Convection The convection heat transfer mode is comprised of two mechanisms. In addition to energy transfer due to random molecular motion (diffusion), energy is also transferred by the bulk, or macroscopic, motion of the fluid. This fluid motion is associated with the fact that, at any instant, large numbers of molecules are moving collectively or as aggregates. Such motion, in the presence of a temperature gradient, contributes to heat transfer. Because the molecules in the aggregate retain their random motion, the total heat transfer is then due to a superposition of energy transport by the random motion of the molecules and by the bulk motion of the fluid. The term convection is customarily used when referring to this cumulative transport, and the term advection refers to transport due to bulk fluid motion. Convection heat transfer may be classified according to the nature of the flow. ˆ forced convection when the flow is caused by external means, such as by a fan, a pump, or atmospheric winds. ˆ Free (or natural) convection, the flow is induced by buoyancy forces, which are due to density differences caused by temperature variations in the fluid. Mixed (combined) forced and natural convection conditions may exist, if 1
Figure 1: Velocity boundary layer on flat plate velocities associated with the flow of are small and/or buoyancy forces are large. 1 Velocity Boundary layer consider flow over the flat plate of When fluid particles make contact with the surface, their velocity is reduced significantly relative to the fluid velocity upstream of the plate, and for most situations it is valid to assume that the particle velocity is zero at the wall. These particles then act to retard the motion of particles in the adjoining fluid layer, which act to retard the motion of particles in the next layer, and so on until, at a distance y = δ from the surface, the effect becomes negligible. The quantity δ is termed the boundary layer thickness, and it is typically defined as the value of y for which u = 0.99. The boundary layer velocity profile refers to the manner in which u varies with y through the boundary layer. With increasing distance from the leading edge, the effects of viscosity penetrate farther into the free stream and the boundary layer grows 2
Figure 2: Thermal boundary layer on flat plate 2 Thermal Boundary layer Fluid particles that come into contact with the plate achieve thermal equilibrium at the plates surface temperature. In turn, these particles exchange energy with those in the adjoining fluid layer, and temperature gradients develop in the fluid. The region of the fluid in which these temperature gradients exist is the thermal boundary layer, and its thickness is δ t Is defined as the value of y for which the ratio (T s T ) (T s T ) = 0.99. At the surface the conductive heat transfer can be given by q s = k d(t s T ) y=0 dy heat transfer rate from Newton s law of cooling is q s = h(t s T ) 3
At surface these two terms are equal so from???? we can write k d(t s T ) y=0 = h(t s T ) dy rearranging we get h k = d(t s T ) dy (T s T ) y=0 Making it dimensionless is known as Nusselt number and is non dimensional temperature gra- hl k dient at the surface. hl k = d(t s T ) dy (T s T ) L y=0 the wall temperature gradient dt/dy y=0, determine the rate of heat transfer across the boundary layer.decreases with increasing x, and it follows that and h decrease with increasing x. 3 Local and Average Convection Coefficients A fluid of velocity V and temperature T flows over a surface of arbitrary shape and of area As. The surface is presumed to be at a uniform temperature, T s. If T s T, convection heat transfer will occur. The surface heat flux and convection heat transfer coefficient both vary along the surface. The total heat transfer rate Q may be obtained by integrating the local flux over the entire surface. 4
Q = q da s A s Using Newtons law of cooling we can write, q = (T s T ) h da s A s If we define h as average heat transfer coefficient we can write h = h da s A s For flat plate we can write h = L 0 h da s 4 Laminar and Turbulent Velocity Boundary Layers Consider flow over a flat plate. On this plate laminar and turbulent flow conditions both occur, with the laminar section preceding the turbulent section. In the laminar boundary layer, the fluid flow is highly ordered and it is possible to identify streamlines along which fluid particles move.the highly ordered behavior continues until a transition zone is reached, across which a conversion from laminar to turbulent conditions occurs. Flow in the fully turbulent boundary layer is, in general, highly irregular and is characterized 5
Figure 3: Boundary layer on flat plate by random, three-dimensional motion of relatively large parcels of fluid. In determining whether the flow is laminar or turbulent, it is frequently reasonable to assume that transition begins at some location x c. This location is determined by the critical Reynolds number. Re c = u x ν = 5 10 5 5 Internal Flow Consider laminar flow in a circular tube of radius ro, where fluid enters the tube with a uniform velocity. We know that when the fluid makes contact with the surface, viscous effects become important, and a boundary layer develops with increasing x. When velocity profile no longer changes with increasing x. The flow is then said to be fully developed, and the distance from the entrance at which this condition is achieved is termed the hydrodynamic entry length. For laminar flow (Re 2300), the hydrodynamic entry length 6
is given by ( x D) laminar 0.05Re Similarly thermal entry length is given by ( x D) laminar 0.05Re P r The properties of the fluid are evaluated at Mean film temperature for external flow and Mean bulk temperature for internal flow. They can be calculated by?? and?? T f = T s + T 2 T b = T in + T out 2 the critical Reynolds number for internal flow is a smooth pipe Re c r = ρ v D h µ 2300 where D h is hydraulic diameter and is given by D h = 4 Area of cross section perimeter 7
6 Prandtl Number TThe Prandtl number is defined as the ratio of the kinematic viscosity, also referred to as the momentum diffusivity, ν, to the thermal diffusivity. It is therefore a fluid property. The Prandtl number provides a measure of the relative effectiveness of momentum and energy transport by diffusion in the velocity and thermal boundary layers, respectively. Figure 4: Boundary layer thickness comparison with different Prandtl numbers Prandtl number of gases is near unity, in which case energy and momentum transfer by diffusion are comparable. In a liquid metal, Pr < 1 and the energy diffusion rate greatly exceeds the momentum diffusion rate. The opposite is true for oils, for which Pr >1. From this interpretation it follows that the value of Pr strongly influences the relative growth of the velocity and thermal boundary layers. P r = ν α 8
7 Natural Convection The egg is cooled by transferring heat by convection to the air and by radiation to the surrounding surfaces. Disregarding heat transfer by radiation, the physical mechanism of cooling a hot egg in a cooler environment can be explained as follows: Figure 5: Cooling of an egg due to Natural convection The temperature of the air adjacent to the egg is higher, and thus its density is lower, since at constant pressure the density of a gas is inversely proportional to its temperature. Thus, we have a situation in which some low-density or light gas is surrounded by a high-density or heavy gas, and the natural laws dictate that the light gas rise. This phenomenon is characterized incorrectly by the phrase heat rises, which is understood to mean heated air rises. The space vacated by the warmer air in the vicinity of the egg is replaced by the cooler air nearby, and the presence of cooler air in the vicinity of the egg speeds up the cooling process. The rise of warmer air and the flow of cooler air into its place continues until the egg is cooled to the temperature of the surrounding air. The motion that results from the continual replacement 9
of the heated air in the vicinity of the egg by the cooler air nearby is called a natural convection current, and the heat transfer that is enhanced as a result of this natural convection current is called natural convection heat transfer. Note that in the absence of natural convection currents, heat transfer from the egg to the air surrounding it would be by conduction only, and the rate of heat transfer from the egg would be much lower. The upward force exerted by a fluid on a body completely or partially immersed in it is called the buoyancy force. The magnitude of the buoyancy force is equal to the weight of the fluid displaced by the body. That is, F b = ρ fluid g V body In heat transfer, the primary variable is temperature, and it is required to express the net buoyancy force in terms of temperature differences. But this requires expressing the density difference in terms of a temperature difference. The property that represents the variation of the density of a fluid with temperature at constant pressure is the volume expansion coefficient (β), defined as: β = 1 ρ ( ) ρ p p and we can show that for an ideal gas β = 1 T Where T is absolute temperature 10
Figure 6: Natural convection on vertical plate The buoyancy force is proportional to the density difference, which is proportional to the temperature difference at constant pressure. Therefore, the larger the temperature difference between the fluid adjacent to a hot (or cold) surface and the fluid away from it, the larger the buoyancy force and the stronger the natural convection currents, and thus the higher the heat transfer rate. The plate is immersed in an extensive, quiescent fluid, and with T s > T the fluid close to the plate is less dense than fluid that is further removed. Buoyancy forces therefore induce a free convection boundary layer in which the heated fluid rises vertically. The resulting velocity distribution is zero as y, as well as at y = 0. Flow regimes in natural convection is governed by a parameter called as 11
Grashof Number Gr = gβ(t s T L 3 c ν 2 Grashof number is a measure of the ratio of the buoyancy forces to the viscous forces acting on the fluid. It serves the same purpose as Reynolds number in forced convection. Rayleigh number will correlate the buoyancy and viscous effects to determine the type of flow in natural convection. Rayleigh number is given by Ra = gβ(t s T )L 3 c να For vertical plates the critical Rayleigh number is 10 9 The characteristics length for vertical and inclined plates is length, for cylinders and spheres is D and for all other geometries it is A s /P 12
8 Dimensionless Numbers Table 1: Dimensionless Numbers Group Definition Interpretation Nusselt number Prandtl Number Grashof number Peclet number Coefficient of friction hl k ν α gβ(ts T L 3 ν 2 V L α τs ρu 2 2 Ratio of convection to pure conduction heat transfer Ratio of the momentum and thermal diffusivities Measure of the ratio of buoyancy forces to viscous forces Ratio of advection to conduction heat transfer rates Dimensionless surface shear stress Some important Correlations for determining Nusselt Number The local Nusselt number at any location x on flat plate is given by Nu x = 0.332Re 0.5 x P r 1 3 LaminarF low Nu x = 0.0296Re 0.8 x P r 1 3 T urbulentf low The average values of Nusselt number can be obtained by integrating the 13
above equation over the length of the plate (ref??) Nu = 0.664Re 0.5 x P r 1 3 LaminarF low Nu = 0.037Re 0.8 x P r 1 3 T urbulentf low In some cases flat plate is sufficiently long for the flow to become turbulent, but enough to disregard laminar region, in such cases average Nusselt number is given by Nu = (0.037Re 0.8 x 871)P r 1 3 For internal laminar flow the conditions could be constant surface heat flux ( q = constant) or constant wall temperature (T s = constant). The Nusselt number in both the cases is constant along the length and is given by Nu = 4.36 ( q = constant) Nu = 3.66 (T s = constant) 14
References [1] Incropera, Frank P., Fundamentals of Heat and Mass Transfer, John Wiley & Sons, 2012 [2] Yunus Cengel, Heat and Mass Transfer, TataMcgraw-Hill Companies, 2012 15