 So, good morning and today we will start a new topic this will be on natural convective heat transfer whatever we did so far you know the external force convection and internal force convection. So, most of you already have done the hydrodynamics part of it before in fluid mechanics you start with the Blasius boundary layer theory for the external force convection or the fully developed internal flows for a channel and for a duct and so on. But in this particular topic natural convective heat transfer definitely you would not have done this in a separate fluid mechanics course because this is one problem where hydrodynamics and heat transfer are coupled together. So, unlike the other external flows and internal force convection so you cannot separate the hydrodynamics from the heat transfer part in the case of natural convection because it is basically the energy equation which drives the momentum in this case. So, today from today the next the focus for the next 5 or 6 hours will be on natural convective heat transfer. So, what is the fundamental physics behind the natural convective heat transfer or the motion the driving force behind natural convection we will look at the example of a simple flat plate which is aligned vertical ok. So, in the case of external force convection it does not matter whether you have a flat plate placed vertically or horizontally since you push the air by external means you create a boundary layer and we do not take into account the effect of gravity in those cases. But when we study natural convection the effect of gravity becomes important in fact the buoyancy is the driving force here. So, it matters what is the orientation of the particular configuration. So, let us look at a flat plate which is oriented vertically and we have gravity acting downwards. So, initially the ambient air is quiescent that means there is no forced convection of anything. So, you just place the flat plate in a quiescent atmosphere and then you heat this plate you can either maintain a constant wall temperature or constant heat flux boundary condition and you can heat this vertical plate. Now, naturally what happens is that the fluid layer which is in contact with this plate will be at a higher temperature compared to the ambient since you are heating the plate. So, let us assume the temperature of the ambient to be T 8 therefore the fluid layer will be at some temperature T which will be greater than T 8. So, now therefore you know that the density is a function of temperature especially very strongly for gases. So, when you look at say quiescent atmosphere now the heated air here will therefore have a density which is different from the density somewhere outside where you have T 8. So the density here rho of T therefore will be less than rho at T 8 so that means you have a lower density air close to the hot surface. Now the tendency of the lower density air is to naturally rise up right. So therefore you will have over a period of time you will see a visually a boundary layer which is actually forming and growing from the leading edge the leading edge here is actually the bottom of the plate here all the way up okay. So we will have a coordinate system X and Y in such a way that X is in the direction of the along the plate and Y is perpendicular to the plate. So our origin will be starting from the bottom this is your X and the perpendicular coordinate is your Y okay. So in the case of therefore the natural convective boundary layer the boundary layer formation happens essentially due to a temperature difference. So this is the starting point of the convection to happen and because of this temperature difference this essentially maintains a density difference between the hot air close to the plate and the ambient quiescent air outside. So this density difference will cause the lighter air which is in contact with the plate to rise up and therefore a boundary layer is formed okay. So now you can very clearly see that the cause for the boundary layer is essentially due to a temperature difference. So the temperature difference is the driving potential in the case of natural convection unlike the pressure gradient in the case of internal force convection or the external flow of air or water in the case of the external boundary layers. So in this case you have temperature difference as the driving potential. So this is a very important aspect of natural convection and therefore intuitively you should understand that the momentum and the energy equations have to be coupled in some way. So unless the energy the information from the energy equation goes into the momentum you cannot actually solve for the boundary layer growth. Now you can also do the same way by reversing the temperature direction that means you can have a cold plate and you can have heated air okay. Suppose you have a plate where your T wall is less than T infinity okay. So your T infinity is somewhere here and T wall is less than T infinity. So what will be the direction of the boundary layer growth in this case from top to bottom because here this density will be a function of temperature which will be lower. So this temperature is lower than T infinity therefore this density will be greater than rho at T infinity. So essentially the heavier fluid has a tendency to go down and therefore you have a boundary layer in this case which essentially goes from top to bottom okay. So now if you want to represent how the velocity and the temperature profile varies at a particular location X location for this case. So unlike the case of external boundary layer where outside you have a bulk motion U infinity in this case this is completely stationary air. So when you want to draw the velocity profile at some location so therefore the velocity has to be 0 at the wall and also 0 at the edge of the boundary layer right. So you have two points where the velocity becomes 0 in this case. So therefore the velocity can attain a maximum somewhere within the boundary layer okay. So obviously this is quite different from the external force convection. So where the velocity can gradually increases from the plate and attains a maxima at the boundary layer. So here you have 0 velocities both at the solid wall and at the boundary edge of the boundary layer and therefore it has to reach a maxima somewhere within the boundary layer we do not know which what is this location we will find it out in the due course and also we do not know what is the value of this maximum velocity right. So let us say this is your U max in the case of external force convection you know that U max is equal to U infinity but in this case we do not know that we have to get it from the solution right and how will the temperature profile look in this case. So you have the maximum temperature here and minimum temperature so this will be similar to your external convection force convection for a flat plate right. So the same way if you draw the velocity profile for this case so it will be 0 at the two edges and then it will peak somewhere we do not know where it peaks and similarly if you draw the temperature profile so this is your plate so you have higher temperature outside and then lower temperature close to the plate okay. So this is how the velocity varies as a function of y and temperature as a function of y correct so we have demonstrated I mean the fundamentals of the motion of convective boundary layer when you have buoyancy in the case where we have a heated plate and a cold plate. Now let us try to derive the governing equations for this case okay we will keep the boundary layer as the example so you have vertical flat plate boundary layer natural convection and let us try to derive the governing equation. So let us assume that the length of this plate to be capital L okay this is one of the characteristic dimensions that we will use in non-dimensionalizing the boundary layer equations but before we go to the boundary layer equation let us first write down what will be the Navier stokes equations two-dimensional steady state Navier stokes equations for the natural convective boundary layer past a vertical flat plate. So how will the continuity equation look what will be the convective continuity equation in this case d by dx of rho u so unlike the external force convection we cannot claim that density is a constant this is an incompressible fluid but density is now a function of temperature and therefore since temperature is a function of position the density becomes a function of space therefore we cannot take rho outside the derivative okay so locally it will look like it is a compressible fluid because the density keeps varying with different position so you cannot therefore directly at a first cut say that this is a incompressible approximation straight away we cannot directly pull density outside the derivative and say it is a constant right however it was shown later by Businesk okay so Businesk made the approximation that it is fairly reasonable to treat density as a constant in the continuity equation and also for the most part of the momentum equation except in the body force term of the momentum equation so the body force this is where the driving potential the density difference emerges as a function of temperature difference so except for the body force term which is the driving potential it is reasonably good enough to approximate the density to be a constant everywhere else the other equations are other parts of the equations provided the temperature differences are small enough if the temperature differences are very large even the Businesk approximation will not be held valid so what we will do is just to start off we will make a approximation as done by Businesk and therefore try to pull density outside as a constant from the continuity equation okay and also the convective part of the momentum equation so therefore if you write down the x momentum and y momentum and invoking the Businesk approximation so let us draw first write down the x momentum equation how does the x momentum equation look so you have let us keep density outside the derivative but let us not divide it by density right away so rho of u du by dx plus v du by dy this is your convective term according to the Businesk approximation he says both in the continuity equation and in the convective part of the convective acceleration part of the momentum equation you can take density to be a constant so this is equal to what are all the terms on the right hand side minus of dp by dx so now this is the momentum in the direction of the plate that is in the vertical direction so you have dp by dx plus u into d square u by dx square d square u by dy square and then what else you have the body force so here definitely the gravitational acceleration cannot be neglected the body force is nothing but the gravitational acceleration g which is acting downwards so therefore we put a minus g right so we have rho g correct right so now once again similar to these external force convective boundary layer if you do a scaling analysis we can order of magnitude analysis we can show that the diffusion in the vertical direction and the diffusion in the y direction here the y direction is the horizontal direction now Cartesian x Cartesian y and is this is much greater than your diffusion along the length of the plate okay so this is the same conclusion that we also got from external force convection and therefore for the sake of simplicity we will only write the diffusion perpendicular to the plate or in the direction across the boundary layer okay so this is the most dominant direction of the viscous diffusion right so therefore we have the x momentum can you write down the y momentum similarly so can we write down the so what do we have for the convective terms in the y momentum u dv by dx plus v dv by dy but we do not have any v velocity in this case we have only u velocity which is varying as a function of y we do not have any v velocity that is velocity perpendicular to the plate length okay so therefore the convective term convective term of the y momentum equation will be 0 on the right hand side you will have pressure gradient term dp by dy there is no diffusion of the v momentum also and there is no body force in the y direction okay so essentially dp by dy is equal to approximately 0 very small the others are very small so we can approximate it to 0 so that means p is not a function of y that means p is a constant along y so this is the same conclusion that we also got for the external force convection okay that means the pressure that you calculate outside the boundary layer the same variation also happens inside the boundary layer so if you draw therefore a line here the pressure at this point outside the boundary layer will be the same as the pressure within the boundary layer okay so that is why your dp by dy is approximately 0 so this also says that the pressure variation that you find outside the boundary layer that is dp by dx here will be the same as what you have dp by dx inside the boundary layer so therefore now since we have these continuity and the x momentum equation we do not have an additional equation for pressure so how do we therefore approximate dp by dx in this case so now therefore we have to calculate dp by dx by writing down the momentum equation outside the boundary layer since we say that dp by dx can be obtained from applying it outside the boundary layer so outside the boundary layer these become the Euler equations okay in this case there is no advection at all so essentially the convective term is completely zero so if you write the momentum equation outside the boundary layer so you end up with minus dp by dx and then what else so you do not have advection term you have you do not have any diffusion term outside there is no viscosity and you have but body force okay but we will distinguish the density within the boundary layer from outside so we will express the density here as rho infinity okay so outside the boundary layer we will use rho infinity here therefore we will write this as minus rho infinity g is equal to 0 so this gives that my dp by minus dp by dx is equal to rho infinity times g so therefore I can find my pressure gradient along the plate by applying the equation outside the boundary layer and I for determine that this is nothing but the gravitational acceleration outside yeah so in the case that inside we do not know how the variation is otherwise we have to solve for this and we have to build another equation to solve for it or we have to use the equation of state correct so in order to therefore simplify it we take it outside the boundary layer and we see that already from momentum equation we get the clue that there is no variation of pressure along why so therefore dp by dx does not matter whether you calculate inside or outside and outside it simply is equal to the gravitational acceleration so if you directly substitute it now you are eliminating dp by dx from the momentum equation therefore now if you substitute for dp by dx you have mu d square u by dy square here so minus dp by dx is rho infinity so therefore you have rho infinity minus rho times g so this will be the body force okay so the effect of rho infinity g is coming from dp by dx and the default body force is rho g so this difference rho infinity minus rho what is this force this is your buoyancy force so this is the net buoyancy force which is now driving the momentum in the natural convection right so if the buoyancy force is 0 then you do not have any boundary layer growth okay the boundary layer growth happens in this case only because of this density difference and what is causing this density difference temperature difference so now this is where business approximation is used in the sense we are ignoring the variation of density as a function of temperature elsewhere except in the body force term okay so now to invoke the business approximation so we will define the coefficient of thermal expansion beta okay so beta is the coefficient of thermal expansion so this is written as minus 1 by rho d rho by dt so what it simply measures is the variation of density of a particular fluid with respect to temperature okay so if this coefficient is high that means you have potential that this fluid can expand or contract very quickly very strongly as a function of temperature okay so the higher the value of beta indicates that the potential for this density difference can be higher okay and these are usually measured as a part of the thermo physical properties just like thermal conductivity specific heat capacity and so on and they are tabulated for different gases okay and for ideal gas what will be the value of beta how do you calculate beta if you make the ideal gas equation of state if you put in the ideal gas equation of state into this so it will come out simply as 1 by t okay so why we are putting a negative sign here because usually the density decreases as your temperature increases okay so in order to make sure that this coefficient is positive okay thermal expansion coefficient is positive you put a negative sign here alright so therefore now if you apply the calculate use a simple finite difference okay assuming a linear variation of density with temperature okay if you want to calculate the variation from some reference temperature T infinity to actual temperature T okay so how will this look you have minus 1 by rho rho minus rho infinity divided by T minus T infinity okay if you assume that small changes in temperature we can assume a linear variation in density okay and therefore we can just approximate the derivative d rho by dt as rho minus rho infinity by T minus T infinity now you can therefore substitute for this buoyancy force rho infinity minus rho from this particular coefficient so therefore what do you get rho minus rho infinity is equal to minus G beta into T minus T infinity or rho infinity minus rho is equal to G beta T minus T infinity so this is basically the relation between the buoyancy force to the driving potential which is the temperature difference so you can therefore substitute for rho infinity minus rho from there so I just take G should not be here I am sorry so once you substitute into this you have G so you have therefore G beta into T minus T infinity also there is a rho here right yeah you have rho here okay rho beta into T minus T infinity okay so therefore now your buoyancy force is now written as a function of temperature difference so this is now what we call as the momentum equation invoking the business caproximation the business caproximation says that the density can be treated as a constant in the advection part whereas you invoke that as a function of temperature in the body force term so now if you divide it throughout by rho so now this looks similar to your external post convection boundary layer equation except the last term which is G beta into T minus T infinity right so this is your buoyancy term or body force term so if your temperature difference is 0 there is no natural convection boundary layer and therefore the boundary layer grows because of this temperature difference so now you can write down the energy equation also how the how does the energy equation look so it will be no different from your external force convective boundary layer equation right Q DT by DX plus V DT by DY is equal to alpha into T square T by DY square we can neglect again heat diffusion in the X direction with respect to Y and if you also neglect the viscous dissipation okay we are talking about small values of occurred numbers so in that case this will be your same as your external laminar force convection past a flat plate okay so this will not change now what is the major difference is the inclusion of the buoyancy term into the momentum equation so now you can see that unlike the the other case where you solve the momentum equation first get the velocity profile so this is how Blasius did first Blasius solve the hydrodynamic part you got the velocity profile then Paul Hausen use that then he solved the energy equation but here the velocity profile itself is coming from the temperature okay so you cannot therefore do it in a serial sequential fashion so all of these has to be simultaneously coupled and solved now this is where the complication comes okay so that means you cannot find a simple sequential segregated solution unlike the case of external force convection so you have to couple all these equations and solve them okay now we will see in the due course of another one or two lectures we will see how to solve these equations one after the other for different boundary conditions but before doing that so now that we derived the governing equations let us try to non-dimensionalize them taking some reference parameters and see what are the non-dimensional numbers that come out okay so I request all of you to scale the all the variables here that means you take your position x and scale it with the length of the plate this will be a non-dimensional x similarly your non-dimensional y and how do you scale velocity here u max because we do not have u infinity but the complication here is we do not know what is u max a priori right this is happening within the boundary layer which comes out of the solution but for the time being you do not worry about it you just assume that u max is your reference okay so we will call this as a reference velocity u subscript r some reference velocity it need not be u max also it can be any other velocity okay so some reference velocity which we do not similarly you are we also okay now we do not have pressure term explicitly so you do not have to worry about non-dimensionalizing the pressure and what about temperature now so again we introduce non-dimensional temperature ? okay when we do the non-dimensionalization let us do it assuming a constant wall temperature so that we can write this as T- T infinity by T wall- T infinity alright so I will give you about 5 5 to 10 minutes time you please substitute this into the governing equation and find out what are the non-dimensional groups so I will write the final expression on the board but you please work it out and check so all of you please check whether you get the same non-dimensional groups so is that okay so you have therefore 1 over Reynolds number here okay and you have 1 over Reynolds number times Prandtl number okay so if you define your Reynolds number now as u r into L by nu some reference velocity times the total plate length so you can therefore write this as 1 over Reynolds number and this is 1 over Reynolds number times Prandtl number what about this now you have an additional non-dimensional group if you see this as no units what is the unit of ? Kelvin inverse okay so this entire thing will be again a non-dimensional group okay now this represents the ratio of two forces the numerator is nothing but the buoyancy force okay the G ? into T wall- T infinity is nothing but the density difference ?- ?infinity and denominator is your inertial force okay so we will now define a non-dimensional number in natural convection this is called the Grashof number so usually denoted as Gr okay this is the ratio of buoyancy force G ? into T wall- T infinity into LQ divided by the viscous force nu square okay so this is basically the ratio of buoyancy and viscous force so you can imagine that this Grashof number is somewhat analogous to the Reynolds number in force convection so there you have inertial force here the inertial force is replaced by the buoyancy force or in fact the inertia here is driven by buoyancy okay and therefore you can write this ratio of buoyancy to viscous from using the Grashof number and Reynolds number because Grashof number is a function of buoyancy and viscous force Reynolds number function of is nothing but inertia and viscous force so therefore you can write G ? L into T wall- T infinity by your square how do you express this in terms of Grashof and Reynolds number turns out to be that this is nothing but Grashof by Reynolds number square okay so therefore this entire non-dimensional group is nothing but the ratio of Grashof to Reynolds number square okay so in the process we have therefore defined a new non-dimensional number which is very much relevant to natural convection which is now called as the Grashof number ratio of buoyancy to viscous force analogous to the Reynolds number now from this can we kind of estimate what is the order of the reference velocity okay so what is the equivalence of Grashof and Reynolds number so you can say that Grashof number to the power half is of the same order as the Reynolds number correct because we have Gr by Re square that means the order of Re square should be of the same order as Gr now therefore you can substitute the expression for Grashof number Reynolds number then calculate what is the order of the expression for calculating the order of U reference okay so this is nothing but G beta T wall- T infinity LQ divided by U square to the power half which is equal to U R L by U so from this what do you get for U R so you have L power 3 by 2-1 which is L power half okay so you have nu nu which cancels therefore U R should be square root of G beta into T wall- T infinity into L okay so to get your reference velocity at least the order of it you can therefore use this particular expression okay otherwise from your you know when you non-dimensionalize it you do not know what is your reference velocity but you can now using the order of magnitude between Grashof and Re square you can therefore come to some reasonably good estimate of reference velocity okay so you can see that this reference velocity is nothing but what is going into your Reynolds number definition and here it is driven by your temperature difference if there is no temperature difference therefore there is no inertia okay the inertia is essentially arising from the buoyancy term which is actually a function of your temperature difference therefore you can classify different regimes now what we have seen is a pure natural convection case but you can also have a case where you can combine your force convection with buoyancy that means you can have a regular bulk velocity do let us say this is your U infinity and the temperature difference is also substantial so that you can have a boundary layer growth now this is a combined effect of both your force convection and natural convection okay so in such a case the same equations are valid right so but in that case your U reference there you can actually use as U infinity because even if you do not have any temperature difference the force convective boundary layer will still exist okay so when you define the Reynolds number in that case you can define Reynolds number using U infinity okay which indicates the external bulk convective motion and the Grashof number is still decided by the buoyancy ratio of buoyancy to viscous forces okay so in that case what is important what are the different regimes and what are the how are the different regimes classified is by the ratio of Grashof by Re square okay so if you are talking about these values much lesser than 1 that means your bulk velocity is now overpowering your buoyancy force so in that case you can ignore natural convection and therefore this is only pure force convection so remember in this case we define Reynolds number as U infinity L by U correct so in this case when Grashof by Re square is very small you ignore your natural convective effects buoyancy effects on the other hand if you are Grashof by Re square is very large that means your buoyancy force is dominating your bulk motion so here this will be your natural convection so by the way the other name for natural convection is called free convection since you do not spend you know you are not putting any effort in driving happening in making this convection happen it happens naturally therefore it is called free convection so the regions where this is significant that is the order of 1 okay so here this is called mixed convection so in the case of mixed convection both the effects of forced convection and natural convection will be equally significant you cannot completely ignore therefore either of them so we have already seen cases of forced convection derived the correlations for Nusselt number and so on similarly for natural convection we will do it now what happens in mixed convection okay so in mixed convection the most simplest way of approaching this is the Nusselt number in mixed convection is simply calculated from independent correlations for forced convection and free convection and we just use some power law to blend these two okay so this is one of the simplest approaches so here the value of M could be either 0.3 or 0.4 okay so we will stop here so tomorrow's class we will look at the different ways of solving the governing equations okay so for first starting with the constant wall temperature case then the constant heat flux case and so on okay thank you