CHAPTER 10 CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS
|
|
- David Webster
- 7 years ago
- Views:
Transcription
1 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 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 Used by permission of McGraw-Hill, Inc. All rights reserved Copyright 003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
2 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.
3 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
4 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 (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 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 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
5 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 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
6 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 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 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 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
7 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 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.
8 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)
9 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 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 10.10 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 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
11 CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS V: (10.17) and values of the dissipation function in different coordinate systems are given in Table 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 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)
12 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 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
13 CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 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
14 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 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.).
15 CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS 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.
16 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,
17 CONSERVATION EQUATIONS AND DIMENSIONLESS GROUPS * 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 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 Also, using the relation of Eq. (9.6), and dimensionless quantities, D m* hd (10.47) L y* y*0
18 TABLE 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
19 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
20 TABLE 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
21 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 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:
22 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 z (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 y x y 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.
23 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 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
24 10.4 CHAPTER TEN In Table 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 )
25 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 )
26 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 )
27 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, R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley, New York, W. M. Rohsenow and H. Y. Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, N.J., W. M. Rohsenow, J. P. Hartnett, and E. N. Ganić, eds., Handbook of Heat Transfer Applications, chap., McGraw-Hill, New York, A. S. Foust, L. A. Wenzel, C. W. Clump, L. Mans, and L. B. Andersen, Principles of Unit Operations, d ed., Wiley, New York, S. Whitaker, Elementary Heat Transfer Analysis, Pergamon, New York, 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, H. Schlichting, Boundary-Layer Theory, 7th ed., McGraw-Hill, New York, B. Gebhart, Heat Transfer, d ed., McGraw-Hill, New York, F. P. Incropera and D. P. DeWitt, Fundamentals of Heat Transfer, Wiley, New York, F. M. White, Viscous Fluid Flow, McGraw-Hill, New York, V. P. Isachenko, V. A. Osipova, and A. S. Sukomel, Heat Transfer, Mir Publishers, Moscow, E. R. G. Eckert and R. M. Drake, Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, New York, J. H. Lienhard, A Heat Transfer Textbook, Prentice-Hall, Englewood Cliffs, N.J., W. C. Reynolds and H. C. Perkins, Engineering Thermodynamics, d ed., McGraw-Hill, New York, W. M. Rohsenow, J. P. Hartnett, and E. N. Ganić, eds, Handbook of Heat Transfer Fundamentals, d ed., chap. 1, McGraw-Hill, New York, * 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.
Basic Equations, Boundary Conditions and Dimensionless Parameters
Chapter 2 Basic Equations, Boundary Conditions and Dimensionless Parameters In the foregoing chapter, many basic concepts related to the present investigation and the associated literature survey were
More informationLecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows
Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3.- 1 Basics: equations of continuum mechanics - balance equations for mass and momentum - balance equations for the energy and the chemical
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationHeat Transfer From A Heated Vertical Plate
Heat Transfer From A Heated Vertical Plate Mechanical and Environmental Engineering Laboratory Department of Mechanical and Aerospace Engineering University of California at San Diego La Jolla, California
More information1 The basic equations of fluid dynamics
1 The basic equations of fluid dynamics The main task in fluid dynamics is to find the velocity field describing the flow in a given domain. To do this, one uses the basic equations of fluid flow, which
More informationdu u U 0 U dy y b 0 b
BASIC CONCEPTS/DEFINITIONS OF FLUID MECHANICS (by Marios M. Fyrillas) 1. Density (πυκνότητα) Symbol: 3 Units of measure: kg / m Equation: m ( m mass, V volume) V. Pressure (πίεση) Alternative definition:
More informationDimensional analysis is a method for reducing the number and complexity of experimental variables that affect a given physical phenomena.
Dimensional Analysis and Similarity Dimensional analysis is very useful for planning, presentation, and interpretation of experimental data. As discussed previously, most practical fluid mechanics problems
More informationBasic Principles in Microfluidics
Basic Principles in Microfluidics 1 Newton s Second Law for Fluidics Newton s 2 nd Law (F= ma) : Time rate of change of momentum of a system equal to net force acting on system!f = dp dt Sum of forces
More informationContents. Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 1
Contents 1. Introduction 2. Fluids 3. Physics of Microfluidic Systems 4. Microfabrication Technologies 5. Flow Control 6. Micropumps 7. Sensors 8. Ink-Jet Technology 9. Liquid Handling 10.Microarrays 11.Microreactors
More informationNatural Convection. Buoyancy force
Natural Convection In natural convection, the fluid motion occurs by natural means such as buoyancy. Since the fluid velocity associated with natural convection is relatively low, the heat transfer coefficient
More informationFluids and Solids: Fundamentals
Fluids and Solids: Fundamentals We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation.
More informationINTRODUCTION TO FLUID MECHANICS
INTRODUCTION TO FLUID MECHANICS SIXTH EDITION ROBERT W. FOX Purdue University ALAN T. MCDONALD Purdue University PHILIP J. PRITCHARD Manhattan College JOHN WILEY & SONS, INC. CONTENTS CHAPTER 1 INTRODUCTION
More informationViscous flow in pipe
Viscous flow in pipe Henryk Kudela Contents 1 Laminar or turbulent flow 1 2 Balance of Momentum - Navier-Stokes Equation 2 3 Laminar flow in pipe 2 3.1 Friction factor for laminar flow...........................
More informationDifferential Balance Equations (DBE)
Differential Balance Equations (DBE) Differential Balance Equations Differential balances, although more complex to solve, can yield a tremendous wealth of information about ChE processes. General balance
More information4. Introduction to Heat & Mass Transfer
4. Introduction to Heat & Mass Transfer This section will cover the following concepts: A rudimentary introduction to mass transfer. Mass transfer from a molecular point of view. Fundamental similarity
More informationFluid Mechanics: Static s Kinematics Dynamics Fluid
Fluid Mechanics: Fluid mechanics may be defined as that branch of engineering science that deals with the behavior of fluid under the condition of rest and motion Fluid mechanics may be divided into three
More informationDistinguished Professor George Washington University. Graw Hill
Mechanics of Fluids Fourth Edition Irving H. Shames Distinguished Professor George Washington University Graw Hill Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok
More informationFundamentals of Fluid Mechanics
Sixth Edition. Fundamentals of Fluid Mechanics International Student Version BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics THEODORE H. OKIISHI Department
More information11 Navier-Stokes equations and turbulence
11 Navier-Stokes equations and turbulence So far, we have considered ideal gas dynamics governed by the Euler equations, where internal friction in the gas is assumed to be absent. Real fluids have internal
More informationCBE 6333, R. Levicky 1 Review of Fluid Mechanics Terminology
CBE 6333, R. Levicky 1 Review of Fluid Mechanics Terminology The Continuum Hypothesis: We will regard macroscopic behavior of fluids as if the fluids are perfectly continuous in structure. In reality,
More informationWhen the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs
More informationScalars, Vectors and Tensors
Scalars, Vectors and Tensors A scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. Examples are hydrostatic pressure and temperature. A vector
More information1. Fluids Mechanics and Fluid Properties. 1.1 Objectives of this section. 1.2 Fluids
1. Fluids Mechanics and Fluid Properties What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids - both liquids and gases.
More informationOUTCOME 3 TUTORIAL 5 DIMENSIONAL ANALYSIS
Unit 41: Fluid Mechanics Unit code: T/601/1445 QCF Level: 4 Credit value: 15 OUTCOME 3 TUTORIAL 5 DIMENSIONAL ANALYSIS 3 Be able to determine the behavioural characteristics and parameters of real fluid
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationRavi Kumar Singh*, K. B. Sahu**, Thakur Debasis Mishra***
Ravi Kumar Singh, K. B. Sahu, Thakur Debasis Mishra / International Journal of Engineering Research and Applications (IJERA) ISSN: 48-96 www.ijera.com Vol. 3, Issue 3, May-Jun 3, pp.766-77 Analysis of
More informationCHEMICAL ENGINEERING AND CHEMICAL PROCESS TECHNOLOGY - Vol. I - Interphase Mass Transfer - A. Burghardt
INTERPHASE MASS TRANSFER A. Burghardt Institute of Chemical Engineering, Polish Academy of Sciences, Poland Keywords: Turbulent flow, turbulent mass flux, eddy viscosity, eddy diffusivity, Prandtl mixing
More informationwww.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x
Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity
More informationHeat transfer in Flow Through Conduits
Heat transfer in Flow Through Conduits R. Shankar Suramanian Department of Chemical and Biomolecular Engineering Clarkson University A common situation encountered y the chemical engineer is heat transfer
More informationLecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2)
Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2) In this lecture How does turbulence affect the ensemble-mean equations of fluid motion/transport? Force balance in a quasi-steady turbulent boundary
More informationOpen channel flow Basic principle
Open channel flow Basic principle INTRODUCTION Flow in rivers, irrigation canals, drainage ditches and aqueducts are some examples for open channel flow. These flows occur with a free surface and the pressure
More informationNUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES
Vol. XX 2012 No. 4 28 34 J. ŠIMIČEK O. HUBOVÁ NUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES Jozef ŠIMIČEK email: jozef.simicek@stuba.sk Research field: Statics and Dynamics Fluids mechanics
More informationCE 204 FLUID MECHANICS
CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 Tuzla-Istanbul/TURKEY Phone: +90-216-677-1630 ext.1974 Fax: +90-216-677-1486 E-mail:
More informationA drop forms when liquid is forced out of a small tube. The shape of the drop is determined by a balance of pressure, gravity, and surface tension
A drop forms when liquid is forced out of a small tube. The shape of the drop is determined by a balance of pressure, gravity, and surface tension forces. 2 Objectives Have a working knowledge of the basic
More informationCBE 6333, R. Levicky 1 Differential Balance Equations
CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,
More informationLecture 6 - Boundary Conditions. Applied Computational Fluid Dynamics
Lecture 6 - Boundary Conditions Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Outline Overview. Inlet and outlet boundaries.
More informationEXAMPLE: Water Flow in a Pipe
EXAMPLE: Water Flow in a Pipe P 1 > P 2 Velocity profile is parabolic (we will learn why it is parabolic later, but since friction comes from walls the shape is intuitive) The pressure drops linearly along
More informationChapter 28 Fluid Dynamics
Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example
More informationSteady Heat Conduction
Steady Heat Conduction In thermodynamics, we considered the amount of heat transfer as a system undergoes a process from one equilibrium state to another. hermodynamics gives no indication of how long
More informationDimensional Analysis
Dimensional Analysis An Important Example from Fluid Mechanics: Viscous Shear Forces V d t / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Ƭ = F/A = μ V/d More generally, the viscous
More informationLecture 8 - Turbulence. Applied Computational Fluid Dynamics
Lecture 8 - Turbulence Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Turbulence What is turbulence? Effect of turbulence
More informationA LAMINAR FLOW ELEMENT WITH A LINEAR PRESSURE DROP VERSUS VOLUMETRIC FLOW. 1998 ASME Fluids Engineering Division Summer Meeting
TELEDYNE HASTINGS TECHNICAL PAPERS INSTRUMENTS A LAMINAR FLOW ELEMENT WITH A LINEAR PRESSURE DROP VERSUS VOLUMETRIC FLOW Proceedings of FEDSM 98: June -5, 998, Washington, DC FEDSM98 49 ABSTRACT The pressure
More informationIterative calculation of the heat transfer coefficient
Iterative calculation of the heat transfer coefficient D.Roncati Progettazione Ottica Roncati, via Panfilio, 17 44121 Ferrara Aim The plate temperature of a cooling heat sink is an important parameter
More informationAbaqus/CFD Sample Problems. Abaqus 6.10
Abaqus/CFD Sample Problems Abaqus 6.10 Contents 1. Oscillatory Laminar Plane Poiseuille Flow 2. Flow in Shear Driven Cavities 3. Buoyancy Driven Flow in Cavities 4. Turbulent Flow in a Rectangular Channel
More informationThe Viscosity of Fluids
Experiment #11 The Viscosity of Fluids References: 1. Your first year physics textbook. 2. D. Tabor, Gases, Liquids and Solids: and Other States of Matter (Cambridge Press, 1991). 3. J.R. Van Wazer et
More informationChapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations
Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.
More informationFLUID DYNAMICS. Intrinsic properties of fluids. Fluids behavior under various conditions
FLUID DYNAMICS Intrinsic properties of fluids Fluids behavior under various conditions Methods by which we can manipulate and utilize the fluids to produce desired results TYPES OF FLUID FLOW Laminar or
More informationLecture 24 - Surface tension, viscous flow, thermodynamics
Lecture 24 - Surface tension, viscous flow, thermodynamics Surface tension, surface energy The atoms at the surface of a solid or liquid are not happy. Their bonding is less ideal than the bonding of atoms
More informationIntegration of a fin experiment into the undergraduate heat transfer laboratory
Integration of a fin experiment into the undergraduate heat transfer laboratory H. I. Abu-Mulaweh Mechanical Engineering Department, Purdue University at Fort Wayne, Fort Wayne, IN 46805, USA E-mail: mulaweh@engr.ipfw.edu
More informationFundamentals of THERMAL-FLUID SCIENCES
Fundamentals of THERMAL-FLUID SCIENCES THIRD EDITION YUNUS A. CENGEL ROBERT H. TURNER Department of Mechanical JOHN M. CIMBALA Me Graw Hill Higher Education Boston Burr Ridge, IL Dubuque, IA Madison, Wl
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationFREESTUDY HEAT TRANSFER TUTORIAL 3 ADVANCED STUDIES
FREESTUDY HEAT TRANSFER TUTORIAL ADVANCED STUDIES This is the third tutorial in the series on heat transfer and covers some of the advanced theory of convection. The tutorials are designed to bring the
More informationHow To Understand Fluid Mechanics
Module : Review of Fluid Mechanics Basic Principles for Water Resources Engineering Robert Pitt University of Alabama and Shirley Clark Penn State - Harrisburg Mass quantity of matter that a substance
More informationLecture L22-2D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for
More informationFree Convection Film Flows and Heat Transfer
Deyi Shang Free Convection Film Flows and Heat Transfer With 109 Figures and 69 Tables < J Springer Contents 1 Introduction 1 1.1 Scope 1 1.2 Application Backgrounds 1 1.3 Previous Developments 2 1.3.1
More informationModule 1 : Conduction. Lecture 5 : 1D conduction example problems. 2D conduction
Module 1 : Conduction Lecture 5 : 1D conduction example problems. 2D conduction Objectives In this class: An example of optimization for insulation thickness is solved. The 1D conduction is considered
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationCFD Simulation of Subcooled Flow Boiling using OpenFOAM
Research Article International Journal of Current Engineering and Technology E-ISSN 2277 4106, P-ISSN 2347-5161 2014 INPRESSCO, All Rights Reserved Available at http://inpressco.com/category/ijcet CFD
More informationChapter 7 Energy and Energy Balances
CBE14, Levicky Chapter 7 Energy and Energy Balances The concept of energy conservation as expressed by an energy balance equation is central to chemical engineering calculations. Similar to mass balances
More informationPhysics Notes Class 11 CHAPTER 2 UNITS AND MEASUREMENTS
1 P a g e Physics Notes Class 11 CHAPTER 2 UNITS AND MEASUREMENTS The comparison of any physical quantity with its standard unit is called measurement. Physical Quantities All the quantities in terms of
More information7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.
7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated
More informationTransport Phenomena. The Art of Balancing. Harry Van den Akker Robert F. Mudde. Delft Academic Press
Transport Phenomena The Art of Balancing Harry Van den Akker Robert F. Mudde Delft Academic Press Delft Academic Press First edition 2014 Published by Delft Academic Press /VSSD Leeghwaterstraat, 2628
More informationEnergy Transport. Focus on heat transfer. Heat Transfer Mechanisms: Conduction Radiation Convection (mass movement of fluids)
Energy Transport Focus on heat transfer Heat Transfer Mechanisms: Conduction Radiation Convection (mass movement of fluids) Conduction Conduction heat transfer occurs only when there is physical contact
More informationPhysics of the Atmosphere I
Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uni-heidelberg.de heidelberg.de Last week The conservation of mass implies the continuity equation:
More informationUnderstanding Plastics Engineering Calculations
Natti S. Rao Nick R. Schott Understanding Plastics Engineering Calculations Hands-on Examples and Case Studies Sample Pages from Chapters 4 and 6 ISBNs 978--56990-509-8-56990-509-6 HANSER Hanser Publishers,
More informationLECTURE 5: Fluid jets. We consider here the form and stability of fluid jets falling under the influence of gravity.
LECTURE 5: Fluid jets We consider here the form and stability of fluid jets falling under the influence of gravity. 5.1 The shape of a falling fluid jet Consider a circular orifice of radius a ejecting
More informationCompressible Fluids. Faith A. Morrison Associate Professor of Chemical Engineering Michigan Technological University November 4, 2004
94 c 2004 Faith A. Morrison, all rights reserved. Compressible Fluids Faith A. Morrison Associate Professor of Chemical Engineering Michigan Technological University November 4, 2004 Chemical engineering
More informationDimensional Analysis
Dimensional Analysis Mathematical Modelling Week 2 Kurt Bryan How does the escape velocity from a planet s surface depend on the planet s mass and radius? This sounds like a physics problem, but you can
More informationCorrelations for Convective Heat Transfer
In many cases it's convenient to have simple equations for estimation of heat transfer coefficients. Below is a collection of recommended correlations for single-phase convective flow in different geometries
More informationVacuum Technology. Kinetic Theory of Gas. Dr. Philip D. Rack
Kinetic Theory of Gas Assistant Professor Department of Materials Science and Engineering University of Tennessee 603 Dougherty Engineering Building Knoxville, TN 3793-00 Phone: (865) 974-5344 Fax (865)
More informationHeat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati
Heat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati Module No. # 04 Convective Heat Transfer Lecture No. # 03 Heat Transfer Correlation
More informationDynamic Process Modeling. Process Dynamics and Control
Dynamic Process Modeling Process Dynamics and Control 1 Description of process dynamics Classes of models What do we need for control? Modeling for control Mechanical Systems Modeling Electrical circuits
More informationXI / PHYSICS FLUIDS IN MOTION 11/PA
Viscosity It is the property of a liquid due to which it flows in the form of layers and each layer opposes the motion of its adjacent layer. Cause of viscosity Consider two neighboring liquid layers A
More informationLecture 5 Hemodynamics. Description of fluid flow. The equation of continuity
1 Lecture 5 Hemodynamics Description of fluid flow Hydrodynamics is the part of physics, which studies the motion of fluids. It is based on the laws of mechanics. Hemodynamics studies the motion of blood
More informationCh 2 Properties of Fluids - II. Ideal Fluids. Real Fluids. Viscosity (1) Viscosity (3) Viscosity (2)
Ch 2 Properties of Fluids - II Ideal Fluids 1 Prepared for CEE 3500 CEE Fluid Mechanics by Gilberto E. Urroz, August 2005 2 Ideal fluid: a fluid with no friction Also referred to as an inviscid (zero viscosity)
More informationHEAT TRANSFER IM0245 3 LECTURE HOURS PER WEEK THERMODYNAMICS - IM0237 2014_1
COURSE CODE INTENSITY PRE-REQUISITE CO-REQUISITE CREDITS ACTUALIZATION DATE HEAT TRANSFER IM05 LECTURE HOURS PER WEEK 8 HOURS CLASSROOM ON 6 WEEKS, HOURS LABORATORY, HOURS OF INDEPENDENT WORK THERMODYNAMICS
More informationKinetic Theory of Gases. Chapter 33. Kinetic Theory of Gases
Kinetic Theory of Gases Kinetic Theory of Gases Chapter 33 Kinetic theory of gases envisions gases as a collection of atoms or molecules. Atoms or molecules are considered as particles. This is based on
More informationLaminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis
Tamkang Journal of Science and Engineering, Vol. 12, No. 1, pp. 99 107 (2009) 99 Laminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis M. E. Sayed-Ahmed
More informationHEAT TRANSFER CODES FOR STUDENTS IN JAVA
Proceedings of the 5th ASME/JSME Thermal Engineering Joint Conference March 15-19, 1999, San Diego, California AJTE99-6229 HEAT TRANSFER CODES FOR STUDENTS IN JAVA W.J. Devenport,* J.A. Schetz** and Yu.
More informationCurrent Staff Course Unit/ Length. Basic Outline/ Structure. Unit Objectives/ Big Ideas. Properties of Waves A simple wave has a PH: Sound and Light
Current Staff Course Unit/ Length August August September September October Unit Objectives/ Big Ideas Basic Outline/ Structure PS4- Types of Waves Because light can travel through space, it cannot be
More informationExperiment 3 Pipe Friction
EML 316L Experiment 3 Pipe Friction Laboratory Manual Mechanical and Materials Engineering Department College of Engineering FLORIDA INTERNATIONAL UNIVERSITY Nomenclature Symbol Description Unit A cross-sectional
More informationThe Viscosity of Fluids
Experiment #11 The Viscosity of Fluids References: 1. Your first year physics textbook. 2. D. Tabor, Gases, Liquids and Solids: and Other States of Matter (Cambridge Press, 1991). 3. J.R. Van Wazer et
More informationA COMPUTATIONAL FLUID DYNAMICS STUDY ON THE ACCURACY OF HEAT TRANSFER FROM A HORIZONTAL CYLINDER INTO QUIESCENT WATER
A COMPUTATIONAL FLUID DYNAMICS STUDY ON THE ACCURACY OF HEAT TRANSFER FROM A HORIZONTAL CYLINDER INTO QUIESCENT WATER William Logie and Elimar Frank Institut für Solartechnik SPF, 8640 Rapperswil (Switzerland)
More informationDimensional Analysis, hydraulic similitude and model investigation. Dr. Sanghamitra Kundu
Dimensional Analysis, hydraulic similitude and model investigation Dr. Sanghamitra Kundu Introduction Although many practical engineering problems involving fluid mechanics can be solved by using the equations
More informationNotes on Polymer Rheology Outline
1 Why is rheology important? Examples of its importance Summary of important variables Description of the flow equations Flow regimes - laminar vs. turbulent - Reynolds number - definition of viscosity
More informationL r = L m /L p. L r = L p /L m
NOTE: In the set of lectures 19/20 I defined the length ratio as L r = L m /L p The textbook by Finnermore & Franzini defines it as L r = L p /L m To avoid confusion let's keep the textbook definition,
More informationPractice Problems on Boundary Layers. Answer(s): D = 107 N D = 152 N. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Nov 22
BL_01 A thin flat plate 55 by 110 cm is immersed in a 6 m/s stream of SAE 10 oil at 20 C. Compute the total skin friction drag if the stream is parallel to (a) the long side and (b) the short side. D =
More informationFor Water to Move a driving force is needed
RECALL FIRST CLASS: Q K Head Difference Area Distance between Heads Q 0.01 cm 0.19 m 6cm 0.75cm 1 liter 86400sec 1.17 liter ~ 1 liter sec 0.63 m 1000cm 3 day day day constant head 0.4 m 0.1 m FINE SAND
More informationExergy Analysis of a Water Heat Storage Tank
Exergy Analysis of a Water Heat Storage Tank F. Dammel *1, J. Winterling 1, K.-J. Langeheinecke 3, and P. Stephan 1,2 1 Institute of Technical Thermodynamics, Technische Universität Darmstadt, 2 Center
More information8.2 Elastic Strain Energy
Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for
More informationCE 6303 MECHANICS OF FLUIDS L T P C QUESTION BANK PART - A
CE 6303 MECHANICS OF FLUIDS L T P C QUESTION BANK 3 0 0 3 UNIT I FLUID PROPERTIES AND FLUID STATICS PART - A 1. Define fluid and fluid mechanics. 2. Define real and ideal fluids. 3. Define mass density
More informationChapter 8: Flow in Pipes
Objectives 1. Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow 2. Calculate the major and minor losses associated with pipe flow in piping networks
More informationSIZE OF A MOLECULE FROM A VISCOSITY MEASUREMENT
Experiment 8, page 1 Version of April 25, 216 Experiment 446.8 SIZE OF A MOLECULE FROM A VISCOSITY MEASUREMENT Theory Viscous Flow. Fluids attempt to minimize flow gradients by exerting a frictional force,
More informationHeat Transfer and Energy
What is Heat? Heat Transfer and Energy Heat is Energy in Transit. Recall the First law from Thermodynamics. U = Q - W What did we mean by all the terms? What is U? What is Q? What is W? What is Heat Transfer?
More informationHEAT AND MASS TRANSFER
MEL242 HEAT AND MASS TRANSFER Prabal Talukdar Associate Professor Department of Mechanical Engineering g IIT Delhi prabal@mech.iitd.ac.in MECH/IITD Course Coordinator: Dr. Prabal Talukdar Room No: III,
More informationFigure 1 - Unsteady-State Heat Conduction in a One-dimensional Slab
The Numerical Method of Lines for Partial Differential Equations by Michael B. Cutlip, University of Connecticut and Mordechai Shacham, Ben-Gurion University of the Negev The method of lines is a general
More informationAdaptation of General Purpose CFD Code for Fusion MHD Applications*
Adaptation of General Purpose CFD Code for Fusion MHD Applications* Andrei Khodak Princeton Plasma Physics Laboratory P.O. Box 451 Princeton, NJ, 08540 USA akhodak@pppl.gov Abstract Analysis of many fusion
More informationFLUID MECHANICS FOR CIVIL ENGINEERS
FLUID MECHANICS FOR CIVIL ENGINEERS Bruce Hunt Department of Civil Engineering University Of Canterbury Christchurch, New Zealand? Bruce Hunt, 1995 Table of Contents Chapter 1 Introduction... 1.1 Fluid
More information