So, far we have considered heat and mass transfer from a wall essentially in force convection situations. The purpose of today's lecture is to consider heat and mass transfer from a wall under natural convection conditions. Natural convection is of importance both in practical equipments as well as in the environment. And therefore, student of convective heat and mass transfer should also know something about natural convection heat transfer as well as mass transfer. So, the topics for today are first of all I will define the problem of natural convection past a vertical wall. Then I will develop constant property laminar boundary layer solutions using similarity method in which initially we shall assume that the vertical wall is at constant wall temperature and there is no suction or blowing. But then we shall allow for variation of wall temperature as well as allow for study the effect of suction and blowing. Likewise we will also develop integral solutions of laminar boundary layers and also of turbulent boundary layers. And finally, we shall consider simultaneous heat and mass transfer in a natural convection boundary layer because natural convection can arise both due to temperature differences between the wall and the surroundings as well as concentration differences between the wall and the surroundings. These differences set up forces which are opposed by gravity and therefore, both buoyancy due to concentration difference and buoyancy due to temperature difference is of importance. So, let us go to the problem definition. So, as I have shown here consider a vertical wall or a vertical plate which is at constant wall temperature T w and let us for the time being suppose that T w is greater than T infinity where T infinity is the temperature of the stagnant surroundings. As a result it the fluid close to the wall will have a smaller density because it heats up and compared to density of the stagnant surrounding the density in this part would be smaller. And as long as the temperature difference persists between T and T infinity there would be a density difference and this density difference would drive the flow up against the gravity force. The density difference sets up an upward flow. So, the flow would come in here and at x equal to 0 there would be 0 boundary layer thickness, but as one moves up the boundary layer would grow of course, assuming a velocity profile which is something of this type. Notice the difference between a force convection boundary layer and a natural convection boundary layer. In both the velocity is 0 of course at the wall, but in natural convection boundary layers the velocity peaks somewhere between the edge of the layer and the wall. And the velocity again falls down to 0 at the edge of the boundary layer and that is because there are no longer any temperature differences beyond that point. And therefore, there is no driving force for the velocity and the velocity assumes the value of the stagnant surroundings which is 0. If however, T w was less than T infinity then natural convection would be as I am showing here now. The natural convection flow will be like that and the velocity profile would be like so and the temperature profile would go something like that T w being less than T infinity. And then we would measure x equal to 0 here and downwards. So, this is the negative bind see if you like with g acting in that direction. The boundary layers thus develop a laminar to begin with, but beyond a certain length the boundary layer becomes unstable and turns turbulent. So, there is actually transition somewhere and then the boundary layer becomes turbulent. So, to begin with let us consider only the laminar boundary layer.  There would be pressure gradient mu d 2 u d y square which is the boundary layer approximation. We are saying that there is no diffusion in the axial direction and minus rho g is the body force due to gravity. Likewise there would be temperature equation or the energy equation in its familiar boundary layer form. Now, from hydrostatics d p infinity by d x would be minus rho infinity into g because the pressure gradient would be simply balanced by the gravity. Therefore, d p infinity rho g will become equal to sorry this should be minus d p infinity by d x minus rho g and therefore this will become rho infinity minus rho into g. Now, if I define beta the coefficient of thermal expansion as 1 over rho into rho minus rho infinity divided by t minus t infinity then you can see that this term rho infinity minus rho g can be written as rho times beta. This will become rho times beta g t minus t infinity and that is what I have written here. So, notice the small error here that the pressure gradient should be minus d p infinity by d x. So, if I define beta equal to minus 1 over rho rho minus rho infinity over t minus t infinity then these two terms would transform to g beta and divide through by density then you will have the momentum equation with nu d 2 u by y square plus g beta t minus t infinity. This is called the buoyancy force term. So, our equations now are the continuity equation, the energy equation and the momentum equation with this and you can see there is a clear coupling between energy equation and the momentum equation through the buoyancy term. Thus, the modified form of the momentum equation and the energy equation must be solved simultaneously along with the continuity equation. Solutions can be obtained by similarity integral or finite difference methods as in force convection. Now, the integral form of the equations can be derived as follows. For example, I can write the left hand side of this in conserved property form as d u square by d x plus d v u by d y equal to all that and if I integrate each term from 0 to delta then you will see that you will get d by d x 0 to delta u square d y which will be the which will follow from the first term. This term which will be d v u by d y would of course, go to 0 both at the at y equal to delta where u is equal to 0 as well as at the wall where u is equal to 0 and therefore, the term would simply vanish. This term would be nu d u by d y at y equal to delta minus nu d u by d y at y equal to 0 and that would be since d u d y at y equal to delta is 0 you will simply get this as minus tau wall by rho plus g v tau by delta into 0 to delta t minus t infinity d y. Likewise, if I write this left hand side in conserved property form as d u t by d x plus d v t by d y and integrate from 0 to delta then I will get d by d x equal to 0 to delta u into t minus t infinity d y equal to q wall over rho c p and the boundary conditions are at y equal to 0 u is equal to 0 and t is equal to t wall at y equal to delta u is equal to again 0 and t is equal to t infinity. So, this would be the integral forms of equations we shall take up these equations when we come to solution, but presently let us turn to solving the equations of this slide by similarity method. So, as usual we define stream function u equal to d psi by d y and v equal to minus d psi by d x and we define a dimensionless temperature t minus t infinity divided by t wall minus t infinity the denominator being constant given constant and then the momentum equation simply would become this is u d psi by d y into d u by d x which is d 2 psi by d y d x minus d psi by d x which is v into d u by d y which is d 2 psi by d y square plus nu times d 2 u by d y square which will be d 3 psi by d y cube plus g beta t w minus t infinity into theta and that would be the momentum equation and likewise this would be the energy equation d psi by d y d theta by d x minus d psi by d y d theta by d y equal to alpha d 2 theta by d y square minus theta d psi by d y d t wall by d x. Now, this term arises out of the u d t by d x term when t w minus t infinity I have retained this term at the moment because later on we are going to consider the case in which t wall itself would vary with x, but presently if t w is 0 then I mean t w is constant then of course, that term would be 0. We shall define as usual similarity variables eta equal to y multiplied by stretching function s x and stream function psi x eta equal to nu times f eta into g x another function of x. Then if we substitute in this then you will get f triple prime g beta t w minus t infinity g s cube nu square theta plus g dash by s f double prime minus f dash square minus g by s square s dash f dash square equal to 0. Now, of course, f dashes here are all derivative with respect to eta and g dash and s dash are derivatives with respect to x. Similarly, the energy equation would read in this like this theta double prime plus Prandtl into g dash by s f theta prime minus g by s t w minus t infinity over t w minus t x into f dash theta equal to 0. Again the two equations are coupled because theta appears in the momentum equation and likewise f appears in the energy equation. These equations can be solved by similarity method if all these factors which I have shown are absolute constant. Remember g is a function of x s is also a function of x t w minus t infinity can also be a function of x, but the group as a whole must be a constant likewise g dash divided by s must also be a constant and g by s square into s dash must also be a constant for similarity solutions to exist. Likewise, g by s t wall minus t infinity d t w by d x must also be an absolute constant.  So, let us see what do these conditions of constant c imply in which would give us the s as a function of x as well as g as a function of x. So, let us first consider d t w by d x equal to 0 and let g dash by s equal to c 1 g beta t w minus t infinity divided by g s cube nu square equal to c 2 and let g s dash by s square is equal to c 3. In other words, what I have done is I have said this is g c 1 this is c 2 and this is c 3 this whole term is c 3 d t w by d x is 0 and therefore that whole term is 0 in the present case. Now, combining expressions for c 1 and c 2 gives that g x should be proportional to x raise to 3 by 4 and s x should be proportional to x raise to minus 1 by 4. So, therefore, if we set quite arbitrarily c 3 equal to minus 1 then it would follow that c 1 equal to 3 and c 2 equal to 1. Now, there is a little algebra here of over a one page to show all this and hence g function would be 4 times grash of number based on x divided by 4 raise to a quarter and s will be again grash of number divided by 4 raise to a quarter divided by x where the grash of number as you know is defined in this fashion g beta t w minus t infinity x cube nu square remember here t w minus t infinity is constant and then the momentum equation would read as f triple prime plus theta plus 3 f f double prime minus 2 f dash square equal to 0 theta double prime plus 3 Prandtl f theta dash equal to 0 these will be the energy equation and the boundary conditions would be f 0 equal to f dash 0 which is the velocity f dash 0 is velocity f infinity f dash infinity would be 0. So, all these are 0s and f 0 equal to 0 implies that the v at the wall is also 0 and the boundary conditions theta would be from its definition theta 0 would be 1 and theta infinity equal to 0. The similarity variable eta which was y into s x would be y divided by x as you can see here into grash of x divided by 4 raise to 1 by 4 and f dash will give us the velocity u x divided by nu divided by 2 root g r x. Now, how are these equations to be solved one can these are coupled equations mind you. So, they have to be solved simultaneously now of course, like in force convection we solve the equations of this type by what was called shooting method, but it turns out to be little bit unstable when we consider natural convection and therefore, unlike in force convection the equations are best solved by finite difference method using what is called as a tri diagonal matrix algorithm which you must have studied in your numerical analysis class. So, very simply what one does is that the momentum equation which is f triple prime plus 3 f f double prime minus 2 f dash square plus theta equal to 0.  So, what I do is I solve for f from which I get f dash and then I solve this equation this second order equation in f dash by finite difference method by finite difference method and this again by finite difference method. So, that is what I have shown on this slide.    Now, these sort of equations that you see here are perfectly suited for applying tri-diagonal matrix algorithm because the diagonal element would always dominate over the neighboring elements neighboring coefficients a e and w whereas, the diagonal element of the matrix would be a e plus a w and therefore, one can get very very fast solutions. Once the solutions are obtained our interest is to obtain theta prime 0 which is the temperature gradient at the wall and that would be 2 times the into theta 2 minus theta 1 over delta eta at eta equal to 0 and likewise if we want to get the resistance at the wall frictional resistance at the wall then it will be f double prime 0 equal to 2 into f dash 2 minus f dash 1 divided by delta eta. Now, the Nusselt number of course is q wall divided by t wall minus t infinity x by k the usual definition and the q wall over t w minus t infinity will be simply minus theta prime 0 grash of x by 4 raise to a quarter. So, once we have solved the temperature and f dash and f distributions we can readily obtain our Nusselt number as well as the skin friction should be be interested although in natural convection generally one is not really interested in skin friction at all because there is no pumping power required in natural convection and therefore, it is of only academic interest because this simply represents the velocity gradient at the wall. So, here are the solutions obtained for t wall equal to constant and I have allowed for a range of Prandtl numbers going from as low as 0.01 which is for liquid metals to as high as 1000 which is for viscous oils around one is the gases and you can see the eta max required is about 7.5. As the Prandtl number increases beyond 1 you can see that the boundary layer is thinner because the temperature gradient occurs much more rapidly near the wall and therefore, you need eta max has to be shortened through to about one at 1000 whereas, in the liquid metals side you get a thicker boundary layer and eta is of the order of 22 and here are the temperature profiles for example, you can see that this is the distribution of velocity f dash and this is for Prandtl equal to 0.1, 1, 10, 100 and 1000. In dimensionless form f dash eta whereas, this is theta versus eta and you can see for 0.1 you are going right up to almost eta of nearly 12 whereas, as the Prandtl number increases the boundary layer thickness reduces as you can see. The Nusselt number on the other hand or Nusselt number you have a Grashof raise to 0.25 is 0.059 at 0.01, 0.164 at 0.1, 0.402 at 0.01 and 0.821 at 10 and 1.54 at 102.5. F double prime 0 of course, goes on reducing as you increase the Prandtl number, but as I said earlier our main interest is not in F double prime 0 it is of academic interest but the main thing is the Nusselt number the manner in which Nusselt number varies. Now similarity solutions are also possible when T w minus T infinity varies as A x to the power of N. Remember now this term which was set to 0 would be finite and if T w minus T infinity varies as A x to the power of N then this term would also be constant and the governing equations would then look like this F triple prime plus N plus 3 F double prime minus 2 N plus 2 F prime square plus theta equal to 0 and theta double prime plus Prandtl N plus 3 F theta prime minus 4 N F dash theta equal to 0. Now you can see if I set N equal to 0 which would be the constant temperature case then the previous equations are readily recovered. I am considering two values of N, N equal to 2 and N equal to 1. N equal to 1 of course, implies linear variation of temperature, but N equal to 2 has a very special significance as you can see for N equal to 2 if I write q wall will be equal to minus k T w minus T infinity theta prime 0 into S and S itself is proportional to x to the power of 0.5 0.25 as you will recall from here as we said S x is proportional to the minus 0.25 and here if T w minus T infinity itself varies as 0.2 x to the power of 0.2 then further raised to 1.25 will give 0.25 x to the power of 0.25 into x to the power of minus 0.25 and therefore, this would result into constant and therefore, N equal to 0.2 represents the constant wall heat flux case and therefore, it is of interest. So, again you will see the values of N u x divided by a Grashof raised to 0.25 are 0.068, 0.189, 0.4579 to 4, 1.705 and 3.03. You will see compared to T wall equal to constant case and therefore, at all Prandtl numbers q wall equal to constant gives you a higher Nusset number a higher Nusset way. This is very much like what we had observed in post convection laminar boundary layers as well. For linear case linear temperature variation case the Nusset number variation is again higher than N equal to 0.2, 0.093, 0.354597, 1.184, 2.178 and 3.8. So, you can get very good solutions and these have usually been found to be very in good agreement with experimental data. Now, let us consider V w the effect of suction and blowing again in the presence of T w minus T infinity varying as A x to the power of N. Now, V is minus d psi by d x and that would transform to minus nu times f g dash plus g f dash y into s dash and therefore, at y equal to 0 I would get the value of V w and this would be f 0. So, f 0 would be simply V w divided by nu g dash and substituting for g dash from over g x relationship you will see f 0 is nothing but minus V x x divided by nu divided by g r x by 4 raise to 0.25 into 1 over N plus 3 or simply V w star minus V w star this is like a Reynolds number. So, I can make it into minus V w star divided by N plus 3 and that would be a constant f w. So, you solve the equations of the previous slide you solve these equations with f 0 equal to minus V w star over N plus 3 where N is specified as well as V w star is specified. Now, here I am giving a solutions for N equal to 0 which is the constant wall temperature case and taking Prandtl equal to 0.7 only the two values and you can see on the suction side V w star would be negative. So, V w star very high suction rate goes on reducing and this is the blowing rate V w star positive 1 2 and 3. Then you can see Nusselt number goes on reducing as suction is reduced and the reduces still further as blowing is increased and this is very much like what we had observed in case of force convection. So, the effect of suction is to enhance heat transfer and for reference I have given for V w star equal to 0 and Prandtl equal to 0.7 Nu x g r x to the power of minus 4 is 0.353. So, you can see when V w star is minus 1 it is 0.664 when it is minus 0.2 it is 3 times and when it is minus 3 it is almost 4 and half times the Nusselt number. So, suction enhances heat transfer and blowing reduces heat transfer. Now for T w equal to constant and V w equal to 0 there are curve fit correlations have been obtained. So, for example, a correlation due to Eid is given here the correlation due to Eid in Britain is given here that due to Ostrich is given here. This Ostrich solution is often found in undergraduate textbooks for T w minus T infinity A x raise to 0.2 which implies constant wall heat flux and V w equal to 0. Here is the correlation for the data obtained by Fuji and Fuji where g r star is now based on Q w x raise to 4 k nu square. So, these are simply for your reference that solutions presented on the previous slides can be correlated in terms of functions of Prandtl number. Now we turn to integral solutions of the laminar boundary layer and as you know integral solutions always require specification of velocity and temperature profile. So, here I am going to define a quantity called u ref some reference velocity I do not know what it is we will say we shall find out in a moment what that is. So, let us say the velocity profile in the boundary layer will be u over u ref y over delta 1 minus y by delta whole square you can see it satisfies the condition that u is equal to 0 at y equal to 0 as well as u equal to 0 at y equal to delta. So, it satisfies both conditions and it also satisfies the idea that the profile would be would peak somewhere between 0 and delta. Likewise, T minus T infinity divided by T w minus T infinity is taken as 1 minus y by delta whole square. So, if you substitute these temperature profiles and use our integral momentum equations which I have given. So, substitute for u here T here as well as T here and carry out the integrations then you will see that the momentum equation transforms to 1 over 105 d u ref square delta divided by dx into g beta by 3 T w minus T infinity delta minus nu u ref by delta this will be the momentum equation and 1 over 30 d u ref by delta u ref delta by dx equal to 2 alpha by delta where alpha is the thermal diffusivity. So, in order to obtain these two equations can be solved by assuming let u ref be equal to c 1 x raise to m and delta be equal to c 2 x raise to n and if you substitute then algebra will show that m equal to 0.5 and n equal to 0.25 satisfies these equations. Now, again this is about 1, 1 and a half page of algebra which I am not presenting here but it is fairly straightforward algebra to do to give m equal to 0.5 and n equal to 0.25 and therefore, you get the result the result is the once you substitute for u ref equal to c 1 x raise to power of m and c 2 x raise to power of n with these values you can show that delta by x would be 3.93 into 0.952 plus Prandtl divided by Prandtl square raise to 0.25 grass of x to the power of minus 0.25 and delta would be proportional to x to the power of 0.25 delta increases with x nu x is 0.508 0.952 plus Prandtl divided by Prandtl square raise to minus 0.25 grass of x to the power of 0.25 and therefore, h x decreases with x the local heat transfer coefficient decreases with x. These are important results they agree fairly well although they look very different from the forms of the curve fit correlations that I have given you in the previous page but they agree fairly well with the correlations that I have shown you. Now, if we write the differential momentum equation now we are looking at the idea of transitional to turbulence before we move on to turbulent. If we rewrite the momentum equation of the differential form equation for y equal to 0 and y greater than delta that is momentum equation that you see here you write it for u equal to 0 and you write it for y equal to 0 and for y greater than delta then in both cases convection terms will be absent at y equal to 0 d p infinity by d x when you replace by d p w by d x and this would be rho minus rho g. So, you will get then u ref d u ref by d x would be equal to minus rho u infinity minus rho w divided by rho into g and that would be equal to g beta t w minus t infinity. So, u ref essentially is nothing but remember this u ref into d u ref by d x would be d u ref square by d x divided by 1 by 2 and therefore, u will be equal to u ref would be simply equal to under root of g beta t w minus t infinity x there should be a factor 2 here inside the and Reynolds x would be equal to u ref x divided by nu. So, correspondingly from this u ref I can imagine a Reynolds number although it is not usually defined and that would equal grass of x to the power of half. So, essentially if we say that the Reynolds number of 10 raise to 5 or so represents transitional then grass of number of 10 raise to 10 should be roughly give you the transition criterion. Now, the generally the transition occurs at 10 raise to 9 to 10 raise to 10 grass of numbers in that range for gases and organic liquids the lower range is found and for Prandtl number greater than 1 that is viscous fluids the transition occurs little bit later. So, in practical work we define a Rayleigh number which is grass which is grass often to Prandtl and usually when Rayleigh number assumes value of 10 raise to 9 we say transition has occurred. So, this is taken as the transitional criterion to solve the temperature boundary layer then again we need velocity profiles and this slide shows what the velocity profiles in turbulent boundary layer are u over u ref equal to y by delta 1 raise to 1 by 7 into 1 minus y by delta raise to 4 again u is equal to 0 at y equal to 0 and u is equal to 0 at y equal to delta and t minus t infinity over t w minus t infinity is equal to 1 minus y by delta raise to 1 by 7. Now, of course, as usual in turbulent boundary layers we cannot depend on these velocity profiles to give us the shear stress because it will simply tend to infinity at y equal to 0 and therefore, we use this form which was also used in laminar boundary layers force convection boundary layers 0.0 225 rho u ref square instead of u infinity we are now replacing by u ref u ref delta by nu raise to minus 0.25 and standard x likewise is 0.0 225 u ref delta by nu minus 0.25 Prandtl raise to minus 2 by 3. So, both tau wall and q wall which are required on the right hand side of the momentum and energy equations are represented in terms of delta. If you assume now u ref equal to c 1 x raise to m and delta equal to c 2 x raise to n and substitute in the integral momentum equations then you find that the two equations are satisfied if m equal to 0.5 which is also the case of laminar boundary layer, laminar natural convection boundary layer, but n is now different n is 0.7 instead of the value we had earlier n value was 0.25 in laminar boundary layer, but in turbulent boundary layer we find that n should be about 0.7 and with these two choices the equations momentum and energy equations are satisfied and as a result if you follow the same algebra that we did for laminar boundary layer we find that the in turbulent boundary layers the nusset x is 0.0295 Prandtl raise to 7 by 15 into grass of x divided by 1 plus 0.494 Prandtl raise to 2 by 3 raise to 4. Now, in this case h x this is h x h into x divided by k. So, h x here turns out to be proportional to x raise to plus 0.2. In other words in turbulent boundary layer the heat transfer local heat transfer coefficient increases with x whereas, in laminar boundary layer the heat transfer coefficient was decreasing with x. Now, this is a very special feature of turbulent boundary layers natural convection turbulent boundary layer that the local heat transfer coefficient actually increases with x as shown here. So, we note this difference and that is of considerable importance. Now, for the entire range of Rayleigh numbers and for T w constant and V w equal to 0 the currently accepted correlations is like this.  So, the currently accepted correlations are n u l equal to 0.68 in Rayleigh number based on l and Rayleigh number as you know is a product of Grashof number and Prandtl number and then these are the correlations given by Churchill and Chu international general of heat and mass transfer 1975. For the lower range of Reynolds number this is this would be the laminar range as we said transition occurs at around 10 to 9 and this would be if the turbulent range is obtained that is the if the plate Grashof number was 10 to 9 and above then we would use that correlation. So, correlations for natural convection from other geometries such as inclined or horizontal plates cylinders cavities etcetera are readily available and you might have already used them in your undergraduate work. I am giving here one reference in cropera and duet fundamentals of heat and mass transfer John Wiley in 1996 which gives a number of correlations for natural convection in different situations. Now, we turn to the case of simultaneous heat and mass transfer that is we are assuming that T w say this is T w and there is omega v w is greater than or less than omega v infinity and T w is greater than all less than T infinity. So, main driving forces are beta times T w minus T infinity which I will call beta T if you like or simply beta T which is thermal cubical expansion, but we can also define beta star for mass transfer into omega v w minus omega v infinity. So, definition of beta and beta star is very parallel to each other. So, again we can show that similarity solutions are possible if T w minus T infinity varies as A x to the power of n and also delta v delta omega v which is omega v w minus omega v infinity in equal to A x to the power of n both should vary as x to the power of n then similarity solutions are possible and you get the momentum equation as f triple prime n plus 3 f f double prime minus 2 n plus 2 f f dash square plus theta plus the buoyancy also due to the concentration difference and the energy equation would read like that and the species transfer the omega v equation would read in this fashion.  The boundary conditions would be f 0 f dash 0 theta 0 phi 0 sorry theta 0 and phi 0 will be 1 and f dash infinity theta infinity and phi infinity will be 0. Now notice from slide 9 that as v w tends to 0 f 0 tends to 0. So, the boundary conditions we are assuming at the moment that we are dealing with a very small mass transfer rate from now eta would be still taken in the as defined as g r x based on temperature difference rather than based on mass transfer difference I mean mass fraction difference. So, these are the set of equations again we can solve them by finite difference method and here is the solution. So, I am taking the case of constant wall temperature the Prandtl number is taken as 0.7 and f beta is equal to 1. Now f beta equal to 1 implies that we have aiding the concentration buoyancy and the thermal buoyancy are aiding each other and in which case the Nusselt number would get defined in this fashion and Sherwood number which is the dimensionless mass transfer coefficient would get defined in this fashion and this is the case of limiting very very small mass transfer at the wall then you will get for different values of s c you get 0.5 0.7 1 5 10 calculated minus theta 0 minus phi dash 0. So, you can get Nusselt numbers out of this you can get Sherwood numbers out of this and notice that I have also printed out here the boundary layer thickness of concentration and boundary layer thickness of temperature. So, when Schmidt number is close to equal to Prandtl number the ratio is 1 when Schmidt number is lower than Prandtl number then the ratio is 1.01 very very close, but the Lewis number now is 1.4 and then when Schmidt number exceeds Prandtl number then you find that the concentration boundary layer thickness is smaller than the temperature boundary layer thickness. Now, this is to be expected, but it is good to notice it from the results as well. Similarly, the results for F beta equal to minus 0.5 now you find that the whereas the thermal buoyancy is trying to take the fluid up the mass transfer buoyancy is trying to take it down. So, you have opposing effects due to opposing buoyancies and you can see that compared to F beta equal to 1 with the aiding buoyancies you find the Nusselt number and Sherwood numbers are all smaller irrespective of the Schmidt number that you use. Now, there are many other variations of now in this all these cases I have used simply constant wall temperature, but there are a variety of cases of finite N and so on and so forth which have been handled in a very specialized book by Y. Jalluria and it is called Natural Convection Heated Mass Transfer Pergamon Press and published in 1980 you can refer to that, but the important thing is that when aiding compared to aiding situation the opposing situation lowers the Nusselt number as well as the Sherwood number finally, those solutions were obtained at low vanishing mass transfer rate. Now, suppose we take F 0 equal to minus V w nu g dash then that will be called minus phi dash 0 S c N plus 3 into B m and minus theta prime 0 Prandtl N plus 3 B h. So, if I use this as the boundary conditions for the momentum equation I can now deal with large mass transfer rates also and for N equal to 0 which is constant wall temperature and I have taken deliberately the case of Prandtl and Schmidt equal and F beta equal to 1 that is aiding buoyances aiding buoyances then you find that for different values of B m from 0 to 0.5 the Sherwood number goes from 0.421, 0.4, 0.381, 0.364, 0.35. So, as B increases the Sherwood number goes on decreasing, but if I calculate g over g star that is Sherwood number for any B divide by Sherwood number for B m equal to 0 then I get 1, 0.951 and so on so forth. Now, if I also calculate L n 1 plus B m divided by B m then I get those values and you can see those are very very close to the Sherwood versus star which are obtained from exact solution to the boundary layer equation and therefore, this data show that Sherwood over Sherwood star equal to L n 1 plus B by B is also verified in natural convection boundary layers as far as as long as properties are constant and this was the result we use extensively in force convection in Reynolds flow modeling, but it is also applicable to the case of natural convection boundary layers and other property corrections can also be applied in the same way that we had applied in Reynolds flow model. So, with this I conclude the lecture on natural convection heat in mass transfer.


Hi, my name is Leon Rowe currency trader and trading coach at trading 180 calm and welcome to this week's supply and demand forests and gold fundamental and technical Analysis if you're a newer one welcome to you and if you're returning an equally one welcome to you, and please don't forget to like subscribe And share press that like button and subscribe button. I'm not even sure whether the YouTube algorithms really That the notification button is actually useful when watching on some videos on YouTube saying Even though you pressed a notification button. It might not actually give you the notification, but press it anyway and If you want to be kept up-to-date with you know the latest trading 180 videos anyways getting into the key events this week and this is from ING and so next week's US inflation numbers will need to be quite surprising for the Fed to deviate from a 75 basis point hike at a September meeting. Yeah, so That's really what the market is looking towards Is it is inflation and also as well the Bank of England? I think they actually delayed one of their scheduled meetings I think the PEP postponement of the Bank of England's scheduled meeting. So really this week is you know for the again developed markets the major markets Is really about US core inflation and it's likely to ride to 6.1 percent Bank of England stick to 50 basis point rate hike despite energy package from list trust and We have GDP July GDP on Monday. I think that's the month for month one the jobs on Tuesday for the UK and inflation on Wednesday as well, so Yeah a few things To watch out for Eurozone, I think also has August CPI numbers as well So that might be something worth watching Eurozone US. Yeah, pretty much everything else is is by the by so a few things to definitely watch this week and Let's get you know into I guess the the technicals and what's happened Fundamentally last week and potentially what may happen in the future Starting off on the dollar index and the dollar index is just a measure of dollar strength against major currencies like the pound the Euro and the yen and some others and Just watching obviously a dollar strength over the past looking back over the past year, you know has just been Kind of predictable when you if you've kept up with what's been happening fundamentally you understand I guess what what that the fact that the US is really One of the better Countries and currencies dealing with you know the crisis As far as or many crisis Events that are going on around the world So Yeah, let's get into the Bloomberg articles to jumbo Fed rate hike as we already know is in play as power sticks to hawkish view so Fed chair Speaks as bets for 75 basis point hike hard and says Fed Has accept and accepts responsibility for price stability price is basically in face inflation. So Basically the Federal Reserve Jerome Powell said officials won't flinch in the battle to curb inflation hardening expectations that they'll deliver a third straight jumbo hike Later this month and we we need to act now forthrightly strongly as we have been doing pal said Thursday in remarks at the Kato Institute's monetary policy conference in Washington My colleagues and I are strongly committed to the project and will keep at it. He spoke With a moderator in a virtual question and answer session. So What that basically means is that they're gonna continue to hike and what does that mean for the dollar and what you know in in trading This is basically a game of probabilities That you know that that is being priced in in so far as You know the fact that the dollars go and higher and higher Is being priced in so it's a you know the money has really been made right is to buy the rumor You know happened was happening and has been happening all this time, right? Just basically buying a rumor and confirming with data now with CPI data, right? Consumer prices index that's going to be a major I guess signal as to Whether the Federal Reserve and I spoke about this last week, right? Whether the Federal Reserve will continue to hike 75 basis points. Yeah, that is going to be key if inflation comes down Yeah, and it's I wouldn't say necessarily disappointing because in fact The US actually want inflation to come down and all central banks want inflation to come down to their 2% target, right? now if Inflation does, you know come down from I think it was a point five eight point six something like that And if let's say for example, it stays around there or you know It starts to plateau or it actually comes out as lower than than the previous number then The Fed are likely to do what maybe hike at 50 basis points because they won't need to To hike aggressively which then would actually should have actually have the effect of The dollar probably selling off and being repriced right revalue because this you know value The the dollar appreciating is based off of you know the market getting ahead of a 75 basis point hike And I'm not saying it can't go higher. Okay, but it probably couldn't potentially this week go higher But the market has already priced in what the 75 basis point hike and the value of the dollar should be now It would have to reprice the dollar yeah, if there's a Smaller Hike right so then you would if and that would be triggered by CPI data That's the CPI data will trigger that and so depending again and whether you know inflation is is continued continuing to rise or if it falls sharply That will determine the what happens really with the dollar this week and one of the things you should Go to on my YouTube page how to find or I'm gonna be going on YouTube and do a search Is my fundamental analysis webinar the three steps to generating profitable forex trade ideas If you don't understand the relationship between inflation and interest rates as well as GDP as well It's the webinar is about you know two hours long and it gives you really a kind of basic foundation as to understanding how you can navigate I guess You know forex and you know pick directions and really kind of predict trends Etc and whether you know it's a market that you want to you should be buying or selling or whether it's a market that you should be staying out of right so Going back to DX why so Yeah, moving forward. I probably expect if again CPI data comes out Positive as far as it's seen as growing then you would probably want to still continue to buy the dollar and pullbacks We do have prices have come down to this this demand zone But also as well the monthly moving fair value Which is basically just a moving fair move in average, it's got the EMA Exponential and a simple move in average, but in the context of understanding value an average or a mean Is is actually what we considered fair value because if this is expensive Right, and it is a bargain price Then if you're looking at you know the last 21 days, which is 21 days in a trading month, which is you know monthly You're looking at where fair fair value is right fair value between An expensive and a bargain area 50% between that is actually fair value Yeah Prices have come down to that fair value price now for me I tend not to look to buy anything or you know anything above at least a monthly Moving fair value because that would be considered expensive right you want to look for Not only demand zones, but on a daily or weekly time frame, but also as well Some confluence in terms of understanding where fair value is And fair value for me has come back down to the dollar So I know now is a decent time to look for you know potential buy trade doesn't mean that you should buy right now Prices go below that then that's gonna be a better bargain, right? So just gives context to You know understanding value and that is really important when it comes to you know fundamental analysis So for me It still continues to be dollar buyers, but I will be and I mean actually some some dollar trades anyway like the euro dollar So I will be Keeping my eye on on the fundamentals as to determine what I should be doing with those trades now Moving on to the dollar yen in the dollar yen Massive move last week, right? I think prices were at the 140s and they've literally just jumped up, you know to the 145s and Very interesting and I think the the Bank of Japan have literally now said That they are very concerned about a devalued yen and the speed at which is devaluing which is really important And so there's talk of potential intervention or they're definitely, you know Looking to push the button Not saying that they will but there's definitely pressure a lot of pressure on them to do that 140 was the line in the sand And obviously prices have gone beyond that so now the Bank of Japan are under a lot of pressure to Potentially step in but they're gonna have to probably, you know wait a little bit just to see If you start to see prices drift higher again, I personally May take a little short not necessarily on the dollar yen, but But something like the euro yen or some other yen crosses So Yeah, as we as it stands i'm still a buyer of the dollar So if you get a you know quite a large pull back now The from from the higher one the 145s where it kind of just peaked Down to the 137s is you looking at what's that about about About 800 pips, right? So that actually is a nice pullback Into a nice demand zone And then we also know as well that at least the monthly fair value would be below that as well So for me, I do like The fact that this area here is a nice area of demand I think bets will be off if the Bank of Japan do come out and say that they are intervening I think for me then I will probably hesitate on on trying to short Any yen Crosses, I think I'd have to just wait for the dust to settle and for for really value to establish itself But until that happens if price is just naturally drift back down to this 137 area Without the bank intervening the bank of japan intervening then I think for me that is a decent That is a decent buy and also as well depends on you know what the dollar is doing From a monetary policy perspective But my buy still for now is to go along on this currency pair Looking at the dollar swiss and the dollar swiss actually has dropped like a stone From last week. So we had a nice supply zone here And then pretty much dropped away, right? There was I guess maybe some risk sentiment going on Um, and uh, yeah, we've come down into this demand zone, which I think is actually really nice But I'm not interested in necessarily buying this currency pair simply because you've got two central banks that are looking to high crates Um, but if you are looking to buy the dollar swiss, I think now is a decent opportunity If you sold the dollar and bought the swiss frank against the dollar, uh, that was a really nice, uh, technical level I did like that, but um That will obviously you're a bit too late for that and if you did get into that that would have been well done, right? um For me, I do think technically I do like this level here the 94 round number I do really like that technically, but I don't like the pair to trade But those are your options for the dollar swiss dollar CAD the canadian dollar hiked rates this week And so, um, you know capping the upside for the Dollar CAD Which basically meant that um, if you were looking to buy the canadian dollar and they hiked by I think it was 75 basis points Um, that was a nice little buy right just before they did and uh, yeah That's it. So, um, make sense. But again for me, you got two Pairs two currencies that are hiking rates and um, if you have if you do have two Currencies that are hiking rates and let's say that you that you can't or you shouldn't trade them Um, because ultimately what you should get is actually a an auction or a ranging market as It's typically known on the uh, uh on the web Well, in fact is actually um an auction and an auctioning market, which is basically, um You know where buyers the sellers in agreement of the exchange rate, right? So this has been, you know an expensive area. This is again an expensive area This has been a bargain area. So, um prices are unlikely to you know, continue trending above that with the Bank of canada Hiking 75 basis points, but there's also a caveat to that right as well because You have to have a really good economy in order to support rate hikes and then you know and to support price Now when we get to the euro, um, you'll see there's a difference between the canadian dollar You know price increasing in value And the euro even though the euro actually did um hike There are their rates by 75 basis points But for me not really a pair that i'm looking to trade But if you are that was really the opportunity for you to get short and buy the Canadian dollar against the u.s. Dollar if you are buying the u.s. Dollar right now I think against the canadian dollar. I think now is a decent time. I think probably for me Maybe a round I think the uh the best area would be the about the one two sevens if it can get down there, right? Nobody knows but if it does and you really think that the u.s. Dollar is a is a bargain against the canadian dollar Then that would be where you're looking to uh to buy uh new zealand dollar u.s. Dollar Uh new zealand dollar struggling at the moment against the u.s. Dollar that is and um Makes sense in a risk-off environment Commodity currencies don't tend to do too well against uh the dollar but um and again for me it wasn't really it's not really a Pair that i'm interested in taking although i am a buyer of the new zealand dollar against some other currencies That did say and been saying for the last couple of weeks that you know If I was going to be a buyer or seller my bias would be to the downside and you're seeing pretty much You know what has uh been happening now. We've got a bit of a pull back into some supply So if you are looking to get short here, that's a decent area. I think the one uh, sorry 0.625 areas I think is probably a nicer zone To get short if you want to get short on that from buying perspective I would probably rather wait for price to I mean there is I guess technically there is a demand zone here because it has made higher highs higher lows So in fact, I will draw it It was a demand zone, but um, I don't think it's necessarily a strong area of demand but Um, it could be right, but in the way that I trade I really want to see some other things just play out before if I was looking to buy this Um, namely I would want prices to really go kind of higher improve that there's demand Then pull back to that that demand zone and then look for to be a buyer if I was looking to be a buyer of the new zealand dollar us dollar Moving on to the pound dollar pound dollar again just keeps going lower, right? And uh, technically, um, I really want to be a buyer of the us dollar And short the the british pound somewhere around here, but we do have Liz trust Who is our new prime minister? her energy bill I get price cap may rescue uk inflation and it has And it may inflation may peak right so economists say freezing power and gas bills will cut inflation, which is basically what they The economy needs right so stimulus now points to higher rates for longer economists say so Economists at Barclays Beerenberg and Bloomberg Economics said trust is packaged to rain and soaring electricity and natural gas prices Would have a sweeping impact on the uk economy and may manage to arrest inflation at july's reading of 10.1 percent so Um, you know the forecast underscore the importance of surging The cost of living squeezed to the outlook of the uk by halting increases in utility bills The government could stop the biggest upward pressure on consumer prices giving giving the both consumers And investors breathing room To adjust to other pressures pushing up the cost of goods and services, right? So if consumers You know as far as like myself paying less on energy and And and food and things like that or energy in more specific Then it can you know help The cost of living crisis. That's basically the point in that and there was something else interesting in here Was it here? I think um Yeah, there was there was talk about, you know Barclays said inflation would fall back to 5 percent in 2023 And it says capping energy prices would provide a welcome help to the bank of england in regaining control of inflation dynamics Montague wrote prolonged hikes next to Inter next year are now less likely in our view, but there was something else in here that I thought I had uh highlighted when was it where was it? Yeah, maybe it wasn't in here, but um, but yeah, so So, uh, that's that's uh, that's pretty much it. I think um Uh, you know, I think the the fact that The the bank of england, um, you know if inflation does start to come down the bank of england They're probably likely to you know hike less um, and In fact, it could improve the economy And so if the data starts to come in that the uk Actually may avoid a recession as unlikely as that seems or maybe not go into a deep recession Then in fact the pound could be a potential buy But for now against the us dollar, which is I think because I'm much better footing I do think any pullbacks uh to that supply zone for me are really decent the 118 levels I think are decent uh shorts Um right now again going long on the pound. You're really going long into um into a historical level Yeah That happened in 2020, but um Um, I don't think for me. I think that's probably more profit-taking as a reaction Um, but who knows right in terms of you know buying a pound for me There's not enough fundamentals for me to buy the pound At this area here because if if the market doesn't like liztrust is um, you know What doesn't think that her policy is going to you know, um Help the economy or as much as you know is Is perceived then the the the pound is still going to be um, it's not going to be a bog in here Is it you know, it could fall further? So For me the path of these resistance is still continuing to the downside and so um against the dollar anyway And that's where my continued bias is And um, if you guys want to you know find out a bit more about fundamental analysis, um I am actually closing up a shop soon as far as enrollment 11th of september is when I'm going to be closing and not only do you get access to you know all the advanced supply and demand Zone strategies that I teach um as well as capture pain relief stock hunts member only discord trade room access which is here um as well as um a fundamental analysis spreadsheet and really you know It's it's all about you know the mentoring because as much as um, you can um look at these things by yourself and watch You know a hundred a thousand videos Unless you have somebody really to hold your hand and guide you through, you know What is going on on a on a daily and weekly basis? Um, it's going to be very difficult for you to kind of make it by yourself There's only so many videos you can watch right how many videos have you watched on technical analysis? But yet you're still here. Um, you know chasing your tail trying to figure out how to be profitable in trading. So Um, you know the mentoring group is not like an other other chat rooms where you can come in and start You know people just chat nonsense. We're very focused On our fundamental analysis and technical analysis and if you do feel that you may need Some coaching and some mentoring Personal as I do have a weekly calls as well live calls on a wednesday These are also recorded This is one of our private pages where Trading 180 videos as well where every wednesday We have a group call. So last one was um on the 7th of september Which was here and we spoke about what is priced in information overload reducing rates and thinking in terms of versus thinking in terms of weakness bank of japan intervention You know the pound of buy and I put a question mark for me It's not but people were asking that question. Um, and lots more as well. That's a two hour Live group call so you can ask me questions Here's one here as well as just, you know, some other daily videos that I produce as well So um and weekly videos that I produce so lots and lots and lots of content on top of, you know, the course You know as well from a from each From each discipline so supply and demand trading course right there capture pain relief right there channels And you've got stop hunts fundamental analysis loads and loads and loads and loads of content So if you do want to join um 11th of september will be a deadline If not and you want to continue to watch my videos, I definitely advise you to go through my, you know, free youtube stuff. Um, and uh And if you want to join next time or not, you know, I definitely will I look forward to working with you if not, then I do wish you all the best in your trading endeavors anyways, let's get back to the charts And uh, let's go to the euro dollar euro dollar Euro dollar i'm actually in short on this um and as are many of the guys in the trading room It's a setup again that not really going to go into on the uh in this videos beyond this video, but From a fundamental perspective um, we did have The euro hike rates, right? And many people are kind of scratching their heads as to, you know, well, they hiked rates So why isn't the euro, you know flying off the shelf? Why aren't investors? You know looking to You know buy the euro and why isn't the euro going higher and Really, I mean, we've we've known this a trade in 180 That it's all really about the the economy and the energy crisis. That is what is really holding the euro back. So The euro trading is not about the energy crisis. Yeah, sorry. It's all about the energy crisis not ECB rate hikes So the euro fell below the dollar parity after the ECB rate decision The ECB is fighting gravity in FX markets says danska bank, right? So danska bank As many as many other banks are the big money And one of the quotes, I guess like I kind of wanted to show you guys was Where was it now? Uh, yeah, it says it said that the Oh, where was it? There's a quote in here. Uh, the ECB is fighting gravity in markets. It's christopher kejar holmholt uh head of FX researcher at danska bank euro dollar has more downsides in so and ECB decision does not change that and um, it says the rate the rate announcements Uh, um, it's in sorry in its race rate announcement that ECB struck a hawkish tone saying that expects to raise rates further um Raise rates further as inflation remains far too high. Um, and it says down here We do not target. No, where was it a euro strategy? So united in their predictions that the way forward is more weakness. Yes, the more weakness for the euro danska banks lohmholt said Fair value for the currency is closer to 90 cents compared with current trading around 99 cents I suspect that the ECB will have to produce these big heights just to hold the euro there So steven barrow an FX strategist as standard banks. So, um, you know bank quotes, you know, these guys are The smartest guys, right? way smarter than us And so if they're saying that potentially, you know, prices could actually fall down to the 90 cent area There's about a thousand pips from here. So, um That all obviously depends on the energy crisis and whether, um You know, russia will turn off the taps and um, there's an article which says a short says germany prepared for russia gas halts over ukraine So again, they have to prepare for these things. There is there is a Something on the horizon and again, nobody knows, right? The euro could be at least a short term buy if there's a resolution with With russia, it doesn't look like it is probably unlikely. But if there is or it's, you know, they managed to kind of Help the economy somehow and you know cap gas prices without having other effects on, you know inflation, etc And the economy then um, then that would be great, but It's it's um, it is an it is a possibility, right? So germany has prepared for russia to largely cut off gas supplies because of the war against ukraine So and it doesn't look like the war is um, doesn't look like the war is uh, is is is going to end anytime soon. So um Yeah, when it comes to The euro for me at this present time With the all the uncertainty around the euro for me. It's just no brainer to go To go short, right? Could I be could I be wrong about this? Yeah, if I'm wrong, it's okay. I'll just lose one trade Right, it's not the biggest deal in the world um, but you know, nobody knows what's gonna happen, but Um, it looks like the path of these positions is to the downside now that could change temporarily um If they do manage to find a way Some sort of resolution and I do think that the euro will rally either way though for now, um, still to the short side if you are going short then any pullbacks to a um, a supply zone daily supply zone like the Around here or the one of freeze. I think of you know, very good levels to look for potential short trades um If you are looking to be a buyer of the euro then any pullbacks into the 99 cent area Is probably going to be your best bet and taking that chance and that gamble right in in the fact that the euro will come out unscathed On the other side of this conflict and the economy um looking at the ozzie dollar and ozzie dollar was saying Last week and week before that. I do think that technically I do like this level Really nice level, but for me it would be very hard to buy The australian dollar against the u.s. Dollar Um, I'm a buyer of the australian dollar just not against the u.s. Dollar. So, um, yeah, it reacted from a technical perspective Which was uh, I wouldn't say surprising, but I think the the australian dollar they did high crates and they had decent GDP numbers as well. So that actually helped with um, you know price Stabilizing here. I guess if you want to call it stabilizing but it was a bargain really here and so, um, you know, uh, if if it was going to go any lower Then that would have meant probably that the australian dollar would have been seen as not a bargain here, right? But obviously, uh, what's helped the australian dollar maintain its bargain Value around here my buyers temporarily. I guess I've got involved in this and um, is because of the recent fundamentals so Um, yeah, I don't really have a a bias either way. Um, in terms of me trying to trade this but Um, if I did if I did then it would still be You know, I would say buying the u.s. Dollar But I'm not looking to, you know get involved in this trade But if you are I think now it's a okay time I think the top end of this, you know, the 70 cent area is probably the best area to look for For short trades or say the best area, but it's a decent area and even better would be The 71 around a 71 cent area If you were looking at getting shorts getting long, obviously you'd have to wait for a pullback Into the back into that, you know 0.675 67 and a half cent area before looking at getting long Aussie yen and Aussie yen again just that the yen is a as um Devalued um, uh this week had an effect on obviously the Australian dollar And um, Australian dollar Japanese yen and now we have Really a decent area to look for any kind of long trades now Again, I think the market really is ignoring. Um Any kind of risk off sentiment so, um Yeah, I think any pullbacks into this zone You know the 95 area, I think it's actually a decent, uh, you know decent buy for the Australian dollar Providing that again the Bank of Japan doesn't come in and start to intervene into markets And then for me, I think all bets are off when it comes to buying or trying to sell the yen And buying anything against the yen until really the dust is settled So any pullbacks into that demand zone for me are decent and Gold right gold Of this, you know Historical level over the past is type several times matter of fact. So for me, I'm not Looking to trade gold so much as I am looking to you know buy in terms of the physical metal But I do think that this is going to be a bit of a long-term play I think as the dollar starts to strengthen if it does strengthen, right? Let's say for example CPI comes out And it goes higher And then the Fed raised by 75 basis points um Then I do think that gold Potentially could still start to drop A little bit more but at these beyond these prices. I think gold is an absolute Steel in the coming future central banks. There was an article I couldn't find it before I recorded it, but there was an article Again another article because I think I posted one earlier in the year about central banks increasing their gold holdings. So that's become another recent report so Central banks, you know, I'm not increasing their gold holdings for for no reason, right? You know, so for me I think I'm just buying any any in and around this area for me is Is decent is definitely a decent prices again Not financial advice not telling you to go out to buy physical gold or trade gold or anything like that If you don't want to trade it, don't trade it, right? If you do, but you're unsure then still stay out, right? but The point being is that for me my bias actually long-term bias is to is to go long gold my short-term bias Is as long as the dollar continues to Strengthen then it just pushes gold down to cheaper prices, right? It just continued to push gold down and made this may start to Break and again the more times the level is touched The less of a bargain it becomes right because it becomes now common price So don't be surprised if you see gold actually break this this level and come down to maybe this longer term Demand zone just below it, which is from April 2020 the 1647 to 1656 area So, uh, so yeah, but if you are looking to potentially get short on gold Then I think the nearest area to look for short trades is going to be beyond that monthly moving fair value and then back down into the Um, I'll back up into that supply zone before looking at, you know, continued short trades anyways guys, that's it for this week again. Don't forget that the trading 180 Enrollment closes tomorrow. So if you do want to join, um, you know feel free to join and reach out to me if you've got any questions and If not, I wish you all the best have a great trading week And I'll speak to you all soon


Hey guys, welcome back to Six Sisters Stuff. Today I'm going to tell you 16 things you didn't know about. Wait! So we are here in Grapevine, Texas at the amazing Great Wolf Lodge. There are so many things you can do here that I'm just going to share with you some of the things you might not have known. You can now find a Great Wolf Lodge pretty close to you because there are 18 locations. You can check out this list to see if there's one close to you. Number two, wolf ears. Every time you go to Great Wolf Lodge and you stay there, every kid gets a pair of wolf ears. Now, you probably already know this, but every time you stay at the Great Wolf Lodge, you get to go to the water park for free. The best part is, the day that you check in, you can go to the water park at one o'clock that day, even before you get your room. And then the second day that you're there, you actually get the water park all day long until it closes. Even when you check out of your room, you get to stay and play at the water park. Now this is one of the biggest indoor water parks I've ever seen and I love it because you don't have to put sunscreen on your kids when you're here. They have huge water slides for teenagers and for kids too. It really is perfect for all ages. Number four, Great Wolf Lodge comes with life jackets and puddle jumpers so it's just one less thing to pack for you. Number five, get there early to the water park. So I stood in line about 20 minutes before the doors opened and this is my favorite place to go. It's up on top, kind of, by the ropes course, but you can overlook everything. So my kids can go play, my baby can sleep. It's just kind of a little secluded area. Number six, this is one of my most favorite things about Great Wolf Lodge is the wrist band key. So in order to get to the water park, you have to have a wrist band. It's also the key to your room. Number seven, now if you have little kids going with you to Great Wolf Lodge, it's worth a little bit of extra money for the kid cabin. It is so fun because it's both bunk beds and there's a little TV in there and my kids and their cousins had a blast. Number eight, you can bring food into the park. That is the best thing about this. So I pack my lunches, sometimes I'll pack my dinner if we're staying for multiple days because you can bring food from the outside in. They also have refrigerators, so if you see, she has a little bottle of milk. This is our second day there. Now with four kids who get hungry very often, it works out well for us so I can bring healthy snacks into the park. We usually eat at Great Wolf Lodge for breakfast and dinner. So this is our pizza from Hungry Wolf Pizza that we bring back to our room to eat. In the night, we love to sneak off to the Bear Paw Sweets and Eats because they have tons of desserts to pick from. Number nine, now I know this might sound funny, but the Great Wolf Lodge kind of has like an underground city full of tons of stuff to do right inside of their hotel. They have things for all ages, even for teenagers. And they have fun shops for the parents too. Number ten, a paw pass. Now you can get different passes that will save you money and you can do lots of different activities if you get one of these paw passes. So when we went, we got the paw pass for our kids and let me show you what's on it. So number eleven, it's called Magite Quest. Now first you get to pick your magic wand from the little shop, then they'll activate it for you and you get to go on a treasure hunt. It literally is the biggest treasure hunt because it covers all eight floors of Great Wolf Lodge. My kids love it. Number twelve, the clubhouse crew. Now if you have some smaller children, they have like a build-a-bear setup where you get to pick your animal, you get to stuff it, but you get to take these animals on an adventure. You just need to find these screens throughout the hotel and it's so fun. The little kids will love this. Now if you have teenagers, I think this is the biggest arcade I have ever seen. And it's not too difficult to win a lot of tickets, that's one thing the girls love about it. Now it doesn't have to be just for teenagers. My husband and I also had a marvelous time playing a lot of these games. It's for all ages. Number fourteen, the candy. So the Great Wolf Candy Company has a wall of candy that the kids can fill up their very own cup full. I think my kids were in complete heaven and then they also got to bring this cup as we went swimming. It was kind of like their own little snack. They absolutely loved it. Number fifteen, every time you go to Great Wolf Lodge, you get a free Shutterfly book. You get a little coupon while you get a free 8x8 photo book so you can record all your memories from Great Wolf Lodge. Number sixteen, breakfast. Now every time we come, we usually do the buffet, the loose mousse cottage kids breakfast because they have everything you can possibly imagine. This is actually the kid area. They have fruits, they have mini pancakes, they have mini waffles, they have donut holes. If you're lucky, your kids might even pick one of the donuts or the cupcakes because really sometimes you just need a cupcake on vacation. Now this buffet breakfast is actually a character breakfast and my kids loved it. Now I'm throwing in a bonus because there are so many free activities you can do at Great Wolf Lodge. Almost every hour they have something planned that your kids can be involved in. From bingo to story time, there's so many different things. Alright you guys, those are my tips for Great Wolf Lodge. If you ever want to go to one, go check out my locations and see where the closest one is to you. Now because my husband is finishing up residency, this was actually our big vacation for the year. It worked perfect for us because we didn't have to fly anywhere and my husband could get back to work while we could still play. Alright guys, see you later!