GFD Lab Demo I Basic fluid properties: pressure, buoyancy, 28ii00

floating, eqn of state, phase change.

Summary

hydrostatic pressure

geopotential

free-surface pressure gauge

U tube oscillator

siphon

milk bottle paradox

floating objects…stability

Archimedes, effect of fluid density

Eqn of state

air

water

nonlinear shape revealed

T,S

heat capacity

air

water

latent heats

evap

freeze

EXPERIMENTS:

momentum flux and pressure

ping-pong balls and dynamic pressure

floating objects on and beneath free surface

U-tube

siphon

weigh glass bulb in cold water and warm water

=> buoyanancy and eqn of state

stir conical flask with cold/warm 2 layers=> shrinking volume

same with salt/fresh layers => nonlinear eqn of state

spray water droplet cloud on liquid. crystal temp sheet

condense/evap cloud due to pressure change (with nuclei)

‘heat pipe’ where water vapor rapidly transports heat

vacuum chamber and its products: clinking water

thermal convection

Introduction

Fluids surround us, and they are particularly well-suited to deliver and distribute the chemicals essential to life. We are carbon- and water-based creatures on an ocean planet, breathing the oxygen/nitrogen atmosphere created by our green co-conspirators.

In this collection of laboratory encounters with fluids, we will attempt to demonstrate the key properties of fluids, particularly those which matter for the oceans and atmosphere. We have found that meeting fluids in the lab, after or during a parallel course on theory, one can use other resources: eye, hand, even artistic senses, and that curiosity can once again come to life.

Pressure and buoyancy

Fluids sit, move and accelerate under the action of forces: both body forces like gravity, and surface-contact forces. A small cube-shaped region of fluid feels such surface forces on its faces, known as ‘pressure’ and ‘viscous stress’. The average (at a point, over all directions) of such forces normal to the cube’s faces is known as pressure, and the net force (the average of the normal vector force on all six faces) is the pressure gradient. It is this particular surface-contact force that tends to compress, the cube, changing its volume slightly. The surface forces tangential to the faces of the cube, which tend to deform its shape, are known as viscous stress.

Pressure is a difficult property to understand. It is a scalar, without direction; perhaps ‘compression’ might be a better name. In a simple gas, the pressure and temperature ‘express’ something about the activity of molecules. Temperature, T (in degrees Celcius, ⁰C, or simply Kelvins, K), is a measure of their mechanical energy, per molecule: this is just kinetic energy, ½ m V² for a monatomic gas. m is the molecular mass and V the speed. Pressure, p (in newton meter^-2, or Pascals), is a component of the momentum flux of all the molecules, or the rate at which momentum moves across a plane within the fluid. This momentum flux conveniently is also proportional to mV². For, a plane, solid wall in contact with the fluid feels molecular impacts; each impact exerts a force normal to the wall whose time integral is equal to twice the molecular momentum, mV₁, in that direction, whatever direction that normal might be. As impacts occur continually, the time-averaged normal force due to many (nV₁ per second) impacts is the momentum flux, nmV₁²; n is the number of molecules per unit volume. nmV₁² is the same as 1/3 rV² assuming that molecular speed in the three directions is equal on average (r, the density in kg meter^-3 is the name we give nm). The similarity of these expressions for pressure and temperature amount to a derivation of the ideal-gas equation of state:

p = rRT

where R = 287.04 J kmol^-1 K^-1 for dry air. In words, the density multiplies the per-molecule kinetic energy to give the total momentum flux, per unit area, with a constant R which would be equal to 1/3 if temperature were really expressed in energy units. A fluid moving past a wall will exert forces along the wall as well, and these ‘viscous stresses’, also have molecular origin. They will be encountered in ‘Stirring and Mixing’, a later lab.

(i) a BB model of fluid pressure

Apparatus:

balance (classic beam balance or modern electronic balance)

metal shot (‘BB’s or small steel ball bearings)

A quick lab demonstration can demonstrate these ideas. Set a cup on the balance and drop a single BB into it, so that it bounces out. Note the temporary deflection of the balance, or (probably unreadable) temporary change in the digital weight reading. Now drop BBs at intervals of about ½ second: experiment to find an interval for which a nearly steady deflection or digital reading is seen. If a digital camera and strobe light are available (see hardware chapter), photograph the process with a strobe light, which will reveal the relevant speeds (fig. ).

This simple, perhaps even tedious, experiment does have a point. Using Newton’s laws, calculate the fall speed of a BB when it hits the cup, and hence estimate the average momentum flux, nmV². See if this indeed is equal (to within experimental error) to the apparent weight registered on the scale. The experiment forces us to think about the continuum model of a fluid: we can’t worry about the details of each molecular impact but we stand back and see their net effect. This is reasonable, because both the molecular diameter [a few tens of angstroms (1A = 10^-10m) or a few tens of nanometers (10^-9m)] and the mean free path of an oxygen or nitrogen molecule, between collisions [roughly 500 A] are so much smaller than the smallest interesting fluid motions which occur at about 1 mm (10^-3m).

(ii) Hydrostatic pressure; U-tube oscillator

Immerse a solid in a liquid. It is compressed by this pressure. But why is the pressure there? In an enclosed box filled with gas it is the confinement that creates pressure. But here the liquid is sitting with a free surface. Why doesn’t it escape?

Of course, it does escape by evaporation. In the vacuum of outer space a beaker of water will turn into vapor and disperse, at least until gravity arrests the water molecules and puts them into some orbit or other. Gravity: that’s it. Gravity is a body force holding the water in a beaker, and only a rare, energetic molecule can break the molecular bonds that maintain the fluid and escape.

So let us focus on the continuum balance between the body force, gravity, and surface-contact force, the pressure gradient. If the fluid is resting (macroscopically of course) there are no viscous stresses, and the entire force balance is the hydrostatic equation:

(1)

where z is the vertical coordinate and g [9.8 meters sec^-2] is the acceleration due to ‘gravity’. Why the ‘’? Because true gravitational force and the apparent centripetal force due to Earth’s rotation combine in the quantity g. {where geopotential?xx}

Hydrostatic balance is the dominant part of the vertical momentum equation for most large-scale motions of the ocean and atmosphere, notably those whose horizontal length scale L and vertical length scale H obey H/L << 1: flat, pancake shaped motions.

The vertical integral of (1),

,

says that the pressure is equal to the weight of fluid overhead (plus whatever pressure is exerted at the top of the fluid). This gives the very useful result that the elevation of water surface acts as a ‘pressure gauge’, for when (2) holds for a uniform density fluid,

p(z) = rg(h-z)

where z=h is the height of the water surface (we assume here that the atmospheric pressure is effectively constant). This graphic result for long gravity waves and flows with a free surface will be exploited in Lab. 3.

It would be wonderful if we could see pressure, perhaps by a change in color of the fluid, but this is the next best thing [in fact, specialized paint exists which changes color in response to pressure; it is used in wind tunnels to give a complete picture of the pressure field on the surface of an airplane in high-speed flow].

Apparatus:

1m. long plastic tubing

Bend the tubing into a U-shape and fill it half-way with water. Holding the two vertical segments together note how the free-surfaces seek each other out. As the U-tube is moved side-to-side oscillations occur, about this mean position. Estimate the natural frequency of the oscillations, comparing with a hydrostatic theory for an idealized U-tube made up of vertical and horizontal arms meeting at right angles: the height increment, h, of either of the two water columns provides a hydrostatic pressure difference 2rgh to accelerate the horizontal arm of fluid. This is the ‘F’ of F=ma. The ‘ma’ is rLd²h/dt². Solving this o.d.e.,

d²h/dt² + (2g/L)h = 0

the natural frequency is (2g/L)^1/2, where L is the length of the horizontal segment. By the hydrostatic assumption, all the inertia of the oscillator is in the horizontal segment, and the vertical acceleration is neglible in the vertical arms. The problem can then be solved more exactly, allowing the vertical acceleration to modify the hydrostatic pressure in the vertical arms; this gives a frequency (2g/(L+2H))^1/2. Thus it is found that only if the horizontal arm is long compared with the vertical, does the hydrostatic theory work. This is the same as ‘H/L << 1’ above. A quick way to the exact result is to use Hamilton’s principle

where , L ºT-V, where T is kinetic energy and V the gravitational potential energy. Here T = ½ r(L+2H)(dh/dt)² and V=2rgh.

Of course a little deception is useful to enliven a laboratory, and carefully filling one arm of the U-tube with salt water and the other with fresh will give a puzzling error in the equality of the respective free surface heights.

Explosive hydrostatic forces. Despite the gentle nature of hydrostatic pressure force in the lab, one should remember that at the bottom of the sea, say 5000 m depth, the pressure is about 5000 decibars or 500 times the atmospheric pressure. This is 5 x 10⁷ Newton meter^-1, or ‘Pascal’. It is great enough to crush oceanographic instruments. Remarkable species of fish and cetacians (whales, dolphins) can move through great changes in pressure without suffering; whales dive rapidly to depths of 800m or more in search of food.

The idea of the hydrostatic pressure field suggests that, in a fluid-filled vessel of complicated shape, the pressure should be the same at all points along a level. This leads to strange, rather uncanny results. Consider the pressure in a flask with a tall, vertical ‘chimney’ (a narrow vertical tube). Since it is proportional to the height of the chimney, the net force on the bottom of the flask, rgAh, is not related to the weight of the water, but exceeds it greatly. A is the area of the bottom, and h the height of the fluid column. Where does this, possibly huge, force come from? Tracing round the container using the hydrostatic relation, we find the pressure is very high also on the sloping sides of the flask. The hydrostatic pressure tries to burst open the flask, with outward force in all directions. This is true, no matter how narrow the chimney is: it could be as thin as a soda straw.

(iii) the siphon

The hydrostatic pressure principle, and Bernoulli’s equation for more general flows, explain the wonders of the siphon: in an inverted U-tube, water rises a great distance before falling down the other arm and exiting. The siphon is such an important tool in the laboratory that it deserves a brief introduction. It is useful for transferring water from source buckets to experimental apparatus, and in providing a continuous inflow to model jets and rivers. Making a siphon work is an exercise in humility. Some rules: wide diameter tubing makes for a fast-flowing siphon, yet is difficult to start (and stop!). Thin tubing works fine but is slow. Start a siphon by filling the tube with fresh water, pinching in firmly near the middle (which will prevent flow everywhere in the tube) and immersing one end in the source water. Both ends should be secured, for example guiding them loosely through a pinch clamp. The normal accident is that the tube is not secured, the inflow end flips up above the water surface and the siphon is ‘broken’. Sucking fluid from one end, the siphon can be reestablished but only at the expense of a large mouthful of (perhaps salty) water. It is not necessary, and may even be dangerous, to suck fluid into tubing. By holding the tube under a water tap or immersed in water, it can filled without oral contact.

When stratified fluid is involved, usually salt- and fresh water, it will be necessary for the outflow to be gentle; a diffuser is described in a later lab, which holds the top on the floor of the tank and spreads the outflow to reduce turbulence.

Large-diameter siphons are useful for rapid draining of tanks. These can be filled in a sink, and carried to the tank, holding an end in each hand. Often a bubble will appear at the high point, stopping the flow. It can in fact be removed by threading a much smaller tube into the large one, and sucking the air bubble out.

Pinch clamps, to control the flow rate in a siphon, are indispensable, yet it is not easy find good ones. Medical apparatus provides a useful family of thumb-roll controlled valves. A small C-clamp works, but is somewhat unstable. Spring clamps, used with thin tubing doubled over, work also. Having a good clamp means that the siphon can be set up and then pinched off, so that an experiment can be started and stopped quickly.

The siphon is rather difficult to get right; wouldn’t it be nice to invent a ‘self-priming’ siphon that would start up whenever the water level exceeded a certain height? Imagine how this might be done. One way is to use capillary fluid-surface effects, which cause fluid to climb up a wick. A piece of string draped over the edge of a beaker of water will eventually drain it out! Try it.

(iv) Buoyancy, floating

Apparatus:

rectangular glass or plexiglass tank (any size)

rectangular blocks of wood and styrafoam

Archimedes principle is based on the assumption that if we place a rigid body in a fluid, the molecular rebounds from its surface will have the same momentum as would the molecular flux from the fluid previously there, and hence the vertical force on the rigid body must equal (minus) the weight of the fluid previously there. This is simply a symmetry argument for an ideal gas, but is a more touchy assumption for a fluid. Perhaps we should just say that if it is not true, we can blame ‘chemical reaction’ between the fluid and solid molecules. But, it seems to work: the net ‘buoyancy’ (the force due to pressure exerted on an immersed object) is equal to the weight of fluid replaced by it. So Archimedes was handed a crown of complex shape and asked to measure its density to determine whether it was gold or some more base metal. He could have weighed it, and then immersed it in water to measure its volume by the amount of water displaced, but a simpler procedure is to weigh it in (W₁) and out of (W₂) water. Now, W₁=gr_crownV and W₂=W₁-gr_waterV, where V is the volume of the crown. Two eqns. in two unknowns, so

r_crown/r_water = W₁/(W₁-W₂)

and the kingdom was saved.

This is the basis of floating and buoyancy. A partially immersed ‘spar buoy’ of mass M has a hydrostatic pressure force supporting it, and if it is displaced upward or downward a distance dz, that force changes by an amount r_waterAdz where A is its cross-sectional area. If then let go, it will oscillate according to

hence, its frequency is (r_waterA/M)^1/2, ignoring the inertia of the water movement (questions of the stability of floating objects, whether they float ‘vertically’ or ‘horizontally’, can be worked out using simple hydrostatic pressure ideas, and have application to the rolling over of icebergs, and the floating of ‘dead-heads’ (logs that float nearly vertically in Puget Sound).

Implicit in all this is hydrostatic vertical force balance. For a fluid at rest, the two active forces (per unit mass) are the pressure force -Ñp/r and the gradient of the geopotential field, -ÑF. F is best thought of as the potential energy of a particle moving in the Earth’s gravity field, yet in the rotating frame of the Earth, giving the two contributions

where G is the gravitational constant, W is rotation rate, r₃ is spherical radius, and r₂ is cylindrical radius measured from the rotation axis. This point-mass approximation works if the Earth is approximately spherical. The true-gravity part dominates near the Earth’s surface, and we write

F » gz

where g = 9.8 m sec^-2, and z is the local vertical coordinate. Note that F=const surfaces are slightly distorted spheres (with an Equatorial bulge, with the equatorial radius, 6378km, exceeding the polar radius by about 21.4 km,) while further out they turn into cylinders. At a radius about 36,000 km above the Equator, the gradient of F vanishes, and there we find geostationary satellites.

On a laboratory rotating table (radial coordinate r), by contrast, the appropriate geopotential is

F_lab = -½ W_table² r² + gz

so that equipotentials (the fluid free surface being one) are just paraboloids of revolution.

Hydrostatic balance is

Ñp/r = -ÑF

which can only be satisfied if p and r are constant along geopotential horizons, F=const. (hence ¶p/¶F = -r)

In a stratified ocean or atmosphere these buoyancy effects are subtle indeed. We showed a column of salt-stratified water in which the density ranged from about 1.2 g cm^-3 to 1.0 g cm^-3. This is enough to float a piece of plexiglass! In the oceans, the density is 1.035 g cm^-3 to within about 2%, and most of this 2% is dynamically inactive, as it is just adiabatic compression. A typical water column has about 2 x 10^-3 fractional change in potential (‘dynamically active’) density from top to bottom. This corresponds to an empirical equation of state r(T,S,p) with ‘slope’ coefficients

Note how small the thermal expansion becomes at low temperature. This means that in subpolar regions thermal convection is reduced in intensity, and ‘haline’ (salt-driven) convection is particularly important. In the tropics, the thermal expansion of water is great, but so too is the evaporation, which leaves more saline water behind when the surface is warmed. Convection can involve both effects.

A perfect gas is much simpler than water, with p=rRT, a is just 1/T at constant pressure; the vertical density profile is exponential, r=r₀ exp(-z/H) for a choice of isothermal, resting atmosphere (the e-folding scale H being RT/g ~ 8 km).

More active pressure. In a steadily flowing fluid, constant-density, inviscid fluid there is a first integral of the momentum equation, showing that B º p/r + ½ |u|² + gz, the Bernoulli function, is conserved along streamlines. Ignoring gravity for now, this shows how the pressure falls as fluid accelerates into a constriction in a channel (with rigid surface). It is somewhat counterintuitive, for we think of jets of fluid ‘blowing’ solid objects in their path, but it must be true, because the pressure gradient that accelerates the fluid along its path is then in the right sense. That force comes from the slow-flow region near the stagnation streamline, however. Low pressure regions exist elsewhere (and can causes houses to ‘explode’ in violent winds). A ping-pong ball tethered to a thread is sucked into a stream of air or water, proving this point (sketch the streamlines to see the region of fast flow). Likewise, the ball suspended in an air jet is stabilized by this effect; if it wanders out, the fast-flow region is on the side nearest the jet, and this sucks the ball back into the stream. You can pick up a piece of paper by blowing an air jet on it!

In GFD the use of hydrostatic balance gives the free surface a special role in interesting flows: the height of the surface is proportional to pressure, and for an unstratified flow, this height profile maps out the horizontal variation of pressure below. We will see many examples of this effect in channel flows later in the term; here we looked at tornado vorticies produced by swirling flow in a cylinder. If (roughly) angular momentum is conserved for rings of fluid moving inward, we have rv=const (v=swirl velocity, r=radius). Now the radial mom. balance is ¶p/r¶r = v²/r, so together we have

p = const/r²

where we use the fact that the atmospheric pressure is nearly constant along the water surface.

With hydrostatic vertical balance, the surface height h is given by

gh= const + p/r = const -const/r².

Now, I said ‘roughly’ above because in fact friction forces in the lower boundary layer are important and a significant part of the radial inflow occurs in that layer (as in a real tornado). Recall the cylinders of dye orbiting round the core, and only slowly flowing out the drain. Nevertheless, a 1/r (irrotational!) swirl profile is not a bad approximation, and in nature leads to wind-speeds in excess of 100 m sec^-1 and very low core pressures. The explosive effects of tornados on buildings can be seen in videos like Cyclone by National Geographic (whereas all the tornadoes in the movie Twister are computer graphics).

In the sunlight you can often see vividly the vortices near the surface of a swimming pool, the deflections of the free-surface acting as lenses.

Stability of floating objects. If a rectangular polyhedron floats on a water surface, will it also float happily, after being rotated 90⁰ to the vertical? A log floats lying on the sea-suface, yet sometimes you see logs floating vertically (‘dead heads’) or at an angle. A very light solid of foam rubber floats either way. The stability can be studied by tipping the solid slightly, and seeing whether it erects itself or tips completely over. This can be worked out by calculating the moment of the hydrostatic pressure about the center of mass of the object: this torque will either restore equilibrium or cause a tip-over. For a rectangular solid the answer is:

Simple convection. A ball of fluid with density less than its surrounding fluid will rise, for the same reason that objects float. Archimedes’ argument holds. Heating a fluid from below will cause this to happen, in great and complex ways. A first encounter with fluid convection involves a gently heated resistor putting a few watts of heat flow into the fluid. You will see at the beginning a ‘mushroom’ shaped buoyant ‘thermal’ rising, followed by a continuous ‘pipe’ of warm fluid. Even with gentle heating this will likely go unstable and produce turbulent convection. A large-scale upwelling is driven, as the thermal drags nearby fluid with it. Eventually the thermal will produce a stratification in the fluid as a whole. Ironically, heating the fluid from below can create stable stratificiation, provided the heat source is not uniformly distributed along the bottom.

Apparatus:

resistor, wired with waterproof seal

d.c. power supply

rectangular tank

shadowgraph (see chap. xx)

……….

(v) Equation of state: air

Natural air is sufficiently close to an ideal gas that the classic equation of state

p = rRT

suffices for most purposes. Water vapor changes the equation slightly, the temperature T above being replaced by T_v = T(1-q+q/e), which is known as the virtual temperature. q is the humidity (the density of water vapor divided by the density of air), and e = 0.62197 is the ratio of average molecular masses of water vapor and air. Van der Waals forces, which are an expression of the finite mean free path and finite radius of interaction as molecules collide, (and finite volume, or ‘excluded volume’ occupied by molecules) make a correction of 1% or less for atmospheric conditions.

Apparatus:

rubber bulb/flask with port (fig )

thermometer

For a quick reality check, insert the thermometer in the port of the flask, and seal by holding it tightly. Squeeze the bulb and observe the temperature change. The change in pressure, assuming no heat loss, is found from

p = A r^g ; g º c_p/c_v, A = exp(h/c_v)

so that dp/p = gdr/r = 1.4 dr/r. Then we expect that dT/T = (1-g)dr/r or about 0.4 times the fractional change in volume. For this apparatus the temperature changes are about 1⁰C.

Ocean water is comprised of H₂O molecules in close proximity, with weak dipolar bonds between them. Polar molecules like this behave very differently than non-polar substances like hydrocarbons. The ‘hydrogen bridge’ is an unusual form of molecule and bond, and water is thus an usual substance. The two hydrogen atoms meet the central oxygen atom with an angle of 104.5⁰. This gives the water molecule an electrostatic dipole moment. It is like a partially ionized substance. If an electric field is applied by putting two plates in water, and holding them at different voltages, the water molecules orient themselves, causing an induced field that opposes that applied: water is a dielectric material.

Most ordinary non-polar substances have solidification and boiling points that increase with molecular weight. Based on its place in this sequence, ice should melt at –100C and water should boil at –80C. The error of 180C in the boiling point is due to the hydrogen bonds that form, holding water molecules in a loose matrix. Hydrogen bonds are known to be weaker than normal covalent bonds (say, in an H₂ molecule) yet stronger than ionic bonds.

Fresh water has a nonlinear equation of state, with maximum density r(T;p) at about 4⁰ C (fig.). Its curvature causes a mixture of hot and cold water to have a density different from the average of the two original densities.

(vi) Water equation of state

Apparatus:

flask (sloping sides)

rubber stopper with hole

glass thermometer that fits into stopper

magnetic stirrer

The shape of its density/temperature/salinity curve is worth exploring. Take a flask with sloping sides (fig ) and fit a stopper with a hole. Insert through the hole a glass tube to act as a ‘chimney’. Fill the flask with equal volumes of hot and cold water, measuring their temperatures. Drop the stir bar into the flask. Insert the stopper so that there is no air gap inside. Press down until water rises about half way up the tube.

Now place the flask on the stirring box. Watch the height of the water column in its glass chimney. It may slowly fall, as the hot water cools (this effect can be minimized by balancing the hot and cold temperatures, or by wrapping insulation round the flask.

Switch on the stirrer, and watch the column height change as the hot and cold waters mix. You can record the height change and calculate the length of line segment between the curved equation of state and the straight line that would be there for a linear equation between the two temperatures. This is quite an accurate way to determine the curvature of the fresh-water equation of state.

A companion experiment investigates the effect of salinity on density: a salty layer of water lies beneath a fresh layer, at the same temperature. When the stirrer is turned on the water column again ‘shrinks’; this says something about the way salt molecules hide among water molecules. A simple fluid would not change in total volume when low-density and high-density regions are mixed together.

These two experiments give us two ‘cuts’ across the equation of state, r as a function of T, S (actually T, S and p, the pressure, but we are staying near atmospheric pressure). The shape of r(T,S; p) is a sloping plane, which is slightly curved (the two figures show the oceanic range of salinity (above) and a larger range of salinity, down to zero, below; note the density maximum at 4C which exists only near zero salinity. Salinity is expressed in kg salt/kg seawater x 10^-3 or ‘parts per thousand’. Oceanic salinities are typically 2 to 3.6 % or 20 to 36 parts per thousand).

In the same apparatus you can explore the hydrostatic paradox described earlier, sometimes known as the milk-bottle paradox. Milk used to be delivered with cream floating on top, and in sloping sided bottles. If the pressure is measured at the bottom of the bottle, then the milk is stirred until homogeneous, how will the pressure change (or will it change)? Because a column in the center of the bottle is on average less creamy after mixing, it will be more dense, and hence the hydrostatic pressure will rise. Because of this pressure rise the net force on the bottom will increase as well (being supplied by an increased force on the sloping sidewall). The same effect says that the pressure at the bottom of a lake with sloping sides will change if the thermocline is mixed away by wind: simple hydrostatics, but nevertheless puzzling.

Use the chimney as a manometer to indicate the pressure at the base of the fluid. This may be done by putting a less dense fluid in the tube; hence use salt water (~ 10% salinity) in the flask. With the same hot and cold water preparation, insert the stopper loosely, so that it is not sealed. Now the height of the fresh water is an indication of the pressure at the bottom of the glass tube. Stir the fluid and note that the pressure changes!

(vi) Full equation of state

An experiment that demonstrates buoyancy and also gives us quantitatively the equation of state is as follows: take a glass bulb of known weight and immerse it in water, hanging it from a thin wire. The wire is attached to a precision scale, sitting on a step-ladder above. Now change the density of the water by heating or cooling with ice. The weight of the bulb will change, according to Archimedes. We have done this experiment for many GFD classes, and it always works...qualtitatively...but is difficult to make it agree precisely with the calculation based on the ‘official’ equation of state. This is often due to air bubbles that form on the bulb and lift it

slightly. Using ice, this year we came closer (fig.; the righthand panel would be a constant for perfect accuracy; the fractional error is small, 4x10^-3 ). The calculation takes the observed temperature and weight of the bulb in water and in air, and derives the density of water. The figure shows an offset which we do not yet understand, but the shape of the curve looks very good. Note the maximum density near 4C. This effect means that fresh-water lakes ‘turn over’ as they cool to 4C, but with further cooling the coldest water remains at the surface. This encourages smooth freezing, good skating, and helps preserve life below. Also, water is anomalous in having its solid phase less dense than its liquid phase. If ice were denser than water, the world would be very different: a far-reaching quirk of Nature.

(vii) Heat capacity and phase change

The heat capacity of a fluid is the heat in Joules required to raise 1 kg. of fluid 1⁰ C. For water at 28⁰ C this is 4000 J kg^-1 K^-1 (see Gill, Appendix 1). For dry air it is

c_p = 7/2 R = 1004.6 J kg^-1 K^-1 ;

the number 7/2 is for diatomic molecules. For a polyatomic molecule, with more than 3 atoms, the number is 4. This leads to a specific heat coefficient for moist air of

c_p = 7/2 R(1 - q +8q/7e)

(Gill, 1982, p43.). These are measured at constant pressure; at constant volume, the specific heat c_v = R – c_p. The specific heat capacity (at constant pressure) of fluids and solids normally exceed that for gases; for air c_p is about 1/4 that of water, With its density some 800 times less, the heat capacity of the entire column of atmosphere is equivalent to that of just 5m of ocean beneath. For this reason the ocean is an active heat source and sink for the atmosphere, and generally more exchange occurs as latent heat of evaporation than as ‘sensible’ conducted heat.

It is difficult to have an intuitive quantitative sense of heating. Perhaps the best visualization is the heating of a tea-kettle, where a 500W stove-top heating unit (500 J sec^-1) takes about 300,000 J. to boil 1 kg. of water initially at 25⁰C, and 10 minutes to do it. As a liquid is heated, it develops an increasing vapor pressure. At at a given temperature and pressure, a vessel of liquid has an equilibrium vapor pressure which describes the escape of water molecules from the surface. At the boiling point, the vapor pressure has risen to equal the atmospheric pressure. Raising the water to the higher energy level requires a great deal of heat,

L_v = 2.50 x 10⁶ -2.3 x 10³ T J kg^-1,

where T is the Celsius temperature. For this reason, it takes longer to boil a tea kettle dry, than the time initially to bring it to a boil.

Energy units are all round us; caloric value of foods, horsepower of automobiles, heating of a liquid, radiation of an electric heater. It takes about 1.5 horsepower to take a hot shower, and a small (250 food ‘calories’) candy bar eaten in 1 minute represents about a million J. of energy/60 seconds, or 22.3 horsepower! (our conversion of food into mechanical work is far from perfectly efficient). (1 hp. = 745.7watts = 745.7J sec ^-1). Note that food ‘calories’ are actually kilocalories; 1 thermodynamic calorie = 4.184 J, yet 1 food ‘calorie’ is 4186.8 J, which is strange. How much power output can the human body produce for a short time? Running upstairs, you can readily calculate your change in gravitational potential energy, and it’s difficult to do much better than 1 horsepower.

The ratio of heat flux carried by water vapor and that carried directly as ‘sensible’ heat (rc_pT) is a crucial part of atmospheric dynamics. Generally in the tropics the latent heat flux upward from the heated sea surface greatly exceeds the sensible: if it were not so, clouds would contain be the dominant vertical motion. The delayed dynamical impact of the latent heating (which appears when the water vapor condenses) makes a peculiar forced fluid dynamics problem (see, e.g., the book Atmospheric Convection by Kerry Emmanuel). At higher latitude, for example in the Labrador Sea, cold, dry winds from the continents flow over the ocean and are warmed; the sensible and latent heat fluxes there are more comparable. Visit the Clausius-Clapyron equation in Gill to learn more.

Apparatus:

rubber bulb/spherical flask with port

This is an apparatus made to illustrate liquid-vapor phase change, the condensation of water vapor into droplets. Inject a drop or two of water through the port, and close the pinch clamp. Squeeze and release the rubber bulb: probably nothing will be visible. The adiabatic temperature rise (squeezed) warms the liquid water; increases its vapor pressure, and puts more moisture into the air. When the bulb is released, pressure drops, adiabatic cooling occurs, and the air can hold less water vapor, which is then prone to condense: but it may not do so unless condensation nucleii are present. The classic mystery of droplet formation is that very small droplets have very great potential energy associated with the curvature of their surface. It is impossible to form very small droplets because of this extreme surface free energy.

Burn a match to produce some smoke, and suck this into the flask through the port. Clamp the port shut again, and squeeze the bulb. Suddenly, upon release, a dense cloud of water droplets forms. The phase change is very rapid; it often may be seen in flow over the wing of an airplane, where moisture condenses in low pressure region on top of the wing. The condensation can be produced repeatedly, and requires only a slight amount of water and smoke.

Apparatus:

vacuum pump with glass chamber

This rather specialized apparatus, if available, produces a wealth of experiments involving very low atmospheric pressures: freezing of water at room temperature, mobility of water vapor, thermal properties of a near vacuum, heat pipes.

Breakable water. It also produces a remarkable demonstration of the properties of water in a sealed glass tube when most of the air is removed. The water breaks apart (evaporating in one place, condensing in another) and gaps open up readily (fig ). As the tube is tipped back and forth, the water ‘clinks’ like metal or glass, as air bubbles and their cushioning effect are then absent. It is interesting that if a small amount of residual air is present, the apparatus acts and ‘feels’ like normal water until it is shocked (accelerated abruptly), whence the phase change and ‘clinking’ are induced. The process of boiling and condensing very quick...effectively instantaneous compared to other atmospheric time scales (but, could a standing sound wave induce a regular pattern of vapor bubbles?).

Note also how a rising bubble in this apparatus grows in size as it comes to the top of the liquid. This is expected, as the hydrostatic pressure varies with depth below the free surface. At atmospheric pressure, the height scale over which the pressure in a water column varies is about 10m (~ 10 decibars ~ 1 bar of pressure). So in a glass of beer, the rising bubbles would not appear to increase in size noticeably (if it were not for the bubbles ‘scavenging’ more gas from the beer as they rise). Here however the pressure is much reduced: you could use the observed bubble expansion to measure the pressure!.

A heat pipe. In an apparatus consisting of two spherical bulbs connected by a tube...a sort of dumb-bell shape, the pressure is similarly reduced to a small value (actually by boiling rather than using a vacuum pump). A small amount of liquid water remains. The water ‘clinks’, hence the pressure is very low. Heat one of the bulbs with a hair-dryer (or your warm hand) while holding the central tube a distance away; you will feel that cold hand warm up quickly. Evaporation sends water vapor along the tube, which condenses and heats your other hand. It does so very rapidly, far more quickly than heat conduction could do. For these reasons, scalding burns caused by steam are especially severe: as the steam condenses on your hand it can produce temperatures far in excess of 100⁰C.

Evaporation/condensation heat transfer is one of the most important processes in atmosphere/ocean dynamics; the delayed ‘heat pipe’ (the delivery of heat evaporated from the sea surface, delivered to the atmosphere at condensation level), provides a complex challenge to numerical models of the atmospheric circulation.

A classic example of evaporative cooling is the ‘sling psychrometer’, the device used to measure the humidity of air. A homely version of this: the temperature of air is measured with a thermometer, and then the thermometer is wrapped in a wet rag, attached to a string and swung round your head. This evaporates water from the rag, and cooling occurs. Eventually the cooling will cease, and the difference between this lower ‘wet-bulb’ temperature and the normal temperature can be used to calculate relative humidity (the temperature difference is 1⁰C for 93 % humidity, 15⁰C for 17% humidity: thus it is not advisable to wear a t-shirt while wind-surfing in Colorado!). Why does the wet-bulb temperature not continue to decrease with continuing evaporation? How would the experiment change if the rag were wetted with alcohol instead of water?

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