Fluid Properties and Units CVEN 311 

Continuum ä All materials, solid or fluid, are composed of molecules discretely spread and in continuous motion. ä However, in dealing with fluid-flow relations on a mathematical basis, it is necessary to replace the actual molecular structure by a hypothetical continuous medium, called the continuum. ä All materials, solid or fluid, are composed of molecules discretely spread and in continuous motion. ä However, in dealing with fluid-flow relations on a mathematical basis, it is necessary to replace the actual molecular structure by a hypothetical continuous medium, called the continuum.

Continuum ä In a continuum, the physical variable at a point in space is the averaged value of the variable in a small sphere. ä How good is the assumption? ä In a continuum, the physical variable at a point in space is the averaged value of the variable in a small sphere. ä How good is the assumption? cm 3x10 10 molecules of air

Dimensions and Units ä The dimensions have to be the same for each term in an equation ä Dimensions of mechanics are ä length ä time ä mass ä force ä temperature ä The dimensions have to be the same for each term in an equation ä Dimensions of mechanics are ä length ä time ä mass ä force ä temperature L T M MLT -2 

Dimensions and Units QuantitySymbolDimensions VelocityV LT -1 AccelerationaLT -2 AreaA L 2 Volume  L 3 DischargeQ L 3 T -1 Pressurep ML -1 T -2 Gravityg LT -2 TemperatureT’  Mass concentrationC ML -3 QuantitySymbolDimensions VelocityV LT -1 AccelerationaLT -2 AreaA L 2 Volume  L 3 DischargeQ L 3 T -1 Pressurep ML -1 T -2 Gravityg LT -2 TemperatureT’  Mass concentrationC ML -3

Dimensions and Units QuantitySymbolDimensions Density  ML -3 Specific Weight  ML -2 T -2 Dynamic viscosity  ML -1 T -1 Kinematic viscosity L 2 T -1 Surface tension  MT -2 Bulk mod of elasticityE ML -1 T -2 QuantitySymbolDimensions Density  ML -3 Specific Weight  ML -2 T -2 Dynamic viscosity  ML -1 T -1 Kinematic viscosity L 2 T -1 Surface tension  MT -2 Bulk mod of elasticityE ML -1 T -2 These are _______ properties! fluid How many independent properties? _____ 4

Definition of a Fluid ä “a fluid, such as water or air, deforms continuously when acted on by shearing stresses of any magnitude.” - Munson, Young, Okiishi Water Oil Air Why isn’t steel a fluid? Water Oil Air Why isn’t steel a fluid?

Fluid Deformation between Parallel Plates Side view Force F causes the top plate to have velocity U. What other parameters control how much force is required to get a desired velocity? Distance between plates (b) Area of plates (A) F b U Viscosity!

Shear Stress change in velocity with respect to distance dimension of Tangential force per unit area Rate of angular deformation rate of shear

Fluid classification by response to shear stress ä Newtonian ä Ideal Fluid ä Ideal plastic ä Newtonian ä Ideal Fluid ä Ideal plastic Newtonian Ideal Fluid Ideal plastic Shear stress  Rate of deformation dy du  1

Fluid Viscosity ä Examples of highly viscous fluids ä ______________________ ä Fundamental mechanisms ä Gases - transfer of molecular momentum ä Viscosity __________ as temperature increases. ä Viscosity __________ as pressure increases. ä Liquids - cohesion and momentum transfer ä Viscosity decreases as temperature increases. ä Relatively independent of pressure (incompressible) ä Examples of highly viscous fluids ä ______________________ ä Fundamental mechanisms ä Gases - transfer of molecular momentum ä Viscosity __________ as temperature increases. ä Viscosity __________ as pressure increases. ä Liquids - cohesion and momentum transfer ä Viscosity decreases as temperature increases. ä Relatively independent of pressure (incompressible) molasses, tar, 20w-50 oil increases _______ increases

Example: Measure the viscosity of water The inner cylinder is 10 cm in diameter and rotates at 10 rpm. The fluid layer is 2 mm thick and 10 cm high. The power required to turn the inner cylinder is 50x10 -6 watts. What is the dynamic viscosity of the fluid? Outer cylinder Thin layer of water Inner cylinder

Solution Scheme ä Restate the goal ä Identify the given parameters and represent the parameters using symbols ä Outline your solution including the equations describing the physical constraints and any simplifying assumptions ä Solve for the unknown symbolically ä Substitute numerical values with units and do the arithmetic ä Check your units! ä Check the reasonableness of your answer ä Restate the goal ä Identify the given parameters and represent the parameters using symbols ä Outline your solution including the equations describing the physical constraints and any simplifying assumptions ä Solve for the unknown symbolically ä Substitute numerical values with units and do the arithmetic ä Check your units! ä Check the reasonableness of your answer Solution

Role of Viscosity ä Statics ä Fluids at rest have no relative motion between layers of fluid and thus du/dy = 0 ä Therefore the shear stress is _____ and is independent of the fluid viscosity ä Flows ä Fluid viscosity is very important when the fluid is moving ä Statics ä Fluids at rest have no relative motion between layers of fluid and thus du/dy = 0 ä Therefore the shear stress is _____ and is independent of the fluid viscosity ä Flows ä Fluid viscosity is very important when the fluid is moving zero

Dynamic and Kinematic Viscosity ä Kinematic viscosity (__) is a fluid property obtained by dividing the dynamic viscosity (__) by the fluid density [m 2 /s] Connection to Reynolds number!

Density and Specific Weight  Density (mass/unit volume)  ä density of water: ä density of air at atmospheric pressure and 15  C:  Specific Weight (weight per unit volume)  ä __________________  Density (mass/unit volume)  ä density of water: ä density of air at atmospheric pressure and 15  C:  Specific Weight (weight per unit volume)  ä __________________ Temperature (C) Density (kg/m 3 ) Temperature (C) Density (kg/m 3 ) 1000 kg/m kg/m 3   =  g = 9806 N/m 3 Specific mass

Perfect Gas Law ä PV = nRT ä R is the universal gas constant ä T is in Kelvin ä PV = nRT ä R is the universal gas constant ä T is in Kelvin Note deviation from the text! Use absolute pressure for P and absolute temperature for T

Bulk Modulus of Elasticity ä Relates the change in volume to a change in pressure ä changes in density at high pressure ä pressure waves ä _________ ä ______ __________ ä Relates the change in volume to a change in pressure ä changes in density at high pressure ä pressure waves ä _________ ä ______ __________ Temperature (C) Bulk Modulus of elasticity (GPa) sound water hammer Water - speed of sound

Vapor Pressure Temperature (C) Vapor pressure (Pa) liquid What is vapor pressure of water at 100°C? 101 kPa Connection forward to cavitation!

Cavitation

Cavitation Damage

 p  R 2 = 2  R  Surface Tension ä Pressure increase in a spherical droplet pR2pR2 2R2R Surface molecules Temperature (C) Surface tension (N/m)

Example: Surface Tension ä Estimate the difference in pressure (in Pa) between the inside and outside of a bubble of air in 20ºC water. The air bubble is 0.3 mm in diameter. R = 0.15 x m  = N/m What is the difference between pressure in a water droplet and in an air bubble? Statics!

Outline the solution ä Restate the goal ä Identify the given parameters and represent the parameters using symbols ä Outline your solution including the equations describing the physical constraints and any simplifying assumptions ä Restate the goal ä Identify the given parameters and represent the parameters using symbols ä Outline your solution including the equations describing the physical constraints and any simplifying assumptions

Viscosity Measurement: Solution Outer cylinder Thin layer of water Inner cylinder r = 5 cm t = 2 mm h = 10 cm P = 50 x W 10 rpm rr 2  rh FrFr