HEAT AND MASS CONVECTION

Transcription

1 HEAT AND MASS CONVECTION. BOUNDARY AYER FOW Heat and mass convection... 1 Heat convection: hat it is... 1 Types of heat convection... A brief on Fluid Mechanics... 3 Continuity equation... 4 Momentum equation... 5 Energy equation... 5 Mass transport equation... 5 Constitutive equations... 6 Introduction to non-dimensional parameters... 7 Boundary layer flo Non-slip condition Boundary layer forced-flo over a flat plate Thermal boundary layer and solutal boundary layer in a forced-flo over a flat plate Reynolds analogy beteen momentum and energy equations Steps to solve heat and mass convection problems... 0 Temperature and pressure effects on fluid properties... 1 Forced and natural convection (aside)... 3 Convection ith phase change (aside)... 3 Heat echangers (aside)... 3 HEAT AND MASS CONVECTION We present here some basic modelling of convective process in Heat and mass transfer. Heat diffusion, mass diffusion, and heat radiation are presented separately. Furthermore, mass convection is only treated here as a spin-off of the heat convection analysis that takes the central focus. Heat convection: hat it is There cannot be any convected heat, since heat is only defined as thermal-energy flo through an impermeable surface due to a temperature difference across. What e call heat convection is the effect of a fluid flo on heat conduction at a fluid-ashed all; i.e. e intend to apply Neton's la of cooling instead of Fourier's la (see Physical transport phenomena in Heat and mass transfer): What is heat (flu) convection? ( ) n q ht T = k T (1) (here n stands for the normal gradient at the all), aiming at substituting the effect of a real flo field by an empirical boundary condition at the all, i.e. ith the convective coefficient h in (1) found from global measurements of temperatures and heat flues in eperiments, instead of by analytically solving the fluid flo (Navier-Stokes' equations) and using (1) to deduce h. Internal thermal energy (not heat) is convected ith the fluid flo, in an amount dependent on a reference energy-level (reference temperature), usually referred to the ambient or sink temperature. When the increase in internal thermal Heat and mass convection. Boundary layer flo page 1

2 energy is due to heat transfer at a source, the energy balance for a fluid flo at constant pressure ithout phase changes and reactions is Q = mc T, hat shos that, the same thermal load can be transported by a high mass-flo-rate flo ith small temperature jump, or by a lo mass-flo-rate flo ith high temperature jump, and that thermal-carrier fluids should have high thermal capacity. Notice that in Fluid Mechanics, there is no Neton's la of cooling, and the only heat-transfer term to be included is Fourier's conduction (and in very special cases thermal radiation emission or absorption through the media). Types of heat convection Heat convection problems may be classified according to: Time variation, as steady or unsteady convection. Only a marginal fraction of applications require transient convection analysis (e.g. hen the onset of natural convection in a fluid layer heated from belo, is studied). Flo origin, as forced convection or natural convection. Forced convection occurs hen the fluid flo is imposed by other agent than the heat-transfer phenomena under study, i.e. by a pump, a fan, or natural convection from other objects. Natural or free convection occurs hen the fluid flo appears as a consequence of the heat-transfer phenomena under study, due to buoyancy forces caused by density gradients in an eternal force field. Natural convection takes place in all heat convection problems under gravity, but hen forced convection is imposed, the latter usually overcomes the former (a combination of the to must be considered at small forcing speeds). Forced convection greatly enhances heat transfer, but demands poer consumption. (According to this division, the internal flo in a heat-pipe, due to capillary pumping, is forced, in spite of not consuming eternal poer. Thermo-capillary convection, like Marangoni convection, is also not considered in these notes.) Flo regime, as laminar flo or turbulent flo. Turbulent flo is the rule in engineering applications, but laminar flo alays eists in some regions, like close to alls and entrance regions. Turbulent convection greatly enhances heat transfer, but increases poer consumption too. Flo topology, as internal flo or eternal flo. Internal flo is hen e focus on the fluid floing inside pipes and ducts, hereas eternal flo is hen e focus on the fluid floing outside pipes and ducts, or around any other solid object. The distinction is not so clear hen one considers a portion of a duct (e.g. a flat plate), or open-channel flo, although all these cases are traditionally considered eternal flo. Some other times, flo topology depends on the detail of the analysis, as in shell-and-tube heat echanger, hen heat transfer can be considered eternal convection of the shell-fluid around the tubes, or internal convection of a shell-ducted flo to the alls (mainly the internal alls), as for compact plates heat echangers. Flo phases, as single-phase or multi-phase flos. An intermediate type is stratified flo (i.e., homogeneous, heterogeneous: stratified, to bulk phases, and disperse). This division is not only important for permanent multi-phase flo, but for vaporising and condensing flos. Heat and mass convection. Boundary layer flo page

3 Flo detail, as detailed heat convection or global heat convection. Most of the times, the empirical approach to convection heat transfer only looks for global values of the convective coefficient around a solid, or along a pipe; but there are cases here temperature variations along the all must be resolved, either during eperiments to compute global h-values by integration, or during analysis to kno if some temperature limit is locally eceeded, and for this purpose a local approach is of interest. Flo compressibility is seldom important in heat convection. Flo reactivity, if any, is considered aside as a distributed energy source or sink. Other important fluid flo divisions, like viscous and inviscid flos, or 1D- D- and 3D- flos, are of little importance in the study of heat transfer by convection, because of the global empirical approach folloed. Thermal boundary regulation. To basic cases are considered: constant all temperature, and constant heat flu, the former being more closely approached in practice (it is simpler to regulate the temperature of the all than the heat flu through it, and there temperature is maintained in phase changes of a pure substance), but the latter being simplest to model, since it means a constant source term in the energy balance (and it is the actual case in counter-flo heat echangers ith similar fluid flos). As a matter of fact, both types of control can be advantageously used, as in hot-ire velocimetry (or hot-ire anemometry), here either the ire temperature is controlled (regulating its electrical resistance R(T) and measuring the required poer as a function of flo speed, Q ), or the supplied poer to the ire is fied, and the steadystate temperature difference beteen ire and fluid measured. The steady-state energy balance, Q = V R = KA( T T ), allos a calibration against relative flo speed, v, hen a cooling la is assumed (e.g. K = a+ b v ). Air is the most ubiquitous fluid in heat convection. All terrestrial animals and most equipment transfer heat to the environment by natural convection in air, ith a typical convection coefficient value in the range h=5..10 W/(m K). For simple cooling/heating load calculations ith ind effects, Duffie and Beckmann (1991) rule h=a+bvind, ith a=3 W/(m K) and b=3 J/(m 3 K) may be a first approimation. Notice that the convective coefficient depends on fluid type, flo type, and geometry. For instance, for natural convection from a plate in air, correlations for h in (1) are (see details in Forced and natural convection) h=a(t T) 1/4, ith a=.4 W/(m K5/4 ) for the upper face of a horizontal surface, a=1.3 W/(m K5/4 ) for the loer face of a horizontal surface (stable vertical gradient), and a=1.8 W/(m K5/4 ) for a vertical surface. A brief on Fluid Mechanics Fluid Physics encompasses the nature of fluids (their structure and properties), the fluid forces ithin and on the boundaries, the transport of mass, momentum and energy, and possible effects of reactive processes, electromagnetic interactions, and so on; Fluid Mechanics concentrates on fluid forces and the transport of momentum. Our interest here is on the transport of energy, but this is linked to the transport of momentum and cannot be studied apart hen fluid flo eists; that is hy e are presenting belo an ad hoc summary of the general equations, a rather complicated formulation ith the three simple objectives: Heat and mass convection. Boundary layer flo page 3

4 To have an overvie of the full equations hich are pre-programmed in computational-fluiddynamic codes (as commercial CFD packages). The user is in charge of selecting the appropriate terms in the equations, and setting the initial and boundary conditions, but the equations are automatically solved. To have an idea of the terms retained and the terms neglected in some simple heat-and-mass transfer problems to be analysed in detail, as the boundary-layer flo, and the pipe flo. To better understand the rational for the grouping of dimensional variables into the traditional non-dimensional parameters. The description of fluid flo makes use of the continuum model, and on the concept of fluid particle, an infinitesimal control system in local thermodynamic equilibrium. Once a reference frame is selected, the motion of fluid particles may be described in to ays: Eulerian description, here the unit volume is fied to the spatial reference frame, and the motion of the particle that at every instant happens to pass by this position, is specified. agrangian description, here the unit volume moves ith the flo relative to the spatial reference frame, and the motion of the same particle at each position, is specified. Passing from the most-intuitive agrangian to the most-used Eulerian description is based on Reynolds transport theorem: dφcm d φ VCV f () t φ = φdv = dv + φ( v va ) nda dv + φv nda CV dt dt t t () VCM ( t) VCV ( t= t0 ) ACV ( t= t0 ) VCV ACV hich says that, for of any conservative property (Φ may be mass, momentum, or energy) in a control mass CM (its value being the integral of the specific function over the volume; e.g. φ ould be mass per unit volume, ρ), the variation ith time in a permeable system can be computed as the integral of the specific function ithin the control volume CV, plus the flu of that variable over the permeable area. Passing from area integrals to volume integral is based on Gauss-Ostrogradski s divergence theorem: vdv = v nda dφ φ φ = φ ( φ ) dt + = + = t t VCV ACV CM dv v nda v dv 0 VCV ACV VCV (3) hich is applied to an infinitesimal volume to get the general equations of Fluid Mechanics: continuity equation, momentum equation, and energy equation. Continuity equation The continuity equation is the mass balance (dmcm/dt=0) for a dv system; ith φ=ρ e get from (3): ρ t + = ( ρv ) 0, or Dρ + ρ v = 0 Dt (4) Heat and mass convection. Boundary layer flo page 4

5 here the convective derivative D() / D t () / t+ v () is often introduced to make the riting more compact. In most heat and mass transfer problems, the continuity equation can be reduced to v = 0 because density changes are usually negligible. Momentum equation d mv dt = F = F + F ), applied to a dv The momentum equation is the linear-momentum balance ( ( ) CM / V A system; ith φ = ρv e get from (3), using the convective derivative: Dv ( ρv ) ρ = + ( ρvv ) = τ + ρg = ( p + ρgz) + τ ' Dt t here τ is the stress tensor (such that the force per unit area of normal vector n is f = τ n ), g is any volumetric force field (e.g. gravity), p is fluid pressure (one third of the trace of the stress tensor), and τ ' the viscous component of the stress tensor. In most heat and mass transfer problems (5) can be reduced to the so-called Boussinesq approimation (constant-density flo ecept for the buoyancy term, proportional to the thermal-epansion coefficient α, named in honour of the French Academician V. J. Boussinesq, ho studied in the late 19th c. convective cooling and turbulence): Dv p = g 1 α( T T0 ) iz + ν v Dt ρ here ν µ/ρ is the kinematic viscosity and µ the dynamic viscosity. Energy equation The energy equation is the energy balance ( ( ) CM / from (3): (5) (5a) d me dt = Q + W ) for a dv system; ith φ=ρe e get De ρ = q + ( τ v) (6) Dt hich in most heat and mass transfer problems is epressed in terms of temperature: DT ρcp = q + αtdp / Dt + τ ': v (6a) Dt Most of the times energy terms other than the accumulation ρcpdt/dt and the heat flu q terms, are grouped under a dissipation variable φ (energy release per unit volume), as seen in Heat and Mass Transfer. Mass transport equation The (global) mass-transport equation is the continuity equation above; hat e deal ith here is the species-balance equation in a miture, (dmi,cm/dt=wi, for any species i, ith Wi being a possible i-species production term by chemical reactions), for a dv system. No, besides substituting φ=ρι mi/v in (3), one has to use in (3) the i-species on velocity, v i, related to the mass-averaged velocity v by the conservation equation ρiv i= ρv, and the definition of diffusion velocity v di v i v, hat yields: Heat and mass convection. Boundary layer flo page 5

6 here j derivative: ρi ρi ρi + ( ρiv i) = i= + ( ρiv ) + ( ρiv di) = + ( ρiv ) + j (7) t t t ρ v i i di is the flu density of species i through a (global) fluid particle. With the convective Dρi + ρi v = i ji Dt (7a) i being a possible i-species production term per unit volume. Constitutive equations To solve a Fluid Mechanics problem, i.e. to find the velocity field, pressure field and temperature field v, pt, t,, besides the initial and boundary conditions of the particular ( ) in terms of position and time ( ) problem at hand, the above balance equations of mass, momentum, and energy, must be supplemented ith some general constitutive equations that relate the additional variables ( ρτ,, eq,, φ, ji, i) to the v, pt,, i.e. the equation of state at equilibrium, ρ=ρ(t,p) and e=e(t,p), and the main one main variables ( ) in Fluid Mechanics, a relation beteen fluid strain-rate and stress, first proposed by Neton in 1687 as, τ = µ v/ y, and in a more general ay named Navier-Poisson's la: T τ = pi + τ ' = pi + µε + µ V µ ( v ) I = pi + µ ( v + ( v) ) + µ V µ ( v ) I 3 3 (8) hich enters into the momentum balance as: ρ = 0 τ = p+ τ ' = p+ µ ( v+ ( v) ) + µ V µ ( v) = τ = p+ µ v (8a) 3 Besides state equations and the strain-rate to stress relation, one needs a heat-rate relation (our ellknon Fourier's la, q = k T, or its etension to multi-component-diffusing systems, q = k T + ρ v h, ith the enthalpy of species i, hi, being deduced from the state equations adopted), i di i plus appropriate dissipation las for φ, plus the mass diffusion rate equations, namely the etended Fick's la and Arrhenius's la: j = D + ct ( ρ / ρ) i i i S ' ν i " ' F ρ i = M ( ν ν ) H G Ba ep M i i i i i I KJ F HG Ea RT here Di is the coefficient of mass diffusion of species i in a given miture due to concentration gradients, cs is the Soret coefficient of mass diffusion due to thermal gradients (usually negligible), i is the mass of species i produced (by unit time and volume of miture) due to chemical reactions, νi'' and νi' the stoichiometric coefficients for the forard and backard reaction considered (one i must be considered for each reactions), and Ba and Ea to empirical Arrhenius coefficients. The kinetic theory of gases provides a simple (although sometimes not very accurate) formulation of all the transport Heat and mass convection. Boundary layer flo page 6 I K J (9) (10)

7 coefficients and equations of state in terms of pressure, temperature and composition, but in practice one usually resorts to tabulated eperimental data. As no reaction is to be considered here, and Soret effects are neglected, the only term entering into the mass-diffusion balance is: j = D ρ i i i In summary, substituting these constitutive relations in the balance equations, the partial differential equations that solve a heat and mass transfer problem, ith fluid flo but nearly constant density, are: (11) Mass balance (continuity): v = 0 (1) Dyi i Species balance: = Di yi + (13) Dt ρ Dv p Momentum balance: = g 1 α( T T0 ) iz + ν v Dt ρ (14) DT φ Energy balance: = a T + (15) Dt ρc p here y ρ / ρ is the mass fraction of species i in the miture. Notice that the i-species diffusivity in i i the miture, Di, the momentum diffusivity (kinematic viscosity) ν, and thermal diffusivity, a=k/(ρcp), all have dimensions of square length divided by time. Finally recall the definition of the convective derivative. D() / D t () / t+ v (), hich reduces to D() / D t = v () in the steady-state case. Introduction to non-dimensional parameters In all fields of physical sciences, but particularly in Fluid Mechanics, and above all in Heat and Mass Transfer, there is such a number of parameters interplaying in each problem, that it is most convenient for us to group them if possible, and there is a general principle applicable for that, namely, the Pi- Buckingham Theorem (ASME, 1915), hich states that a physical equation ith N variables hose magnitudes can be epressed in terms of M independent physical units, is equivalent to a non-dimensional physical equation ith N M non-dimensional variables. Before attempting that grouping, hoever, some remarks are appropriate. Firstly, the number of independent physical units, M, is not a universal invariant but a universal agreement, and the metre-unit (m) could be totally skipped if lengths ere measured ith the second-unit (s) and the universal la for the speed of light in vacuum c=1 assumed, instead of giving dimensions to this universal invariant, c= m/s. Secondly, and most important, if orking ith non-dimensional variables is so advantageous, hy most physical subjects are learnt using dimensional magnitudes? The anser is that e, humans, ant to compare every magnitude ith our on measurements: lengths ith the length of our arm span, masses ith the mass of a stone e can thro, times ith our heart period, and so on, and each of our anthropocentric units e introduce, contributes to one of the M-basic magnitudes mentioned above (seven in the SI: m, kg, s, K, A, cd, and mol). Heat and mass convection. Boundary layer flo page 7

8 For the grouping of dimensional variables to get non-dimensional parameters, one may follo an ad hoc approach. For instance, in thermal convection studies, one may reason that the convective coefficient h must be a function of the fluid properties (k,ρ,cp,µ), and the characteristic fluid-velocity gradient v/, i.e. h=h(k,ρ,cp,µ,v/), and say that the combinations Nu h / k, Re ρ v / µ, and Pr µ c / k, are 'the usual choice', or 'the standard rule', but someone might ask hy not another combination, and if there is a rational behind. Adding that the rational is to compare heat convected (h T) against heat conducted if the fluid as quiescent (k T/), to compare change of momentum ((ρv)v) against viscous stress (µ(v/)), and to compare momentum diffusivity (ν µ/ρ) against thermal diffusivity (a=k/(ρcp)), may seem enough justification already. But the most conclusive eplanation of hy those parameters and their meaning, comes from an order-of-magnitude analysis of the general equations presented above, both the balance equations and the boundary conditions, namely: Thermal boundary condition at a all. From the definition of heat convection coefficient: T h q h( T T ) = k nt h T k Nu (16) k hat teaches that a non-dimensional parameter, Nu, can be defined to measure the ratio of heat flu transferred ith convection to that ithout convection; it is named Nusselt number in honour of the great thermal engineer Wilhelm Nusselt, ho introduced it in his 1915 pioneering article "The Basic as of Heat"; the 'number' ending is the traditional designation of non-dimensional parameters (no physical units, just the number). In spite of heat convection being alays greater than a corresponding heat conduction, Nu may be smaller than unity if one choose for it a length smaller than the boundary-layer thickness (e.g. hen using the diameter for fine ires). Mass balance. From the continuity equation: p v v y v = = y (17) hat teaches that, if there is a change of fluid speed along one direction (v/), it must be a balancing change of fluid speed along another direction (the flo must be at least to-dimensional); i.e., in a onedimensional flo (in Cartesian coordinates), the speed cannot change along a streamline ( v/ =0). Notice that the fact that the to velocity-gradients be of equal magnitude does not mean that the longitudinal flo must be of the same order as the transversal flo, the paradigmatic case being the boundary-layer flo to be analysed belo, here the longitudinal flo v is dominant, i.e. vy<<v, but still vy/ y= v/. Momentum balance. From the longitudinal momentum equation, assuming gravity effects irrelevant, and epanding the convective derivative: Heat and mass convection. Boundary layer flo page 8

9 Dv p v v p v 1 v v p 1 v Dt t Sr Re = + ν v + v + ν + + ith ρ ρ ρ Sr v Re ν (18) vt hich can be interpreted in the folloing ay. At least to terms in (18) must be of the same order of magnitude; it is important then to compare each other, and for that purpose several non-dimensional ratios are defined: the Strouhal number to measure the ratio of convective forces per unit volume (ρv /) to inertia forces per unit volume (ρv/t), the Reynolds number to measure the ratio of convective forces per unit volume (ρv /) to viscous forces per unit volume (µv/ ), and so on. Several other non-dimensional parameters are used in heat and mass transfer, as the ratio of momentum diffusivity to thermal diffusivity, named Prandtl number Pr=ν/a, the ratio of momentum diffusivity to species diffusivity, named Schmidt number Sc=ν/Di, and so on, all of hich ill be introduced at due time (Tables 1 and give a compilation), but no e turn to the details of fluid flo. Table 1. Main non-dimensional parameters in convective heat transfer. Parameter Definition Meaning Nusselt number h Nu k Ratio of convective heat flu to conductive heat flu. Ratio of momentum diffusivity to thermal diffusivity. Prandtl number Also, thickness ratio beteen velocity-boundary-layer and thermal-boundary-layer. Reynolds number Grashof number Rayleigh number Peclet number Stanton number Strouhal number Ra ν Pr a v Re ν 3 α g T Gr ν 3 α g T = Gr ν a Ratio of flo convection-inertia stress to viscous stress. Ratio of fluid-buoyancy stress to viscous stress. Pr Ratio of fluid-buoyancy stress to viscous and thermal stresses. v Ratio of flo convection-inertia stress to viscous and Pe = = RePr a thermal stresses. h Nu St = = Ratio of heat convection flo convection to flo ρvc Re Pr convection. ω Sr = Ratio of flo convection-inertia stress to viscous stress. v Table. Main non-dimensional parameters in convective mass transfer. Parameter Definition Meaning Sherood number Ratio of convective mass flu, m i A= hm( ρi, ρi, ), to diffusive h m Sh = mass flu, m i A= Di ρi. Notice that if the convection term is Di ritten as m i A= hm( yi, yi, ) (i.e. including the density in the convective coefficient), then Sh hm ( ρdi). Smidth number ν Ratio of momentum diffusivity to solutal diffusivity. Sc = Also, thickness ratio beteen velocity-boundary-layer and solutalboundary-layer. Di Heat and mass convection. Boundary layer flo page 9

10 eis number a e = = D i Sc Pr Ratio of momentum diffusivity to thermal diffusivity. Also, thickness ratio beteen velocity-boundary-layer and thermalboundary-layer. We focus no on the fluid-mechanics near alls, and ill follo on ith the analysis of thermal and solutal effects there, but the theory of boundary layers can be applied to other interesting cases like miing layers, here to parallel streams ith different speed, or different temperature, or different composition, meet together. For instance, it can be deduced that, in the laminar regime, similarly to the thickness of the viscous boundary-layer, δ, groing parabolically ith distance,, as δ/ Re 1/, the thickness of a thermal boundary-layer (either attached to a all, or free-floing beteen to fluids at different temperature), δt, gros as δt/ (RePr) 1/, and the thickness of a solutal boundary-layer (either attached to a all, or free-floing beteen to fluids ith different composition), δs, gros as δs/ (ReSc) 1/. Boundary layer flo Heat and Mass Transfer by convection, focuses on heat and mass flos at alls; that is hy fluid flo near a solid all (boundary layer flo) is so important. Here, the general layout of flo fields at high Reynolds-number-flos (those found in most practical problems) should be recalled: the hole fluid flo can be divided in: a) the main nearly-inviscid flo, here viscous effects can be neglected, and b) some thin boundary-layer flos here viscous effects are concentrated; a seminal approach in Fluid Mechanics, first introduced by. Prandtl in We focus no on boundary layers attached to alls; free boundary layers, as the miing layer just mentioned above, or other more complicated shear flos like jets and akes, present similar behaviour: an initial laminar region that gets unstable at a transition region (here aves appear), ith turbulence development further donstream. Non-slip condition The local equilibrium assumption means that, if the observer considers very small systems (e.g. fluid particles, let say 10-6 m in size), ith not-too-small time scales (let say 10-3 s), they can be assumed to be at equilibrium, since those times are larger than the relaation time (hich is proportional to size, since its inertia is proportional to its volume, and the forcing is proportional to its surface). Thus, the velocity field cannot have discontinuities, neither ithin the fluid, not at the boundaries, and thus fluid particles in contact ith solid alls must be in mutual equilibrium, i.e. have the same velocity (hat implies the nonslip condition, but also the non-detachment condition), the same temperature, and the same chemical potential for each of the species present (not the same concentration, obviously). Boundary layer forced-flo over a flat plate The boundary layer forced-flo over a flat plate is a canonical fluid-mechanics problem here a uniform flo ith velocity u, meets a flat solid sharp edge aligned ith the flo (Fig. 1). In absence of thermal and solutal effects, the presence of the plate at zero incidence only introduces a mechanical perturbation, a shear stress in the direction of flo, due to the non-slip condition, hich, at a constant separation from the plate, causes a deceleration of the flo and, as a consequence of the continuity equation, a small Heat and mass convection. Boundary layer flo page 10

11 transversal outards flo that makes the region affected groing. The region affected starts at the entry border and gros along the length of the plate, ith longitudinal velocity groing from u=0 to u=u across the layer (and small transversal velocities); e may arbitrarily set the thickness of the boundary layer, δ, as that here u=0.99u, and e ant to kno its groth rate, δ(); let us advance that, after some length, the orderly shear flo (laminar flo) transforms (after some transition region) into a lessordered turbulent-flo ith random velocity-fluctuations, ith a thicker boundary layer and a much thinner laminar sub-layer close to the all (Fig. 1). Fig. 1. Structure of the boundary layer flo over a flat plate. The equations governing the flo over a flat plate, assumed steady, incompressible, and ithout gravity effects, are the folloing: Mass balance (continuity equation (4) ith ρ=constant): u v u v v = 0 + = 0 y δ (19) here an order-of-magnitude analysis has also been performed. Assuming the thickness of the boundary layer to be much smaller than the length of the plate under consideration, i.e. δ<<, the continuity equation shos that transversal velocities are much smaller than longitudinal velocities. Notice that e have assumed the flo to be to-dimensional (really it is quasi-one-dimensional), but, hen the flo becomes turbulent, three-dimensional random motions appear. Momentum balance (equation (5) ith ρ=constant and p =0): u u u u u u δ ν u + v = ν + ν Dv p y y u δ = + ν v Dt ρ v v 1 p v v δ u p u + v = + ν + y ρ y y y (0) The order-of-magnitude analysis of the transversal momentum-balance shos that transversal pressure variations are negligible (proportional to δ/), and the longitudinal momentum balance shos that the thickness ratio, δ/, is of order (u/ν) -1/, i.e. δ/ Re -1/, hich is a most important result, to be compared ith the eact solution, δ/=4.9re -1/, first developed by Blasius (see belo). More precisely, the height at hich u=0.99u, gros parabolically as δ=4.9(ν/u) 1/, e.g., for air ith ν= m /s moving at u=10 m/s, the boundary-layer thickness after =1 m from the leading edge is δ=6 mm. Heat and mass convection. Boundary layer flo page 11

12 Another consequence of (0) can be found applicable to the all vicinity: since u y=0=0, thence µ u/ y y=0= p (equal zero for a flat plate), for both the laminar and the turbulent cases! One of Prandtl's students, P. Blasius, found in 1908 the eact solution by introducing a self-similar variable, η y(u/(ν)) 1/, that transforms the PDE-system into an ordinary differential equation in the auiliary function (the stream function, such that u= ψ/ y and v= ψ/ ), ψ(η)=(uν) 1/ f(η), ith f(η) (u/u)dη, the equation being: 3 d f d d d 3 0 d f f d f f f η = η η dη dη η = 0 η = + = 0, ith = 0, = 0, = 1 (1) hich, although not analytically integrable, has a universal solution easily computed numerically, and shon in Fig.. The longitudinal speed fraction u/u= f/ η asymptotically gros from 0 at the plate to 1 at infinity, attaining a precise value of 0.99 for η=4.9 (sometimes rounded to η=5, here u/u=0.99). Instead of the eact solution to the boundary-layer equations, an integral approimation may be good enough, i.e., a solution to integral forms of the mass and momentum equations (first developed by von Kármán in 1946), instead of a detailed solution at each point. et u/u=f(y/δ) be the proposed fitting function (ith δ() the unknon thickness), already verifying the boundary conditions f(0)=0, f(1)=1, and f'(1)=0); another condition may be added, because the longitudinal momentum equation (0) at y=0 is 0= u/ y, as said above, and thus f''(0)=0, but this is not much important. The integrated equations are obtained for the rectangular control volume of idth d and height H sketched in Fig. 1: δ ( ) d m = ρuy d + ρu ( H δ(), ) m = m out δ( ) δ( ) d 0 d d u u dy uu dy δ ( ) d d d ρ = ρ µ u y 0 0 y= 0 ρ d ρ ( δ (), ) out µ d y 0 y= 0 p = u y+ u H p = m u () hich can be more eplicitly formulated in terms of f u/u and η y/δ() as: d ν df df dδ ν df (( 1 f ) f ) dy = ( f 1) ηdη = d u dy dη d u δ dη δ ( ) 1 (3) 0 y= 0 0 η = 0 No, an eplicit f(η) ill yield an eplicit δ() from (3). For instance, if e try the simplest polynomial verifying the three boundary conditions above, f=η η (i.e. u/u=(y/δ) (y/δ) ), e get the differential equation (/15)δ/d=ν/(uδ), hich, ith the condition δ(0)=0 (the layer starts at the leading edge), finally yields the result sought: δ()=(30ν/u) 1/, i.e. the boundary-layer-thickness gros parabolically ith the distance to the edge. We can no found the un-stretched longitudinal velocity field, u(,y)=u(y/δ() (y/δ()) ), the longitudinal velocity slope at the all, u/ y y=0=(u/δ) f/ η η=0=(u/δ)=u /(30νu) 1/, the all shear stress τ=µ u/ y y=0, and the drag Heat and mass convection. Boundary layer flo page 1

13 coefficient, cf, is defined by τ=cfρu /, hich is cf=(4/30 1/ )/(u/ν) 1/. The transversal velocity profile can be found from continuity equation (19) as v(,y)= ( u/ )dy=(5/6) 1/ u(u/ν) 1/ (3η η 3 ). If e add the fourth boundary condition stated above, u/ y y=0=0, e need a cubic polynomial, hich is f=(3η η 3 )/, and ne eplicit values can be found as just shon.. Table 3 gives a summary of those results, and Fig. the corresponding velocity profiles. Notice, by the ay, that the transversal velocity v(y) at a stage gros in a S-shape from 0 (ith dv/dy=0) to a maimum v(δ)=0.86u/(re) 1/ (0.86 for Blasius solution; 0.91=(5/6) 1/ ith the simple fitting above), and remains ith that value outside the boundary layer, contrary to some intuitive reasoning telling that it should vanished outside the boundary layer in the undisturbed flo; the eplanation is that, ith the incompressible fluid model, there is no undisturbed flo, the perturbations travelling in all directions instantaneously, and the m out contribution alays eists (in reality, the hole boundary-layer model relies on the δ<< assumption, not valid near the leading edge and far outside the boundary layer). Table 3. Comparison of different solutions to the laminar boundary layer flo over a flat plate. Code (*) Solution, f f u/u Thickness coeff., a, in δ/ a/re 1/ Slope coeff., b, in u/ y bre 1/ u/ Friction coeff.,** c, in cf,=c/re 1/ Coeff.,d, in v/u=d/re 1/ 4 (y/δ) (y/δ) =5.48 =0.365 =0.730 = (3(y/δ) (y/δ) 3 )/ = = = =0.870 sin(π(y/δ)/) π (4 π) ( 4 π ) 8 (4 π ) ( π) 8 π =4.80 =0.38 =0.655 = Eact sol.. (1) 4.9*** *Codes for Fig.. **Defined from τ=µ u/ y y=0=cfρu /. ***Eact solution hen u/u=0.99 (it etends from y=0 to, hereas the others etend from y=0 to δ); in many books, this 4.9 is rounded to 5.0. Fig.. Non-dimensional velocity profiles inside the boundary layer. Four models are shon for the longitudinal velocity profile, u(,y) (see details in Table 3), and only the eact profile for the transversal velocity profile, v(,y), ith Re u/ν. Heat and mass convection. Boundary layer flo page 13

14 Notice that the choice of reference frame modifies the epression of the velocity profile, and, for instance, the profile u/u=(y/δ) (y/δ) (number 4 in Fig. ) refers to the origin at the plate, hereas if the origin is set at the free-end of the boundary layer, the same profile ould read u/u=1 (y/δ). By the ay, the latter origin is more convenient for fully-developed flo in pipes and to-dimensional ducts, here the boundary layers meet at the centre and, ith the origin there, the epression u/u0=1 (y/δ) is valid for the hole duct; notice the change from u to u0, the speed at the centre line, hich is (3/)-times the average speed in to-dimensional ducts, and tice the average speed in circular pipes. This parabolic velocity profile (named Poiseuille flo) becomes more uniform in turbulent flo, here it can be approimated by a higher-poer la u/u=1 (y/δ) n ith n beteen 6 an 10 (n=7 is the most common). Besides the normal boundary-layer thickness defined ith u(y)=0.99u, to other related variables are sometimes used to quantify boundary-layer thickness: the displacement thickness δ * () defined by * δ ( 1 u) ( u u( y) ) dy, and the momentum thickness θ() defined by θ ( 1 u) uy ( )( u uy ( )) dy. For a laminar boundary layer over a flat plate ith no pressure gradient, δ * δ/3 and θ δ/15. All the models developed above, only apply to laminar boundary layer flo over a flat plate. In practice, there is an initial length ith laminar flo for both sharp and rounded blunt leading edges (i.e. provided there is not flo separation at the edge), folloed on by a transition region starting at some such that Re=(0.3..1) 10 6 (ith very smooth plates, laminar flos up to Re= have been achieved), and finally ending in a turbulent flo donstream. For most engineering problems it is assumed that the transition region is abrupt, and that the laminar region spans from =0 to = ν/u (corresponding to a standard critical value of Re= ), and the turbulent one starts there and etends beyond (it actually depends on plate roughness and turbulence level of the entry-flo. Turbulent thickness cannot be analytically modelled (cross-coupling velocity terms appear in the momentum equation, Neton's la of friction τ=µ u/ y y=0 is no longer valid, and so on), and empirical correlations, based on the momentumenergy Reynolds analogy (eplained belo), are used; traditional correlations are presented in Table 4, in comparison ith their laminar counterparts. Table 4. Comparison of laminar boundary-layer characteristics model ith turbulent ones (see Table 3). Velocity profile ayer thickness Friction coefficient* aminar** u y δ Re< = 1 = c 1 f, =, c 1 f, = 1 u δ Re Re Re Turbulent*** <Re< u u y = 1 δ 7 δ = c 15 f, =, c 15 f, = 15 Re Re Re *ocal friction coefficient is defined by τ()=cf,ρu /, hereas global friction coefficient is defined by (1/) τ()d=cf,ρu /. **Blasius laminar velocity profile code 4 in Table 3 and Fig., but ith eact coefficients for δ and cf. Notice the change of y-coordinate origin and sense from Table 3 and Fig.. ***Prandtl turbulent model as calibrated by Schlichting; -coordinate origin at transition point. Notice that turbulent thickness (Table 4) is not defined so precisely as laminar thickness, here the separation at hich u(y)=0.99u, is neat; in turbulent boundary layers, large eddies are created and burst, Heat and mass convection. Boundary layer flo page 14

15 causing typical protuberances up to 1.δ and depressions don to 0.5δ. In any case, it can be concluded that turbulent thickness is alays larger than laminar thickness and gros quicker. aminar-to-turbulent transition (TT) depends a lot in the pressure gradient in non-flat surfaces (to be studied aside); even more, there can be a turbulent-to-laminar transition in the strongly favourable pressure gradient that occurs in a converging nozzle (relaminarization). A general arning on using empirical correlations is to be careful about the application range: all empirical correlations are limited in scope, and the most accurate, the narroer their applicability range. Thermal boundary layer and solutal boundary layer in a forced-flo over a flat plate Analogous to the velocity boundary layer due to the jump from the non-slip condition to the free-stream flo, a thermal-boundary-layer appears if there is a difference from all-temperature to free-flotemperature, and a solutal boundary layer appears if there is a difference from all-concentration of a solute to its free-flo concentration. The governing balance equations for the general case of flo-, thermal-, and solutal-boundary layers are: u v v = 0 + = 0 y D v p u u u = + ν v u + v = ν Dt ρ y y D D φ T T T T = a T + u + v = a t ρcp y y Dy y y y Dt ρ y y i i i i i = Di yi + u + v = Di (4) (5) (6) (7) Boundary conditions can be layout in a similar ay to the velocity boundary layer above-eplained, hat shos that in the case of Pr ν/a=1, the function (T T)/(T T) has the same shape as the alreadyknon u/u profile, and, in the case of Sc ν/di=1, the function (yi yi)/(yi yi) has the same shape as the already-knon u/u profile; see Table 3 for several approimations. The main goal in heat convection is founding h (or Nu, in non-dimensional variables), hich ith the above thermal-boundary-layer model yields: T T y = f k T / y T / y T T q y 0 h δ = y= 0 f ( η ) δ η η = 0 h = Nu = = = 0.33 Re T T T T k T T (8) here the eact Blasius solution is used (see Table 3); using instead the simplest model (T T)/(T T) =u/u=(y/δ) (y/δ) )=1 (1 y/δ), one gets Nu = 0.6 Re. Instead of the local Nusselt number, the global-average value over a hole plate of length, Nu (1/) Nud=Nu= can be used. Heat and mass convection. Boundary layer flo page 15

16 As above, the turbulent case cannot be analytically solved, but the equivalence beteen thermal boundary-layer and velocity boundary-layer (for Pr=1), allos to compute the temperature gradient at the all in terms of the velocity gradient at the all, much easier to measure, hat gives Nu=cfRe/, called Reynolds analogy, although a modified Reynolds analogy, named Reynolds-Colburn or Colburn-Chilton analogy, is commonly used (see belo). An entirely similar ith the convection of species i in a miture, here the mass-convection coefficient, hm, and Sherood number, Sh, are defined in terms of the mass-flo-rate of species i at the interface, ji, as: h Sh Re ρi ρi y = f j Di ρi / y / i i i y 0 h ρ m i y ρ ρ δ = y= 0 f ( η ) m = = = = 0.33 ρi ρi ρi ρi Di ρi ρi δ η η = 0 (9) The problem no is to find the solution for the thermal boundary layer in the laminar case but for Pr 1, and for the solutal boundary layer hen Sc 1. Besides, e may ant to consider thermal or solutal convection to start somehere donstream, at =0, and not precisely at the leading edge of the plate, =0. For Pr>1, thermal diffusivity (i.e. penetration) is smaller than momentum diffusivity (a<ν), and consequently the thermal boundary layer, δt, is thinner than the velocity boundary layer, δ. et us measure the ratio by ζ δt/δ. The integral method, applied before to the velocity boundary layer, for the rectangular control volume of idth d and height H sketched in Fig. 1, gives no: δ ( ) d m = ρuy d + ρu H δ(), m = m ( ) out δ( ) δ( ) d 0 d d u u dy uu dy δ ( ) d d d ρ = ρ µ u y 0 0 y= 0 ρ d ρ ( δ (), ) out µ δ( ) δ( ) d y 0 y= 0 d d T ρuedy ρuedy δ ( ) d T d = d k 0 0 y y= 0 ρ d ρ ( δ(), ) out d y 0 y= 0 p = u y+ u H p = m u e= ue y + u e H e= m e k (30) Using the simplest approimation u/u=f(η)=(y/δ) (y/δ) (see Table 3), putting energy proportional to temperature, e=ct, a corresponding profile (T T)/(T T)=(y/δΤ) (y/δτ) =f(η/ζ), and ζ δt/δ, one gets: δ( ) δ( ) d u d ν u ( u( u u )) dy f ( )( f ( ) 1d ) y d = ν η η = y d u y 0 y= 0 0 y= 0 δt( ) δ ( ) T d T d η a T ( ut ( T) ) dy a f( ) f 1 dy d = η = y d ζ u ( T T ) y 0 y= 0 0 y= 0 (31) Heat and mass convection. Boundary layer flo page 16

17 here the energy integral is limited to δt because T T is zero outside. Performing the substitutions and integration, (31) yields: ( ) d δ ν ν = δ ( ) = 30 d 15 u δ ( ) u dζ 4 3 ( 0 6ζ ) + 5ζ ζ ζ d 5ζ ( ) δ ( ) ζ ( ) δ ( ) a d = Pr = d u ζ ( ) δ ( ) (3) i.e., the momentum integral gives the thickness la for the velocity boundary layer, hich is substituted in the energy integral to get, either a constant δt/δ-relation if only the leading terms are kept, namely 5ζ 3 =4/Pr, or, if an initial condition ζ(0)=0 is imposed: 3/4 1/ /3 0 1/3 δt ζ ( ) = 0.93Pr 1 = 0.93Pr δ (33) If, instead of model 4 (see Table 3), the more refined model 1 is used, the only changes are the change in the coefficients: δ=4.64(ν/u) 1/ instead of δ=5.48(ν/u) 1/, and 1.0 instead of 0.93 in (33). Besides the thermal thickness, the slope of the thermal profile at the all is important; ith model 4, for hich e found du/dy y=0=0.365(νu) 1/, no e get, for 0=0, dt/dy y=0=0.39(t T)Re 1/ Pr 1/3 /, hich is not too far from the most precise coefficient found by Pohlhausen in 191 (0.33 instead of (70) 1/6 =0.39). In the traditional non-dimensional form, retaining the possibility of the temperature jump starting somehere donstream, at =0, one has: 3/4 1/3 / y= 0 1/3 1/ h T y Nu = = Pr Re k T T (34) Again, the global-average value of the Nusselt number over a hole plate of length, is often used; for 0=0, Nu (1/) Nud=Nu=. Notice that the above result comes from a global energy balance in the hole of the thermal layer, thus, an integral average of the fluid properties must be used, and not just their values at all conditions; these 'film averaged' values are usually computed just as the algebraic mean of the values at all conditions and at bulk conditions (here the undisturbed conditions), i.e. Tfilm (T+T)/. Pohlhausen correlation (34), although deduced for Pr>1, has been found to be accurate for 0.6<Pr<60, but not enough for the very high Prandtl numbers ehibited by some oils and silicones, and for Pr<<1 typical of liquid metals. An etension to Pohlhausen correlation in the hole range of Prandtl numbers as made by Churchill and Ozoe in 1973 in the form: Heat and mass convection. Boundary layer flo page 17

18 Nu 0.34 Pr Re = Pr 1/3 1/ 3 14 (35) valid also for the case hen the heat flu density at the all is kept constant (instead of the all temperature), if the coefficient 0.34 is changed to 0.46, and the coefficient is changed to From the local Nusselt number Nu, the local convective coefficient is deduced, h=knu/. For practical ork it may be good enough to use a global convective coefficient h to be deduced from a global Nusselt number Nu. Notice, hoever, that h is the average value of h, but Nu is not the average of Nu; e.g., from (34) ith 0=0: 1/ 1 1 1/3 u d 0.33 d = = = ν Nu Nu Pr Nu 1/ 1 1 knu 1 k 1/3 u = d = d = 0.33 d = = ν (36) h h Pr h Reynolds analogy beteen momentum and energy equations Reynolds analogy is based on the similarity beteen momentum, heat, and mass transfer from the general balance equations: Dv DT 1 v 1 T 1 yi Dt Dt Dt v a T D y p= 0 φ = 0 0 D i = yi = ν v, = a T, = Di yi = = ν i i (37) It follos from (37) that, if ν=a=di, then the scaled functions ould be identical, u(,y)/ u=t(,y)/ T= yi(,y)/ yi (ith u u 0, T T T0, and u yi yi0), their slopes at the all identical too, and thus, ith the definition of the Fanning factor, cf, and the convective coefficients, for laminar flos, one gets: h k T T Nu Nu, h = = T Nu k T T T Nu 1 τ µ u u uc = = = c u uc c f = = Re f f f 1 1 ρu ρu u ν u ν (38) i.e. Nu=(cf/)Re, knon as Reynolds analogy (he deduced it in 1874), and valid for laminar flos (to apply Neton's la of friction) and Pr=Sc=1 (for ν=a=di). The influence of Pr 1 (and/or Sc 1) can be retained by stretching the transversal dimension differently for each function in (36) to absorb the respective coefficient; i.e., no the functions hich are identical are, u(,y/ν)/ u=t(,y/a)/ T= yi(,y/di)/ yi, and the slopes hich are identical are. Heat and mass convection. Boundary layer flo page 18

19 u(,y/ν)/ u=ν u(,y)/ u= T(,y/a)/ T=a T(,y)/ T= yi(,y/di)/ yi=di yi(,i)/ yi, Thence, (38) becomes: h k T T Nu Nu, h = a = a T Nu k T T a a T 1 τ µ u u uc = = = c u uc c f = ν = ν ν f f f 1 1 u ν ρu ρu ν u ν Nu Re Pr (39) i.e. Nu=(cf/)RePr. Hoever, if e compare the eact solutions to the laminar boundary layer obtained above (Eq. (34) and Table 4), e obtained a more accurate Pr-correction to Reynolds analogy: 0 = 0 1/3 1/ Nu = 0.33 Pr Re cf, 1/3 Nu c f, 0.66 Nu = RePr St = /3 c Pr Re f, Pr 1 Re = (40) hich is knon as Chilton-Colburn analogy (1934); St is a combined parameter named Stanton number. Although (40) has been developed only for a laminar-boundary-layer flo over a flat plate, it applies ith good accuracy to both laminar and turbulent flos over flat plates in the 0.6<Pr<60, and even to any turbulent flo ith pressure gradients, but not to laminar flos ith p 0 (i.e. Colburn-Chilton analogy can be applied to any turbulent flo, but only to laminar flos over flat plates, not to laminar flos in pipes or around bodies). A compilation of heat-transfer correlations in forced convection over a flat plate is presented in Table 5. A note on correlations for turbulent flo is required: the local Nusselt number Nu is used to get the local convective coefficient using h=knu/ ith measured from the leading edge of the plate (not from the start of the turbulent layer), hereas the global Nusselt number Nu is used to get the global convective coefficient using h=knu/ hich includes both the laminar zone and the turbulent zone: tr tr 1/ 4/ k 1/3 u k 1/3 u h hd h,lam d h,turbd = 0.33 Pr d 0.03Pr d = + = + = ν ν 0 0 tr 0 tr 1/3 1/3 k Pr 6 1/ 5 4/5 4/5 Retr = k Pr 4/5 = 0.33Retr ( Re Retr ) h = ( ( Re 3600) ) = 4 1/3 k Pr 4/5 h = ( 0.037Re 870) (41) Table 5. Heat transfer correlations in forced convection over a flat plate. Flo regime Correlation aminar Re<Retr= If 0.6<Pr<500 (modified Pohlhausen equation): (i.e. <tr= ν/u) 3/4 1/3 1/3 1/ 0 Nu = 0.33Pr Re 1 (coefficients shon are for constant Tall; for constant qall, change Heat and mass convection. Boundary layer flo page 19

20 coeff to 0.45, coeff to 1.3) Turbulent <Re<10 8 (either constant Tall or constant qall). The global value includes the laminar contribution at Re< /3 1/ Nu = 0.66 Pr Re 0 1 If Pr<0.05 (liquid metals): Nu 0.56( ) 1/ = PrRe Nu = 1.1( PrRe ) 1/ If 0.6<Pr<60: Nu = Pr Re 1/3 4/5 ( ) Nu = Re Pr 3/4 4/5 1 3 An entirely similar analogy can be applied to mass convection: Sh Sc Re c f, = (4) /3 Sc valid for 0.6<Sc<3000 and both laminar and turbulent flos. Notice, by the ay, that other types of friction factors different to the Fanning friction factor, cf, defined in (37) are often used in some cases, particularly in pipe flo, namely the Darcy friction factor, λ=4cf (named f sometimes). Steps to solve heat and mass convection problems To solve a heat or mass convection problem, the folloing steps are usually folloed: Make a quick order-of-magnitude analysis using typical values as from Table 6. Geometry characterisation, i.e. try to reduce the geometry to a canonical case (e.g. flat all, cylinder, tube bank...). Reference conditions characterisation, i.e. identify type of fluid, and estimate fluid properties at estimated film-averaged conditions. Reynolds number computation, to discern the flo type. Selection of the appropriate non-dimensional correlation (local or global). Table 6. Order of magnitude of convection coefficient, h, for typical configurations. Configuration Typical value of h [W/(m K)] Typical range, h [W/(m K)] Natural convection in air (<1 m/s) Forced convection in air (>5 m/s) Natural convection in ater (<0.1 m/s) Forced convection in ater (>0.5 m/s) Natural convection boiling in ater Forced convection boiling in ater Natural convection condensation in ater Forced convection condensation in ater Heat and mass convection. Boundary layer flo page 0