Bernoulli, Macroscopic Models

Bernoulli, Macroscopic Models Pascal’s Principle For an incompressible fluid, any change in pressure at one point is transmitted undiminished to all other points in the fluid Consider piston-liquid interface: weight of lead (=mg) results in force on piston, which pushes down on liquid piston does not move, though: incompressible fluid The pressure at P is pp = pext + ½gh If we then add some more lead shot, we increase external pressure by a change in external pressure but the density, gravity, and the height is not changed The pressure change at p is: change in pressure = change in external pressure Independent of h: the change in pressure must be felt equally throughout the fluid Pascal’s Principle: Hydraulic Lift What’s the added pressure from Fa just beneath the left piston? By Pascal’s principle, this will also be the added pressure just beneath the right piston: pa = pb Therefore (including atm pressure for completeness): pa = p0 + Fa=Aa pb = p0 + Fb=Ab Fluids in Motion: A1v1 = A2v2 is called the “equation of continuity” Nothing more than “what goes in must go out” Av = R(v) is called the volume flow rate and is constant Multiplying by the density gives the mass flow rate, Rm = ½Av which is also constant Along a tube, fluid flows faster in narrow regions (small cross-sectional area A) than in wide regions (bigger A) A hose nozzle reduces the cross-sectional area where water emerges from the hose; the water comes out faster than it was flowing in the hose itself “Still waters run deep” - where a river flows more slowly, it is deeper Next time you turn on a faucet, notice how the diameter of the water stream narrows as it travels down from the faucet Bernoulli’s Principle: Forces Consider a small element of fluid By the equation of continuity it must speed up as it passes through the narrow neck It must therefore experience acceleration and a net force, so the pressure behind must be higher than the pressure ahead As it exits the neck, it must slow back down - must have higher pressure in front of it to make it decelerate Conclusion: faster moving fluids have lower pressure Bernoulli’s Principle Bernoulli’s principle applies to any 2 points on a “streamline” Working definition: connects any 2 points through which a flowing fluid particle could pass Bernoulli’s Equation Qualitative derivation: consider a fluid moving in a tube as shown. Generally speaking, Pressure can do work on the fluid The velocity of the fluid can change (obeying continuity condition v1A1 = v2A2), changing its kinetic energy The height of the fluid can change, changing its potential energy Giving: 1 1 p1 + ½v12 + ½gy1 = p2 + ½v22 + ½gy2 2 2 Ideal Gases We’ve been looking at fluids up until now We’ve modeled these materials as ideal fluids, and we have been able to learn a lot by doing that Now we’ll turn our attention to ideal gases The most obvious difference is removing a key assumption about ideal fluids: unlike ideal fluids, ideal gases are compressible The Ideal Gas Law Gases are the simplest macroscopic systems, especially if they are nicely behaved: “ideal gases” The ideal gas law: pV = nRT R = 8.31 J/mol*K n = number of moles = N/N(A) 1 mole is N(A) = 6.02 x 10^23 basic particles (atoms or molecules) of a substance Relates four “state” variables of a gas in “thermal equilibrium” State variable: a parameter that describes the current state of the system (e.g., volume, pressure, temperature…) Thermal equilibrium: state variables are not changing On the face of it, it looks like pV = nRT is nearly useless If it only holds in thermal equilibrium, it cannot allow is to relate two different states of a system As soon as we start to move from one state to another, like by changing the pressure, we’ll move the system out of thermal equilibrium, and pV = nRT no longer holds Loophole: if we change things slowly enough, at any point in time we can stop the process, and the law will hold A process that is nearly always in thermal equilibrium, even though it’s state variables may be changing, is called a quasi-static process We will assume all processes are quasi-static (unless otherwise stated) The pV Diagram We show the states of an ideal gas by plotting pressure vs. volume Usually assume sealed system, so n, the number of moles, is fixed with p and V known, we can find T Note: for quasi-static processes paths in (p,V) space are reversible we can get back to where we started Infinite number of paths in pV space