HW No. 5 7-26 0 7-32 given 7-41 7.9 KW

Introduction to Fluid Mechanics Fluid Mechanics is concerned with the behavior of fluids at rest and in motion Distinction between solids and fluids: According to our experience: A solid is “hard” and not easily deformed. A fluid is “soft” and deforms easily. Fluid is a substance that alters its shape in response to any force however small, that tends to flow or to conform to the outline of its container, and that includes gases and liquids and mixtures of solids and liquids capable of flow. A fluid is defined as a substance that deforms continuously when acted on by a shearing stress of any magnitude.

Units of Force: Newton’s Law F=m.g SI system: Base dimensions are Length, Time, Mass, Temperature A Newton is the force which when applied to a mass of 1 kg produces an acceleration of 1 m/s2. Newton is a derived unit: 1N = (1Kg).(1m/s2) EE system: Base dimensions are Length, Time, Mass, Force and Temperature The pound-force (lbf) is defined as the force which accelerates 1pound-mass (lbm), 32.174 ft/s2.

Properties of Fluids Fundamental approach: Study the behavior of individual molecules when trying to describe the behavior of fluids Engineering approach: Characterization of the behavior by considering the average, or macroscopic, value of the quantity of interest, where the average is evaluated over a small volume containing a large number of molecules Treat the fluid as a CONTINUUM: Assume that all the fluid characteristics vary continuously throughout the fluid

Measures of Fluid Mass and Weight Density of a fluid, r (rho), is the amount of mass per unit volume of a substance: r = m / V For liquids, weak function of temperature and pressure For gases: strong function of T and P from ideal gas law: r = P n/Ru T where R = universal gas constant, M=mol. weight R= 8.314 J/(g-mole K)=0.08314 (liter bar)/(g-mole K)= 0.08206 (liter atm)/(g-mole K)=1.987 (cal)/(g-mole K)= 10.73 (psia ft3)/(lb-mole °R)=0.7302 (atm ft3)/(lb-mole °R)

Measures of Fluid Mass and Weight Specific volume: u = 1 / r Specific weight is the amount of weight per unit volume of a substance: g = w / V = r g Specific Gravity (independent of system of units)

Fluid Statics The word “statics” is derived from Greek word “statikos”= motionless For a fluid at rest or moving in such a manner that there is no relative motion between particles there are no shearing forces present: Rigid body approximation

Definition of Pressure Pressure is defined as the amount of force exerted on a unit area of a substance: P = F / A Fluid statics Chee 223

Pascal’s Laws Pascals’ laws: Pressure acts uniformly in all directions on a small volume (point) of a fluid In a fluid confined by solid boundaries, pressure acts perpendicular to the boundary – it is a normal force.

Direction of fluid pressure on boundaries Furnace duct Pipe or tube Heat exchanger Pressure is due to a Normal Force (acting perpendicular to the surface) It is also called a Surface Force Dam

Units for Pressure Unit Definition or Relationship 1 pascal (Pa) 1 kg m-1 s-2 1 bar 1 x 105 Pa 1 atmosphere (atm) 101,325 Pa 1 torr 1 / 760 atm 760 mm Hg 1 atm 14.696 pounds per sq. in. (psi)

Hydrostatic forces on plane surfaces Case 1: Horizontal surface exposed to a gas F P=constant everywhere F = P . A Case 2: Horizontal surface exposed to a liquid F P=rgh+Patm h P = Patm P=constant along the horizontal surface F = P . A

Hydrostatic forces on plane surfaces Case 3: Vertical surface exposed to air F Pressure varies linearly with height (see also equation 2.4): P=rgh However, because r of gases is very low, the dependence is very week Therefore we can assume that P=constant everywhere F = P . A

Hydrostatic forces on plane surfaces Case 4: Vertical surface exposed to liquid Example: The lock gate of a canal is rectangular, 20 m wide and 10 m high. One side is exposed to the atmosphere and the other side to the water. What is the net force on the lock gate? Vertical rectangular wall (wall width = W) F H h P Here the pressure varies linearly with depth (see also equation 2.4): P=rgh

Buoyancy H Laws of buoyancy discovered by Archimedes: A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces A floating body displaces its own weight in the fluid in which it floats Free liquid surface F1 h1 The upper surface of the body is subjected to a smaller force than the lower surface  A net force is acting upwards H h2 F2

Buoyancy where r = the fluid density The net force due to pressure in the vertical direction is: FB = F2- F1 = (Pbottom - Ptop) (DxDy) The pressure difference is: Pbottom – Ptop = r g (h2-h1) = r g H From (2.8): FB = r g H (DxDy) Thus the buoyant force is: FB = r g V (2.8) (2.9) where r = the fluid density

Newton’s law of Viscosity (6.6) m [=N/m2 . s=Pa . s]: Viscosity n = m/r : Kinematic viscosity [=m2/s] Newtonian fluids: Fluids which obey Newton’s law: Shearing stress is linearly related to the rate of shearing strain. The viscosity of a fluid measures its resistance to flow under an applied shear stress.

Effect of temperature on viscosity Viscosity is very sensitive to temperature The viscosity of gases increases with temperature: Power-law Sutherland equation The viscosity of liquids decreases with temperature:

Non-Newtonian fluids Non-Newtonian fluids: Fluids which do not obey Newton’s law: Shearing stress is not linearly related to the rate of shearing strain. Bingham plastics Shear thinning Shear thickening The study of these materials is the subject of rheology

The Balance Equation / Mass Balances – Continuity Equation

The Mass Balance - Rate of mass flow into CV Surroundings, S Control Volume (CV) Accumulation or depletion Loss through outlet Addition through inlet Boundary, B (Control Surface, CS) The law of conservation of mass states that mass may be neither created nor destroyed. With respect to a control volume (CV): Rate of mass flow into CV Rate of mass efflux from CV Rate of accumulation of mass into CV - = (3.1)

Conservation of Mass- Continuity Equation The mass balance (equation 3.1) can be written: (3.2) where: Rate of addition of mass into the system Rate of removal of mass from the system Rate of accumulation of mass in the system denotes mass flow rate [mass/time=kg/s]

Steady and Unsteady Flows If there is no accumulation of mass (dm/dt = 0) then the system is at steady-state conditions The fluid flow in general is a function of location and time f(x, y, z, t). If the flow at every point in the fluid is independent of time, the flow is at steady state conditions. For a steady-state flow: dX/dt = 0, where X is a physical quantity (for example mass, temperature or velocity)

Mass and Volumetric Flow Rates Recall from definition of density: (3.3) In terms of “rates”: where = The mass flow rate: The mass of fluid flowing past a section per unit time [=kg/s] Q = The volumetric flow rate :The volume of fluid flowing past a section of a pipe or channel per unit time [=m3/s] (3.4) (3.5)

Conservation of Mass- Continuity Equation For steady-state conditions equation 3.2 combined with 3.3, 3.5 becomes: or (3.7) (3.6) Aout, rout, Vout Inlet Outlet Ain, rin, Vin

Conservation of Mass- Continuity Equation For incompressible fluids (r=const) from (3.6) and (3.7): (3.8)

Conservation of Mass- Continuity Equation For multiple inlets – outlets, starting from (3.2): If there is no accumulation of mass (dm/dt = 0) then the system is at steady-state conditions (3.7)

Conservation of Mass- Continuity Equation For incompressible flow: (3.8)

Summary The first of the fundamental laws of fluid flow, the conservation of mass, has been considered. It is also called the continuity equation. The continuity equation has been derived using the “Control Volume”, or “integral method of analysis”, by applying the general balance equation to a finite control volume. Significant simplification of the continuity equation can be made by using the steady state and incompressibility assumptions.

Bernoulli’s Equation- Mechanical Energy Balance

Simplifications (5.3) In terms of “heads” (units of length) (5.4) Assumptions: Steady, incompressible, frictionless (inviscid) flow No heat transfer or change in internal energy, no shaft work Single input-output Bernoulli’s equation (5.1) reduces to: (5.3) In terms of “heads” (units of length) (5.4)

The Bernoulli equation can be expressed in three different ways as follows: energies: pressures: heads:

at the exit of the tube so that P1 = P2 = Patm Analysis We take point 1 to be at the free surface of water in the bottle and point 2 at the exit of the tube so that P1 = P2 = Patm (the bottle is open to the atmosphere and water discharges into the atmosphere), V1  0 (the bottle is large relative to the tube diameter), and z2 = 0 (we take point 2 as the reference level). Then the Bernoulli Equation simplifies to 0.25 in 1.5 ft 2 ft 1 2

Substituting, the discharge velocity of water and the filling time are determined as follows: (a) Full bottle (z1 = 3.5 ft): (b) Empty bottle (z1 = 2 ft):

Flow in Pipes – Fluid Friction

The Pressure-Drop Experiment Q L

Laminar vs. Turbulent Flow Laminar flow: Fluid flows in smooth layers (lamina) and the shear stress is the result of microscopic action of the molecules. Turbulent flow is characterized by large scale, observable fluctuations in the fluid and flow properties are the result of these fluctuations.

Reynolds Number The Reynolds number can be used as a criterion to distinguish between laminar and turbulent flow: (6.1) For flow in a pipe Laminar flow if Re < 2100 Turbulent flow if Re > 4000 Transitional flow if 2100< Re < 4000 For very high Reynolds numbers, viscous forces are negligible: inviscid flow For very low Reynolds numbers (Re<<1) viscous forces are dominant: creeping flow

Case 1: Laminar Flow

Laminar flow: Velocity and Shear stress profiles

Fully Developed Flow Flow in the entrance region of a pipe is complex. Once the velocity profile no longer changes, we have reached fully developed flow. Mathematically dV/dx = 0 Typical entrance length, 20 D < Le < 30 D

Case 2: Turbulent Flow

Turbulent Flow When the flow is turbulent the velocity and pressure fluctuate very rapidly. The velocity components at a point in a turbulent flow field fluctuate about a mean value.

Turbulent Flow – Velocity Profile For turbulent flow in tubes the time-averaged velocity profile can be expressed in terms of the power law equation. n =7 is usually a good approximation. where Vc is the velocity at The centerline

Friction factor: The Moody Chart The Moody Chart (Figure 6.10 textbook) provides a convenient representation of the functional dependence f = f(Re, e/D) For laminar flow: f = 16 / Re For turbulent flow: (6.18) Colebrook formula (6.19a ) (6.19b ) For turbulent flow, with Re<105 and for hydraulically smooth surfaces: (6.20) Blasius formula

To find f: Calculate Re If Re<2100 (laminar flow) find f from eq. (6.18) If Re>2100 (turbulent flow) Estimate roughness, e Find f from Moody chart

Calculation of Friction Factor

Other considerations: Non circular conduits Define hydraulic diameter, Dh: Then use Reynolds number based on hydraulic diameter, Dh:

Non circular conduits Pipe of circular cross-section Annulus (inside diameter D1, outside D2) Rectangular conduit (area ab)