Convection Boundary Layer flow

convective boundary layer eddy diffusivity and natural convection boundary layer equations and mixed convection boundary layer flow heat transfer
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c06.qxd 3/6/06 10:54 AM Page 347 CHAPTER 6 Introduction to Convectionc06.qxd 3/6/06 10:54 AM Page 348 348 Chapter 6  Introduction to Convection Thus far we have focused on heat transfer by conduction and have considered convection only to the extent that it provides a possible boundary condition for con- duction problems. In Section 1.2.2 we used the term convection to describe energy transfer between a surface and a fluid moving over the surface. Convection includes energy transfer by both the bulk fluid motion (advection) and the random motion of fluid molecules (conduction or diffusion). In our treatment of convection, we have two major objectives. In addition to obtaining an understanding of the physical mechanisms that underlie convection transfer, we wish to develop the means to perform convection transfer calculations. This chapter and the supplementary material of Appendix D are devoted primarily to achieving the former objective. Physical origins are discussed, and relevant dimensionless parameters, as well as important analogies, are developed. A unique feature of this chapter is the manner in which convection mass trans- fer effects are introduced by analogy to those of convection heat transfer. In mass transfer by convection, gross fluid motion combines with diffusion to promote the transfer of a species for which there exists a concentration gradient. In this text, we focus on convection mass transfer that occurs at the surface of a volatile solid or liq- uid due to motion of a gas over the surface. With conceptual foundations established, subsequent chapters are used to develop useful tools for quantifying convection effects. Chapters 7 and 8 present methods for computing the coefficients associated with forced convection in external and internal flow configurations, respectively. Chapter 9 describes methods for determining these coefficients in free convection, and Chapter 10 considers the problem of convection with phase change (boiling and condensation). Chapter 11 develops methods for designing and evaluating the performance of heat exchangers, devices that are widely used in engineering practice to effect heat transfer between fluids. Accordingly, we begin by developing our understanding of the nature of convection. 6.1 The Convection Boundary Layers The concept of boundary layers is central to the understanding of convection heat and mass transfer between a surface and a fluid flowing past it. In this section, veloc- ity, thermal, and concentration boundary layers are described, and their relationships to the friction coefficient, convection heat transfer coefficient, and convection mass transfer coefficient are introduced. 6.1.1 The Velocity Boundary Layer To introduce the concept of a boundary layer, consider flow over the flat plate of Figure 6.1. When fluid particles make contact with the surface, they assume zero velocity. 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   from the surface, the effect becomes negligible. This retardation of fluid motion isc06.qxd 3/6/06 10:54 AM Page 349 6.1  The Convection Boundary Layers 349 u ∞ Free stream δ (x) u ∞ u Velocity y τ boundary δ layer FIGURE 6.1 τ Velocity boundary layer development on a flat plate. x associated with shear stresses  acting in planes that are parallel to the fluid velocity (Figure 6.1). With increasing distance y from the surface, the x velocity component of the fluid, u, must then increase until it approaches the free stream value u . The sub-  script  is used to designate conditions in the free stream outside the boundary layer. The quantity  is termed the boundary layer thickness, and it is typically defined as the value of y for which u  0.99u . The boundary layer velocity profile refers to  the manner in which u varies with y through the boundary layer. Accordingly, the fluid flow is characterized by two distinct regions, a thin fluid layer (the boundary layer) in which velocity gradients and shear stresses are large and a region outside the boundary layer in which velocity gradients and shear stresses are negligible. With increasing distance from the leading edge, the effects of viscosity penetrate fur- ther into the free stream and the boundary layer grows ( increases with x). Because it pertains to the fluid velocity, the foregoing boundary layer may be referred to more specifically as the velocity boundary layer. It develops whenever there is fluid flow over a surface, and it is of fundamental importance to problems involving convection transport. In fluid mechanics its significance to the engineer stems from its relation to the surface shear stress  , and hence to surface frictional effects. For exter- s nal flows it provides the basis for determining the local friction coefficient  s C  (6.1) f 2 u /2  a key dimensionless parameter from which the surface frictional drag may be deter- mined. Assuming a Newtonian fluid, the surface shear stress may be evaluated from knowledge of the velocity gradient at the surface u    (6.2)  s y y0 where  is a fluid property known as the dynamic viscosity. In a velocity boundary layer, the velocity gradient at the surface depends on the distance x from the leading edge of the plate. Therefore, the surface shear stress and friction coefficient also depend on x. 6.1.2 The Thermal Boundary Layer Just as a velocity boundary layer develops when there is fluid flow over a surface, a thermal boundary layer must develop if the fluid free stream and surface temperatures differ. Consider flow over an isothermal flat plate (Figure 6.2). At the leading edge the temperature profile is uniform, with T(y)  T . However, fluid particles that  come into contact with the plate achieve thermal equilibrium at the plate’s surfacec06.qxd 3/6/06 10:54 AM Page 350 350 Chapter 6  Introduction to Convection u ∞ Free stream T δ (x) ∞ t T ∞ Thermal y boundary FIGURE 6.2 T δ t layer Thermal boundary layer development on an isothermal flat x T s plate. 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 t typically defined as the value of y for which the ratio (T  T)/(T  T )  0.99. s s  With increasing distance from the leading edge, the effects of heat transfer penetrate further into the free stream and the thermal boundary layer grows. The relation between conditions in this boundary layer and the convection heat transfer coefficient may readily be demonstrated. At any distance x from the leading edge, the local surface heat flux may be obtained by applying Fourier’s law to the fluid at y  0. That is, T  q  k (6.3)  s f y y0 The subscript s has been used to emphasize that this is the surface heat flux, but it will be dropped in later sections. This expression is appropriate because, at the surface, there is no fluid motion and energy transfer occurs only by conduction. Recalling Newton’s law of cooling, we see that  q  h(T  T ) (6.4) s s  and combining this with Equation 6.3, we obtain k T/y f y0 h  (6.5) T  T s  Hence, conditions in the thermal boundary layer, which strongly influence the wall temperature gradient T/y , determine the rate of heat transfer across the y0 boundary layer. Since (T  T ) is a constant, independent of x, while  increases s  t with increasing x, temperature gradients in the boundary layer must decrease with increasing x. Accordingly, the magnitude of T/y decreases with increasing y0  x, and it follows that q and h decrease with increasing x. s 6.1.3 The Concentration Boundary Layer When air moves past the surface of a pool of water, the liquid water will evaporate, and water vapor will be transferred into the air stream. This is an example of convec- tion mass transfer. More generally, consider a binary mixture of chemical species A 3 and B that flows over a surface (Figure 6.3). The molar concentration (kmol/m ) of species A at the surface is C , and in the free stream it is C . If C differs from A,s A, A,s C , transfer of species A by convection will occur. For example, species A could A,c06.qxd 3/6/06 10:54 AM Page 351 6.1  The Convection Boundary Layers 351 u ∞ Mixture of A + B C A, ∞ Free stream δ (x) c C A, ∞ Concentration y boundary FIGURE 6.3 δ C c A layer Species concentration boundary layer development on a flat x C plate. A, s be a vapor that is transferred into a gas stream (species B) due to evaporation at a liq- uid surface (as in the water example) or due to sublimation at a solid surface. In this situation, a concentration boundary layer will develop that is similar to the velocity and thermal boundary layers. The concentration boundary layer is the region of the fluid in which concentration gradients exist, and its thickness  is typically defined c as the value of y for which (C  C )/(C  C )  0.99. With increasing dis- A,s A A,s A, tance from the leading edge, the effects of species transfer penetrate further into the free stream and the concentration boundary layer grows. Species transfer by convection between the surface and the free stream fluid is determined by conditions in the boundary layer, and we are interested in determin- ing the rate at which this transfer occurs. In particular, we are interested in the 2  molar flux of species A, N (kmol/s  m ). It is helpful to recognize that the molar A flux associated with species transfer by diffusion is determined by an expression that is analogous to Fourier’s law. For the conditions of interest in this chapter, the expression, which is termed Fick’s law, has the form C A 1  N  D (6.6) A AB y where D is a property of the binary mixture known as the binary diffusion coeffi- AB cient. At any point corresponding to y  0 in the concentration boundary layer of Figure 6.3, species transfer is due to both bulk fluid motion (advection) and diffusion. However, at y  0 there is no fluid motion (neglecting the often small velocity normal to the surface caused by the species transfer process itself, as is discussed in Chapter 14), and thus species transfer is by diffusion only. Applying Fick’s law at y  0, we see that the molar flux at the surface at any distance from the leading edge is then C A  N  D (6.7)  A,s AB y y0 The subscript s has been used to emphasize that this is the molar flux at the surface, but it will be dropped in later sections. Analogous to Newton’s law of cooling, an equation can be written that relates the molar flux to the concentration difference across the boundary layer, as  N  h (C  C ) (6.8) A,s m A,s A, 1 This expression is an approximation of a more general form of Fick’s law of diffusion (Section 14.1.3) when the total molar concentration of the mixture, C  C + C , is a constant. A Bc06.qxd 3/6/06 10:54 AM Page 352 352 Chapter 6  Introduction to Convection where h (m/s) is the convection mass transfer coefficient, analogous to the convec- m tion heat transfer coefficient. Combining Equations 6.7 and 6.8, it follows that D C /y AB A y0 h  (6.9) m C  C A,s A, Therefore, conditions in the concentration boundary layer, which strongly influence the surface concentration gradient C /y , also influence the convection mass A y0 transfer coefficient and hence the rate of species transfer in the boundary layer. 6.1.4 Significance of the Boundary Layers For flow over any surface, there will always exist a velocity boundary layer and hence surface friction. Likewise, a thermal boundary layer, and hence convection heat transfer, will always exist if the surface and free stream temperatures differ. Similarly, a concentration boundary layer and convection mass transfer will exist if the surface concentration of a species differs from its free stream concentration. The velocity boundary layer is of extent (x) and is characterized by the presence of velocity gradients and shear stresses. The thermal boundary layer is of extent  (x) t and is characterized by temperature gradients and heat transfer. Finally, the concen- tration boundary layer is of extent  (x) and is characterized by concentration gradi- c ents and species transfer. Situations can arise in which all three boundary layers are present. In such cases, the boundary layers rarely grow at the same rate, and the val- ues of ,  , and  at a given location are not the same. t c For the engineer, the principal manifestations of the three boundary layers are, respectively, surface friction, convection heat transfer, and convection mass trans- fer. The key boundary layer parameters are then the friction coefficient C and the f heat and mass transfer convection coefficients h and h , respectively. We now turn m our attention to examining these key parameters, which are central to the analysis of convection heat and mass transfer problems. 6.2 Local and Average Convection Coefficients 6.2.1 Heat Transfer Consider the conditions of Figure 6.4a. A fluid of velocity V and temperature T  flows over a surface of arbitrary shape and of area A . The surface is presumed to be s at a uniform temperature, T , and if T  T , we know that convection heat transfer s s  will occur. From Section 6.1.2, we also know that the surface heat flux and convec- tion 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. That is, q  qdA (6.10)  s A sc06.qxd 3/6/06 10:54 AM Page 353 6.2  Local and Average Convection Coefficients 353 V, T ∞ u , T q" ∞ ∞ dA s q" A , T s s A , T s s L x dx (a) (b) FIGURE 6.4 Local and total convection heat transfer. (a) Surface of arbitrary shape. (b) Flat plate. or, from Equation 6.4, q  (T  T ) hdA (6.11) s  s A s Defining an average convection coefficient h for the entire surface, the total heat transfer rate may also be expressed as q  h A (T  T ) (6.12) s s  Equating Equations 6.11 and 6.12, it follows that the average and local convection coefficients are related by an expression of the form 1 h   hdA (6.13) s A s A s Note that for the special case of flow over a flat plate (Figure 6.4b), h varies only with the distance x from the leading edge and Equation 6.13 reduces to L 1 h   hdx (6.14) L 0 6.2.2 Mass Transfer Similar results may be obtained for convection mass transfer. If a fluid of species molar concentration C flows over a surface at which the species concentration is A, maintained at some uniform value C  C (Figure 6.5a), transfer of the species A,s A, by convection will occur. From Section 6.1.3, we know that the surface molar flux and convection mass transfer coefficient both vary along the surface. The total molar transfer rate for an entire surface, N (kmol/s), may then be expressed as A N  h A (C  C ) (6.15) A m s A,s A, where the average and local mass transfer convection coefficients are related by an equation of the form 1 h  h dA  (6.16) m m s A s A sc06.qxd 3/6/06 10:54 AM Page 354 354 Chapter 6  Introduction to Convection V, C A,∞ N" u , C A ∞ A, ∞ dA s N" A A , C s A, s A , C s A, s L x dx (a) (b) FIGURE 6.5 Local and total convection species transfer. (a) Surface of arbitrary shape. (b) Flat plate. For the flat plate of Figure 6.5b, it follows that L 1 h  h dx (6.17)  m m L 0 2  Species transfer may also be expressed as a mass flux, n (kg/s  m ), or as a A mass transfer rate, n (kg/s), by multiplying both sides of Equations 6.8 and 6.15, A respectively, by the molecular weight  (kg/kmol) of species A. Accordingly, A  n  h (   ) (6.18) A m A,s A, and n  h A (   ) (6.19) A m s A,s A, 3 2 where  (kg/m ) is the mass density of species A. We can also write Fick’s law on A a mass basis by multiplying Equation 6.7 by  to yield A  A  n  D (6.20)  A,s AB y y0 Furthermore, multiplying the numerator and denominator of Equation 6.9 by  A yields an alternative expression for h : m D  /y AB A y0 h  (6.21) m    A,s A, To perform a convection mass transfer calculation, it is necessary to determine the value of C or  . Such a determination may be made by assuming thermody- A,s A,s namic equilibrium at the interface between the gas and the liquid or solid phase. One implication of equilibrium is that the temperature of the vapor at the interface is equal to the surface temperature T . A second implication is that the vapor is in a s saturated state, in which case thermodynamic tables, such as Table A.6 for water, may be used to obtain its density from knowledge of T . To a good approximation, s the molar concentration of the vapor at the surface may also be determined from the vapor pressure through application of the equation of state for an ideal gas. That is, 2 Although the foregoing nomenclature is well suited for characterizing mass transfer processes of interest in this text, there is by no means a standard nomenclature, and it is often difficult to recon- cile the results from different publications. A review of the different ways in which driving poten- tials, fluxes, and convection coefficients may be formulated is provided by Webb 1.c06.qxd 3/6/06 10:54 AM Page 355 6.2  Local and Average Convection Coefficients 355 p (T ) sat s C  (6.22) A,s T s where  is the universal gas constant and p (T ) is the vapor pressure correspond- sat s ing to saturation at T . Note that the vapor mass density and molar concentration are s related by    C . A A A 6.2.3 The Problem of Convection The local flux and/or the total transfer rate are of paramount importance in any con- vection problem. These quantities may be determined from the rate equations, Equations 6.4, 6.8, 6.12, and 6.15, which depend on knowledge of the local (h or h ) and average (h or h ) convection coefficients. It is for this reason that determi- m m nation of these coefficients is viewed as the problem of convection. However, the problem is not a simple one, for in addition to depending on numerous fluid proper- ties such as density, viscosity, thermal conductivity, and specific heat, the coeffi- cients depend on the surface geometry and the flow conditions. This multiplicity of independent variables is attributable to the dependence of convection transfer on the boundary layers that develop on the surface. EXAMPLE 6.1 Experimental results for the local heat transfer coefficient h for flow over a flat x plate with an extremely rough surface were found to fit the relation 0.1 h (x)  ax x 1.9 where a is a coefficient (W/m • K) and x (m) is the distance from the leading edge of the plate. 1. Develop an expression for the ratio of the average heat transfer coefficient h x for a plate of length x to the local heat transfer coefficient h at x. x 2. Show, in a qualitative manner, the variation of h and h as a function of x. x x SOLUTION Known: Variation of the local heat transfer coefficient, h (x). x Find: 1. The ratio of the average heat transfer coefficient h(x) to the local value h (x). x 2. Sketch of the variation of h and h with x. x x Schematic: Boundary layer –0.1 h = ax x  (x) t T ∞ T s xc06.qxd 3/6/06 10:54 AM Page 356 356 Chapter 6  Introduction to Convection Analysis: 1. From Equation 6.14 the average value of the convection heat transfer coeffi- cient over the region from 0 to x is x 1 h  h (x)  h (x) dx  x x x x 0 Substituting the expression for the local heat transfer coefficient 0.1 h (x)  ax x and integrating, we obtain x x 0.9 1 a a x 0.1 0.1 0.1 h  ax dx  x dx   1.11ax   x x x x  0.9 0 0 or h  1.11h  x x 2. The variation of h and h with x is as follows: x x 1.5a h = 1.11h x x 1.0a –0.1 h = ax x 0.5a 0 01 234 x (m) Comments: Boundary layer development causes both the local and average coefficients to decrease with increasing distance from the leading edge. The average coefficient up to x must therefore exceed the local value at x. EXAMPLE 6.2 A long circular cylinder 20 mm in diameter is fabricated from solid naphthalene, a common moth repellant, and is exposed to an airstream that provides for an aver- age convection mass transfer coefficient of h  0.05 m/s. The molar concentration m 6 3 of naphthalene vapor at the cylinder surface is 5  10 kmol/m , and its mole- cular weight is 128 kg/kmol. What is the mass sublimation rate per unit length of cylinder? SOLUTION Known: Saturated vapor concentration of naphthalene. Find: Sublimation rate per unit length, n (kg/s • m). A 2 • h , h (W/m K) x xc06.qxd 3/6/06 10:54 AM Page 357 6.2  Local and Average Convection Coefficients 357 Schematic: n' A C = 0 A, ∞ h = 0.05 m/s m L Air –6 3 C = 5 × 10 kmol/m A, s D = 20 mm Assumptions: 1. Steady-state conditions. 2. Negligible concentration of naphthalene in free stream of air. Analysis: Naphthalene is transported to the air by convection, and from Equation 6.15, the molar transfer rate for the cylinder is N  h DL(C  C ) A m A,s A, With C  0 and N  N /L, it follows that A,  A A 6 3  N  (D)h C    0.02 m  0.05 m/s  5  10 kmol/m A m A, s 8  N  1.57  10 kmol/s  m A The mass sublimation rate is then 8   n   N  128 kg/kmol  1.57  10 kmol/s  m A A A 6  n  2.01  10 kg/s  m  A EXAMPLE 6.3 At some location on the surface of a pan of water, measurements of the partial pres- sure of water vapor p (atm) are made as a function of the distance y from the sur- A face, and the results are as follows: 8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.10 p (atm) A Determine the convection mass transfer coefficient h at this location. m,x y (mm)c06.qxd 3/6/06 10:54 AM Page 358 358 Chapter 6  Introduction to Convection SOLUTION Known: Partial pressure p of water vapor as a function of distance y at a partic- A ular location on the surface of a water layer. Find: Convection mass transfer coefficient at the prescribed location. Schematic: p = 0.02 atm A, ∞ Air y p , T A, ∞ ∞ p = 0.10 atm y 3.0 mm A, s p , T A, s s Water Tangent at y = 0 x p (y) A Assumptions: 1. Water vapor may be approximated as an ideal gas. 2. Conditions are isothermal. Properties: Table A.6, saturated vapor (0.1 atm  0.101 bar): T  319 K. Table s 3/2 A.8, water vapor–air (319 K): D (319 K)  D (298 K)  (319 K/298 K)  AB AB 4 2 0.288  10 m /s. Analysis: From Equation 6.21 the local convection mass transfer coefficient is D  /y  AB A y0 h  m,x    A,s A, or, approximating the vapor as an ideal gas p   RT A A with constant T (isothermal conditions), D p /y  AB A y0 h  m,x p  p A,s A, From the measured vapor pressure distribution p (0  0.1) atm A   33.3 atm/m  y (0.003  0) m y0 Hence 4 2 0.288  10 m /s (33.3 atm/m) h   0.0120 m/s m,x (0.1  0.02) atm Comments: From thermodynamic equilibrium at the liquid–vapor interface, the interfacial temperature was determined from Table A.6.c06.qxd 3/6/06 10:54 AM Page 359 6.3  Laminar and Turbulent Flow 359 6.3 Laminar and Turbulent Flow In the discussion of convection so far, we have not addressed the significance of the flow conditions. An essential first step in the treatment of any convection problem is to determine whether the boundary layer is laminar or turbulent. Surface friction and the convection transfer rates depend strongly on which of these conditions exists. 6.3.1 Laminar and Turbulent Velocity Boundary Layers Boundary layer development on a flat plate is illustrated in Figure 6.6. In many cases, laminar and turbulent flow conditions both occur, with the laminar section preceding the turbulent section. For either condition, the fluid motion is characterized by veloc- ity components in the x- and y-directions. Fluid motion away from the surface is necessitated by the slowing of the fluid near the wall as the boundary layer grows in the x-direction. Figure 6.6 shows that there are sharp differences between laminar and turbulent flow conditions, as described in the following paragraphs. In the laminar boundary layer, the fluid flow is highly ordered and it is possible to identify streamlines along which fluid particles move. From Section 6.1.1 we know that the boundary layer thickness grows and that velocity gradients at y  0 decrease in the streamwise (increasing x) direction. From Equation 6.2, we see that the local surface shear stress  also decreases with increasing x. The highly ordered s behavior continues until a transition zone is reached, across which a conversion from laminar to turbulent conditions occurs. Conditions within the transition zone change with time, with the flow sometimes exhibiting laminar behavior and some- times exhibiting the characteristics of turbulent flow. Flow in the fully turbulent boundary layer is, in general, highly irregular and is characterized by random, three-dimensional motion of relatively large parcels of fluid. Mixing within the boundary layer carries high-speed fluid toward the solid surface and transfers slower-moving fluid farther into the free stream. Much of the Streamline v u u ∞ y, v x, u Turbulent region u u ∞ ∞ Buffer layer Viscous sublayer x c x Laminar Turbulent Transition FIGURE 6.6 Velocity boundary layer development on a flat plate.c06.qxd 3/6/06 10:54 AM Page 360 360 Chapter 6  Introduction to Convection mixing is promoted by streamwise vortices called streaks that are generated inter- mittently near the flat plate, where they rapidly grow and decay. Recent analytical and experimental studies have suggested that these and other coherent structures within the turbulent flow can travel in waves at velocities that can exceed u , interact nonlin-  early, and spawn the chaotic conditions that characterize turbulent flow 2. As a result of the interactions that lead to chaotic flow conditions, velocity and pressure fluctuations occur at any point within the turbulent boundary layer. Three different regions may be delineated within the turbulent boundary layer as a function of distance from the surface. We may speak of a viscous sublayer in which transport is dominated by diffusion and the velocity profile is nearly linear. There is an adjoin- ing buffer layer in which diffusion and turbulent mixing are comparable, and there is a turbulent zone in which transport is dominated by turbulent mixing. A comparison of the laminar and turbulent boundary layer profiles for the x-component of the velocity, provided in Figure 6.7, shows that the turbulent velocity profile is relatively flat due to the mixing that occurs within the buffer layer and turbulent region, giving rise to large velocity gradients within the viscous sublayer. Hence,  is generally larger in the tur- s bulent portion of the boundary layer of Figure 6.6 than in the laminar portion. The transition from laminar to turbulent flow is ultimately due to triggering mechanisms, such as the interaction of unsteady flow structures that develop natu- rally within the fluid or small disturbances that exist within many typical boundary layers. These disturbances may originate from fluctuations in the free stream, or they may be induced by surface roughness or minute surface vibrations. The onset of turbulence depends on whether the triggering mechanisms are amplified or atten- uated in the direction of fluid flow, which in turn depends on a dimensionless grouping of parameters called the Reynolds number, u x  Re  (6.23) x  where, for a flat plate, the characteristic length is x, the distance from the leading edge. It will be shown later that the Reynolds number represents the ratio of the iner- tia to viscous forces. If the Reynolds number is small, inertia forces are insignificant relative to viscous forces. The disturbances are then dissipated, and the flow remains laminar. For a large Reynolds number, however, the inertia forces can be sufficient to amplify the triggering mechanisms, and a transition to turbulence occurs. u u y y = 0, lam y y = 0, turb u u ∞ ∞ _ u(y) u(y) FIGURE 6.7 Comparison u u y y = 0 of laminar and turbulent y y = 0 velocity boundary layer profiles for the same free 3 Laminar Turbulent stream velocity. 3 Since velocity fluctuates with time in turbulent flow, the time-averaged velocity, u, is plotted in Figure 6.7.c06.qxd 3/6/06 10:54 AM Page 361 6.3  Laminar and Turbulent Flow 361 In calculating boundary layer behavior, it is frequently reasonable to assume that transition begins at some location x , as shown in Figure 6.6. This location is c determined by the critical Reynolds number, Re . For flow over a flat plate, Re x,c x,c 5 6 is known to vary from approximately 10 to 3  10 , depending on surface rough- ness and the turbulence level of the free stream. A representative value of u x  c 5 Re   5  10 (6.24) x,c  is often assumed for boundary layer calculations and, unless otherwise noted, is used for the calculations of this text that involve a flat plate. 6.3.2 Laminar and Turbulent Thermal and Species Concentration Boundary Layers Since the velocity distribution determines the advective component of thermal energy or chemical species transfer within the boundary layer, the nature of the flow also has a profound effect on convective heat and mass transfer rates. Similar to the laminar velocity boundary layer, the thermal and species boundary layers grow in the streamwise (increasing x) direction, temperature and species concentration gradients in the fluid at y  0 decrease in the streamwise direction, and, from Equations 6.5 and 6.9, the heat and mass transfer coefficients also decrease with increasing x. Just as it induces large velocity gradients at y  0, turbulent mixing promotes large temperature and species concentration gradients adjacent to the solid surface as well as a corresponding increase in the heat and mass transfer coefficients across the transition region. These effects are illustrated in Figure 6.8 for the velocity boundary layer thickness  and the local convection heat transfer coefficient h. Because turbulence induces mixing, which in turn reduces the importance of con- duction and diffusion in determining the thermal and species boundary layer thick- nesses, differences in the thicknesses of the velocity, thermal, and species boundary layers tend to be much smaller in turbulent flow than in laminar flow. As is evident in Equation 6.24, the presence of heat and/or mass transfer can affect the location of the transition from laminar to turbulent flow x if the density or dynamic viscosity c of the fluid is dependent on temperature or species concentration. h, δ h (x) δ (x) u , T ∞ ∞ T s FIGURE 6.8 x c Variation of velocity boundary layer thickness x Laminar Turbulent  and the local heat transfer coefficient h for Transition flow over an isothermal flat plate.c06.qxd 3/6/06 10:54 AM Page 362 362 Chapter 6  Introduction to Convection EXAMPLE 6.4 Water flows at a velocity u  1 m/s over a flat plate of length L  0.6 m. Consider two  cases, one for which the water temperature is approximately 300 K and the other for an approximate water temperature of 350 K. In the laminar and turbulent regions, experi- mental measurements show that the local convection coefficients are well described by 0.5 0.2 h (x)  C x h (x)  C x lam lam turb turb where x has units of m. At 300 K, 1.5 1.8 C  395 W/m  K C  2330 W/m  K lam,300 turb,300 while at 350 K, 1.5 1.8 C  477 W/m  K C  3600 W/m  K lam,350 turb,350 As is evident, the constant, C, depends on the nature of the flow as well as the water temperature because of the thermal dependence of various properties of the fluid. Determine the average convection coefficient, h, over the entire plate for the two water temperatures. SOLUTION Known: Water flow over a flat plate; expressions for the dependence of the local convection coefficient with distance from the plate’s leading edge, x; and approxi- mate temperature of the water. Find: Average convection coefficient, h. Schematic: Laminar Turbulent u ∞ T s y δ (x) t x c L x Assumptions: 1. Steady-state conditions. 5 2. Transition occurs at a critical Reynolds number of Re  5  10 . x,c 1 3 6 Properties: Table A.6, water (T  300 K):   v  997 kg/m ,   855  10 f 2 1 3 6 2 N  s/m . Table A.6 (T  350 K):   v  974 kg/m ,   365  10 N  s/m . f Analysis: The local convection coefficient is highly dependent on whether lami- nar or turbulent conditions exist. Therefore, we first determine the extent to which these conditions exist by finding the location where transition occurs, x . From c Equation 6.24, we know that at 300 K,c06.qxd 3/6/06 10:54 AM Page 363 6.3  Laminar and Turbulent Flow 363 5 6 2 Re  x,c 5  10  855  10 N  s/m x    0.43 m c u 3  997 kg/m  1 m/s while at 350 K, 5 6 2 Re  x,c 5  10  365  10 N  s/m x    0.19 m c u 3  974 kg/m  1 m/s From Equation 6.14 we know that L x L c 1 1 h  hdx  h dx  h dx    lam turb   L L 0 0 x c or x L c C C 1 lam turb 0.5 0.8 h  x  x     L 0.5 0.8 0 x c At 300 K, 1.5 1.8 1 395 W/m  K 2330 W/m  K 0.5 0.5 h   (0.43 ) m   0.6 m 0.5 0.8 0.8 0.8 0.8 2  (0.6  0.43 ) m  1620 W/m  K   while at 350 K, 1.5 1.8 1 477 W/m  K 3600 W/m  K 0.5 0.5 h   (0.19 ) m   0.6 m 0.5 0.8 0.8 0.8 0.8 2  (0.6  0.19 ) m  3710 W/m  K   The local and average convection coefficient distributions for the plate are shown in the figure below. 8000 h, 300 K h, 350 K 6000 h, 350 K 4000 h, 300 K 2000 0 0 0.2 0.4 0.6 x (m) 2 • h, h, (W/m K)c06.qxd 3/6/06 10:54 AM Page 364 364 Chapter 6  Introduction to Convection Comments: 1. The average convection coefficient at T  350 K is over twice as large as the value at T  300 K. This strong temperature dependence is due primarily to the significant shift of x that is associated with the smaller viscosity of the water at c the higher temperature. Careful consideration of the temperature dependence of fluid properties is crucial when performing a convection heat transfer analysis. 2. Spatial variations in the local convection coefficient are significant. The largest local convection coefficients occur at the leading edge of the flat plate, where the laminar thermal boundary layer is extremely thin, and just downstream of x , where the turbulent boundary layer is thinnest. c 6.4 The Boundary Layer Equations We can improve our understanding of the physical effects that determine boundary layer behavior and further illustrate its relevance to convection transport by consid- ering the equations that govern boundary layer conditions, such as those illustrated in Figure 6.9. The velocity boundary layer results from the difference between the free stream velocity and the zero velocity at the wall, while the thermal boundary layer results from a difference between the free stream and surface temperatures. The fluid is considered to be a binary mixture of species A and B, and the concentration bound- ary layer originates from a difference between the free stream and surface concen- trations (C  C ). Illustration of the relative thicknesses (    ) in Figure A, A,s t c 6.9 is arbitrary, for the moment, and the factors that influence relative boundary layer development are discussed later in this chapter. Our objective in the next section is to obtain the differential equations that gov- ern the velocity, temperature, and species concentration fields that are applicable to C u A, ∞ ∞ N" A, s dy T ∞ y dx q" s Mixture of A + B τ s x T ∞ C A, s T s V δ δ δ t c C A, ∞ Thermal Concentration Velocity boundary boundary boundary layer layer layer FIGURE 6.9 Development of the velocity, thermal, and concentration boundary layers for an arbitrary surface.c06.qxd 3/6/06 10:54 AM Page 365 6.4  The Boundary Layer Equations 365 boundary layer flow with heat and species transfer. Section 6.4.1 presents the lami- nar boundary layer equations, and Appendix E gives the corresponding equations for turbulent conditions. 6.4.1 Boundary Layer Equations for Laminar Flow Motion of a fluid in which there are coexisting velocity, temperature, and concentration gradients must comply with several fundamental laws of nature. In particular, at each point in the fluid, conservation of mass, energy, and chemical species, as well as Newton’s second law of motion, must be satisfied. Equations representing these These equations are requirements are derived by applying the laws to a differential control volume situated derived in Section 6S.1. in the flow. The resulting equations, in Cartesian coordinates, for the steady, two- dimensional flow of an incompressible fluid with constant properties are given in Appendix D. These equations serve as starting points for our analysis of laminar boundary layers. Note that turbulent flows are inherently unsteady, and the equations governing them are presented in Appendix E. We begin by restricting attention to applications for which body forces are neg- ˙ ligible (X  Y  0), there is no thermal energy generation (q  0), and the flow is ˙ nonreacting (N  0). Additional simplifications may be made by invoking approx- A imations pertinent to conditions in the velocity, thermal, and concentration bound- ary layers. The boundary layer thicknesses are typically very small relative to the size of the object upon which they form, and the x-direction velocity, temperature, and concentration must change from their surface to their free stream values over these very small distances. Therefore, gradients normal to the object’s surface are much larger than those along the surface. As a result, we can neglect terms that rep- resent x-direction diffusion of momentum, thermal energy, and chemical species, relative to their y-direction counterparts. That is 3, 4: 2 2 2 2 2 2  C  C  u  u  T  T A A (6.25) 2 2 2 2 2 2 x y x y x y By neglecting the x-direction terms, we are assuming that the corresponding shear stress, conduction flux, and species diffusion flux are negligible. Furthermore, because the boundary layer is so thin, the x-direction pressure gradi- ent within the boundary layer can be approximated as the free stream pressure gradient: p dp   (6.26) x dx The form of p (x) depends on the surface geometry and may be obtained from a  separate consideration of flow conditions in the free stream. Hence, the pressure gradient may be treated as a known quantity. With the foregoing simplifications and approximations, the overall continuity equation is unchanged from Equation D.1: u v   0 (6.27) x y This equation is an outgrowth of applying conservation of mass to the differential, dx  dy  1 control volume shown in Figure 6.9. The two terms represent the net out- flow (outflow minus inflow) of mass in the x- and y-directions, the sum of which must be zero for steady flow.c06.qxd 3/6/06 10:54 AM Page 366 366 Chapter 6  Introduction to Convection The x-momentum equation (Equation D.2) reduces to: 2 dp u u 1   u u  v     (6.28)  2 x y dx y This equation results from application of Newton’s second law of motion in the x-direction to the dx  dy  1 differential control volume in the fluid. The left-hand side represents the net rate at which x-momentum leaves the control volume due to fluid motion across its boundaries. The first term on the right-hand side represents the net pressure force, and the second term represents the net force due to viscous shear stresses. The energy equation (Equation D.4) reduces to 2 2 T T  T  u u  v    (6.29) 2 c   x y p y y This equation results from application of conservation of energy to a differential control volume in the flowing fluid. Terms on the left-hand side account for the net rate at which thermal energy leaves the control volume due to bulk fluid motion (advection). The first term on the right-hand side accounts for the net inflow of ther- mal energy due to y-direction conduction. The last term on the right-hand side is what remains of the viscous dissipation, Equation D.5, when it is acknowledged that, in a boundary layer, the velocity component in the direction along the surface, u, is much larger than that normal to the surface, v, and gradients normal to the surface are much larger than those along the surface. In many situations this term may be neglected relative to those that account for advection and conduction. However, aerodynamic heating that accompanies high-speed (especially supersonic) flight is a noteworthy situation in which this term is important. The species conservation equation (Equation D.6) reduces to 2 C C  C A A A u  v  D (6.30) AB 2 x y y This equation is obtained by applying conservation of chemical species to a differ- ential control volume in the flow. Terms on the left-hand side account for net trans- port of species A due to bulk fluid motion (advection), while the right-hand side represents the net inflow due to y-direction diffusion. Equations 6.27 through 6.30 may be solved to determine the spatial variations of u, v, T, and C in the different laminar boundary layers. For incompressible, constant A property flow, Equations 6.27 and 6.28 are uncoupled from Equations 6.29 and 6.30. That is, Equations 6.27 and 6.28 may be solved for the velocity field, u(x,y) and v(x,y), without consideration of Equations 6.29 and 6.30. From knowledge of u(x,y), the velocity gradient (u/y) could then be evaluated, and the wall shear stress y0 could be obtained from Equation 6.2. In contrast, through the appearance of u and v in Equations 6.29 and 6.30, the temperature and species concentration are coupled to the velocity field. Hence u(x,y) and v(x,y) must be known before Equations 6.29 and 6.30 may be solved for T(x,y) and C (x,y). Once T(x,y) and C (x,y) have been A A obtained from such solutions, the convection heat and mass transfer coefficients may