condensation and films forming on vertical surfaces what we'll be doing in the next segment is we're going to go back to this representation that we have here and this was a starting point that nusselt used and he started with an object or a model that looked like this and he came up with equations and he went through dimensional analysis and came up with a way to be able to get the convective heat transfer coefficient so that's what we're going to be doing in the next two segments which will give us a form of the convective heat transfer coefficient it won't be perfect because it's based on a theoretical model and then we'll move in in the next lecture to the correlations that are actually used and so that's where we're going to be going