Some Observations on Natural or Free Convection

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1 Some Oservations on Natural or Free Convection Sometimes a single phsical process in nature can explain a variet of events. Free convection is one such process. It functions ecause heated fluids, due to their lower densit, rise and cooled fluids fall. A heated fluid will rise to the top of a column, radiate heat awa and then fall to e reheated, rise and so on. Gasses, like our atmosphere, are fluids, too. A packet of fluid can ecome trapped in this ccle. When it does, it ecomes part of a convection cell. Convection cells can form at all scales. The can e millimeters across or larger than Earth. The all work the same wa. The convection that ou are most likel to have oserved is in cumulonimus clouds or "thunderheads." These towering vertical clouds can e seen to evolve over a few minutes. The tops of the clouds have a sort of cauliflower appearance as warm moist air rises through the center of the cloud. The moisture in the cloud condenses as it cools. The air gives up some of its heat to the cold high altitude air and egins to fall. As the air falls along the exterior of the cloud, it returns to warmer low altitudes where it can e caught up in the rising column of air in the center of the cloud. This fountain-like cell can form alongside other cells, and a packet can move etween cells. Hail forms when water droplets, carried the strong updrafts, freeze, fall through the cloud and are caught in the updraft again. An additional laer of water freezes around the ice all each time it makes a trip up through the cloud. Eventuall, the hail ecomes too heav to e carried up anmore, so it falls to the ground. Large hailstones, when cut apart, show multiple laers, indicating the numer of vertical trips the stone made while it was caught in the convection cell. Convection also occurs on the Sun. A high resolution white light image of the Sun shows a pattern that looks something like rice grains. Ver large convection cells cause this granulation. The right center of each cell is the top of a rising column of hot gas. The dark edges of each grain are the cooled gas eginning its descent to e re-heated. These granules are the size of Earth and larger. The constantl evolve and change. Thunderheads and granulation are large-scale examples of convection. Fortunatel, there are examples of convection that fit into a classroom. An excellent example can e seen in hot Japanese Miso (soean paste) soup. The interior of the roth is hot. The surface of the soup is exposed to cool air. Hot packets of fluid rise out of the interior of the soup to the surface where the give off heat. Now cooled, the fall down into the owl to e re-heated. Left alone, the soup will dissipate its heat in this wa (and through conduction with the sides of the owl) and reach room temperature. The soean paste granules and other ingredients will highlight the convection cells vividl. As students gaze into their soup, the will see the rising and descending columns of fluid. The cells will evolve and change their positions. Dark ottomed owls show the effect est. If the soup is stirred up, students can oserve the cells reform. Of course, the demonstration material can e consumed at the conclusion of the demonstration. Free onvection acts as descried in the examples aove where gravit's effects are present (so that warm, low densit fluids can rise and cool, high densit fluids can fall). What happens in the weightlessness of space where up (rise) and down (fall) have no meaning? ChE - Lecture 15 1

2 How do We Descrie Free convection? Free convection is driven densit differences. In order to descrie the process, we can express the change in densit as a function of temperatuer in terms of a Talor s Series T = T + T T T +... The coefficient of volumetric expansion is β = 1 T so that the Talor s series can e expressed more simpl. T = T 1 + β T T +... Role of uoanc on the flow field If there is a gravitational field, then there is a uoanc force that acts on the fluid to impart motion to the fluid. The induced motion would e influenced the viscost or viscous drag. Such motion we term natural convection or free convection. In a column of fluid of height L, with a cross section A (normal to x-axis), the od force acting across the ends of the column is BAL. If the od force were unopposed drag or viscous forces the momentum of the fluid would e of order ( u) 2. The uoanc driven velocit u would e no larger than u = O gl 1 /2 ChE - Lecture 15 2

3 One can express the velocit in terms of a Renolds Numer Re = u L ν = gl ν 2 The natural numer to descrie natural convection is the Grashof Numer, a merasure of the uoanc forces to inertial forces. ut recognizing that ehaves leads to a simpler expression. Gr = gl ν 2 = β T Gr = gl ν 2 β T Buoanc-induced Flow in a Confined Space Suppose two vertical plates located 2 apart, One palte is heated and maintained at T 1 The second plate is set at T 2. Suppose as well that all the phsical properties are independent of temperature. The temperature field does not change apprecial. Now the energ equation looks like L: 1 /2 v + v z z = α 2 T T z 2 The velocit field is one dimensional so that v 0,v z () = εv z0,v z0 () ChE - Lecture 15

4 Then the temperature equation simplifies to d 2 T d 2 = 0 where T = T 1 at = T = T 2 at = + and the solution can e expressed as T = T m 1 2 T T = T 1 T 2 ; T m = 1 2 T 1 + T 2 The Navier-Stokes equation descriing the flow field in the z-direction is d 2 = dp dz + g For this mode, we make the Boussinesq approximation that is all properties are T-independednt save densit. d 2 = dp dz + g βg T T Now if we examine the perturation expansion of v z applied to the N-S eqaution, we oserve that since v z = v z0 + εv z 1 The leading term of the perturation expansion is the solution to 0 = = dp dz + g That is, in the zeroth approximation, there is no flow in the z-direction as a result of the temperature field... the pressure field is given the hdrostatic law... the pressure field is alanced the gravitational force. However, in the first approximation, the velocit field is governesd n the alance of viscous forces and uoanc forces. 1 d 2 = βg T T ChE - Lecture 15 4

5 The conservation equations are actuall coupled, ut in this first approximation, we ma treat them as uncoupled µ d 2 1 v z = βg T d 2 m T 1 2 T The oundar conditions are v z = 0 at = - and at = + The solution is simple. where v z1 = βg 2 T 12µ A A = 6 T m T T 2 + A However, there should e no net flowhat is : v z d = 0 This demands that A = 0, so v z1 = βg 2 T 12µ We can make the velocit dimensionless v z 1 µ = 1 12 Gr Gr is the Grashof numer and is a measure of the uoanc forces to the viscous forces. ChE - Lecture 15 5