SGChapt01 - I. FLUID MECHANICS I.1 Basic Concepts &...

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Unformatted text preview: I. FLUID MECHANICS I.1 Basic Concepts & Definitions: Fluid Mechanics - Study of fluids at rest, in motion, and the effects of fluids on boundaries. Note: This definition outlines the key topics in the study of fluids: (1) fluid statics (fluids at rest), (2) momentum and energy analyses (fluids in motion), and (3) viscous effects and all sections considering pressure forces (effects of fluids on boundaries). Fluid - A substance which moves and deforms continuously as a result of an applied shear stress. The definition also clearly shows that viscous effects are not considered in the study of fluid statics. Two important properties in the study of fluid mechanics are: Pressure and Velocity These are defined as follows: Pressure - The normal stress on any plane through a fluid element at rest. Key Point: The direction of pressure forces will always be perpendicular to the surface of interest. Velocity - The rate of change of position at a point in a flow field. It is used not only to specify flow field characteristics but also to specify flow rate, momentum, and viscous effects for a fluid in motion. | v v I-1 | e-Text Main Menu | Textbook Table of Contents | Study Guide I.4 Dimensions and Units This text will use both the International System of Units (S.I.) and British Gravitational System (B.G.). A key feature of both is that neither system uses gc. Rather, in both systems the combination of units for mass * acceleration yields the unit of force, i.e. Newton’s second law yields S.I. 1 Newton (N) = 1 kg m/s2 B.G. 1 lbf = 1 slug ft/s2 This will be particularly useful in the following: Concept Expression Units kg/s * m/s = kg m/s2 =N & mV momentum slug/s * ft/s = slug ft/s2 = lbf ρgh manometry kg/m3*m/s2*m = (kg m/s2)/ m2 =N/m2 slug/ft3*ft/s2*ft = (slug ft/s2)/ft2 = lbf/ft2 dynamic viscosity µ N s /m2 = (kg m/s2) s /m2 = kg/m s lbf s /ft2 = (slug ft/s2) s /ft2 = slug/ft s Key Point: In the B.G. system of units, the mass unit is the slug and not the lbm. and 1 slug = 32.174 lbm. Therefore, be careful not to use conventional values for fluid density in English units without appropriate conversions, e.g., ρw = 62.4 lb/ft3 For this case the manometer equation would be written as ∆P=ρ g h gc | v v I-2 | e-Text Main Menu | Textbook Table of Contents | Study Guide Example: Given: Pump power requirements are given by & Wp = fluid density*volume flow rate*g*pump head = ρ Q g hp For ρ = 1.928 slug/ft3, Q = 500 gal/min, and hp = 70 ft, Determine: The power required in kW. 3 & Wp = 1.928 slug/ft3 * 500 gal/min*1 ft /s /448.8 gpm*32.2 ft/s2 * 70 ft & Wp = 4841 ft–lbf/s * 1.3558*10-3 kW/ft–lbf/s = 6.564 kW Note: We used the following: 1 lbf = 1 slug ft/s2 to obtain the desired units Recommendation: Properties of the velocity Field Two important properties in the study of fluid mechanics are Pressure and Velocity The basic definition for velocity has been given previously, however, one of its most important uses in fluid mechanics is to specify both the volume and mass flow rate of a fluid. I-3 | v v 1.5 In working with problems with complex or mixed system units, at the start of the problem convert all parameters with units to the base units being used in the problem, e.g. for S.I. problems, convert all parameters to kg, m, & s; for BG problems, convert all parameters to slug, ft, & s. Then convert the final answer to the desired final units. | e-Text Main Menu | Textbook Table of Contents | Study Guide Volume flow rate: & Q = ∫ V ⋅ n dA = ∫ Vn dA cs cs where Vn is the normal component of velocity at a point on the area across which fluid flows. Key Point: Note that only the normal component of velocity contributes to flow rate across a boundary. Mass flow rate: & m = ∫ ρV ⋅n d A = ∫ ρV d A n cs cs NOTE: While not obvious in the basic equation, Vn must also be measured relative to any flow area boundary motion, i.e., if the flow boundary is moving, Vn is measured relative to the moving boundary. This will be particularly important for problems involving moving control volumes in Ch. III. | v v I-4 | e-Text Main Menu | Textbook Table of Contents | Study Guide 1.6 Thermodynamic Properties All of the usual thermodynamic properties are important in fluid mechanics P - Pressure (kPa, psi) T- Temperature ( C, F) ρ ñ Density (kg/m3, slug/ft3) o o Alternatives for density γ - specific weight = weight per unit volume (N/m3, lbf/ft3) H2O: γ = 9790 N/m3 = 62.4 lbf/ft3 Air: γ=ρg γ = 11.8 N/m3 = 0.0752 lbf/ft3 S.G. - specific gravity = ρ / ρ (ref) where: ρ (ref) = ρ (water at 1 atm, 20˚C) for liquids = 998 kg/m3 = ρ (air at 1 atm, 20˚C) for gases = 1.205 kg/m3 Example: Determine the static pressure difference indicated by an 18 cm column of fluid (liquid) with a specific gravity of 0.85. ∆P = ρ g h = S.G. γ h = 0.85* 9790 N/m3 0.18 m = 1498 N/m2 = 1.5 kPa I.7 Transport Properties Certain transport properties are important as they relate to the diffusion of momentum due to shear stresses. Specifically: µ ≡ coefficient of viscosity (dynamic viscosity) {M / L t } ν ≡ kinematic viscosity ( µ / ρ ) 2 {L /t} | v v I-5 | e-Text Main Menu | Textbook Table of Contents | Study Guide This gives rise to the definition of a Newtonian fluid. Newtonian fluid: A fluid which has a linear relationship between shear stress and velocity gradient. dU dy The linearity coefficient in the equation is the coefficient of viscosity µ . τ =µ Flows constrained by solid surfaces can typically be divided into two regimes: a. Flow near a bounding surface with 1. significant velocity gradients 2. significant shear stresses This flow region is referred to as a "boundary layer." b. Flows far from bounding surface with 1. negligible velocity gradients 2. negligible shear stresses 3. significant inertia effects This flow region is referred to as "free stream" or "inviscid flow region." An important parameter in identifying the characteristics of these flows is the Reynolds number = Re = ρV L µ This physically represents the ratio of inertia forces in the flow to viscous forces. For most flows of engineering significance, both the characteristics of the flow and the important effects due to the flow, e.g., drag, pressure drop, aerodynamic loads, etc., are dependent on this parameter. | v v I-6 | e-Text Main Menu | Textbook Table of Contents | Study Guide ...
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