# GasLab Heat Box ### 1 collaborator 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 is illustrates the relationship between temperature and pressure in
a fixed volume gas container.

This model is part of the Connected Mathematics "Making Sense of Complex Phenomena" Modeling 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.

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. In this model, two turtles are aimed so that they will collide at the origin.
3. An angle of collision for the particles is chosen, as if they were two solid balls that hit, and this angle describes the direction of the line connecting their centers.
4. The particles exchange momentum and energy only along this line, conforming to the conservation of momentum and energy for elastic collisions.
5. Each particle is assigned its new speed, heading and energy.
6. If a particle finds itself on or very close to a wall of the container, it "bounces" --- that is, reflects its direction and keeps its same speed.

As the walls of the box are heated, the sides of the walls will change color from a deep red (cool) to a bright red, to pink to a pale pink white (hot). The walls contain a constant heat value throughout the simulation. If ONE-SIDE? is set to ON, only the left wall will be heated, while the other three walls remain yellow.

The exact way particles gain energy from the walls of the box is as follows:

1. Particles check their state of energy.
2. They hit or bounce off the wall.
3. They find wall energy and recalculate their new energy.
4. They change their speed and direction after the wall hit.

## HOW TO USE IT

### Initial settings

• NUMBER-OF-PARTICLES: number of particles within in the box

### Buttons

The SETUP button will set these initial conditions.
The GO button will begin the simulation.

### Other Settings

• OUTSIDE TEMPERATURE: temperature of the outside of the box and the wall of the box.
• ONE SIDE?: heats only the left wall if enabled. the other walls are colored yellow, and do not affect the energy of the particles that bounce into it.
• COLLIDE?: Turns collisions between particles on and off.

### Monitors

• PRESSURE: the pressure of the gas particles in the box
• WALL HITS PER PARTICLE: number of times that each particle hit the walls
• AVERAGE SPEED: average speed of the particles.
• AVERAGE ENERGY: average kinetic energy of the particles.

### Plots

• SPEED COUNTS: plots the number of particles in each range of speed.
• SPEED HISTOGRAM: speed distribution of all the particles. The gray line is the average value, and the black line is the initial average.
• ENERGY HISTOGRAM: distribution of energies of all the particles, calculated as m*(v^2)/2.
• PRESSURE VS. TIME: plots average pressure of the inside of the box over time.
• TEMPERATURE VS. TIME: plots particle temperature inside the box over time and wall temperature over time.
• WALL HITS PER PARTICLE: plots average wall hits per particle over time.

## THINGS TO NOTICE

How does adding heat to the box walls affect the pressure?

How does adding heat to the wall affect the particle behavior?

How does the particle behavior or system response change with only one wall heated instead of all walls heated?

Does the system reach an equilibrium temperature faster when the wall is heated or cooled the same amount in comparison to the temperature of the particles?

## THINGS TO TRY

Try to get the inside temperature to reach the outside temperature. Is this possible?

Try to increase the wall hits per particle.

## EXTENDING THE MODEL

Give the wall a mass and see how that affects the behavior of the model.

Close off the right side of the box. Create two valves on either side to the wall that allow the user to "spurt" particles into the chambers to see how number of particles affects pressure.

Vary the width and length of the box, does this effect how fast the particle temperature changes?

## NETLOGO FEATURES

Notice how the collisions are detected by the turtles and how the code guarantees the same two particles do not collide twice. What happens if we let the patches detect them?

## 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:

• Wilensky, U. (2002). NetLogo GasLab Heat Box model. http://ccl.northwestern.edu/netlogo/models/GasLabHeatBox. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.
• Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.

Copyright 2002 Uri Wilensky. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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.

## Comments and Questions

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
box-edge                        ;; distance of box edge from axes
pressure                        ;; pressure in the box
pressure-history
zero-pressure-count             ;; how many zero entries are in 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
walls                           ;; agentset containing patches that are the walls of the box
init-avg-speed init-avg-energy  ;; initial averages
avg-speed avg-energy            ;; current averages
fast medium slow                ;; current counts
percent-slow percent-medium percent-fast   ;;  percentage of current counts
outside-energy
]

breed [ particles particle ]
breed [ flashes flash ]

flashes-own [birthday]

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

to setup
clear-all
set-default-shape particles "circle"
set-default-shape flashes "plane"
set max-tick-delta 0.1073
;; box has constant size...
set box-edge (max-pxcor - 1)
;;; 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
set zero-pressure-count 0
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 percent-medium (medium / (count particles)) * 100
set percent-slow (slow / (count particles)) * 100
set percent-fast (fast / (count particles)) * 100
set avg-speed  mean [speed] of particles
set avg-energy mean [energy] of particles
end

to go
[set pcolor box-color]
ask particles [ bounce ]
ask particles [ move ]
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 ]
set outside-energy outside-temperature
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-patch patch-ahead 1
let new-px [pxcor] of new-patch
let new-py [pycor] of new-patch
;; if we're not about to hit a wall, we don't need to do any further checks
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 [heated-wall?] of patch new-px new-py  ;; check if the patch ahead of us is heated
[

set energy ((energy +  outside-energy ) / 2)
set speed sqrt (2 * energy / mass )
recolor
]

ask patch new-px new-py
[ 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)) ]
]

;; PHASE 5: final updates

;; 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

;; reports color of box according to temperature and position
;; if only one side is heated, the other walls will be yellow

to-report box-color
ifelse heated-wall?
[report scale-color red outside-temperature -200 600]
[report yellow]
end

;; reports true if there is a heated wall at the given location

to-report heated-wall?
ifelse one-side?
[
ifelse ((pxcor = (- box-edge)) and (abs pycor < box-edge))
[report true]
[report false]
]
[
if (( abs pxcor = box-edge) and (abs pycor <= box-edge)) or
((abs pycor = box-edge) and (abs pxcor <= box-edge))
[report true]
]
report false
end

;;;
;;; drawing procedures
;;;

;; draws the box

to make-box
set walls patches with [ ((abs pxcor = box-edge) and (abs pycor <= box-edge)) or
((abs pycor = box-edge) and (abs pxcor <= box-edge)) ]
[ set pcolor box-color ]
end

;; creates initial particles

to make-particles
create-particles number-of-particles
[
setup-particle
set speed random-float 20
set energy (0.5 * mass * speed * speed)
random-position
recolor
]
calculate-tick-delta
end

to setup-particle  ;; particle procedure
set speed 10
set mass 1.0
set energy (0.5 * mass * speed * speed)
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

;;; plotting procedures

to-report init-particle-speed
;;the initial speeds in this model cannot be faster than 20 at setup
report 20
end

to-report particle-mass
;; each particle has a mass of 1
report 1
end

;; histogram procedure

to draw-vert-line [ xval ]
plotxy xval plot-y-min
plot-pen-down
plotxy xval plot-y-max
plot-pen-up
end

; Copyright 2002 Uri Wilensky.
; See Info tab for full copyright and license.
```

There are 15 versions of this model.

Uri Wilensky over 9 years ago Updated to NetLogo 5.0.4 Download this version
Uri Wilensky almost 10 years ago Updated version tag Download this version
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Uri Wilensky over 10 years ago Updated to NetLogo 5.0 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 Heat Box Download this version

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