Fluids Bernoulli s equation
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1 Chapter 11 Fluids Bernoulli s equation
2 11.9 Bernoulli s Equation W NC = ( P 2! P 1 )V W NC = E 1! E 2 = 1 mv 2 + mgy ( )! ( 1 "v 2 + "gy ) ( P 2! P 1 ) = 1 "v 2 + "gy NC Work yields a total Energy change. ( )! ( 1 mv 2 + mgy ) Equating the two expressions for the work done, ( P 2! P 1 )V = 1 mv 2 + mgy mgy 2 ( )! ( 1 mv 2 2 ) BERNOULLI S EQUATION m =!V Rearrange to obtain Bernoulli's Equation In steady flow of a nonviscous, incompressible fluid, the pressure, the fluid speed, and the elevation at two points are related by: P 1 + 1!v 2 +!gy = P 2 + 1!v 2 +!gy 2 2 2
3 11.10 Applications of Bernoulli s Equation Conceptual Example 14 Tarpaulins and Bernoulli s Equation When the truck is stationary, the tarpaulin lies flat, but it bulges outward when the truck is speeding down the highway. Account for this behavior. Bernoulli s Equation P 1 + 1!v 2 +!gy = P 2 + 1!v 2 +!gy Relative to moving truck v 1 = 0 under the tarp v 2 air flow over top P 1 = P 2 + 1!v P 1 > P 2
4 11.10 Applications of Bernoulli s Equation Lift force of an airplane wing Venting keeps trap filled with water The curve ball
5 11.10 Applications of Bernoulli s Equation Example 16 Efflux Speed The tank is open to the atmosphere at the top. Find and expression for the speed of the liquid leaving the pipe at the bottom. P 1 = P 2 = P atmosphere (1!10 5 N/m 2 ) v 2 = 0, y 2 = h, y 1 = 0 P 1 + 1!v 2 +!gy = P 2 + 1!v 2 +!gy !v 2 =!gh 2 1 v 1 = 2gh
6 Clicker Question 11.3 Fluid flows from left to right through the pipe shown. Points A and B are at the same height, but the cross-sectional area is bigger at point B than at A. The points B and C are at two different heights, but the cross-sectional area of the pipe is the same. Rank the pressure at the three locations in order from lowest to highest. Bernoulli's equation: P 1 + 1!v 2 +!gy = P 2 + 1!v 2 +!gy a) P A > P B > P C b) P B > P A = P C c) P C > P B > P A d) P B > P A & P B > P C e) P C > P A & P C > P B
7 Clicker Question 11.3 Fluid flows from left to right through the pipe shown. Points A and B are at the same height, but the cross-sectional area is bigger at point B than at A. The points B and C are at two different heights, but the cross-sectional area of the pipe is the same. Rank the pressure at the three locations in order from lowest to highest. Bernoulli's equation: P 1 + 1!v 2 +!gy = P 2 + 1!v 2 +!gy a) P A > P B > P C b) P B > P A = P C c) P C > P B > P A d) P B > P A & P B > P C P A + 1!v 2 = P 2 A B + 1!v 2 2 B Since v A > v B : P B +!gy B = P C +!gy C Since y C > y B : P B > P A P B > P C e) P C > P A & P C > P B Pipe area grows: v A > v B
8 11.11 Viscous Flow Flow of an ideal fluid. Flow of a viscous fluid. FORCE NEEDED TO MOVE A LAYER OF VISCOUS FLUID WITH CONSTANT VELOCITY The magnitude of the tangential force required to move a fluid layer at a constant speed is given by: F =!Av y!, is the coefficient of viscosity SI Unit: Pa "s; 1 poise (P) = 0.1 Pa "s POISEUILLE S LAW (flow of viscous fluid) The volume flow rate is given by: Pressure drop in a straight uniform diamater pipe.
9 11.11 Viscous Flow Example 17 Giving and Injection A syringe is filled with a solution whose viscosity is 1.5x10-3 Pa s. The internal radius of the needle is 4.0x10-4 m. The gauge pressure in the vein is 1900 Pa. What force must be applied to the plunger, so that 1.0x10-6 m 3 of fluid can be injected in 3.0 s? P 2! P 1 = 8"LQ # R 4 ( )( m) ( 1.0 $10!6 m s) = 1200 Pa # ( 4.0 $10-4 m) 4 = 8 1.5$10!3 Pa %s P 2 = ( P 1 )Pa = ( )Pa = 3100 Pa F = P 2 A = ( 3100 Pa) ( 8.0!10 "5 m 2 ) = 0.25N
10 Chapter 12 Temperature and Heat
11 12.1 Common Temperature Scales Temperatures are reported in degrees-celsius or degrees-fahrenheit. Temperature changes, on the other hand, are reported in Celsius-degrees or Fahrenheit-degrees: 1 C! = 5! F! 180 = 5 $ " # 9% & Convert F to C : C = 5 (F! 32) 9 Convert C to F : F = 9 C
12 12.2 The Kelvin Temperature Scale Kelvin temperature
13 12.2 The Kelvin Temperature Scale A constant-volume gas thermometer. absolute zero point = o C
14 12.3 Thermometers Thermometers make use of the change in some physical property with temperature. A property that changes with temperature is called a thermometric property.
15 12.4 Linear Thermal Expansion NORMAL SOLIDS
16 12.4 Linear Thermal Expansion LINEAR THERMAL EXPANSION OF A SOLID The length of an object changes when its temperature changes: Change in length proportional to original length and temperature change.!l = " L 0!T coefficient of linear expansion Common Unit for the Coefficient of Linear Expansion: 1 C = ( )!1! C!
17 12.4 Linear Thermal Expansion
18 12.4 Linear Thermal Expansion Example 3 The Buckling of a Sidewalk A concrete sidewalk is constructed between two buildings on a day when the temperature is 25 o C. As the temperature rises to 38 o C, the slabs expand, but no space is provided for thermal expansion. Determine the distance y in part (b) of the drawing.!l = " L o!t ( ) $1 = % 12 #10 $6 C &'! = m ( ) 13 C! ( )* 3.0 m ( ) y = = m ( m) 2! ( m) 2
19 12.4 Linear Thermal Expansion Example 4 The Stress on a Steel Beam The beam is mounted between two concrete supports when the temperature is 23 o C. What compressional stress must the concrete supports apply to each end of the beam, if they are to keep the beam from expanding when the temperature rises to 42 o C? Stress = F A = Y!L L 0 with!l = " L 0!T = Y"!T ( ) 12 #10 $6 ( C! ) $1 = 2.0 #10 11 N m 2 = 4.7 #10 7 N m 2 % &' ( )* 19C! ( ) Pressure at ends of the beam, 4.7!10 7 N m 2 " 170 atmospheres (1!10 5 N/m)
20 12.4 Linear Thermal Expansion THE BIMETALLIC STRIP Temperature control with bimetalic strip
21 12.4 Linear Thermal Expansion Conceptual Example 7 Expanding Cylinders As the cylinders are heated to the same, but higher, temperature, cylinder C falls off, while cylinder A becomes tightly wedged to cylinder B. Which cylinder is made from which material? Diameter change proportional to!.! Pb >! Brass >! Fe Lead ring falls off steel, brass ring sticks inside. Linear themal expansion!l = " L 0!T Volume themal expansion
22 12.5 Volume Thermal Expansion Example 8 An Automobile Radiator The radiator is made of copper and the coolant has an expansion coefficient of 4.0x10-4 (C o ) -1. If the radiator is filled to its 15-quart capacity when the engine is cold (6 o C), how much overflow will spill into the reservoir when the coolant reaches its operating temperature (92 o C)? ( ) #1!V coolant = $ 4.10 "10 #4 C %&! = 0.53 liters ( ) #1!V radiator = $ 51"10 #6 C %&! = liters ( ) '( 15 liters) 86 C ()! ( ) '( 15 liters) 86 C ()!!V expansion = (0.53 " 0.066) liters = 0.46 liters
23 12.5 Volume Thermal Expansion Expansion of water.
24 12.6 Heat and Internal Energy DEFINITION OF HEAT Heat is energy that flows from a highertemperature object to a lower-temperature object because of a difference in temperatures. Hot cup SI Unit of Heat: joule (J) The heat that flows from hot to cold originates in the internal energy of the hot substance. It is not correct to say that a substance contains heat. You must use the word energy or internal energy. Cold cup
25 12.7 Heat and Temperature Change: Specific Heat Capacity Temperature of an object reflects the amount of internal energy within it. But objects with the same temperature and mass can have DIFFERENT amounts of internal energy! SOLIDS AND LIQUIDS (GASES ARE DIFFERENT) HEAT SUPPLIED OR REMOVED IN CHANGING THE TEMPERATURE OF A SUBSTANCE. The heat that must be supplied or removed to change the temperature of a substance is c, is the specific heat capacity of the substance Common Unit for Specific Heat Capacity: J/(kg C o )!T > 0, Heat added!t < 0, Heat removed GASES The value of the specific heat of a gas depends on whether the pressure or volume is held constant. This distinction is not important for solids.
26 12.7 Heat and Temperature Change: Specific Heat Capacity Example 9 A Hot Jogger Specific means per unit mass In a half-hour, a 65-kg jogger produces 8.0x10 5 J of heat. This heat is removed from the body by a variety of means, including sweating, one of the body s own temperature-regulating mechanisms. If the heat were not removed, how much would the body temperature increase? Q = mc!t!t = Q mc = 8.0 "10 5 J 65 kg $ % ( ) ( ) 3500J kg #C! & ' = 3.5 C! OTHER UNITS for heat production 1 cal = joules (calorie) 1 kcal = 4186 joules ([kilo]calories for food)
27 Clicker Question 12.1 Four 1-kg cylinders are heated to 100 C and placed on top of a block of paraffin wax, which melts at 63 C. There is one cylinder made from lead, one of copper, one of aluminum, and one of iron. After a few minutes, it is observed that the cylinders have sunk into the paraffin to differing depths. Rank the depths of the cylinders from deepest to shallowest.. Q = mc!t a) lead > iron > copper > aluminum b) aluminum > copper > lead > iron c) aluminum > iron > copper > lead d) copper > aluminum > iron > lead e) iron > copper > lead > aluminum
28 Clicker Question 12.1 Four 1-kg cylinders are heated to 100 C and placed on top of a block of paraffin wax, which melts at 63 C. There is one cylinder made from lead, one of copper, one of aluminum, and one of iron. After a few minutes, it is observed that the cylinders have sunk into the paraffin to differing depths. Rank the depths of the cylinders from deepest to shallowest.. Q = mc!t a) lead > iron > copper > aluminum b) aluminum > copper > lead > iron c) aluminum > iron > copper > lead d) copper > aluminum > iron > lead e) iron > copper > lead > aluminum
Fluids Bernoulli s equation
Chapter 11 Fluids Bernoulli s equation 11.9 Bernoulli s Equation W NC = ( P 2! P 1 )V W NC = E 1! E 2 = 1 mv 2 + mgy 2 1 1 ( )! ( 1 "v 2 + "gy 2 2 2 ) ( P 2! P 1 ) = 1 "v 2 + "gy 2 1 1 NC Work yields a
Fluids Bernoulli s equation conclusion
Chapter 11 Fluids Bernoulli s equation conclusion 11.9 Bernoulli s Equation W NC = ( P 2! P 1 )V W NC = E 1! E 2 = 1 mv 2 + mgy 2 1 1 ( )! ( 1 "v 2 + "gy 2 2 2 ) ( P 2! P 1 ) = 1 "v 2 + "gy 2 1 1 NC Work
Fluids Bernoulli s equation conclusion
Chapter 11 Fluids Bernoulli s equation conclusion 11.9 Bernoulli s Equation W NC = ( P 2! P 1 )V W NC = E 1! E 2 = 1 mv 2 + mgy 2 1 1 ( )! ( 1 "v 2 + "gy 2 2 2 ) ( P 2! P 1 ) = 1 "v 2 + "gy 2 1 1 NC Work
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