UNIT 4 HEAT TRANSFER BY CONVECTION

Transcription

1 UNIT 4 HEAT TRANSFER BY CONVECTION 4.1 Introduction to convection 4. Convection boundar laers 4..1 Hdrodnamic boundar laer over flat plate 4.. Thermal boundar laer 4..3 Concentration boundar laer 4.3 Dimensional analsis of convection 4.4 Boundar laer equations Conservation of mass-continuit equation 4.4. Momentum equation Energ equation Conservation of species concentration Boundar laer similarit. 4.5 Eact laminar boundar laer solution Blasius solution for continuit and momentum equation in Hdrodnamic boundar laer region Pohlhausen solution for energ equation 4.6 Approimate integral boundar laer solutions Integral momentum equation 4.6. Integral energ equation 4.7 Analogies between momentum and heat transfer 4.8 Empirical relations for turbulent flow over flat plate 4.9 Some other important empirical relations for forced convection Introduction to free convection 4.11 Governing equations for natural convection in boundar laer region 4.1 Non dimensionalising the set of governing equations for free convection 4.13 Some important correlations for free convection. Previous ear Gate questions Previous ear IES conventional questions

2 4.1 INTRODUCTION TO CONVECTION So far we have studied energ transfer b means of conduction which does not involve an macroscopic bulk transport of matter. We have considered convection as a boundar phenomenon to provide a possible boundar condition for solving conduction problems. In this chapter we deal with basic and phsical mechanism of convection energ transfer along with stud of various parameters affecting convection. We know that convection involves flow of fluid and convection heat transfer rate depends upon flow characterstics of fluid and various fluid properties. Man heat transfer problem involves diffusion of particles due to concentration gradient between two mediums involving convection heat transfer apart from bulk fluid motion. For eample flow of air over a water bod involves evaporation (diffusion) of water vapour molecules into the unsaturated air stream. This diffusion of particles in convection phenomena is known as mass transfer. There are man analogies between mass transfer and heat transfer and hence both these phenomenans are studied in correlation with each other. 4. CONVECTION BOUNDARY LAYERS From the knowledge of fluid mechanics, we know that for flow of fluid over a surface, there eist a hdrodnamic boundar laer characterized b velocit gradient and shear stresses. Apart from hdrodnamic (velocit) boundar laer, there ma eist two other tpes of boundar laers namel Thermal boundar laer and concentration boundar laer. Consider an eample of air flow across a perspiring human bod. Apart from hdrodnamic boundar laer formed on surface of human bod, we observe that a thermal boundar laer as well as concentration boundar laer are also formed on surface of human bod. Various metabolic activities going in human bod generates heat continuousl in the bod. This heat is required to be dissipated to the surrounding at the same rate at which it is generated to maintain an optimum temperature of bod. This heat dissipation is generall due to evaporative, convective and radiative heat loss. Radiation heat loss is generall neglected. If ambient air temperature is less than bod temperature, heat will be convected to the moving air due to temperature difference and a thermal boundar laer will be formed characterized b temperature gradient. Similarl sweat on the human bod will evaporate into unsaturated ambient air and a concentration boundar laer will be formed characterized b concentration gradients. Note that thermal boundar laer would not have formed if there was no temperature difference between the bod and ambient air. Similarl no concentration boundar laer would have formed if the ambient air was saturated (Air having relative humidit 1 is called saturated air) 4..1 HYDRODYNAMIC BOUNDARY LAYER OVER FLAT PLATE When a real fluid flow post a solid object, fluid particles adjacent to the surface will have same velocit as that of surface. That is the fluid particles adjacent to the solid surface sticks to it and this condition is known as no slip condition. For a static surface, the fluid particles adjacent to the surface will have zero velocit. From Figure 4.1 it can be seen that velocit of fluid laer just above the static laer will be equal to free stream velocit U at the leading edge of flat plate. As the fluid moves downstream awa from leading edge, motionless laer because of viscous force between two adjoining laers. In other words we can sa that viscous force starts penetrating the

3 laers of fluid above static laer and velocit gradients eists in a region close to the u boundar with decreasing in downstream direction. The region of flow above 0 stationar object in which effect of viscous shearing forces causes retardation of fluid motion is called hdrodnamic or velocit boundar laer. Another effect of viscous force is surface drag, or skin friction drag. The force eerted b a flowing fluid over a surface in flow direction is called surface drag. Boundar laer thickness is a function of coordinate direction and can be measured as the distance from the boundar to the point where velocit is 99 % of free stream velocit. Flow in the boundar laer region is known as viscous flow due to pressure of viscous shear forces. However flow above the boundar laer region have negligible viscous force and uniform free stream velocit U. This region of flow is known as invisaid flow region. U u u u u > > > Boundar laer 0 A 0 B 0 C 0 D U U U A U Static plate U() B C D Figure 4.1: A, B, C, D represents various points of static fluid laer adjacent to plate Hdrodnamic boundar laer has 3 distinct flow regions. a. Laminar flow region Near the leading edge, the flow is highl organized characterized b movement of adjacent fluid laers sliding past one another in straight lines parallel to solid surface without an lateral miing of fluid particles. b. Transition flow region This is a flow region characterized b the process of laminar flow turning turbulent. As the flow velocit increases, tendenc for lateral miing of fluid particles also increases and the laminar flow slowl turns into unorganized flow with cross current perpendicular to the direction of flow and eddies or swirls of fluid. c. Turbulent flow region This is the last stage of flow in highl unorganized manner which occurs at high velocit characterized b velocit fluctuations. Generall, the highl viscous fluids (e.g. oils) with low flow velocities have laminar flow and fluids with low viscosit and high flow velocities have turbulent flow characterstics. Osborne

4 Renolds demonstrated transition from laminar to turbulent in an eperiment conducted in 1880 and established a governing parameter, the dimensionless Renold number. Renold number will be discussed later in detail. Distance from leading edge of the plate at which transition from laminar to turbulent flow occurs is known as critical length and it is represented as as shown in figure 4.. cr Laminar boundar Laer region Transition region Turbulent boundar laer region cr Figure 4.: Hdrodnamic Boundar laer over a flat plate Shear force per unit area (Shear stress) developed at surface of plate due to resistance offered b surface of plate to the flowing fluid is given b u Where is a fluid propert known as dnamic viscosit. A commonl used dimensionless parameter, local skin friction coefficient is developed to determine relationship between frictional force and shear stress developed at the surface s C f U / Note that velocit gradients are maimum at the surface near the leaching edge and progressivel decreases along with distance from the leading edge in laminar boundar laer region. Hence shear stress and skin friction coefficient var with distance from leading edge. 4.. THERMAL BOUNDARY LAYER A thermal boundar laer is developed over a surface due to temperature difference between flowing fluid and surface. When a fluid flow past an isothermal surface, fluid particles adjacent to the surface comes in thermal equilibrium with the surface and attains surface temperature T. s Thermal energ then diffuses in the adjoining fluid laers and a temperature gradient is developed in a similar fashion as velocit gradient in hdrodnamic boundar laer with temperature varing from Ts at the surface to T sufficientl far from surface. The region of fluid over the surface in which temperature gradient eist is called as thermal boundar laer. Thermal boundar laer thickness at an location from the leading edge of surface is the normal distance from the surface for which the ratio Ts T / Ts T Figure 4.3 represents development of thermal boundar laer on an isothermal surface t

5 T T U t t T T T s a. Ts > T b. Ts < T Figure 4.3. Thermal boundar laer over a flat isothermal surface Local heat transfer coefficient h at an distance from the leading edge is given b k f T / h Ts T 4..3 CONCENTRATION BOUNDARY LAYER Consider a case of air flowing over dr ice (solid CO ). A sublimation process takes place as solid Co directl converts into gaseous form and diffuses into ambient air. The diffusion process occurs due to concentration gradient of gaseous Co particles at the surface and that in ambient air. As the air flow past dr ice, air particles adjoining to the surface of dr ice comes in chemical equilibrium with dr ice and attains a molar concentration C. gaseous Co molecules than diffuses into adjoining air laer and a concentration gradient is developed similar to velocit and temperature gradient in hdrodnamic and thermal boundar laer respectivel. The region of air over the surface of dr ice in which concentration eists is called as concentration boundar laer. Concentration laer thickness at an location from leading edge of surface is the normal distance from the surface for c which ratio Cs C / Cs C 0.99 where C is the molar concentration of Co in free stream of air. Figure 4.4 represents development of concentration boundar laer over a flat surface. T s s C U c C c C s Figure 4.4. concentration boundar laer

6 The surface of flow of commodit concenteration boundar laer associated with diffusion process due to concentration gradient can be defined b Ficki s law which is analogus to Fourier s law of conduction c N kdiffa Where is diffusion coefficient of a medium which is a measure of rate of diffusion in a k diff medium. Molar flu at the surface is given b Ns c kdiff A 0 Note that concentration gradient at the surface is maimum near the leading edge and progressinl decreases along distance from leading edge in laminar boundar laer region 4.3 DIMENSIONAL ANALYSIS OF CONVECTION Dimensional analsis involve grouping of various variables involved in phsical phenomena of heat and mass transfer in convection into a dimensionless group to reduce the total number of working variables. Consider a hot bod placed in cooler ambient air in a room. The bod will slowl cool down due to convection heat transfer to surroundings. According to newton s law of cooling, the heat transfer rate is given as Q hats T Where Ts T is the temperature difference between surface and surroundings, A is the surface area eposed to convection and h is convection heat transfer coefficient. Heat transfer rate Q var with change in heat transfer coefficient h. h depends upon various flow characterstics and fluid properties. For eample if fan is switched on in the room, heat transfer rate will increase rapidl. If the hot bod is kept in water instead of air, the cooling rate of bod will increase due to higher specific heat of water. Specific heat represents heat absorbing capacit of fluid. If the ambient fluid is replaced b a fluid with relativel lower viscosit and densit, heat transfer rate b convection will increase due to lower resistance in flow of fluid over a solid surface. Fluid laer adjacent to the surface is static and therefore heat transfer from solid surface to static fluid laer is b means of conduction. Hence heat transfer rate will be increased if the ambient fluid is replaced b a fluid with comparativel higher thermal conductivit (For eample water replaced b mercur). Hence we see that h is a function of various variables and can be written as h f,,c,k,v,l where Dnamic viscosit of fluid Densit of fluid c Specific heat of fluid k Thermal conductivit of fluid V Free stream velocit of fluid L Characterstic dimension of solid surface It can be premised that the functional relationship is: f,,c,k,v,l,h

7 There are 7 phsical quantities and 4 fundamental units M,L,T,. Therefore there are terms. Taking k,v,l and as Repeating variables, terms can be written as Equating the eponents of fundamental dimensions on both sides M : 0 a b 1 : 0 b Therefore a 1 a b c d 1 kv a 3 1 b 1 c d 3 M T ML T MLT LT L ML L : 0 a b c d 3 T : 0 a 3b c b 0 c1 d1 Putting values of a,b,c and d 1 st term VL 1 k V L The term is called REYNOLDS NUMBER (Re) 1 Following the same procedure, we obtain c k The term is called Prandtl number P hl 3 k The term is called Nusselt number (Nu) 3 The functional relation (4.3) becomes VL c hl f,, 0 k k hl VL c f, k k Nu f Re,Pr It is usual practice to rewrite the correlation (4.4) in the form a Nu Re P r b The constant C and the eponents a and b are evaluated through eperiments for various flow conditions. Singnificance of various dimensionless groups are a. Renolds number Renolds number is the ratio of momentum force to viscous force on the fluid. VL Re

8 The Renolds number can be written as VL V V Momentum force / Area Re V V Viscous force/area L V represents shear stress developed in a laminar fluid flow due to viscosit for linear L velocit profile. V represents momentum force per unit area. M L M L ML V L T LT L L T 3 ML 1 MLT Momentum force V T L L Area Renolds number indicates the relative domination of momentum force and viscous force in a fluid flow. At low Renolds number, the viscous effect dominates and the fluid flow is laminar. At high Renolds number, the momentum force dominates and the flow is turbulent. The Renolds number at which transition from laminar to turbulent flow takes place is called critical Renolds number. Critical Renolds number for flow in a pipe 5 is 000 and for flow over a flat plate is 510. b. Prandtl number Prandtl number is the ratio reflecting the relative rates or abilit of the fluid to diffuse momentum and heat into the fluid c / Kimematic viscosit/momentum diffusivit Pr k k / c Thermal diffusivit Prandtl number is related to the relative growth of velocit and thermal boundar laer. Prandtl number greater than 1 indicates relativel faster rate of momentum diffusion as compared to thermal diffusion and therefore hdrodnamic boundar laer thickness will be less than thermal boundar laer thickness for a given value of distance from the leading edge of plate. For Pr = 1, thermal and momentum diffusion rates are equal and hence the hdrodnamic and thermal boundar thickness are equal over entire length as plate. 1 Pr t Liquid metal mercur : Pr << 1 Oils : Pr >> 1 coolants due to rapid diffusion of heat. Liquid metals are generall used as c. Nusselt Number : Nusselt number represents the abilit of fluid to transfer heat through convection relative to heat transfer through conduction. Nusselt number is a dimensionless convection heat transfer coefficient generall utilized to measure value of h from various empirical correlations given in net units. hl h AT hat Qconvection Nu k k / L AT kat Qconduction L Nusselt number can also be defined as ratio of conduction resistance to convection resistance of a fluid

9 hl L / ka Rconduction Nu k 1/ ha Rconvection Heat transfer rate b convection and thus Nusselt number has higher value for forced convection as compared to free convection for same solid fluid convective interface. Nusselt number v/s Biot number hl f Nu k f hl f Bi ks Where subscript and represents fluid and subscript s represents solid. Nusselt number involves thermal conductivit of fluid where as biot number involves thermal conductivit of solid at solid fluid convective interface. Since hf >> k f, therefore nusselt number has a value less than 0.1 for lumped heat capacit sstem. Other dimensionless groups of importance in convection analsis are coefficient of friction c, f Stanton number St (also called modified nusselt number) and grash of number (Gr). Coefficient of friction has alread been discussed in section 4..1 and other two dimensionless numbers will be discussed later on. 4.4 BOUNDARY LAYER EQUATIONS In chapter we have developed governing differential heat diffusion equations to find temperature distribution for a given set of boundar conditions. This is done b appling conservation of energ to a differential control volume b assuming heat conducting material to be stationar without an movement of macroscopic matter across the boundaries of differential control volume. However convection involves bulk transport of fluid matter across the boundaries of differential control volume along with energ interactions. In this section we derive differential equations that govern the fluid flow in boundar laer region. The resulting governing equations of velocit, temperature and species concentration are derived b appling conservation of mass, conservation of momentum, conservation of energ and conservation of species concentration. We restrict our attention to -dimensional, stead state incompressible fluid flow in laminar boundar laer region in this section. Deriving the governing equations for unstead state or compressible fluid flow or in turbulent boundar laer region is beond the scope of this book. Consider a control volume of ABCD of dimensions d d with unit width perpendicular to the plane of paper ling in a common region of thermal boundar laer, laminar hdrodnamic boundar laer and concentration boundar laer as shown in figure 4.5.

10 D' A' C ' D d B ' d A C B Control volume U T, V, C C HBL T CBL TBL HBL CBL TBL Ts Hdrodnamic boundar laer Concentration boundar laer Thermal boundar laer Figure: 4.5 Development of HBL, CBL and TBL CONSERVATION OF MASS-THE CONTINUITY EQUATION: The continuit equation governs the conservation of mass b equating the mass flow rate of fluid entering and leaving the control volume. The -component of the velocit of flow u in the boundar laer region varies along direction but is assumed constant for a infinitesimall small differential length AD in direction. Mass flow rate can be epressed as m densit velocit area D ' C' u V dd Cs ud A' D B' C u U d d A B vd Figure 4.6. Mass flow rate through control volume The flow rate of mass entering the left face ADD A of the control volume is given b u d1 The flow rate of mass leaving the right face BCC B of the control volume is given b u u d d1

11 Similarl the flow rate of mass entering the bottom face ABB A is given b v d 1 v the flow rate of mass leaving the top face is v d d 1 There is no accumulation of mass in the control volume for a stead flow and thus the mass flow rate entering and leaving the control volume are equal. The continuit equation can be epressed as u v ud vd u d d v d d u v dd dd u v This is the mass continuit equation for two-dimensional, stead state, incompressible fluid flow in rectangular coordinates Momentum Equation Momentum equation is based on Newton s second law of motion which states that rate of change of momentum of a fluid element in a particular direction in control volume is equal to net force acting on control volume in that direction. Momentum of mass of fluid entering the control volume in direction is m u ud u u d Where m is the mass of the fluid flowing in -direction. The momentum efflu through the right face in -direction is u mu mud ud u ud ud u d u d d Let the mass flowing to the control volume through the bottom face in direction be m. Since we are concerned onl with momentum in -direction, therefore the momentum influ from the bottom face in -direction is m u vd u u v d And the momentum effu from the top face in -direction is mu mud v d u v d ud v u u v d u d d v d d and

12 D m u C m u d mu Control Volume m u m u d A mu Figure 4.7 Momentum forces acting on control volume in -direction The resultant change in momentum in -direction Total momentum of mass leaving the control volume Total momentum of mass entering the control volume. u u u u d u d d u v d u d d v d d u d u v d u v u u d d u d d v d d u u u v u v d d u d d We know that u/ v/ 0 from continuit equation and therefore the net momentum transfer to the control volume in -direction becomes: u u u v d d...(4.9) A fluid motion is subjected to several forces which results in variation of fluid velocit and hence momentum. The various forces that influence the motion of fluid are Gravit Force ( F ) due to weight of fluid acting in vertical downward direction, Pressure Force ( F ) is g eerted on the fluid mass due to difference in pressure between two point, Viscous Shear Force ( F ) due to viscosit, Turbulence Force ( F ) due to random movement of fluid v particles between adjacent fluid laers, Surface Tension Force ( due to cohesive propert of fluid and Compressibilit Force ( ) acting at the free surface ) due to elastic propert of compressible fluids. From Newton s second law of motion F g F p F v F T F s F u u e = u v d d Taking into consideration onl viscous shear forces u u Fv u v d d...(4.10) B T FS F e p

13 D u u ê d ú d C Control Volume A u d Figure 4.7: Viscous shear forces acting on control volume in -direction The shearing force due to viscosit at the lower face of the control volume is u (d 1) The shearing force due to viscosit at the upper face of the control volume is u u u u dd ê dúd d d Neglecting the shearing force in -direction, the net viscous force in the direction of motion is u u u u Fv d d d d d...(4.11) From equation (4.10) and (4.11) u u u d d u v d d u u u u v...(4.1) Since term / can be written as kinematic viscosit ( ), we obtain u u u u v...(4.13) This is the force or momentum equation in the laminar boundar laer region in -direction for a stead, incompressible fluid flow Energ Equation For a control volume of dimensions d d1 in the boundar laer, the quantities of energ transfer across the boundaries of control volume have been shown in Figure 4.8. Conduction heat transfer in -direction is neglected and conduction is considered onl in the -direction. B

14 Viscous heat generation u d d d T T k d ê ú ê ú E E 1 D Control Volume C E A B T kd Figure 4.8 Differential control volume for energ conservation Rate of energ entering in the control volume in -direction (E ) m specific heat temperature u d c T u T Rate of energ leaving the control volume in -direction (E ) u d d c T d T u Neglecting the product of small quantities (E ) c d êut u d T d ú Net rate of energ transfer in direction due to convection T u E E 1 c u T d d...(4.14) ê ú Similarl net rate of energ convection in the -direction T E1 E v dc T c d v T v vt v d T d c ê v T d d...(4.15) ú ê ú Net rate of conduction heat transfer in -direction T T T k d êk d dú ê ú T k d d...(4.16) Due to viscosit of flowing fluid, viscous heat generation For stead state conditions, net accumulation of energ in the control volume is zero and thus the algebraic sum of all the energies crossing the boundaries of control volume equals zero. Thus 1 E 1 u T u T v T u c u T v T d d k d d d d 0 d d...(4.17) T T u v T u c êu v T ú k 0...(4.18)

15 We know that (4.18) can be written as: u/ v/ 0 from continuit equation and therefore the equation T T k T u u v...(4.19) c c If the heat generation due to viscous effects is neglected, the energ conservation equation (4.19) takes the form; T T k T T u v...(4.0) c T T T u v...(4.1) CONSERVATION OF SPECIES CONCENTRATION In analog with momentum equation in hdrodnamic boundar laer and energ equation in thermal boundar laer, species concentration equation in concentration boundar laer can be written as c c c u v k diff Where C is the concentration of component in concentration boundar laer region varing from C at the surface to in free stream. s C BOUNDARY LAYER SIMILARITY Momentum equation governing fluid flow in Hdrodnamic Boundar laer region u u v u u Energ equation governing fluid flow in thermal boundar laer region u T v T T Species concentration equation governing fluid flow in concentration boundar laer region c c c u v k diff Eamination of these three equations simultaneousl leads to the conclusion that terms on left hand side of equation represents bulk transport of matter into or out of control volume which are know as advection terms. Terms on the right hand side represents diffusion aspect of convection. Kinematic viscosit represents momentum diffusivit in hdrodnamic boundar laer, thermal diffusivit represents diffusion of thermal energ into thermal boundar laer and represents diffusion of species in concentration boundar laer k diff due to concentration difference. These diffusion propert leads to development of respective boundar laers. Hence a strong similarit is observed in momentum, energ and concentration equations which can be analsed b non dimensionalisation of these governing equations.

16 Non dimensionalisation involves dividing all independent variables (such as space coordinates) and dependent variables (such as velocit, temperature and concentration) b a relevant constant quantit * and * L L Here space coordinates are divided b constant quantit L (Characterstic dimension of solid surface) to obtain independent dimensionless parameters * and *. Variables with asterisks denote dimensionless variables. Similarl dimensionless dependent variables obtained are u U* V v V* V T Ts T* T T Substituting these dimensionless variables into momentum, energ and concentration equations, we obtain Momentum equation: Energ equation: s C Cs C* C C s * * * * u * v u * * * U v VL * * T * * T * T * * * U v VL * * * * c * c kdiff c Concentration equation: U v * * * VL VL Note that represent dimensionless Renolds number, represents dimensionless Prandtl number and another dimensionless parameter is obtained which is known as Schmidt number. Therefore the three governing equation can be written as For, value of Prandtl number becomes equal to 1 and momentum equation and energ equation becomes similar. Thus velocit profile and temperature profile becomes identical. In other words we can sa that Prandtl number represents ratio reflecting relative growth of velocit and thermal boundar laers. For Pr = 1, Thermal and momentum diffusion rates are equal and hence hdrodnamic boundar laer coincides with thermal boundar laer. k diff * * * * U * V 1 U * * * Re L * * * * T * T 1 T * * * Re L Pr * * * * C * C 1 C * * * Re S L c Momentum equation: U V Energ equation: U V Concentration equation : U V

17 For k diff, Schmidt number becomes equal to 1. Thus momentum equation and concentration equation becomes similar and therefore velocit and concentration profiles are identical. It can be analsed that hdrodnamic boundar laer and concentration boundar laer coincides if the momentum diffusivit is equal to species diffusivit and Schmidt number represents relative growth of hdrodnamic and concentration boundar laers Similarl, the three boundar laer coincides if Pr Sc 1. Pr Sc k k diff diff 1 kdiff can be defined as another dimensionless parameter known as Lewis number (Le). k diff Pr Sc Le 1 represents the case where all three governing equations becomes similar and velocit, temperature and concentration profiles become identical.... Join us to learn more