**F1**** ****Flow of incompressible fluid and
gases assumed to be incompressible**

The *incompressibility** *of
fluids and the *compressibility** *of gases are the reason for the separation of
problems of the *equlibrium** *of fluids and gases*.** *However, *flows**
*of fluids and gases can be treated together, since also a
gas can be *considered* to be incompressible as long as
its flow velocity remains *below** *a certain level (small compared with the velocity
of sound in the gas under consideration). Its change in volume is
less than 0.5% and practically irrelevant as long as the velocity
is less than 0.5% of the *velocity
of sound*. Then its motion differs in *no essential aspect* from that of an incompressible fluid (Ludwig Prandtl
1875-1933). In what follows, it is assumed that the velocities of
gases are *below* this limit.

So far, we have only dealt with
the pressure in a fluid at *rest*. As soon as the pressure of a mass at
rest causes acceleration of the mass, a fluid *flows** *and the pressure at a definite
location depends on the local velocity: It *increases* as the velocity *decreases* and *decreases**
*as the velocity* **increases*.

How to follow a flow?* *It is difficult to follow an individual fluid particle,
whence we will only investigate the *changes in time* of the velocity and the pressure in a fluid at a given
point in space. At each point in space, you find a velocity in
magnitude and direction and, in general, these *quantities change* from point to point. They are
represented by *stream
lines*: Curves the
tangents of which at each point have the direction of the
velocity. An area of the shape of a tube, which is formed by
stream lines, is called a *stream
tube*, its content of
fluid a *stream thread*.

A stream thread
flows in a stream tube like in a channel with solid walls - the
fluid moves only *along** *the stream lines, not at right
angle to them. If there enters or leaves at *each* instant an equal amount of fluid, that
is, the velocity does not anywhere depend on the time, the flow
is called *stationary*. The *strength*
of the flow - the amount of fluid passing through a cross-section
in a second - is then the same at each cross-section along the
thread, whence the velocities at different locations are
inversely proportional to the cross-section: The *smaller* the area of the section, the *larger* in the same ratio is the velocity.

If you imagine the
space filled by the fluid subdivided into thin stream tubes, you
have an instantaneous image of the flow; naturally, it can change
all the time. Only during *stationary* flow does the flow pattern not depend
on the time, that is, the flow tubes do not change their shapes.
In general, in an elementary treatment, attention is concentrated
on a single stream thread: You either deal with one exhibiting
mean conditions or you consider the entire flow as a collection
of threads and then average the results. *Changes in density *can be neglected, since the
compressibility of the fluid is much too small to demand
attention. An elementary treatment also neglects *friction*. Water has so little friction that the
results can be applied to real conditions.

**Equation of ****Daniel Bernoulli**** 1738**

What can we
discover about the pressure in a stationary fluid flow? Consider in the flow tube of Fig. 225,
which we imagine to be like a downward channel, the fluid between
the cross-sections *A* and *B **and follow* it over the time interval *dt*;
during this time, the cross-section *A *moves to *A'*,
the cross-section* B *to *B'*. Let *f*_{A}*
*be the area of *A* and the velocity in it *v*_{A},
and for *B* correspondingly *f*_{B}
and* v*_{B}. Since the flow is *stationary*, one has *f*_{A}*·v*_{A}*
*= *f*_{B}*·v*_{B}.
We now ask: How changes the
kinetic energy* *during the time interval* dt*? *This change** *is equal to the work performed during *dt* by
the force which is causing the motion.

We shall first
find the expression for the *change
of the kinetic energy*
of the fluid mass and *assume* - certainly approximately correctly -
that all particles, which at a definite instant of time lie
between cross-sections (*A* and* B*) of the stream
tube, which are very close to each other, will also during any
arbitrary later time interval lie between two such cross-sections
(*A*' and* B*'). According to this assumption, the mass which was lying between *A* and* B*
reaches the space between *A'* and* B'*; in the
space between *A*' and* B*
nothing changes during
the time *dt*, as every fluid particle is only *replaced* by another one which takes its
velocity; however, the fluid between *A*' and* B*
does not act on the *outside*. (By the above assumption, you make it
possible, to take into consideration *only* the
events at *those
cross-sections*.)

The *change* of kinetic energy during *dt* is
reflected in the change of the velocity (from *v*_{A}*
*to *v*_{B}) of the fluid masses
which pass during *dt* through the cross-sections *A*
and* B*. If the cross-section *A* has the area *f*_{A}*
*and the velocity in it* *is *v*_{A},
there passes during *dt* the volume *f*_{A}*v*_{A}*dt
*with the mass (*s*/*g)f*_{A}*v*_{A}*dt
*(*s* being the weight of unit volume) and
the kinetic energy (*s*/*g)f*_{A}*v*_{A}*dtv*_{A}^{2}.
The corresponding expression at the cross-section *B* is (*s*/*g)f*_{B}*v*_{B}*dtv*_{B}^{2}.
The kinetic energy of the fluid mass, lying initially between *A*
and* B, *changes during *dt by *

(*s*/*g)f*_{A}*v*_{A}*dtv*_{A}^{2}
- (*s*/*g)f*_{B}*v*_{B}*dtv*_{B}^{2}.

Next, consider the
work done. It comprises the work which *gravity* and the *pressure*
have performed on the fluid. For this purpose, we need again only
consider the events at the cross-sections *A* and* B*.
Gravity performs on a particle of weight *q*, which sinks
from the height *z* (above the horizontal plane of
reference) to the height *z*', the work *q·*(*z
- z*'). The *work* done on the entire fluid mass is
therefore S*q·*(*z - z*'), where the
sum is taken over *all* fluid particles. However, the particles
between *A*' and *B *do not partake in
the action, whence we
extend the sum only over the fluid particles between *A*
and *B*.

Let *z*_{A}
and *z*_{B} denote the *heights *of the centres of gravity of the fluid
masses, which pass through the cross-sections *A* and *B
*during *dt*,* *and *q*_{A}
and *q*_{B}* *the amounts of
fluid, then S*qz - *S*qz' = q*_{A} *z
*_{A} *- q*_{B} *z*_{B}*.
*Thus, the work done by gravity during* dt *is:
*s* ·*f*_{A}*v*_{A}*dt·z*_{A}*
- **s* ·*fv*_{B}*dt·z*_{B}*.
*Letting *p*_{A}* *and *p*_{B}*
*denote the pressures per unit area at* A *and *B*
and taking into consideration that the pressure at *B* is
directed in the opposite direction to that at *A, *the
work of the *pressures** *during *dt* is: *p*_{A}*f*_{A}
·*v*_{A}*dt - p*_{B}*f*_{B}
·*v*_{B}*dt. *Hence follows the
equation

(*s*/*g)f*_{B}*v*_{B}*dtv*_{B}^{2}-(*s*/*g)f*_{A}*v*_{A}*dtv*_{A}^{2}=*s* ·*f*_{A}*v*_{A}*dt*·*z*_{A}*-**s* *f*_{B}*v*_{B}*dt*·*z*_{B}+*p*_{A}*f*_{A}*v*_{A}*dt-p*_{B}*f*_{B}*v*_{B}*dt.
*

Each term contains
*dt* and *f*_{B}*v*_{B}*
*or* f*_{A}*v*_{A},
which are equal to each other. Dividing by and shifting terms with
negative signs to the other side, we find:

*v*^{2}_{B}/2*g
+ p*_{B}*/**s** + z*_{B}* =
v*^{2}_{A}/2*g + p*_{A}/*s** + z*_{A }or *v*^{2}/2*g
+ p/**s** + z = *const;

letting *s* = *r**g,
(**r *is the mass per unit volume), we obtain:

*r**v*^{2}/2*+**r**gz+p=*const,

the form in which
we will use this equation below. It is *Bernoulli**'s equation**, *the basic equation of
Hydrodynamics.

The term *v*^{2}*/2g* is called the *velocity height*, because it is exactly the height by
which a body falls to reach the velocity *v*, the term *p/**s** *the *pressure height*, the height of a fluid column which by
its weight generates the pressure *p*, *z* the *location height*, the height of the point of
consideration of the fluid above a horizontal reference plane.
Hence Bernoulli's equation says: The sum of velocity,
pressure and location heights is the *same* at
each point along a stream line. In practice, you do not only
compute with this formula and relationship for single stream
lines, but also for stream tubes with final cross-sections by
averaging the true heights.

You can apply to
the fluid flow between *A* and* B *(Fig. 225) the
Momentum Theorem of Mechanics and investigate the change of
momentum, which the fluid, *initially*
contained in the bounded
space, experiences during unit time when it is moved *steadily*. Here too you only need to consider the
momentum at the cross-sections *A* and *B*, in
order to find the *Momentum
Theorem of Hydrodynamics*: The resultant of the forces, acting on a bounded space
filled with fluid, equals in magnitude and direction the excess
of the momentum of the exiting fluid over that of the entering
fluid during unit time.

You can now draw important
conclusions regarding the *force
*exerted by the fluid
on solid bodies without detailed knowledge of the velocity
distribution. For example, you can compute the force, exerted by
the fluid on a *submerged
and moving body* as
well as treat other tasks which are too difficult for elementary
treatment. We will not discuss the momentum theorem in detail and
turn now to the problems of *pressure
and velocity in moving fluids*, a treatment of which is possible on the basis of
Bernoulli's equation.

**Consequences and application of ****Bernoulli****'s equation (Pressure and
velocity in fluid flows)**

Through every cross-section in a *steady flow* passes at
each instant the same amount
of fluid, whence the quantity which passes through a
cross-section per second is *inversely
proportional* to the area of the
cross-section. What about
the pressure at different cross-sections?

Let *f* and *f*_{0}*
*be the areas of two cross-sections, to which correspond the
pressures *p* and *p*_{0}, the velocities *v*
and *v*_{0}, and the heights *z* and* z*_{0},
related to some reference plane. Bernoulli's Equation then yields
for steady fluid flow

*p* - *p*_{0}+
*r**g*(*z - z*_{0}) + (*v*^{2}*
- v*^{2}_{0}) = 0.

Since *fv = f*_{0}*v*_{0},
you can introduce the *cross-section
*into this equation. The computation
yields the result (Fig. 225a): In a {smaller/larger}
cross-section - that is, where the velocity is {larger/smaller} -
the pressure is (smaller/larger}.

But you can also perceive this *without computation* (Fig.
226): Through a, horizontal tube, narrowing towards the right
hand side, flows steadily towards the right hand
side an incompressible, frictionless fluid. There must pass every
second through each cross-section the same amount of fluid.
Therefore the *velocity** *of the fluid's particles is smallest at *L *and
grows steadily towards *R*. The fluid particles can
receive this acceleration
only through the pressure forces acting on
them. In order that at an instant the cylindrical fluid quantity *F*
will be accelerated towards the right hand side, a larger
pressure must act at its back* A* than at its front *B** ,
*that is, the

We find the same pressure
distribution (decrease of pressure from *L *to* R*)
by an analogous argument regarding the reverse flow of the fluid.
The fluid particle *F* then moves slower from *R*
to *L. *Fluid particles can therefore only experience
acceleration during motion from points of higher pressure to
points of lower pressure or, more briefly stated, when the
pressure drops in the direction in which they move.

If a wall, along
which a flow passes, has a wavy form (Fig. 226b), the *cross-section *of the flow is more slow at *B* and faster at *T*,
whence the flow generates *excess
pressure** *at*
B,* *deficit
pressure *at *T*.
However, everywhere along the wall the pressure is directed in
such a way that each curve would be enlarged, if the wall were
flexible. This is characterized by a weakly waved, thin, really
deflectable wall, both sides of which have a flow along them
(Fig. 226c); excess and deficit pressures at both sides of the
wall *support* one another. This process explains the *flutter of flags* in wind, the *generation of waves *on the surface of water, when the wind
reaches a certain velocity, etc.

You see the
distribution of pressure, which follows from Bernoulli's Equation
at a tube (Fig. 225a)
with *gentle narrowing and widening** *of its cross-section (for the
reason of gentle) after attaching manometer tubes: The shorter
and longer fluid columns indicate the smaller and larger
pressures at these locations.

A widening tube
with flow through it is employed in water-, steam- and
gas-jet-appliances. For example, at the narrow end (Fig. 230) ,
the flow provides suction for air; the rising pressure at the
widening end takes the sucked-in air away. Among these, you have Bunsen's water-jet-air-pump of Fig. 227, suction installations
like that of Fig. 228 and the Bunsen-burner of
Fig. 229, in which the gas jet exiting at *A* sucks air
through the opening *L *with which it mixes before being
lighted. Others are the inhalation instrument and injectors, for
example, the steam-jet-injector (Fig. 230).

Langmuir 1916 built such an airpump, the *suction
velocity* of which exceeded that of all
pumps built earlier (Fig.231). The vapour rises from the heated
mercury *A* through the pipe *B *and sucks air
through the *ring-shaped
slot **S *around the exit *L*
from the receiver shut at *F*. The sucked air flows away
in the the vapour stream. You first remove air with an auxiliary
pump (*pre-vacuum pump*) from the receiver until the pressure drops to 0.05 -
0.1 mm.You then activate the main pump. The vapour jet, entering
the pre-vacuum space *C*, spreads lin tufts.

Unless you provide precautions
against the *vapour *hitting the wall of the tube, into which it entered, and
forming by deposition mercury drops, these would again vaporize
and reduce the suction. Langmuir avoids this by *energetic
condensation* of the particles
approaching the wall (flow of cool water through *K*_{1}*K*_{2}),
a special design of the pre-vacuum space *C* for very fast
removal of the condensation heat. According to him, this is most
important, whence also the name *condensation pump**.*The
pump sucks 1500 - 3000 cm³/sec.

**Measurement of pressure and flow
velocity**

Bernoulli's Equation relating the pressure and
velocity of a flow permits *computation
of ***1) **the*
**velocity*, if you can measure the *pressure*,*
*and **2) **the
*pressure*, if you can measure the *v**elocity*. If there exists in a uniform fluid
flow (with velocity *v*_{0}) an obstacle, there arises in front of
it *retardation* of fluid particles within which the
flow spreads in all directions (Fig. 232) At the centre of the
impounding, the *stagnation
point* *P*,
the velocity *v* vanishes.

At the stagnation point, Bernoulli's
Equation yields *p=p+v*²*·**r*/2, where *p*_{0}*
*is the undisturbed pressure in the same horizontal plane.
You call the increase in pressure *p* - *p*_{0}*
**stagnation **or** velocity **or** dynamical pressure**. *The *total pressure** p* is measured with the Pitot Tube (Fig. 233), named after its inventor Henri Pitot in 1730 (Histoire de l'Academie des Sciences, 1732, p
376), one leg of which is placed facing the flow, the other
vertically upwards. In the first leg, the flow stops, whence the
pressure rises. The fluid arriving at velocity *v* raises
the water level above the free surface by *D*, which is in agreement with the
velocity height *v*²/2*g* in
Bernoulli's Equation.

The *static
pressure* *p* is measured with a
*pressure probe* (Fig. 234); in the sidewards cuts, the openings of
which are *parallel *to the direction of the flow, and inside the probe the
fluid is practically at rest. The pressure in the probe, since is
it constant in the section at rest and must change in the
interface steadily into that in the moving fluid, is equal to the
pressure in the passing fluid. The stagnation pressure *p - p *is
measured with Prandtl's stagnation instrument which combines the Pitot Tube and
Pressure Sonde (Fig. 235). In order to simplify the reading of
the, as a rule, small differences in height *D*, which would be
very inconvenient just above the free surface of the water, the
fluid is sucked in both tubes simultaneously to a convenient
height.

The *flow velocity *(air, water) in a pipe line is measured with a *stagnation disk* (Fig. 236). A venturi nozzle is also employed for its
measurement (Fig. 237); you measure (manometrically) the
difference between the (lower) pressure at the *narrowest *section
*f*_{0}, which arises from the increase of
velocity, and the pressure at the *not reduced** *opening
*f* in__ __front.

**Outflow velocity (T****orricelli****'s Law). Discharge quantity**

Bernoulli's Equation also yields a *theoretical expression* for the velocity, at which a fluid
escapes through the lower section of an open vessel (Fig. 238),
given the constant *height* (*z - z*) and the *pressure difference* at the cross-sections. If the velocity
at *A* is so small that it can be neglected compared with
that at *B*, it is assumed that the *pressure* at the lower section is equal to that
at the upper section and, if the vertical distance between the
sections is *h*, then

*r**gh - **r**v*^{2}·/2 = 0, whence *v*
*= *(2gh)^{1/2},

a formula which Torricelli discovered in 1641. The fluid escapes
down below as if it had *fallen freely* from the height *h
*above. The velocity depends only on the height *h*,
and not on the direction in which the jet escapes. - If *p*
and *p*_{0}* *are not *equal*, that *p - p*_{0} is an
excess pressure, acting on the free surface, and *v*_{0}
if the mean velocity of the free surface, that is

*v = *{*v²*_{0}*
+ *2[*gh *+ (*p - p*_{0})/2}^{1/2}.

A jet which
escapes from a circular opening (the form of it influences that
of the jet) is behind the smallest cross-section like a solid,
transparent cylinder, further on it becomes turbid and displays
now and then nodes and bulges. These *nodes *and
*bulges *(Felix
Savart 1791-1841) can be
shown to be an optical illusion: The jet does not form there a
contiguous structure, it decomposes as a result of the ever
increasing velocity into drops, which change during the fall
their shape and oscillate about a spherical one (Fig. 239). By
eye, you cannot distinguish the drops, which follow rapidly one
another, and you *see* nodes and bulges.

The *independence* of the escape velocity of the direction
suggests experiments, which establish the formula *v* *=
*(2gh)^{1/2}. You let the fluid escape horizontally
from a vessel (Fig. 240). With water columns of 2 - 6 m height,
the *observed *escape velocities have been found to be
very close to their calculated values, but *always* somewhat *smaller *due
to fluid friction at the wall of the vessel and the air, and due
to *internal friction* of the fluid.

According to the formula *v* *=
*(2gh)^{1/2}, the velocity *v *depends only on
*h*, all fluids *should
escape* from an opening
equally fast, as long as *h* is constant (like all bodies
fall equally fast and reach in the course of time the same
velocity). However, this does not apply to fluids. In fact, we
have silently assumed that the work, performed by gravity on the
escaping fluid, converts *completely* into kinetic energy, that is, we have
not taken into account that the fluid particles undergo *friction*, one with another, and along the wall
of the vessel, whence their motion is retarded. The more viscous
a fluid, the larger is the impediment and the larger must be *h*,
so that *v* reaches the same value (the smaller is *v*
as *h* stays constant). Hence, the *actual escape velocity *differs the more from the theoretical
value, the more viscous is the fluid, for example, more for
castor oil than for alcohol.

If you *measure
*in grams the amount of fluid , which
escapes from a vessel during 1 second, and compute how much
should escape, corresponding to the formula *v* *= *(2gh)^{1/2},
the size of the opening and the specific weight, you will find in
the mean that only *two
thirds* actually
escape, because during the computation you have assumed that the
opening *a b *and the jet (Fig. 241) have everywhere the
same cross-section. Actually, the jet *contracts *near the opening; Its smallest
cross-section at *c d* is only 0.6 of that at the opening
(*vena contracta*). In fact, the fluid flows from all
sides to the opening, as is readily confirmed experimentally. The
individual stream tubes only become parallel at a certain
distance from the opening.

*Surface tension* has here a remarkable role. At the opening, the outer
layer of fluid particles acts like an *elastic ring *around the jet. For example, the
influence of the surface tension manifests itself, if you
vaporize alcohol or ether near the jet . The escape velocity will
then* increase* noticeably, because the surface tension
is reduced.

A modification of the outlet (Fig. 242) changes *considerably* the shape of a jet and the quantity discharged; a short cylinder stops the jet from
contraction and makes the discharge quantity close to that
computed. A long tube has a greater effect.
Torricelli's formula than *fails* completely and you must rely on experiments. Beyond
all, frictional resistance becomes important: It increases with the
surface of the tube and the velocity of escape. One *part* of
the pressure, exerted on the fluid, is then spent on overcoming
this resistance. Hence the escape velocity becomes smaller than
what it would have been, if the *entire *pressure had
generated velocity. For example, if water escapes from a vessel
through a long tube, it escapes with a much smaller velocity than
would correspond to the pressure height; you notice that from the
parabola which the fluid describes on leaving the tube.

The pressure acting on moving water at a given
location in a tube is the smaller the further it is away from the
vessel. You see this when you install along the tube *a b *thin,
vertical tubes (Fig. 243). As long as the tube is closed at *b*,
the water in these tubes has the same height - namely that in the
vessel itself; as soon as you open *b*, you obtain the
shown distribution.

**Escape of a gas under pressure**

As long as in a space, filled with gas,
the pressure everywhere is the same, there is no *flow*. According to the *Kinetic Theory*, there exists a state of equilibrium,
although not one of rest: Through a plane, which you imagine
having been placed through the space, pass each second as many
particles from *A* to *B* (Fig. 244) as from *B*
to* A*, so that the particles do not form anywhere *clusters* at the expense of another location: The
*centre of gravity* of all particles remains at rest. This
prevails as long as the pressure throughout the space is *equal*. However, if for some reason or other
the pressure in *A* becomes larger than in *B, *a
partition (in place of the imagined plane) experiences an *excess pressure*. If you puncture it, the gas flows from
*A* to *B* until the pressure on both sides is the
same. Hence the content of gas in *B* is increased at the
expense of *A*, that is, more gas particles pass from *A*
to *B *than from* B* to *A*: The centre of
gravity of the gas is also shifted from *A* towards *B*.

The velocity *v*, at
which a gas *escapes* from a space, is measured by the
volume, which escapes in a given time interval from an opening of
given size; *v* depends on the density *r* of the gas and the excess pressure *p
- p*_{0}. This equation only applies while *p - p*_{0}
is very small and the opening, through which the gas escapes, is
very small compared with the size of the space, containing the
gas. Nevertheless, the formula is valuable: The
relationship between the *escape
velocity* *v*
and the *gas density* *r *allows
you to measure the density without having to know the escape
velocity.

**Measurement of the density of
gases through their escape velocity (****Bunsen****)**

If *v*,* v*_{1}*
*and *s*, *s*_{1}* *are, respectively, the velocities and
densities of two gases which escape from vessels under the excess
pressure *p - p*_{0} = *P*, then

*v **= *Ö 2·*P/**s *and *v*_{1}*= *Ö 2·*P/**s*_{1},
whence *v*^{2}*/v*_{1}^{2}*
= **s*_{1}*/**s**,*

that is, the gas densities are *inversely proportional* to the *squares
of the escape velocities*. For example,
if *v*_{1 }*= *2*v,* then *s*_{1 }=
*s*/4, whence the gas which escapes twice as fast as
another gas (under the same conditions) has one quarter of the
specific weight of the first gas.

Thomas Graham 1805-1869 has derived this result experimentally and Bunsen has based on it a method for the *measurement of the density* of gases; however, he does not measure (by means of a
special apparatus) the escape velocity, but the *time*,
which is required by equal quantities of both gases for their
escape from a vessel. Since the gas with double the density
requires only half the time as the other gas, you have the
result: The densities of the gases are related one to another as
the *squares of the time
intervals **t* and *t*_{1},
during which equal amounts of gas escape:

*s*_{1}*/**s** = t*^{2}_{1}/*t*^{2}*.*

For example, if a certain
amount of air (density *s*) requires *t = *36.9 sec and the
same amount of carbolic acid (density *s*_{1}) requires *t*_{1}= 45.3
sec, then

*s*_{1}*/**s** = *(45.3/36.9)^{2}
= 1.507,

that is, the density of carbolic
acid *s*_{1 }is 1.507 that of air. The method is *reliable*
and requires only a few cm³ of gas, an advantage which is
valuable for gas measurements. It is especially convenient for *industrial purposes* (measurement of domestic gas).

**Diffusion of gas through porous
bodies**

The velocity of
escape of a gas from a vessel depends also greatly on the *size of the opening*. The conditions, under which the
formula above applies, also include that the opening is in very *thin wall*, that is, the gas escapes only though
an opening in a very thin wall, so that gas really escapes
through an *opening*, not through a channel. (In Bunsen's
apparatus, it escapes through a hole in a thin platinum foil.)

If a gas flows
through a capillary, the formula does not apply; you have a *new law* (O.E.Meyer 1834-1909), if the length of the
capillary is approximately 4000 times its diameter. If part of
the wall is a porous plate (graphite, unglazed clay), so that the
gas can escape through a large number of capillary channels, it
diffuses through the plate. Through a porous subdivision between
two gases (Fig. 244 above), the gases diffuse into each other *without there having to be a
pressure difference*.

**Graham****'s Law of diffusion rate**

The rate at which
a gas passes through a porous plate is, other things being equal, *inversely proportional to the
density *(Thomas Graham); for example, for oxygen, which is 16
times as dense as hydrogen, it is only one quarter of the
velocity of hydrogen.

You can demonstrate the difference between the diffusion
rates of *air* and *hydrogen** *by means of an unglazed clay cell (Fig. 245):
You conduct the hydrogen through an air tight fitted pipe and the
clay cell until all the air has been ejected; you then close the
tap. The clay wall borders inside against hydrogen, outside
against air; the hydrogen diffuses much faster than the air,
whence the *pressure
sinks* inside, as is shown by the
rising of the fluid.

The difference between diffusion rates can also be employed to *separate* from each other two* mixed *gases with
different densities. Thus, you can reduce the content of

nitrogen in atmospheric air - a mixture of 21%
oxygen and 79% nitrogen, the
densities of which are interrelated like 16:14, - by conducting
the air through an *unglazed
clay tube* through a vacuum (Fig. 246).
This method, changed and improved according to the the purpose it
is to serve, enabled *separation
of the isotopes* of neon (Gustav Hertz
1932).

The mercury-vapour-pump of Wolfgang Gaede
1878-1945 1913 employs diffusion and is the protoype of the pumps of modern high *vacuum
techniques*. The air to be removed and
the mercury vapour diffuse into each other and the mercury vapour
entering into the air is condensed rapidly in a vacuum chamber.

The principle of the pump is shown in Fig. 247:
*C *is a porous clay wall; on its left hand side is the *mercury vapour* entering through the tube *A B, *on its right
hand side the *air *which communicates through the tube *E* with the
recipient. To start with, the pressure in the space is reduced
with the aid of an auxiliary pump to 0.1 mm mercury column
(pre-vacuum). The air diffuses from the right hand side to the
left hand side through *C *and flows away in the vapour
jet *A B, *the mercury vapour diffuses from the left hand
side to the right hand side and condenses immediately, because
the gas trap *D* is submerged in *liquid air*.
This is the basic idea.

In the pump (Fig.
248), Gaede employs instead of a clay wall with its pores a *steel pipe*
*C*, through which the mercury valour passes and which has
a slit *S **as
wide as the pores** *- that is,
approximately the* **mean
path length *of the molecules. Through
the slit, air diffuses from the outside inwards (into the vapour
stream) - it flows away with the vapour - and mercury vapour from
the inside outwards (into the *vacuum space*); the
refrigeration section *K*_{1}*K*_{2}*
*condenses it immediately. The pressure in the recipient
becomes eventually 10^{-6} to 10^{-8} mm mercury
(Gaede's
rotating pump 7.10^{-5}); theoretically, there exists no
lower bound for the possible vacuum. The highest suction velocity
of the pump is 80 cm/sec (rotating pump 130) and it remains
constant also at the lowest pressures. It is substantially higher
in the mercury-vapour-jet- condensation-pump.