CHAPTER 10 CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS CONSERVATION EOUATIONS IN FLUID MECHANICS, HEAT TRANSFER, AND MASS TRANSFER Each time we try to solve a new problem related to momentum and heat and mass transfer in a fluid, it is convenient to start with a set of equations based on basic laws of conservation for physical systems. These equations include 1. The continuity equation (conservation of mass). The equation of motion (conservation of momentum) 3. The energy equation (conservation of energy, or the first law of thermodynamics) 4. The conservation equation for species (conservation of species) These equations are sometimes called the equations of change, inasmuch as they describe the change of velocity, temperature, and concentration with respect to time and position in the system. The first three equations are sufficient for problems involving a pure fluid (a pure substance is a single substance characterized by an unvarying chemical structure). The fourth equation is added for a mixture of chemical species, i.e., when mass diffusion with or without chemical reactions is present. The control volume: When deriving the conservation equations, it is necessary to select a control volume. The derivation can be performed for a volume element of any shape in a given coordinate system, although the most convenient shape is usually assumed for simplicity (e.g., a rectangular shape in a rectangular coordinate system). For illustration purposes, different coordinate systems are shown in Fig. 10.1. In selecting a control volume, we have the option of using a volume fixed in space, in which case the fluid flows through the boundaries, or a volume containing a fixed mass of fluid and moving with the fluid. The former is known Adapted in part from Handbook of Heat Transfer Fundamentals, chap. 1, by W. M. Rohsenow, J. P. Hartnett, and E. N. Ganić, eds. Copyright 1985. Used by permission of McGraw-Hill, Inc. All rights reserved. 10.1 Copyright 003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
10. CHAPTER TEN FIGURE 10.1 Coordinate systems: (a) rectangular, (b) cylindrical, (c) spherical. as the eulerian viewpoint, and the latter is the lagrangian viewpoint. Both approaches yield equivalent results. The partial time derivative B/t: The partial time derivative of B(x,y,z,t), where B is any continuum property (e.g., density, velocity, temperature, concentration, etc.), represents the change of B with time at a fixed position in space. In other words, B/t is the change of B with t as seen by a stationary observer. Total time derivative db/dt: The total time derivative is related to the partial time derivative as follows: db B dx B dy B dz B (10.1) dt t dt x dt y dt z where dx/dt, dy/dt, and dz/dt are the components of the velocity of the moving observer. Therefore, db/dt is the change of B with time as seen by the moving observer. Substantial time derivative DB/ : This derivative is a special kind of total time derivative where now the velocity of the observer is just the same as the velocity of the stream; i.e., the observer drifts along with the current: DB B B B B u v w (10.) t x y z where u, v, and w are the components of the local fluid velocity V. The substantial time derivative is also called the derivative following the motion. The sum of the last three terms on the right-hand side of Eq. (10.) is called the convective contribution because it represents the change in B due to translation. The use of the operator D/ is always made when rearranging various conservation equations related to the volume element fixed in space to an element following the fluid motion. The operator D/ also may be expressed in vector form: D (V ) (10.3) t Mathematical operations involving are given in many textbooks. Applications of in various operations involving the conservation equations are given in Refs. 1 and. Table 10.1 gives the expressions for D/ in different coordinate systems.
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.3 TABLE 10.1 Substantial Derivative in Different Coordinate Systems Rectangular coordinates (x, y, z): D u v w t x y z Cylindrical coordinates (r,, x): D v vr vz t r r z Spherical coordinates (r,, ): D v v vr t r r r sin 1. The Equation of Continuity. For a volume element fixed in space, ( V) (10.4) t net rate of mass efflux per unit volume The continuity equation in this form describes the rate of change of density at a fixed point in the fluid. By performing the indicated differentiation on the right side of Eq. (10.4) and collecting all derivatives of on the left side, we obtain an equivalent form of the equation of continuity: D ( V) (10.5) The continuity equation in this form describes the rate of change of density as seen by an observer floating along with the fluid. For a fluid of constant density (incompressible fluid), the equation of continuity becomes V 0 (10.6) Table 10. gives the equation of continuity in different coordinate systems.. The Equation of Motion (Momentum Equation). The momentum equation for a stationary volume element (i.e., a balance over a volume element fixed in space) with gravity as the only body force is given by V t rate of increase of momentum per unit volume ( V)V rate of momentum gain by convection per unit volume P pressure force on element per unit volume rate of momentum gain by viscous transfer per unit volume g gravitational force on element per unit volume (10.7) Equation (10.7) may be rearranged, with the help of the equation of continuity, to give
10.4 CHAPTER TEN TABLE 10. Equation of Continuity in Different Coordinate Systems Rectangular coordinates (x, y, z): Cylindrical coordinates (r,, z): Spherical coordinates (r,, ): (u) () (w) 0 (A) t x y z 1 1 (rv r) (v ) (v z) 0 (B) t r r r z 1 1 1 (r v r) (v sin ) (v ) 0 (C) t r r r sin r sin Rectangular coordinates (x, y, z): Cylindrical coordinates (r,, z): Spherical coordinates (r,, ): Incompressible flow u v w 0 (D) x y z 1 1 v vz (rv r) 0 (E) r r r z 1 1 1 v (r v ) (v sin ) r 0 (F) r r r sin r sin DV P g (10.8) The last equation is a statement of Newton s second law of motion in the form mass acceleration sum of forces. These two forms of the equation of motion, Eqs. (10.7) and (10.8), correspond to the two forms of the equation of continuity, Eqs. (10.4) and (10.5). As indicated above, the only body force included in Eqs. (10.7) and (10.8) is gravity. In general, electromagnetic forces also may act on a fluid. 4 The scalar components of Eq. (10.8) are listed in Table 10.3, and the components of the stress tensor are given in Table 10.4. For the flow of a newtonian fluid with varying density but constant viscosity, Eq. (10.8) becomes DV 1 P ( V) V g (10.9) 3 If and are constant, Eq. (10.9) may be simplified by means of the equation of continuity ( V 0) to give for a newtonian fluid
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.5 DV P V g (10.10) This is the famous Navier-Stokes equation in vector form. The scalar components of Eq. (10.10) are given in Table 10.5. For 0, Eq. (10.8) reduces to Euler s equation: DV P g (10.11) which is applicable for describing flow systems in which viscous effects are relatively unimportant. TABLE 10.3 Equation of Motion in Terms of Viscous Stresses [Eq. (10.8)]* x direction Rectangular coordinates (x, y, z) y direction z direction r direction x u u u u P xx yx zx u v w g (A) t x y z x x y z y v v v v P xy yy u v w zy g (B) t x y z y x y z z w w w w P xz yz zz u v w g (C) t x y z z x y z vr vr v vr v vr r z v v t r r r z direction v v v v v v v vr vz t r r r z z direction r Cylindrical coordinates (r,, z) P 1 1 r zy (r rr) g r (A) r r r r r z 1 P 1 1 z (r r) g (B) r r r r z vz vz v vz vz P 1 1 z zz vr vz (r rz) g z (C) t r r z z r r r z
10.6 CHAPTER TEN TABLE 10.3 Equation of Motion in Terms of Viscous Stresses (Continued) [Eq. (10.8)]* Spherical coordinates (r,, ) r direction vr vr v vr v vr v v P 1 vr (r rr) t r r r sin r r r r 1 1 r (r sin ) g r (A) r sin r sin r direction v v v v v v vrv v cot 1 P vr t r r r sin r r r 1 1 1 r cot (r r) ( sin ) g (B) r r r sin r sin r r direction v v v v v v vvr vvcot 1 P vr t r r r sin r r r sin 1 1 1 r cot (r r) g (C) r r r r sin r r * Components of the stress tensor for newtonian fluids are given in Table 10.4. This equation also may be used for describing nonnewtonian flow. However, we need relations between the components of and the various velocity gradients; in other words, we have to replace the expressions given in Table 10.4 by other relations appropriate for the nonnewtonian fluid of interest. The expressions for for some nonnewtonian fluid models are given in Ref.. See also Ref.4. TABLE 10.4 Components of the Stress Tensor for Newtonian Fluids* Rectangular coordinates (x, y, z) u xx ( V) (A) x 3 v yy ( V) (B) y 3 w zz ( V) (C) z 3 u v xy yx (D) y x v w yz zy (E) z y w u zx xz (F) x z u v w ( V) (G) x y z
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.7 TABLE 10.4 Components of the Strress Tensor for Newtonian Fluids* (Continued) Cylindrical coordinates (r,, z) 1 v vr r r 3 vz z 3 v 1 vr rr r v 1 vz z r vz vr vr rr ( V) (A) r 3 ( V) (B) ( V) (C) zz r (D) r r (E) z z zr rz (F) r z 1 1 v vz ( V) (rv r) (G) r r r z Spherical coordinates (r,, ) 1 v vr r r 3 1 v vr v cot r sin r r 3 v 1 vr rr r sin v 1 v r sin r sin 1 vr v vr rr ( V) (A) r 3 ( V) (B) ( V) (C) r (D) r r (E) r r r (F) r sin r r 1 1 1 v ( V) (r v ) (v sin ) r (G) r r r sin r sin *It should be noted that the sign convention adopted here for components of the stress tensor is consistent with that found in many fluid mechanics and heat-transfer books; however, it is opposite to that found in some books on transport phenomena, e.g., Refs., 3, and 5.
10.8 CHAPTER TEN As mentioned before, there is a subset of flow problems, called natural convection, where the flow pattern is due to buoyant forces caused by temperature differences. Such buoyant forces are proportional to the coefficient of thermal expansion, defined as 1 (10.1) T P TABLE 10.5 Equation of Motion in Terms of Velocity Gradients for a Newtonian Fluid with Constant and, Eq. (10.10) x direction Rectangular coordinates (x, y, z) y direction z direction r direction u u u u P u u u x u v w g (A) t x y z x x y z v v v v P v v v y u v w g (B) t x y z y x y z w w w w P w ww w z u v w g (C) t x y z z x y z vr vr v vr v vr r z v v t r r r z direction v v v v v v v vr vz t r r r z z direction Cylindrical coordinates (r,, z) r r r P 1 1 v v v (rv ) g r (A) r r r r r r z r r 1 P 1 1 v v v (rv g (B) r r r r r r z vz vz v vz vz P 1 vz 1 vz vz r z z v v r g t r r z z r r r r z (C)
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.9 TABLE 10.5 Equation of Motion in Terms of Velocity Gradients for a Newtonian Fluid with Constant and, Eq. (10.10) (Continued) Spherical coordinates (r,, ) r direction v v v v v v v v v t r r r sin r direction r r r r r P vr v v cot v vr g r (A) r r r r r sin v v v v v v vrv v cot r v t r r r sin r r direction 1 P vr v cos v v g (B) r r r sin r sin v v v v v v vvr vv vr cot t r r r sin r r 1 P v vr cos v v g (C) r sin r sin r sin r sin For spherical coordinates the Laplacian is 1 1 1 r sin r r r r sin r sin where T is absolute temperature. Using an approximation that applies to low fluid velocities and small temperature variations, it can be shown,3 that Then Eq. (10.8) becomes P g g(t T ) (10.13) DV g(t T ) (10.14) buoyant force on element per unit volume The preceding equation of motion is used for setting up problems in natural convection when the ambient temperature T may be defined. 3. The Energy Equation. For astationary volume element through which a pure fluid is flowing, the energy equation reads
10.10 CHAPTER TEN 1 (u V ) V(u 1 V ) q (V g) t rate of gain of energy per unit volume rate of energy input per unit volume by convection rate of energy input per unit volume by conduction rate of work done on fluid per unit volume by gravitational forces PV ( V) q (10.15) rate of work done on fluid per unit volume by pressure forces rate of work done on fluid per unit volume by viscous forces rate of heat generation per unit volume ( source term ) The left side of the preceding equation, which represents the rate of accumulation of internal and kinetic energy, does not include the potential energy of the fluid, since this form of energy is included in the work term on the right side. Equation (10.15) may be rearranged, with the aid of the equations of continuity and motion, to give,6 Du q P( V) V: q (10.16) A summary of V: in different coordinate systems is given in Table 10.6. For a newtonian fluid, TABLE 10.6 Summary of Dissipation Term V: in Different Coordinate Systems Rectangular coordinates (x, y, z): v w w u yz zx u v w u v V: xx yy zz xy x y z y x Cylindrical coordinates (r,, z): (A) z y z z 1 vz v vz vr z rz vr 1 v vr vz v 1 vr V: rr zz r r r r r z r r r (B) r z r z Spherical coordinates (r,, ): v 1 vr v v 1 vr v r r r r r r r sin r 1 v 1 v v cot vr 1 v vr 1 v vr v cot V: rr r r r r sin r r (C) r r sin r
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.11 V: (10.17) and values of the dissipation function in different coordinate systems are given in Table 10.7. Components of the heat flux vector q kt are given in Table 10.8 for different coordinate systems. Often it is more convenient to work with enthalpy rather than internal energy. Using the definition of enthalpy, i u P/, and the mass conservation equation [Eq. (10.5)], then Eq. (10.16) can be rearranged to give TABLE 10.7 The Viscous Dissipation Function Rectangular coordinates (x, y, z): u w u v w u v w v u w v x y z x y y z Cylindrical coordinates (r,, z): (A) z x 3 x y z r r z r 1 vz v vr vz 1 1 v vz r v 1 v v v v 1 v r r r r z r r r (rv ) (B) r z z r 3 r r r z Spherical coordinates (r,, ): vr 1 v vr 1 v vr v cot r r r r sin r r r 1 vr v 1 1 1 v r v 1 v sin v 1 v r r r r r sin r sin r (r v ) (v sin ) r sin r r 3 r r r sin r sin (C) TABLE 10.8 Scalar Components of the Heat Flux Vector q Rectangular (x, y, z) Cylindrical (r,, z) Spherical (r,, ) T T T q x k (A) qr k (D) qr k (G) x r r T q y k y (B) 1 T q k r (E) 1 T q k r (H) T q z k z (C) T qz k z (F) 1 T q k r sin (I)
10.1 CHAPTER TEN Di DP kt q (10.18) For most engineering applications, it is convenient to have the equation of thermal energy in terms of the fluid temperature and heat capacity rather than the internal energy or enthalpy. In general, for pure substances, 3 p Di i DP i DT 1 DP DT (1 T) c (10.19) P T T P where is defined by Eq. (10.1). Substituting this into Eq. (10.18), we have the following general relation: For anideal gas, 1/T, and then DT DP c kt T q (10.0) p DT DP c kt q (10.1) p Note that c p need not be constant. We could have obtained Eq. (10.1) directly from Eq. (10.18) by noting that for an ideal gas, di c p dt, where c p is constant, and thus Di DT c p For anincompressible fluid with specific heat c c p c v,wegoback to Eq. (10.16) (du cdt)toobtain DT c kt q (10.) Equations (10.16), (10.18), and (10.0) can be easily written in terms of energy (heat) and momentum fluxes using relations for fluxes given in Tables 10.4, 10.6, and 10.8. The energy equation given by Eq. (10.) (with q 0 for simplicity) is given in Table 10.9 in different coordinate systems. For solids, the density may usually be considered constant, and we may set V 0, and Eq. (10.) reduces to T c kt q (10.3) t which is the starting point for most problems in heat conduction. The Energy Equation for a Mixture. The energy equations in the preceding section are applicable for a pure fluid. A thermal energy equation valid for a mixture of chemical species is required for situations involving simultaneous heat and mass transfer. For a pure fluid, conduction is the only diffusive mechanism of heat flow; hence Fourier s law was used, which resulted in the term kt. More generally, this term may be written q, where q is the diffusive heat flux, i.e., the heat flux relative to the mass average velocity. More specifically, for a mixture, q is now made from three contributions: (1) ordinary conduction, described by Fourier s
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.13 TABLE 10.9 The Energy Equation (for Newtonian Fluids of Constant and k)* Rectangular coordinates (x, y, z): w u v u w v w T T T T T T T u v p c u v w k t x y z x y z x y Cylindrical coordinates (r,, z): (A) z y x z x z y T T v T T 1 T 1 T T p r z c v v k r t r r z r r r r z vz vr 1 vr v r z z r z v 1 v v v 1 v v r r z r r (B) r z r r r Spherical coordinates (r,, ): 1 T vr 1 v vr 1 v vr v cot r sin r r r r sin r r v 1 vr 1 vr v r r r r sin r r sin v 1 v T T v T v T 1 T 1 T cp vr k r sin t r r r sin r r r r sin r r r sin r sin * The terms contained in braces { } are associated with viscous dissipation and may usually be neglected except in systems with large velocity gradients. (C) law, kt, where k is the mixture thermal conductivity; () the contribution due to interdiffusion of species, given by i j i i i ; and (3) diffusional conduction (also called the diffusion-thermo effect or Dufour effect 1,7 ). The third contribution is of the second order and is usually negligible: ii i q k T j i (10.4) Here j i is a diffusive mass flux of species i, with units of mass/(area time), as mentioned before. Substituting Eq. (10.4) in, for example, Eq. (10.18), we obtain the energy equation for a mixture: Di DP kt jii i q (10.5) i For anonreacting mixture the term (ij i i i )isoften of minor importance. But
10.14 CHAPTER TEN when endothermic or exothermic reactions occur, the term can play a dominant role. For reacting mixtures, the species enthalpies T 0 ii i i cpi dt T 0 must be written with a consistent set of heats of formation i 0 i at T 0. 8 4. The Conservation Equation for Species. For a stationary control volume, the conservation equation for species is C i t rate of storage of species i per unit volume (C i V) j i r i (10.6) net rate of convection of species i per unit volume net rate of diffusion of species i per unit volume production rate of species i per unit volume Using the mass conservation equation, the preceding equation can be rearranged to obtain Dm i j i r i (10.7) where m i is mass fraction of species i, i.e., where m i C i /, where is the density of the mixture, i C i, and C i is a partial density of species i (i.e., a mass concentration of species i). The conservation equation of species also can be written in terms of mole concentration and mole fractions, as shown in Refs., 3, and 7. The mole concentration of species i is c i C i /M i, where M i is the molecular weight of the species. The mole fraction of species i is defined as x i c i /c, where c i c i.asisobvious, i m i 1 and i x i 1. Equations (10.6) and (10.7) written in different coordinate systems are given in Ref.. 5. Use of Conservation Equations to Set Up Problems. For a problem involving fluid flow and simultaneous heat and mass transfer, equations of continuity, momentum, energy, and chemical species, Eqs. (10.5), (10.8), (10.18), and (10.7), are a formidable set of partial differential equations. There are four independent variables: three space coordinates (say, x, y, z) and a time coordinate t. If we consider a pure fluid, then there are five equations: the continuity equation, three momentum equations, and the energy equation. The accompanying five dependent variables are pressure, three components of velocity, and temperature. Also, a thermodynamic equation of state serves to relate density to the pressure, temperature, and composition. (Notice that for natural-convection flows, the momentum and energy equations are coupled.) For a mixture of n chemical species, there are n species conservation equations, but one is redundant, since the sum of mass fractions is equal to unity. A complete mathematical statement of a problem requires specification of boundary and initial conditions. Boundary conditions are based on a physical statement or principle (for example, for viscous flow, the component of velocity parallel to a stationary surface is zero at the wall; for an insulated wall, the derivative of temperature normal to the wall is zero; etc.).
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.15 A general solution, even by numerical methods, of the full equations in the four independent variables is difficult to obtain. Fortunately, however, many problems of engineering interest are adequately described by simplified forms of the full conservation equations, and these forms can often be solved easily. The governing equations for simplified problems are obtained by deleting superfluous terms in the full conservation equations. This applies directly to laminar flows only. In the case of turbulent flows, some caution must be exercised. For example, on an average basis a flow may be two-dimensional and steady, but if it is unstable and as a result turbulent, fluctuations in the three components of velocity may be occurring with respect to time and the three spatial coordinates. Then the remarks about dropping terms apply only to the time-averaged equations. 7,8 When simplifying the conservation equation given in a full form, we have to rely on physical intuition or on experimental evidence to judge which terms are negligibly small. Typical resulting classes of simplified problems are Constant transport properties Constant density Timewise steady flow (or quasi-steady flow) Two-dimensional flow One-dimensional flow Fully developed flow (no dependence on the streamwise coordinate) Stagnant fluid or rigid body Terms also may be shown to be negligibly small by order-of-magnitude estimates. 7,8 Some classes of flow that result are Creeping flows: Inertia terms are negligible. Forced flows: Gravity forces are negligible. Natural convection: Gravity forces predominate. Low-speed gas flows: Viscous dissipation and compressibility terms are negligible. Boundary-layer flows: Streamwise diffusion terms are negligible. DIMENSIONLESS GROUPS AND SIMILARITY IN FLUID MECHANICS AND HEAT TRANSFER Modern engineering practice in the fields of fluid mechanics and heat transfer is based on a combination of theoretical analysis and experimental data. Often the engineer is faced with the necessity of obtaining practical results in situations where, for various reasons, physical phenomena cannot be described mathematically or the differential equations describing the problem are too difficult to solve. An experimental program must be considered in such cases. However, in carrying the experimental program, the engineer should know how to relate the experimental data (i.e., data obtained on the model under consideration) to the actual, usually larger, system (prototype). A determination of the relevant dimensionless parameters (groups) provides a powerful tool for that purpose.
10.16 CHAPTER TEN The generation of such dimensionless groups in heat transfer and fluid mechanics (known generally as dimensional analysis) is basically done (1) by using differential equations and their boundary conditions (this method is sometimes called a differential similarity) and () by applying the dimensional analysis in the form of the Buckingham pi theorem. The first method (differential similarity) is used when the governing equations and their boundary conditions describing the problem are known. The equations are first made dimensionless. For demonstration purposes, let us consider the relatively simple problem of a binary mixture with constant properties and density flowing at low speed, where body forces, heat source term, and chemical reactions are neglected. The conservation equations are, from Eqs. (10.6), (10.10), (10.16), and (10.7), Mass V 0 (10.8) Momentum DV P V (10.9) Thermal energy c DT k T (10.30) Dm 1 Species D m 1 (10.31) Using L and V as characteristic length and velocity, respectively, we define the dimensionless variables and also x y z x* y* z* (10.3) L L L V V* (10.33) V t t* (10.34) L/V P P* (10.35) V T T T* w (10.36) T T w m1 m1,w m* (10.37) m m 1, 1,w where the subscript refers to the external free-stream condition or some average condition and the subscript w refers to conditions adjacent to a bounding surface across which transfer of heat and mass occurs. If we introduce the dimensionless quantities, Eqs. (10.3) to (10.37), into Eqs. (10.8) to (10.31), we obtain, respectively,
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.17 * V* 0 (10.38) DV* 1 *P* * V* (10.39) * Re DT* 1 Ec * T* * (10.40) * Re Pr Re Dm* 1 * m* (10.41) * Re Sc Obviously, the solutions of Eqs. (10.38) to (10.41) depend on the coefficients that appear in these equations. Solutions of Eqs. (10.38) to (10.41) are equally applicable to the model and prototype (where the model and prototype are geometrically similar systems of different linear dimensions in streams of different velocities, temperatures, and concentration), if the coefficients in these equations are the same for both model and prototype. These coefficients, Pr, Re, Sc, and Ec (called dimensionless parameters or similarity parameters), are defined in Table 10.10. Focusing attention now on heat transfer, from Eq. (9.), using the dimensionless quantities, the heat-transfer coefficient is given as or in dimensionless form, kt* h (10.4) L y* y*0 hl T* Nu (10.43) k y* y *0 where the dimensionless group Nu is known as the Nusselt number. Since Nu is the dimensionless temperature gradient at the surface, according to Eq. (10.40) it must therefore depend on the dimensionless groups that appear in this equation; hence Nu ƒ (Re, Pr, Ec) (10.44) 1 For processes where viscous dissipation and compressibility are negligible, which is the case in many industrial applications, we have Nu ƒ (Re, Pr) (forced convection) (10.45) In the case of buoyancy-induced flow, Eq. (10.9) should be replaced with the simplified version 9 of Eq. (10.14), and following a similar procedure, we obtain Nu ƒ (Gr, Pr) (natural convection) (10.46) 3 where Gr is the Grashof number, defined in Table 10.10. Also, using the relation of Eq. (9.6), and dimensionless quantities, D m* hd (10.47) L y* y*0
TABLE 10.10 Summary of the Chief Dimensionless Groups* Group Symbol Definition Physical significance (interpretation) Main area of use Biot number Bi hl Biot number (mass transfer) Coefficient of friction (skin friction coefficient) Eckert number Ec V c (T T ) Euler number Eu P V k s Ratio of internal thermal resistance of solid to fluid thermal resistance Heat transfer between fluid and solid Bi D hl Ratio of the internal species transfer Mass transfer between fluid and solid D resistance to the boundary-layer species transfer resistance c f w Dimensionless surface shear stress Flow resistance V / D p w Kinetic energy of the flow relative to the boundary-layer enthalpy difference Ratio of friction to velocity head Fluid friction Forced convection (compressible flow) Fourier number Ratio of the heat conduction rate to the V gl Fo t L rate of thermal energy storage in a solid Fourier number (mass Fo D Ratio of the species diffusion rate to the transfer) L rate of species storage Froude number Fr Graetz number Gz c VD D p Re Pr L kl Grashof number Gr 3 Unsteady-state heat transfer Unsteady-state mass transfer Ratio of inertial to gravitational force Wave and surface behavior (mixed natural and forced convection) Ratio of the fluid stream thermal capacity to convective heat transfer Forced convection g TL Ratio of buoyancy to viscous forces Natural convection Colburn j factor (heat transfer) Colburn j factor (mass transfer) j H St Pr /3 Dimensionless heat transfer coefficient Forced convection (heat, mass, and momentum transfer analogy) j D St D Sc /3 Dimensionless mass transfer coefficient Forced convection (heat, mass, and momentum transfer analogy) 10.18
Jakob number Ja c (T T ) Knudsen number Kn L Lewis number Le Sc D Pr Mach number Ma V a Nusselt number Nu hl k l pl w sat Péclet number Pe c VL Péclet number (mass transfer) i Prandtl number Pr c v Rayleigh number Ra 3 Reynolds number Re VL Schmidt number Sc D glg p Ratio of sensible heat absorbed by the liquid to the latent heat absorbed Ratio of molecular mean free path to characteristic dimension Ratio of molecular thermal and mass diffusivities Ratio of the velocity of flow to the velocity of sound Basic dimensionless convective heat transfer coefficient (ratio of convection heat transfer to conduction in a fluid slab of thickness L) Re Pr k Dimensionless independent heat transfer parameter (ratio of heat transfer by convection to conduction) Pe D VL Dimensionless independent mass transfer Re Sc D coefficient (ratio of bulk mass transfer to diffusive mass transfer) Ratio of molecular momentum and k thermal diffusivities Gr Pr Modified Grashof number (see interpretation for Gr and Pr) p g TL Boiling Low-pressure (low-density) gas flow Combined heat and mass transfer Compressible flow Convective heat transfer Forced convection Mass transfer Forced and natural convection Natural convection Ratio of inertia to viscous forces Forced convection; dynamic similarity Ratio of molecular momentum and mass diffusivities Mass transfer 10.19
TABLE 10.10 Summary of the Chief Dimensionless Groups* (Continued) Group Symbol Definition Physical significance (interpretation) Main area of use Sherwood number Sh hl D Strouhal number Sr Lƒ V Stanton number St Nu h Re Pr c V Stanton number (mass transfer) D St D Sh h D Weber number We V L Re Sc V p Ratio of convection mass transfer to diffusion in a slab of thickness L Ratio of the velocity of vibration Lƒ to the velocity of the fluid Convective mass transfer Flow past tube (shedding of eddies) Dimensionless heat transfer coefficient Forced convection (ratio of heat transfer at the surface to that transported by fluid by its thermal capacity) Dimensionless mass transfer coefficient Convective mass transfer Ratio of inertia force to surface tension force Droplet breakup; thin-film flow *In these dimensionless groups, L designates characteristic dimension (e.g., tube diameter, hydraulic diameter, length of the tube or plate, slab thickness, radius of a cylinder or sphere, droplet diameter, thin-film thickness, etc.). Physical properties are usually evaluated at mean temperature unless otherwise specified. Note: D D 1 (D 1 is also a commonly used symbol for binary diffusion coefficient; D iƒ is theh multicomponent diffusion coefficient). When species 1 is in very small concentration, the symbol D 1m is occasionally used, 7 representing an effective binary diffusion coefficient for species 1 diffusing through the mixture. In some engineering texts, the symbol St is also used for this group. 10.0
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.1 or L m* h Sh (10.48) D D y* y*0 This parameter, termed the Sherwood number, is equal to the dimensionless mass fraction (i.e., concentration) gradient at the surface, and it provides a measure of the convection mass transfer occurring at the surface. Following the same argument as before [but now for Eq. (10.41)), we have Sh ƒ (Re, Sc) (forced convection, mass transfer) (10.49) 4 The significance of expressions such as Eqs. (10.44) to (10.46) and (10.49) should be appreciated. For example, Eq. (10.45) states that convection heat-transfer results, whether obtained theoretically or experimentally, can be represented in terms of three dimensionless groups, instead of seven parameters (h, L, V, k, c p,, and ). The convenience is evident. Once the form of the functional dependence of Eq. (10.45) is obtained for a particular surface geometry (e.g., from laboratory experiments on a small model), it is known to be universally applicable; i.e., it may be applied to different fluids, velocities, temperatures, and length scales, as long as the assumptions associated with the original equations are satisfied (e.g., negligible viscous dissipation and body forces). Note that the relations of Eqs. (10.44) and (10.49) are derived without actually solving the system of Eqs. (10.8) and (10.31). References 7 to 1 cover the preceding procedure with more details and also include many different cases. It is important to mention here that once the conservation equations are put in dimensionless form, it is also convenient to make an order-of-magnitude assessment of all terms in the equations. Often a problem can be simplified by discovering that a term that would be very difficult to handle if large is in fact negligibly small. 7,8 Even if the primary thrust of the investigation is experimental, making the equations dimensionless and estimating the orders of magnitude of the terms are good practice. It is usually not possible for an experimental test to include (simulate) all conditions exactly; a good engineer will focus on the most important conditions. The same applies to performing an order-of-magnitude analysis. For example, for boundary-layer flows, allowance is made for the fact that lengths transverse to the main flow scale with a much shorter length than those measured in the direction of main flow. References 7, 11, and 13 cover many examples of the order-ofmagnitude analysis. When the governing equations of a problem are unknown, an alternative approach of deriving dimensionless groups is based on use of dimensional analysis in the form of the Buckingham pi theorem. 3,5,9,1,14 The Buckingham pi theorem proves that in a physical problem including n quantities in which there are m dimensions, the quantities can be arranged into n m independent dimensionless parameters. The success of this method depends on our ability to select, largely from intuition, the parameters that influence the problem. The procedure is best illustrated by an example. Example 10.1. The discharge through a horizontal capillary tube is thought to depend on the pressure drop per unit length, the diameter, and the viscosity. Find the form of the equation. The quantities with their dimensions are as follows:
10. CHAPTER TEN Quantity Symbol Dimensions Discharge Q L 3 t 1 Pressure drop/length p/l ML t Diameter D L Viscosity ML 1 t 1 Then p F Q,, D, 0 l Three dimensions are used, and with four quantities there will be one parameter: y p 1 x1 z1 Q D l Substituting in the dimensions gives 3 1 x1 y1 z1 1 1 0 0 0 (L T ) (ML T ) L ML t MML The exponents of each dimension must be the same on both sides of the equation. With L first, and similarly for M and t, 3x y z 1 0 1 1 1 y 1 0 1 x y 1 0 1 1 from which x 1 1, y 1 1, z 1 4, and After solving for Q, Q D4 p/l pd 4 Q C l from which dimensional analysis yields no information about the numerical value of the dimensionless constant C; experiment (or analysis) shows that it is /18. In the preceding example, if kinematic viscosity had been used in place of dynamic viscosity, an incorrect formula would have resulted. Example 10.. A fluid-flow situation depends on the velocity V; the density ; several linear dimensions l, l 1, and l ; pressure drop p; gravity g, viscosity ; surface tension ; and bulk modulus of elasticity E. Apply dimensional analysis to these variables to find a set of parameters.
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.3 F(V,, l, l, l, p, g,, E) 0 1 Since three dimensions are involved, three repeating variables are selected. For complex situations, V, p, and l are generally helpful. There are seven parameters: x1 y1 z1 1 V l p x y z V l g x3 y3 z3 3 V l x4 y4 z4 4 V l x5 y5 z5 5 V l E l 6 l1 l 7 l By expanding the quantities into dimensions as in the first example, we have p gl E l l 1 3 4 5 6 7 V V Vl V l V l l 1 p gl E l l ƒ,,,,,, 0 V V Vl V l V l l and 1 It is convenient to invert some of the parameters and to take some square roots: p V Vl V l l ƒ 1,,,,, 0 V gl E/ l 1 l The first parameter, usually written p/(v /), is the pressure coefficient; the second parameter is the Froude number Fr; the third is the Reynolds number Re; the fourth is the Weber number We; and the fifth is the Mach number Ma. Hence After solving for pressure drop, p l l ƒ 1, Fr, Re, We, Ma,, 0 V l l 1 l l p V ƒ Fr, Re, We, Ma,, l l in which ƒ 1,ƒ must be determined from analysis or experiment. By selecting other repeating variables, a different set of pi parameters could be obtained. Forexample, knowing in advance that the heat-transfer coefficient in fully developed forced convection in a tube is a function of certain variables, that is, h ƒ(v,,, c p, k, D), we can use the Buckingham pi theorem to obtain Eq. (10.45), as shown in Ref. 3. However, this method is carried out without any consideration of the physical nature of the process in question; i.e., there is no way to ensure that all essential variables have been included. However, as shown above, starting with the differential form of the conservation equations, we have derived the similarity parameters (dimensionless groups) in rigorous fashion. 1
10.4 CHAPTER TEN In Table 10.10 those dimensionless groups which appear frequently in fluid-flow, heat-, and mass-transfer literature have been listed. The list includes groups already mentioned above as well as those found in special fields of heat transfer. Note that although similar in form, the Nusselt and Biot numbers differ in both definition and interpretation. The Nusselt number is defined in terms of thermal conductivity of the fluid; the Biot number is based on the solid thermal conductivity. NOMENCLATURE Symbol Definition, SI Units (U.S. Customary Units) A heat-transfer area, m (ft ) a acceleration, m/s (ft/s ) a speed of sound, m/s (ft/s) C mass concentration of species, kg/m 3 (lb m /ft 3 ) c specific heat, J/(kg K) [Btu/(lb m F)] c p specific heat at constant pressure, J/(kg K) [Btu/(lb m F)] c v specific heat at constant volume, J/(kg K) [Btu/(lb m F)] D tube inside diameter, diameter, m (ft) D diffusion coefficient, m /s (ft /s) Ec Eckert number (Table 10.10) e emissive power, W/m [Btu/(h ft )] e b blackbody emissive power, W/m [Btu/(h ft )] F force, N (lb ƒ) ƒ frequency of vibration (Table 10.10), s 1 ƒ 1,ƒ,ƒ 3,ƒ 4 denotes function of Eqs. (10.44) to (10.46) and (10.49) Gr Grashof number (Table 10.10) g, g gravitational acceleration (magnitude and vector), m/s (ft/s ) h heat-transfer coefficient, W/(m K) [Btu/(h ft F)] h D mass-transfer coefficient, m/s (ft/s) i enthalpy per unit mass, J/kg (Btu/lb m ) i lg latent heat of evaporation, J/kg (Btu/lb m ) i 0 heat of formation, J/kg (Btu/lb m ) j, j mass diffusion flux of species (magnitude and vector), kg/(s m ) [lb m /(h ft )] k thermal conductivity, W/(m K) [Btu/(h ft F)] L length, m (ft) M mass, kg (lb m ) m mass fraction of species [Eq. (10.7)] Nu Nusselt number (Table 10.10) P pressure: Pa, N/m (lb f/ft )
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.5 Pr Prandtl number (Table 10.10) P pressure drop, Pa, N/m (lb f /ft ) q heat-transfer rate, W (Btu/h) q heat flux (vector), W/m [Btu/(h ft )] q heat flux, W/m [Btu/(h ft )] q volumetric heat generation, W/m 3 [Btu/(h ft 3 )] Re Reynolds number (Table 10.10) r radial distance in cylindrical or spherical coordinate, m (ft) r volumetric generation rate of species, kg/(s m 3 ) [lb m /(h ft 3 )] Sc Schmidt number (Table 10.10) Sh Sherwood number (Table 10.10) St Stanton number (Table 10.10) T temperature: C, K (F, R) T temperature difference, C (F) t time, s u velocity component in the axial direction (x direction) in rectangular coordinates, m/s (ft/s) u internal energy per unit mass, J/kg (Btu/lb m ) V, V velocity (magnitude and vector), m/s (ft/s) v velocity component in the y direction in rectangular coordinates, m/s (ft/s) v r velocity component in the r direction, m/s (ft/s) v z velocity component in the z direction, m/s (ft/s) v velocity component in the direction, m/s (ft/s) v velocity component in the direction, m/s (ft/s) w velocity component in the z direction in rectangular coordinates, m/s (ft/s) x rectangular coordinate, m (ft) y rectangular coordinate, m (ft) z rectangular or cylindrical coordinate, m (ft) Greek thermal diffusivity, m /s (ft /s) coefficient of thermal expansion, K 1 (R 1 ) emissivity angle in cylindrical and spherical coordinates, radians (degrees) molecular mean free path, m (ft) dynamic viscosity, Pa s [lb m /(h ft)] kinematic viscosity, m /s (ft /s) density, kg/m 3 (lb m /ft 3 )
10.6 CHAPTER TEN surface tension (Table 10.10), N/m (lb /ft) f shear stress, N/m (lb /ft ) f shear stress tensor, N/m (lb /ft ) f dissipation function (Table 10.7), s angle in spherical coordinate system, rad (degrees) Subscripts a surroundings aw adiabatic wall cr critical ƒ fluid g gas (vapor) i species i l liquid m mean s solid sat saturation t total w wall x x component y y component z z component component component 1 species 1 in binary mixture of 1 and free-stream condition Mathematical Operation Symbols 1 d/dx derivative with respect to x, m (ft 1 ) /t partial time derivative operator, s 1 d/dt total time derivative operator, s 1 D/ substantial time derivative operator, s 1 del operator (vector), m 1 (ft 1 ) laplacian operator, m (ft )
CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 10.7 REFERENCES* 1. W. M. Kays and M. E. Crawford, Convective Heat and Mass Transfer, d ed., McGraw- Hill, New York, 1980.. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. 3. W. M. Rohsenow and H. Y. Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, N.J., 1961. 4. W. M. Rohsenow, J. P. Hartnett, and E. N. Ganić, eds., Handbook of Heat Transfer Applications, chap., McGraw-Hill, New York, 1985. 5. A. S. Foust, L. A. Wenzel, C. W. Clump, L. Mans, and L. B. Andersen, Principles of Unit Operations, d ed., Wiley, New York, 1980. 6. S. Whitaker, Elementary Heat Transfer Analysis, Pergamon, New York, 1976. 7. D. K. Edwards, V. E. Denny, and A. F. Mills, Transfer Processes: An Introduction to Diffusion, Convection, and Radiation, d ed., Hemisphere, Washington, and McGraw- Hill, New York, 1979. 8. H. Schlichting, Boundary-Layer Theory, 7th ed., McGraw-Hill, New York, 1979. 9. B. Gebhart, Heat Transfer, d ed., McGraw-Hill, New York, 1971. 10. F. P. Incropera and D. P. DeWitt, Fundamentals of Heat Transfer, Wiley, New York, 1981. 11. F. M. White, Viscous Fluid Flow, McGraw-Hill, New York, 1974. 1. V. P. Isachenko, V. A. Osipova, and A. S. Sukomel, Heat Transfer, Mir Publishers, Moscow, 1977. 13. E. R. G. Eckert and R. M. Drake, Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 197. 14. J. H. Lienhard, A Heat Transfer Textbook, Prentice-Hall, Englewood Cliffs, N.J., 1981. 15. W. C. Reynolds and H. C. Perkins, Engineering Thermodynamics, d ed., McGraw-Hill, New York, 1977. 16. W. M. Rohsenow, J. P. Hartnett, and E. N. Ganić, eds, Handbook of Heat Transfer Fundamentals, d ed., chap. 1, McGraw-Hill, New York, 1985. * Those references listed here but not cited in the text were used for comparison of different data sources, clarification, clarity of presentation, and, most important, reader s convenience when further interest in subject exists.