GasLab Pressure Box

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Uri Wilensky (Author)

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WHAT IS IT?

This model is one in a series of GasLab models. They use the same basic rules for simulating the behavior of gases. Each model integrates different features in order to highlight different aspects of gas behavior.

The basic principle of the models is that gas particles are assumed to have two elementary actions: they move and they collide - either with other particles or with any other objects such as walls.

This model simulates the behavior of gas particles trapped in a container with a fixed volume, such as inflating a bike tire. The number of particles in the box can be changed by adding particles through a valve on the side.

This model is part of the Connected Chemistry curriculum, which is part of the Modeling Across the Curriculum (MAC) Project.

HOW IT WORKS

The particles are modeled as hard balls with no internal energy except that which is due to their motion. Collisions between particles are elastic. Particles are colored according to speed --- blue for slow, green for medium, and red for high speeds.

Coloring of the particles is with respect to one speed (10). Particles with a speed less than 5 are blue, ones that are more than 15 are red, while all in those in-between are green.

Particles behave according to the following rules:

1. A particle moves in a straight line without changing its speed, unless it collides with another particle or bounces off the wall.
2. Two particles "collide" if they find themselves on the same patch (the world is composed of a grid of small squares called patches).
3. A random axis is chosen, as if they are two balls that hit each other and this axis is the line connecting their centers.
4. They exchange momentum and energy along that axis, according to the conservation of momentum and energy. This calculation is done in the center of mass system.
5. Each turtle is assigned its new velocity, energy, and heading.
6. If a turtle finds itself on or very close to a wall of the container, it "bounces" -- that is, reflects its direction and keeps its same speed.

Pressure is defined as the force per unit area (or length in this two-dimensional model). Pressure is calculated by adding up the momentum transferred to the walls of the box by the particles when they bounce off and divided by the length of the wall, which they hit.

Initially, the particles are not moving, and cannot hit the wall. Therefore the initial pressure is zero. As the particles start moving, they all first have the same speed in random directions. As the particles repeatedly collide, they exchange energy and head off in new directions, and the speeds are dispersed --- some particles get faster, some get slower.

HOW TO USE IT

Initial settings

• BOX-SIZE: The size of the box as a percentage of the world-width
• INITIAL-NUMBER-PARTICLES: the number of gas particles in the box when the simulation starts

The SETUP button puts in the initial conditions you have set with the sliders. Be sure to wait until the SETUP button stops before pushing GO.

The GO button runs the code again and again. This is a "forever" button.

COLLIDE?: determines whether or not the particles collide among themselves
ADD PARTICLES: pressing this button adds particles to the container through a valve, before and while the model is running
NUMBER-TO-ADD: number of particles that are added to the box with each press of the "ADD PARTICLES" button

Monitors

• NUMBER OF PARTICLES: the number of particles in the box
• PRESSURE: the pressure of the gas particles in the box
• WALL HITS PER PARTICLE:
• AVERAGE SPEED: average speed of the particles.
• AVERAGE ENERGY: average kinetic energy of the particles.

Plots

• PRESSURE: plots the pressure in the box
• PARTICLE COUNT: plots the number of particles in the box
• WALL HITS PER PARTICLE: plots the number of times a particle hits a wall every big tick

THINGS TO NOTICE

What is happening to the particles in the box as a new group of particles is injected? How is this related to the way pressure changes over time?

Can you observe collisions with the walls as they happen (you can pendown a particle or slow down the model)? For example, do the particles change their color? Direction?

Can you relate what you can see happening to the particles in the box with changes in pressure? The average speed? The average energy?

Why does the pressure fluctuate, even though the number of particles is the same? How long does it take for the pressure to stabilize? Does the number of particles in the box affect this time?

In what ways is this model an incorrect idealization of the real world?

THINGS TO TRY

Try different settings, especially the extremes. Are the particles behaving in a similar way? How does this effect the pressure? The energy?

How are the number of particles and the pressure in the box related to each other? Can you quantify this relationship? Does it change if collisions among particles are removed?

Does the same relationship hold for the number of wall hits per particle and the number of particles? Why? How are the two relationships connected?

Can you make the pressure graph smooth? Can you do it in more than one way?

The ideal gas equation shows that when keeping a constant temperature and volume, increasing the number of particles increases the pressure as well. Keeping a constant temperature means that the average kinetic energy of the particles, does not change, nor does their average speed. How is this related to the behavior of particles that you have seen so far? Are there any other observations that could be made to test this relationship between the number of particles and the pressure?

Notice that the pressure does not go up immediately after particles are added. What is the reason for this delay? Try adding particles into the box in different ways to increase or decrease this delay. Can you think of real-life situations, with a similar delay?

Sometimes, when going up in an elevator, airplane or up a mountain we feel a 'popping' sensation in our ears. This is associated with changes in pressure. Can you relate between this model and these changes in pressure? Are the temperature and volume constant in this situation?

EXTENDING THE MODEL

What would happen if particles were added continuously rather than instantly? Add a switch that allows for particles to continuously be added to the box.

Can you "puncture" the box, so that particles will escape?

What would happen if the box were heated? How would the particles behave? How would this affect the pressure? Add a slider and code that increases the temperature inside the box.

If you could change the shape of the box, so that the volume remains the same: Does the shape of the box make a difference in the way the particles behave, or the values of pressure?

CREDITS AND REFERENCES

This model was developed as part of the GasLab curriculum (http://ccl.northwestern.edu/curriculum/gaslab/) and has also been incorporated into the Connected Chemistry curriculum (http://ccl.northwestern.edu/curriculum/ConnectedChemistry/)

HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

This model and associated activities and materials were created as part of the project: MODELING ACROSS THE CURRICULUM. The project gratefully acknowledges the support of the National Science Foundation, the National Institute of Health, and the Department of Education (IERI program) -- grant number REC #0115699. Additional support was provided through the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT -- NSF (REPP & ROLE programs) grant numbers REC #9814682 and REC-0126227.

Click to Run Model

```globals
[
tick-delta                         ;; how much we advance the tick counter this time through
max-tick-delta                     ;; the largest tick-delta is allowed to be
init-particle-speed particle-mass  ;; default particle settings
box-edge                           ;; distance of box edge from axes
pressure
pressure-history
wall-hits-per-particle     ;; average number of wall hits per particle
length-horizontal-surface  ;; the size of the wall surfaces that run horizontally - the top and bottom of the box
length-vertical-surface    ;; the size of the wall surfaces that run vertically - the left and right of the box

init-avg-speed init-avg-energy  ;; initial averages
avg-speed avg-energy            ;; current averages
fast medium slow                ;; current counts

]

breed [ particles particle ]
breed [ flashes flash ]

flashes-own [birthday]

particles-own
[
speed mass energy          ;; particle info
wall-hits                  ;; # of wall hits during this cycle ("big tick")
momentum-difference        ;; used to calculate pressure from wall hits
last-collision
]

to setup
clear-all
set-default-shape particles "circle"
set-default-shape flashes "plane"
set init-particle-speed 10.0
set particle-mass 1.0
set max-tick-delta 0.1073
;; box size is determined by the slider
set box-edge (round (max-pxcor * box-size / 100))
;;; the length of the horizontal or vertical surface of
;;; the inside of the box must exclude the two patches
;; that are the where the perpendicular walls join it,
;;; but must also add in the axes as an additional patch
;;; example:  a box with an box-edge of 10, is drawn with
;;; 19 patches of wall space on the inside of the box
set length-horizontal-surface  ( 2 * (box-edge - 1) + 1)
set length-vertical-surface  ( 2 * (box-edge - 1) + 1)
make-box
make-particles
set pressure-history [0 0 0]  ;; plotted pressure will be averaged over the past 3 entries
update-variables
set init-avg-speed avg-speed
set init-avg-energy avg-energy
reset-ticks
end

to update-variables
set medium count particles with [color = green]
set slow   count particles with [color = blue]
set fast   count particles with [color = red]
set avg-speed  mean [speed] of particles
set avg-energy mean [energy] of particles
end

to go
[ if collide? [check-for-collision] ]
if floor ticks > floor (ticks - tick-delta)
[
ifelse any? particles
[ set wall-hits-per-particle mean [wall-hits] of particles ]
[ set wall-hits-per-particle 0 ]
[ set wall-hits 0 ]
calculate-pressure
update-variables
update-plots
]
calculate-tick-delta
ask flashes with [ticks - birthday > 0.4]
[ die ]
display
end

to calculate-tick-delta
;; tick-delta is calculated in such way that even the fastest
;; particle will jump at most 1 patch length in a tick. As
;; particles jump (speed * tick-delta) at every tick, making
;; tick length the inverse of the speed of the fastest particle
;; (1/max speed) assures that. Having each particle advance at most
;; one patch-length is necessary for it not to "jump over" a wall
;; or another particle.
ifelse any? particles with [speed > 0]
[ set tick-delta min list (1 / (ceiling max [speed] of particles)) max-tick-delta ]
[ set tick-delta max-tick-delta ]
end

;;; Pressure is defined as the force per unit area.  In this context,
;;; that means the total momentum per unit time transferred to the walls
;;; by particle hits, divided by the surface area of the walls.  (Here
;;; we're in a two dimensional world, so the "surface area" of the walls
;;; is just their length.)  Each wall contributes a different amount
;;; to the total pressure in the box, based on the number of collisions, the
;;; direction of each collision, and the length of the wall.  Conservation of momentum
;;; in hits ensures that the difference in momentum for the particles is equal to and
;;; opposite to that for the wall.  The force on each wall is the rate of change in
;;; momentum imparted to the wall, or the sum of change in momentum for each particle:
;;; F = SUM  [d(mv)/dt] = SUM [m(dv/dt)] = SUM [ ma ], in a direction perpendicular to
;;; the wall surface.  The pressure (P) on a given wall is the force (F) applied to that
;;; wall over its surface area.  The total pressure in the box is sum of each wall's
;;; pressure contribution.

to calculate-pressure
;; by summing the momentum change for each particle,
;; the wall's total momentum change is calculated
set pressure 15 * sum [momentum-difference] of particles
set pressure-history lput pressure but-first pressure-history
[ set momentum-difference 0 ]  ;; once the contribution to momentum has been calculated
;; this value is reset to zero till the next wall hit
end

to bounce  ;; particle procedure
;; get the coordinates of the patch we'll be on if we go forward 1
let new-px [pxcor] of new-patch
let new-py [pycor] of new-patch

;; if we're not about to hit the edge of the wall, we don't need to do any further checks
;; note: we check the new coordinates, and not the color, to make sure particles
;; do not go into the nozzle
if (abs new-px != box-edge and abs new-py != box-edge)
[ stop ]
;; if hitting left or right wall, reflect heading around x axis
if (abs new-px = box-edge)
set wall-hits wall-hits + 1
;;  if the particle is hitting a vertical wall, only the horizontal component of the speed
;;  vector can change.  The change in velocity for this component is 2 * the speed of the particle,
;;  due to the reversing of direction of travel from the collision with the wall
set momentum-difference momentum-difference + (abs (dx * 2 * mass * speed) / length-vertical-surface) ]
;; if hitting top or bottom wall, reflect heading around y axis
if (abs new-py = box-edge)
set wall-hits wall-hits + 1
;;  if the particle is hitting a horizontal wall, only the vertical component of the speed
;;  vector can change.  The change in velocity for this component is 2 * the speed of the particle,
;;  due to the reversing of direction of travel from the collision with the wall
set momentum-difference momentum-difference + (abs (dy * 2 * mass * speed) / length-horizontal-surface)  ]

;; if we are at the entrance of the nozzle, do not make a flash
if (new-py = 0 and new-px = (- box-edge))
[stop]

[ sprout-flashes 1 [
set color pcolor - 2
set birthday ticks
]
]
end

to move  ;; particle procedure
if patch-ahead (speed * tick-delta) != patch-here
[ set last-collision nobody ]
jump (speed * tick-delta)
end

to check-for-collision  ;; particle procedure
;; Here we impose a rule that collisions only take place when there
;; are exactly two particles per patch.  We do this because when the
;; student introduces new particles from the side, we want them to
;; form a uniform wavefront.
;;
;; Why do we want a uniform wavefront?  Because it is actually more
;; realistic.  (And also because the curriculum uses the uniform
;; wavefront to help teach the relationship between particle collisions,
;; wall hits, and pressure.)
;;
;; Why is it realistic to assume a uniform wavefront?  Because in reality,
;; whether a collision takes place would depend on the actual headings
;; of the particles, not merely on their proximity.  Since the particles
;; in the wavefront have identical speeds and near-identical headings,
;; in reality they would not collide.  So even though the two-particles
;; rule is not itself realistic, it produces a realistic result.  Also,
;; unless the number of particles is extremely large, it is very rare
;; for three or more particles to land on the same patch (for example,
;; with 400 particles it happens less than 1% of the time).  So imposing
;; this additional rule should have only a negligible effect on the
;; aggregate behavior of the system.
;;
;; Why does this rule produce a uniform wavefront?  The particles all
;; start out on the same patch, which means that without the only-two
;; rule, they would all start colliding with each other immediately,
;; resulting in much random variation of speeds and headings.  With
;; the only-two rule, they are prevented from colliding with each other
;; until they have spread out a lot.  (And in fact, if you observe
;; the wavefront closely, you will see that it is not completely smooth,
;; because some collisions eventually do start occurring when it thins out while fanning.)

if count other particles-here = 1
[
;; the following conditions are imposed on collision candidates:
;;   1. they must have a lower who number than my own, because collision
;;      code is asymmetrical: it must always happen from the point of view
;;      of just one particle.
;;   2. they must not be the same particle that we last collided with on
;;      this patch, so that we have a chance to leave the patch after we've
;;      collided with someone.
let candidate one-of other particles-here with
[who < [who] of myself and myself != last-collision]
;; we also only collide if one of us has non-zero speed. It's useless
;; (and incorrect, actually) for two particles with zero speed to collide.
if (candidate != nobody) and (speed > 0 or [speed] of candidate > 0)
[
collide-with candidate
set last-collision candidate
ask candidate [ set last-collision myself ]
]
]
end

;; implements a collision with another particle.
;;
;; THIS IS THE HEART OF THE PARTICLE SIMULATION, AND YOU ARE STRONGLY ADVISED
;; NOT TO CHANGE IT UNLESS YOU REALLY UNDERSTAND WHAT YOU'RE DOING!
;;
;; The two particles colliding are self and other-particle, and while the
;; collision is performed from the point of view of self, both particles are
;; modified to reflect its effects. This is somewhat complicated, so I'll
;; give a general outline here:
;;   1. Do initial setup, and determine the heading between particle centers
;;      (call it theta).
;;   2. Convert the representation of the velocity of each particle from
;;      speed/heading to a theta-based vector whose first component is the
;;      particle's speed along theta, and whose second component is the speed
;;      perpendicular to theta.
;;   3. Modify the velocity vectors to reflect the effects of the collision.
;;      This involves:
;;        a. computing the velocity of the center of mass of the whole system
;;           along direction theta
;;        b. updating the along-theta components of the two velocity vectors.
;;   4. Convert from the theta-based vector representation of velocity back to
;;      the usual speed/heading representation for each particle.
;;   5. Perform final cleanup and update derived quantities.

to collide-with [ other-particle ] ;; particle procedure
;;; PHASE 1: initial setup

;; for convenience, grab some quantities from other-particle
let mass2 [mass] of other-particle
let speed2 [speed] of other-particle

;; since particles are modeled as zero-size points, theta isn't meaningfully
;; defined. we can assign it randomly without affecting the model's outcome.
let theta (random-float 360)

;;; PHASE 2: convert velocities to theta-based vector representation

;; now convert my velocity from speed/heading representation to components
;; along theta and perpendicular to theta
let v1t (speed * cos (theta - heading))
let v1l (speed * sin (theta - heading))

;; do the same for other-particle
let v2t (speed2 * cos (theta - heading2))
let v2l (speed2 * sin (theta - heading2))

;;; PHASE 3: manipulate vectors to implement collision

;; compute the velocity of the system's center of mass along theta
let vcm (((mass * v1t) + (mass2 * v2t)) / (mass + mass2) )

;; now compute the new velocity for each particle along direction theta.
;; velocity perpendicular to theta is unaffected by a collision along theta,
;; so the next two lines actually implement the collision itself, in the
;; sense that the effects of the collision are exactly the following changes
;; in particle velocity.
set v1t (2 * vcm - v1t)
set v2t (2 * vcm - v2t)

;;; PHASE 4: convert back to normal speed/heading

;; now convert my velocity vector into my new speed and heading
set speed sqrt ((v1t ^ 2) + (v1l ^ 2))
set energy (0.5 * mass * speed ^ 2)
;; if the magnitude of the velocity vector is 0, atan is undefined. but
;; speed will be 0, so heading is irrelevant anyway. therefore, in that
;; case we'll just leave it unmodified.
if v1l != 0 or v1t != 0
[ set heading (theta - (atan v1l v1t)) ]

;; and do the same for other-particle
set speed sqrt ((v2t ^ 2) + (v2l ^ 2))
set energy (0.5 * mass * (speed ^ 2))
if v2l != 0 or v2t != 0
[ set heading (theta - (atan v2l v2t)) ]
]

;; now recolor, since color is based on quantities that may have changed
recolor
[ recolor ]
end

to recolor  ;; particle procedure
ifelse speed < (0.5 * 10)
[
set color blue
]
[
ifelse speed > (1.5 * 10)
[ set color red ]
[ set color green ]
]
end

;;;
;;; drawing procedures
;;;

;; draws the box

to make-box
ask patches with [ ((abs pxcor = box-edge) and (abs pycor <= box-edge)) or
((abs pycor = box-edge) and (abs pxcor <= box-edge)) ]
[ set pcolor yellow ]
;; make nozzle
ask patches with [ pycor = 0 and pxcor < (1 - box-edge) ]
[
set pcolor yellow - 5  ;; trick the bounce code so particles don't go into the inlet.
ask patch-at 0  1 [ set pcolor yellow ]
ask patch-at 0 -1 [ set pcolor yellow ]
]
end

;; creates initial particles

to make-particles
create-particles initial-number-of-particles
[
setup-particle
random-position
recolor
]
calculate-tick-delta
end

;; adds new particles into the box from the left nozzle

if n > 0
[ create-particles n
[ setup-particle
setxy (- box-edge) 0
rt 45 - random-float 90
recolor
]
calculate-tick-delta
]
end

to setup-particle  ;; particle procedure
set speed init-particle-speed
set mass particle-mass
set energy (0.5 * mass * speed ^ 2)
set last-collision nobody
set wall-hits 0
set momentum-difference 0
end

;; place particle at random location inside the box.

to random-position ;; particle procedure
setxy ((1 - box-edge) + random-float ((2 * box-edge) - 2))
((1 - box-edge) + random-float ((2 * box-edge) - 2))
end

```

There are 15 versions of this model.

Uri Wilensky over 9 years ago Updated to NetLogo 5.0.4 Download this version
Uri Wilensky over 10 years ago Updated to version from NetLogo 5.0.3 distribution Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Model from NetLogo distribution Download this version
Uri Wilensky over 12 years ago Model from NetLogo distribution Download this version
Uri Wilensky over 12 years ago GasLab Pressure Box Download this version

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