Week 9: Lecture 42: Natural convection 1

Hello everyone. Uh welcome to this module. The module is on natural convection. Until now we were discussing force convection as part of the topic on convective heat transfer and this one and the next module will cover basics of natural convection that you would need to know as part of undergraduate heat transfer. So we have seen in the introductory slides of the course what natural convection is. Natural convection essentially is what you see in nature at all points of time. the uh how you know uh water evaporates and gets converted to the whole cycle of cloud formation and rain is natural convection. The surface of the ocean becomes hot the water evaporates and then goes and this entire cycle is established. If I want to cool a cup of coffee or a hot potato, we just leave it. The easiest way to do it is to leave it in a bowl or something and allow it to cool naturally. The word naturally means it's just that you're not doing anything to cause this cooling to happen. It is just happening because of the difference in temperature between the object and the surrounding. And this figure essentially shows the mechanism. You have a hot surface. The air surrounding it is going to become hot by virtue of the body giving energy to the air. This hot air becomes light. So it rises up. In the process, it brings in the cooler air from there to come to the surface. And this is how the convective currents are established. So motion that results from continuous replacement of heated air in the vicinity of the hot object in this case the egg by cooler air which is surrounding it is basically the natural convection current. So heat transfer that is enhance enhance as a result of this natural convection current is the natural convection heat transfer. Now this is going to happen independent of the fact whether the body is at 300 kel and the surrounding is at 295 kel or the body is at 500 kel and the surrounding at is at 295 kel whatever be the temperature difference natural convection is going to happen for sure it could you could have higher or the stronger natural convection happening when the delta t is large But when delta t is small also natural convection is going to happen. The magnitude of heat transfer that is going to be there for natural convection would reduce significantly when my delta t's are small. That's the only change. But you cannot say natural convection doesn't exist. Natural convection may become negligibly small numerically valuewise but it is always going to exist. And we also saw in the introductory slides that whenever there is radi natural convection envi radiation is going to be there natural convection will always be there in that. So we had uh made a comment in the earlier part also uh natural convection and radiation may have equal weight in a typical radiation kind of problem. So when you're having natural convection kind of problem radiation also will be important. So we'll see that through one example at the end of the modules. Okay. Who drives natural convection? Density different. Who drives the density difference? Temperature. So temperature difference is essential for heat transfer only that in this case it leads to a density different which causes the currents to be established. Then of course gravity has to be present because gravity is what is going to uh show this increase decrease in temperature and uh because gravity is acting the weight of the fluid or the medium which is going to be lighter because of higher temperature and you have you get the current because the cooler air is heavier and it comes down. Gravity is aiding this part. So in fact one very favorite question that we typically have is I have two two plates which are for example so this is the direction of gravity and I have two cases one which is the high temperature T hot here and T cold here the other case is uh T cold here and T hot here where will natural convection be established in both cases G is acting vertically downward. This hot air here will be hot in this region. Cold air here will be cold. Hot air tends to rise up. So there is no reason for it to come down. This is not going to happen. Whereas here the hot air will rise up, the cold air will come down. So just because your delta t is there, here also the delta t is there. Here also the delta t is there. same value of delta t but here the delt this is gravity is aiding this and because gravity is aiding this natural convection gets established here no natural convection this is something which I think we need to understand just because a delta t is there doesn't mean you're going to have natural convection the direction of gravity becomes very very significant here the direction of gravity is here it is not aiding the convective current. Okay. So this is something we need to emphasize uh that mere presence of delta t is not sufficient. It has to be aided by gravity. When there is no gravity in space, no natural convection heat transfer in a spacecraft. In heat transfer studies, the primary variable is the temperature. Therefore, we need to express the net buoyancy force which is occurring because of the density difference in terms of temperature difference. And this is how your momentum equation which has density and the energy equation would get linked unlike force convection where the momentum equation and energy equation were not linked other than by the velocity term. Here you would have the density different which is the buoyancy force in the momentum term having a temperature term already. We'll see that in the derivation. So temperature occurs even in the momentum equation in natural convection. Whereas in force convection temperature was not a part of the momentum equation. That's the major difference here and therefore the solution methodology also would be diff different. Okay. So density difference is to be expressed in terms of temperature difference which requires the knowledge of a property that represents variation of density of a fluid with temperature at constant pressure. How density changes with temperature is important. This property which takes care of this variation is called as the volutric expansion coefficient beta. And all of us know what this volumetric coefficient is. It is a measure of change in volume of the substance with temperature at constant pressure. So dv by dt at constant pressure divided by the v or you can look at it as dv by v divided by dt. Change in volume per unit volume due to change in temperature dt process occurring at constant pressure. And because volume and density are inversely proportional to each other, you get a minus sign when you replace the V's by row minus 1 by row D row by DT at constant pressure. The figure shows a substance with a large volumetric coefficient and this one with a small one 20° 100 kPa 1 kg. This one would go up significantly. This one would go up very less for 1°ree change in temperature. Okay. Now here we come to the general equation of motion and formal definition of the non-dimensional quantities. What are we looking at? We are looking at this derivation for a vertical plate. Why vertical plate? Why not horizontal? For the simple reason that if I have a horizontal plate, the boundary layer develops here. Gravity is going to act in the downward direction. The currents cannot be established. Whereas if the plate is vertical, gravity acting vertically downward, these convective currents are going to be established like this and the boundary layer would be formed. Think of the agarati or incense stick when it is burning even when you keep it horizontal the plumes the smoke will go vertical. Okay. So the nature of natural convection is to rise is to go up. So we look at the derivation also for a vertical flat plate gravity is acting in the negative y direction and what there are a few things that we need to look at from terminology point of view. One the fluid the the surrounding for example when you light the agarati or incense stick the surrounding fluid which is the air is not agitated it is whatever is there in the room. So we typically call it as quieten fluid. The surrounding fluid is so means it is stationary kind of thing. Then we also have the flow is laminina two-dimensional steady fluid is Newtonian and the properties are constant. What do I mean by that? This very important assumption properties are constant. All properties including density would be constant except the fact that the density difference wherever density difference is going to occur. It is between the density of the fluid near inside the boundary layer and outside which is at t infinity. This concept is very important. So just explain this. This is surrounding fluid which is at t infinity. This plate is at t wall. The density here is going to be different than the surrounding density here. This is a row infinity. This is some row. Only this difference is what is going to cause my buoyancy force. Gravity is going to act this way. This density difference is going to cause the buoyancy force. So when I say properties are constant, yes they are constant except this changes in density which is responsible for the buoyancy force to occur. This variation is taken care by what we call as the booziness approximation. Very important terminology. Booziness approximation is nothing but the fact that the density difference between the fluid inside or the region where there is activity inside and outside of the boundary layer which gives rise to buoyancy force and sustains the natural convection flow. Okay, I'll repeat again. The density though is constant. If we say constant properties but the buoy if density is constant you will not have buoyancy force. So what we are saying is yes when you write the equation continuity etc you can take row is constant but when the force balance is occurring the buoyancy force which is a result of deltat t is resulting in a row difference between the fluid here which is the inside the boundary layer and the surrounding quascent fluid which is at t infinity. This density difference is needed to give you the buoyancy force which is going to sustain the natural convection. This approximation is called as booziness approximation. And one more thing here to look at is your coordinate system. Gravity is acting vertically downward. Let me just go back and draw this again for you. Gravity is acting vertically downward. This is my plate. This is my gravity. The boundary layer is formed like this. This is the quieten fluid t infinity. Okay. And these are my coordinate axis. This is your x-axis and this is my yaxis. Uh sorry yaxis. Sorry. Always get this incorrect. This is my x-axis and this is my y-axis. All the action is happening in the y direction. This is a stationary plate and the velocity distribution is zero velocity here. No slip condition. So velocity would rise here like this. And external flow for example when I had external flow this was u infinity. So the boundary layer would have free stream velocity at this location u infinity. Whereas here this is U is approximately zero. Quascent fluid Q U I may be wrong with the spelling but uh quiscent Q U I S. So it's almost stationary fluid. So the velocity here also has to be zero. So it will reach a maximum at the center somewhere and come back to zero. This is my velocity distribution in case of natural convection temperature T wall T wall greater than T infinity. So I would have T infinity here T wall here. Okay that's what is drawn in the slide. V is zero. Y direction velocity V is zero temperature TS or T wall and the maximum occurs inside the boundary layer unlike the external force convection. Okay. Okay. Continuity equation is this one usual du by dx plus dv by dy equal to zero. The momentum equation is exactly what it is. Row into we are assuming 2D. So row into udu by dx plus v du by dy is minus dp by dx plus mu into the viscous term. Y momentum equation is exactly what we would write but additional term is going to be there as minus ro g. This is the body force term which is there in the y momentum equation because the y direction is vertical. Gravity is acting this way. And therefore you have this additional term in the energy equation. You have u dt by dx + v dt by dy is equal to alpha d² t by dx² + alpha d² t by dy². Exactly what we had derived earlier. The only change is this extra term which is your buoyancy force term and the fact that your properties are constant is all fixed here. The rows are all disappearing except that you have the additional term ro g. And now how these two get coupled we are going to see how the energy equation and the momentum equation get coupled that we will see also one other approximation or whatever we have done in force convection scaling the bulk of the action bulk of the action is happening in the vertical direction nothing is happening here so u which is representing the x component of the velocity V which is representing the Y component of the velocity. Action is in the Y direction. So V is much more important than U unlike external flow. Please remember this. Don't make this mistake. The action is because of gravity which is causing the fluid to rise. Therefore V velocity becomes important compared to U. So U much less than V. Therefore, complete X momentum equation vanishes. I I hope I'm clear. X momentum there in force convection, the Y momentum would vanish. Here the X momentum vanishes because U is much much smaller than V. So I'm left with continuity Y momentum equation with the body force term and the energy equation. Let's go ahead and look at this. Now the y momentum equation is written and we know by scaling analysis maybe I'll just write this also y momentum equation row into u dv by dx plus v dv by dy is equal to minus dp by dy + mu into d² v by dx² + d² v by dy 2 - ro g. This is my y momentum equation. Now u is much much less than v. Therefore x momentum equation vanishes. This we saw earlier. Also look at the scaling part. So let's use a slightly different color. Y scales as L, X scales as delta. Right? Therefore, d² v by scales as v / delta² d² v by d y² is going to scale something as v / l² delta much less than l. Therefore, d² v by dx² is much more important than d² v by dy² and that's what is written. Yeah, d² v by dx² much much greater than this and therefore the d² v by dy² term would drop off. Okay, so we lost that term. Then we also said that the pressure dp by dy. So those logic of whatever we did pressure imposed pressure etc all of them remain the same here except that this is now P infinity row infinity quent fluid this is being impressed on the boundary layer this pressure is being impressed on the boundary layer and dp by dy because please remember dp by dx is zero when I go to x momentum equation The inertial force term go to zero. The viscous force term go to zero. Therefore, dp by dx equal to zero. It means pressure is not a function of x. So that is what my x momentum equation is going to give me. X momentum x momentum vanishes dp by dx equal to zero. That means p not a function of x. This is a consequence of x momentum. Therefore, P is only a function of Y. So, this can be written as DP by DY. Now, this DP by DY is not what is it? Change in pressure with respect to the Y direction. The what is that? So, this pressure is not a function of X. It's only a function of Y. And the fact that we saw earlier also is that the inside the boundary layer the imposed pressure is because of the surrounding. So dp by dy is nothing but dp infinity by dy because p is not a function of x. So x is this way p infinity is here that is what is the total pressure. So dp by dy is nothing but dp infinity by dy. Substituting that there we would get using the approximation or using the hydrostatic force relationship dp by dy or dp by dz is equal to minus row or p is equal to row gz. You know from that we get dp by dy is minus row infinity g. This is important hydrostatics. and we are going to replace this. Okay. So the pressure term gets replaced by this. So when I put that in here, I would get row u dv by dx plus v dv by dy equal to mu d² v by dx² because y y term went away minus row g minus of minus row infinity g. So plus row infinity minus row into g. Okay, this quantity is nothing but the body force tumboy term. And who who is the cause for this? This row infinity is virtue of the temperature of the surrounding medium T infinity. So row infinity and T infinity are coupled. Row and T are coupled. So deltat T leads to delta row which is what we are going to see. The flow is driven by the density field which is generated by the temperature field. That's what we are seeing. So once I write the momentum equation now simplified momentum equation. So simplified y momentum row into u dv by dx plus v dv by dy is equal to mu d² v by dx² + row infinity minus row into g correct row infinity minus row into G. Now we have the definition of beta which is -1x row d by dt at constant pressure. We will expand this as minus 1 by row row infinity minus row divided by t infinity minus t. So this can be substituted. So I'll go back and do beta is equal to -1 by row d row by dt at constant pressure which would be - 1 by row row infinity - row divided by t infinity minus t at constant pressure. Therefore, row infinity minus row is equal to minus row beta t infinity minus t which is row beta t minus t infinity row infinity minus row is this. Now I substitute this here I would get row into u dv by dx plus v dv by dy is equal to mu d² v by dx² + row beta t minus t infinity row beta t minus t infinity of course I forgot the g sorry into g okay okay so this is my x sorry y momentum equation after all the approximations and the use of the definition of the volutric coefficient okay now put these two together this is the booziness approximation put all this together you can manipulate the rows and this become new and this is your governing equation So my equation here becomes u dv by dx plus v dv by dy is equal to new d² v by dx² + beta into t minus t infinity into g. All the rows get divided. This is my y momentum. energy equation. Let me write u dt by dx plus v dt by dy is equal to alpha d² t by dx² + d² t by dy 2. Again this term this term we would drop off because of delta much much less than L. And therefore you would get u dt by dx plus v dt by dy equal to alpha d² t by dx². This is my energy equation. This is my y momentum equation. Of course, I have my continuity equation. du by dx plus dv by dy equal to zero. This is also there. Okay. So what do I have? This is the momentum and the energy. And I'll just complete in a minute. Putting all these together. Continuity y momentum x momentum vanished. Y momentum energy equation. What are the boundary conditions? I'll just go back to my diagram. The boundary conditions are as follows. At x =0, u =0, v =0, no slip condition. Correct. At this wall, no slip condition. at x = delta u =0 quiet fluid nothing is changing here so I have d² v by dx² so second derivative in v therefore two boundary conditions right then what else do we have so u is equal to0 v =0 at x= 0 V =0 at X= to delta I wrote but it is X tends to infinity at X tends to infinity T is T infinity quant fluid temperature at the wall temperature is T surface or T wall so this system with these boundary condition technically can be solved more on this in the next module thank