text:
## Thursday, December 25, 2014

### Single Rotation rule with frame interpolation

TL;DR: See the online simulator of the Single Rotation cellular automaton, with interpolation of individual cell movement. frames. Its primary purpose is a nice visual effect, for experiments with reversible cellular automata see online simulator, described in the previous blog post.

In case it is not working for you (Since it is HTML5 canvas application, I did not pay much attention to the old browser support), here is a demonstration video.

Single Rotation rule is a very simple reversible cellular automaton that acts on a grid of cells, that are either dead (0) or alive (1). On each generation, the grid is divided into 2x2 blocks, and if the block contains exactly one alive cell, it is rotated by 90°. Thus "single rotation".

## Cell tracking

Single Rotation rule preserves total number of the alive cells. Moreover, from the definition of the rule it is clear that it is possible to track the position of every individual alive cell (something that is impossible in the Conway's Game of Life, where cells born and die but not move). The simulator implements this idea.

## Interpolation

On each generation, alive cells always move by 1 grid unit in some direction, making animation jerky. To make it smoother, this simulator interpolates intermediate coordinates of the cells, using Lanczos resampling.

## Oscillation filtering

A single cell in the "Single Rotation" rule would constantly rotate with period 4. To make animation less annoying, a possibility to filter higher frequencies of the cell movement is added: "filter" setting. For example, when "filter" setting is 4, every oscillation with period 4 or less is replaced by the constant average value, and rotating cells visually stand still. Higher filter setting allow to cease oscillations of higher period. This is also done by Lanczos filter. Note that computational complexity is proportional to the the filter setting.

Here are some of my other pages, where you can find information about the reversible cellular automata and Single Rotation rule.

Source code is available on Github, under the permissive MIT license. It is written in CoffeeScript.

## Sunday, November 16, 2014

### The scale of the Universe, in log-polar coordinates

Have you enjoyed the awe-inspiring interactive flash animation by, Cary and Michael Huang The Scale of the Universe? If not, then do it before reading. I enjoyed it a lot, and when few days ago I've seen it again in the G+ stream, an idea appeared in my mind: make a log-polar image of it.

Logarithmic polar coordinate system is a natural way to show things of different scale on the same diagram. Previously I've already used it to make a photographic image of the Earth: Logarithmic zoom at the Palace square, Saint Petersburg, and played with Google Maps: Log-polar system applied to Google Maps, so I already had some python scripts to do the thing. So I took the Python, and after a night of scripting and Gimp'ing got this! (Linked image is 800x18000 pixels)

In case you would like better resolution, here is the full 2000x48000 version.

## Monday, November 10, 2014

### Crystal growing: Rochelle salt

Few more nice crystals grew today, this time it is Rochelle salt, or sodium potassium tartrate tetrahydrate: $$NaK C_4H_6O_6 \cdot 4H_2O.$$

This compound is very well known among crystal growers because of its availability and ability to form big and clear crystals easily, so I won't say much about the growing process. The usual method with slow evaporation was used, with seed crystal suspended on a thin thread. The crystals on the photo grew only in 3 weeks.

### Preparation

Rochelle salt can be prepared from the cream of tartar (unfortunately, it is not sold in my country, so I ordered some on eBay), and baking soda. Some instructions on preparation can be found on the Dom's crystal growing page (UPD: the site is down, but web archive have it), though I've simplified that recipe by using baking soda directly, without the heating step. Simply stir some cream of tartar powder in hot water and add small portions of baking soda until it stops fizzing. The reaction is: $$K H C_4H_6O_6 + NaHCO_3 \rightarrow K Na C_4H_6O_6 + H_2O + CO_2 \uparrow.$$

### Safety

This compound is not toxic and, in fact, edible (but not tasty at all). I would not recommend consuming big quantities though.

### More photos

More photos of these crystals are in the gallery Crystal growing: Rochelle salt.

### Update: piezoelectric effect

Rochelle salt is well known for its property to generate electricity, when mechanical force is applied to the crystal: piezoeffect. Actually, the voltage is so big that it can make flash a light emitting diode, directly connected to the crystal.

I took 2 pieces of aluminum foil and tightly attached them to the crystal sides with thread and adhesive tape. Wires are connected to these foil pieces with another piece of the same tape, and a pair of LEDs in opposite direction is connected to wires.

Then, in the dark room, I can see dim but visible red flashes, when the crystal is hit along the vertical axis (blue arrow on the schematic). To protect crystal from smashing, I put a small wooden piece between the crystal and the "hammer" (actually teaspoon).

So cool. Not that I ever doubted the existence of this effect, but it is really pleasure to observe such effects on a crystals, grown "from scratch", at the corner of the table.

## Thursday, October 23, 2014

### Fourier transform of the coprime numbers map

Disclaimer: the idea is not mine, I've found a low-resolution image and the idea behind it here.

### Co-prime integers

A pair of integers is called co-prime, if their greatest common denominator is 1. For example, 4 and 21 are co-prime, while 14 and 21 are not.

First let's draw a map of co-prime pairs. The image below shows co-prime pairs with white color.

The map above clearly shows some regularities, but it is actually aperiodic. It is known that its average density is $$6 / \pi^2$$, or about 61%.

### Fourier transform

But what if you take a Fourier transform of this nearly periodic pattern? The result is the following mind-blowing image.

Fourier transform discovers periodicity in the source data, so applying it to some data points with nearly periodic patterns can often produce interesting results. For example, see Fourier transform of the Hilbert curve images. Still, the 3-dimensional look of the resulting image is something totally unexpected for me. Below is the same image, but rendered for a bigger fragment of the plane, 2048x2048.

Finally, for those who like it big, here is a 4096x4096 image, 15 Mb large. Open the link and click "Download" to get the whole file.

### Extensions

The idea can be extended by building the co-prime matrix not for integers but for some general integer sequence. This gallery contains some of the spectra, obtained for different sequences: Fibonacci numbers, $$n^2+1$$, $$n^3+3$$, $$n^2-n+41$$ (Euler polynomial with many primes), partition function values and tribonacci numbers. Curious fact: the spectrum for the Fibonacci numbers is very similar visually to the original integers spectrum. But there is a difference, the image below shows two spectra superimposed, with yellow color for integers, and blue color for Fibonacci numbers.

### Source code

The images above were generated by a simple Python script below. See it here, if the gist fails to load. To run it, you will need to install Numpy and PIL (Python Imaging Library). The code itself is fairly straightforward, in fact, the most complicated part is the adaptive calculation of the brightness diapason.

## Friday, October 3, 2014

### Cellular automata on hyperbolic {5,4} tiling, strobing rules

In the previous post on hyperbolic cellular automata I presented a program for simulating them (Pentagrid). One of the slightly disappointing discoveries I made using this tool was the fact that totalistic binary automata on the {5;4} hyperbolic tiling are either "explosive" (almost every initial configuration grows with time, that's class 3 by the Wolfram's classification), or "limited", unable to support spaceships (classes 1 or 2).

That's because if the minimal number of alive neighbors required to the cell birth is 3 or more, then due to the geometry of the {5;4} tiling, a compact pattern can't grow "outside". This prohibits spaceships and possibility of infinite growth in such rules. However, if 2 alive neighbors are enough to create a new alive cell, then 2-cell pattern will grow infinitely, and the rule becomes "explosive". (However, some of the explosive rules support spaceships!)

## Strobing rules

There is one exception though: "strobing" rules, where a cell spontaneously becomes alive if it has 0 alive neighbors, and dies, if all its neighbors are alive.

In the standard B/S notation, strobing rules on the {5,4} tiling have B0 and don't have Sa (where "a" stands for 10 neighbors). Under such rules, empty field becomes completely filled in the next generation, and then it completely dies out one generation later.

In these rules, if a pattern starts with finitely many live cells on a background of dead cells, it will continue to have finitely many live cells on a background of dead cells in every even step, but in the odd steps the pattern is reversed: there are finitely many dead cells on a background of infinitely many live cells
(From the Growth and Decay in Life-Like Cellular Automata).

Naive simulation of such rules would either require making field finite or consume infinite amount of memory, but a simple trick could be used for effective simulation: consider evolution of the difference between the "flashing" background and the cells. This can be done by replacing one strobing rules with a pair of complementary stable rules without B0 part; where first rule is applied on even generations, and the second is on odd. For example rule B0157/S01 is replaced by the pair: B234689a/S23456789a, B59a/S359a.

It is possible to find such strobing rule, that the first rule of its non-strobing representation is "explosive", and the second is "stable". Careful choice of these rules allows to find that "edge of chaos" between stability and chaotic explosion, where interesting things could occur.

Below are some results of my small exploration of such rules. I used brute-force search in the rule space, searching for the rules that do not grow exponentially fast, but produce patterns away from the original random seed.

## Rules with spaceships

### Rule B0157/S01

The rule is stable, random initial patterns evolves into spaceships and few very rare oscillators. This is the first rule where I discovered a spaceship. The spaceship has 7 cells in one of its configuration. It has period 4 and velocity c/2.

The spaceship virtually the only common pattern in this rule, but there are some rare oscillators too. One of them is 5-star:

### Rule B0157A/S01

This rule is very similar to the previous. The same spaceship is present, but the 5-star oscillator has different period

### Rule B01357/S01

This rule has a slightly different spaceship, made of 5 cells. Also, this rule is on the "explosive" side of the "edge of chaos". Though most small patterns are not growing infinitely, most big random patterns show gradual population growth. I has not identified a smallest pattern, that causes this behavior, probably, there are some natural replicators, puffers or rakes.

5-star is stable in this rule, but there are different oscillators.

### Rule B01357A/S01

This rule has the same 32-spaceship and several rare oscillators. Same as the previous, B01357/S01 shows slow infinite growth for large initial patterns. No minimal pattern with infinite growth identified.

### Rule B01347/S013

The spaceship in this rule has different configuration, I dubbed it L-ship. It has period 8 and velocity c/2, same as the rest spaceships. This rule is stable, random initial pattern quickly evolves into some spaceships and very rare oscillators.

Oscilators are rarely produced

### Rule B01367A/S013

This rule is stable, with c/2 spaceship which is the same as L-ship above, but because of the different intermediate steps, this spaceship does not coincides with its own mirror image.

## Rules without spaceships

There also are some rules exhibiting slow growth, but without apparent spaceships or puffers. I haven't identified the mechanism of growth in them.

### Rule B01568A/S013

No spaceships visible, but random initial patterns tends to grow slowly. Some oscillators are present.

### Rule B01568/S01

Same as above. No spaceships found, slow but steady growth, oscillators.

### Rule B01478A/S01

Same as above. No spaceships found, slow but steady growth. Has 2 common oscillator patterns: 3-celled corner and 5-star.

## Conclusion

Some properties are common for all of the explored strobing rules (that are the small part of the whole rule space).

1. "Static" patterns are rare. This is natural, because the patterns that appear static in the simulation (hence quotes), are actually inverting on each step, together with background.
2. The set of naturally occurring patterns is very limited. Usually, there is one or no spaceship with velocity c/2, and a few significantly rare oscillators. Some oscillators can be constructed manually, but almost never form naturally.
3. Discovered spaceships have clear resemblance. The same velocity c/2 regardless the period, the same method of moving by growing a block of 2 cells on each even generation.

## Open questions

Some questions I don't have an answer:

1. Are there any other spaceship in these rules? Geometry of the hyperbolic plane seems to complicate formation of the "wide" spaceships, but "long" ones should be possible.
2. What pattern causes very slow but infinite growth in some of the rules? Some patterns are visible, but they are relatively big and hard to grasp.
3. Are there any significantly different rules with spaceships?

## Software

To produce images and animations, self-written Pentagrid simulator was used. It is open source, and runs on every desktop platform supported by Java (Linux and Windows are tested). To run the program, download the pentagrid-0.x.jar file and run it with java -jar (usually double-click in Windows). Install Java runtime, if not installed (it is safe if you don't enable java plug-in in the browser). The program has very basic GUI and controlled by the hot keys, so read the readme first.

## Sunday, September 21, 2014

### Logarithmic zoom at the Palace square, Saint Petersburg.

This post continues the previous one: Log-Polar coordinate system applied to Google maps. Now I made an attempt to extend the set of Google maps images with own photos, to make an image showing things at scales from millimeters to tens of thousands of kilometers.

The picture above is in logarithmic polar coordinates, with center at the specific point on the Palace square in Saint-Petersburg. I put a few coins on a bricks of the square to slightly animate the view and provide some scale reference.

On this image, vertical coordinate shows distance from the center of the coin. Horizontal coordinate corresponds to angle. Since the central coin is round, it looks like a yellow band at the bottom of the image.

It is composed from many sources. Maps were taken from the Google, and the close-up photos were taken by me and processed with Gimp to remove perspective distortion. To convert regular photos to log-polar coordinate system a self-written script: log-zoom (Github) was used.

Single image of reasonable size in the regular coordinates can't show details of such different scales, but it can be done with animation:

## Source photos

Because bricks are arranged in circles, on the log-polar image they appear in straight rows, gradually decreasing as the scale of the image grows. Removing perspective distortion and adjusting the pieces in Gimp was a real pain...

Three coins on the central brick. Center of the log-polar coordinate system is in the center of one of them.

Finally, a design of the central coin. The text around the border says "50 years since first manned space flight".

## Sunday, August 17, 2014

### Crystal growing: table salt (sodium chloride)

Table salt, or sodium chloride (NaCl) is a widely available chemical. It also forms nice transparent cubic crystals that are, unlike copper compounds, stable on dry air and not toxic. No surprise that almost every crystal growing book for kids recommends table salt as a good material to try. The instructions are innocuously easy: make a saturated solution, put a small seed crystal into it, and let it evaporate slowly. And voila!

But this almost never works. With some patience, you can make good transparent cubes... few millimeters in size. Often, you won't get even this. This method works perfectly with copper sulphate, alum, many other salts, but not with NaCl. I guess, those who write all these books are just citing other's words and never tried to follow their own instructions.

Those why tried to grow them, talk about salt differently:

... Small (sub-millimetre) clear cubes with smooth faces will grow on the bottom of your glass dish or jar, or on any thread suspended in the jar (see figure on the right). Larger crystals tend to develop hopper faces, or even more erratic growth habits ...
Dom's crystal growing page
There are several factors that are causing the problem. Firstly, NaCl is prone to forming Hopper faces. With some substances like bismuth, the sometimes leads to a beautiful results, but not with table salt. You'll get dendrites, white crystalline crust, hollow flat rectangles - anything but not the transparent cubes, drawn in the books. NaCl also crystallizes too readily, using every smallest crack or dust particle as a seed.

So, without some trick you'll never get big and (relatively) transparent crystals like ones at the photo above. I'd like to share a trick I have found by an accident. Not sure that it is the best method for growing NaCl crystals at home, but it worked for me, at least.

The trick was to use not the pure NaCl solution (as it is usually recommended), but a mixed solution of salt and iron (III) chloride FeCl3. The latter is a brown crystalline solid, highly soluble in water. It is used by amateurs in the printed circuit board production, and can be bought in the most shops for electronic hobbyists.

By some mysterious reason, presence of FeCl3 changes the way salt crystals grow: they are growing bigger and more transparent. Iron chloride itself is not going into the crystal, though occasionally, small portions of the ferrous solution are getting included in the transparent crystals, creating yellow spots inside.

## Growing instructions

I used the following procedure, you could improve it according to your taste.

• Prepare solution: dissolve some FeCl3 in the warm water. Solution must have intense yellow color. Then add salt in excess, and let it dissolve for several hours. If all salt dissolved, add more.
• Prepare a seed crystal: take thin fishing line or some filament, put it in a drop of solution and let it dry. Some very small crystals will stick to the filament. Remove all of them except 1 or 2. (Instead of growing crystal on a thread, it can also be grown on the bottom of the container).
• Fill a clean glass or plastic container with solution, and put the seed into it. Cover the solution (not tightly), to slow down evaporation and avoid dust.
• Wait patiently. If salt crust is forming at the edge, don't let it grow too large, move your crystals to a new clean container. Also, crust can be dissolved by a drop or two of pure water. During the growth, concentration of FeCl3 grows gradually, so add small amounts of pure salt solution when appropriate.
Crystal growth is not fast. In my case, growing centimeter-size crystals required about 6 months.

## Crystal properties

Salt crystals are hard and very stable on air, because they don't contain crystalline water (unlike the easier to grow copper(II) sulphate, which gradually loses transparency if exposed to the air). Their basic form is cube, though crystals growing on the bottom of a jar would be shorter along the vertical axis. Also, growing in the solution with FeCl3 is making their faces somewhat uneven, as if they are made of lots of smaller cubes.

## Safety

Iron(III) chloride is not a toxic compound, though it can irritate skin on contact. Don't taste it though, technical grade chloride may contain significant amount of more dangerous impurities. It is also strong corrosive agent that can easily etch copper, iron and many other metals. Don't put it into a metallic container, use glass or plastic. Iron(III) chloride also makes permanent stains on clothes and tables, so handle it carefully.

Wash the crystals with pure saturated salt solution and wipe them well, and they will be entirely safe to handle.

## Further ideas

I've discovered the effect of iron chloride accidentally, forgetting a bottle with solution opened for several weeks. When I found it again, it contained some transparent salt crystals, much bigger and cleaner than I would expect. Possibly, there are other compounds that work as growth stabilizers, what are them? My intuition suggests that other metal chlorides and glycerol are good candidates.

## Sunday, July 20, 2014

### Frequency-based CA visualization (using Single Rotataion as an example)

In the binary cellular automata, cells can have only 2 states: ON and OFF. This, monochrome images are the most natural choice to visualize such automata. While informative, this visualization method produces somewhat dull images, especially when spaceships don't have easily recognizable form. My first idea to make it a bit fancier was to add some motion blur, emphasizing spaceships, and it worked well:

Motion blur is effectively a low-pass filter applied to the original BW image. The next idea was to use a combination of several filters to make image more colorful. In my final application, 3 filters were used:

• Low-pass filter, detecting cells that are constantly ON. This filter was assigned to the red channel.
• High-pass filter, detecting cells that are flashing with period 2 (ON-OFF-...). Green channel was used for this filter.
• Band filter (2-nd order), detecting cells, flashing with period 4. If was used for blue.
Together, these 3 filters produce images like this:

The picture above shows the Single Rotation rule, where random reactions produce many different kinds of spaceships. Interestingly, this colorization scheme allows to distinguish different spaceships by their colors. Finally, by applying these filters to the animation, you'll get the video above. In this video, the field is initialized with random block at the center (at first it appears red because of high cell concentration), and a vertical bar of 2 cells. In the "Single Rotation" rule, such bar of cells is an example of indestructible still life. After the simulation is started, dense random block starts dissipating, emitting multiple spaceships of different kinds. Single Rotation rule supports a lot of them!

For those who prefer higher resolutions, a HD version is available. Full screen and "HD" quality are recommended.

This colorization algorithm can be applied to any other cellular automata too. For example, here is Conway's Life. Note how still lifes are colored in red, and p2 oscillators are green:

## Software

All sources for this post are published at Github:

To use them, clone the repository above, build C++ modules, executing
$make and then run $ python make_color_animation.py
This will create a sequence of animation frames, that can later be converted to video, using ffmpeg. I used the following command line for conversion (not sure it is the best one though):
ffmpeg -i frame%04d.png -movflags faststart -c:v\
libx264 -preset veryfast -bf 2 -g 15 -crf 23\
-pix_fmt yuv420p -r 30 animation.mp4

At the time of writing, settings for the script above are done inside the source code.

## Algorithm

Three discrete time filters are used to detect flashing frequencies:
$$\begin{eqnarray} x_{t+1} &=& k x_t + s_t,\\ y_{t+1} &=& - k y_t + s_t,\\ z_{t+1} &=& i z_t + s_t. \end{eqnarray}$$
Where $$k \approx 0.99$$ is a decay coefficient, $$s_t$$ is a t-th state of the cell (0 or 1), i is a complex unity. Then, RGB color for the pixel is obtained by formulas:
$$\left( \begin{array}{c}r\\g\\b \end{array} \right) = k_\gamma rgb',$$ $$rgb' = \mathbf{K} \left( \begin{array}{c}x \\ |y| \\ |z| \\ \end{array} \right),$$
where K is color transformation matrix and $$k_\gamma$$ is gamma correction coefficient. The latter is calculated like this:
$$k_\gamma = \max(r', g', b')^{\gamma-1}.$$
where $$\gamma$$ is the gamma correction value (in the video above, 0.2 was used). For the top picture, the following colorization matrix was used:
$$\mathbf{K}=\left(\begin{array} 18.7& 0 & 0 \\ 0 & 94 & -15.7 \\ 0 & 0 & 68 \end{array}\right).$$

## Sunday, July 13, 2014

### Single Rotation Spaceships (205 of them)

The “Single Rotation” rule (it's my personal naming, it could be already known under the different name) is a very simple reversible cellular automaton, acting on a field with Margolus neighborhood. In this rule, each 2x2 block is rotated by 90° if it has exactly one alive cell. All other blocks remain unchanged. Despite its simplicity, this rule supports a multitude of different pattern types, and is especially interesting for me because of its spaceships. There are hundreds of spaceships with different periods.

Live version of this animation can be seen in the online simulator.

Even simple observations showed that random reactions in this rule naturally produce more than ten different spaceships. Of course, ten was not enough so I tried to find more of them. This post is a digest of what I have found.

## Spaceship search method

The method to search for spaceships was simple: take an empty field and put a random initial block in its center. During evolution, random reaction would produce multiple spaceships, which are then removed from the field and automatically classified. Typical parameters, used for search:

• Field size: 128x128.
• Initial random block size: 80x80.
• Initial pattern density: 40%.
• Simulation time: 30000 generations.
Several days of search brought a list of 205 different spaceships. To identify them, 108'720'436 moving patterns were analyzed. Here are the top 10 most frequent spaceships from this list:

Posting the whole list of 205 spaceships here would blow up the post, so I've uploaded it to a separate page. You can download the data for this list in different formats:

• JSON library for the online simulator. To use it in the simulator, copy-paste it to the bottom text area on the “Library” pane of the simulator, and click “Import from JSON”
• CSV file.

## Digesting the data

### Size distribution

All discovered spaceships have 4 to 9 cells. The distribution of their sizes shows good agreement with power law:

Power law approximation (dashed line on the plot) gives the following distribution: $$P_n \approx \frac{0.679} {{17.55}^{n-4}},$$ where Pn is a probability of a randomly produced spaceship having weight n. For 10-cell spaceships, this formula gives expected probability P10=2.33·10-8. Thus, the expected number of 10-cellers is 2 or 3, while none were discovered. Either I have bad luck, or the actual probability is lower, or maybe these spaceships have too large periods, and were not detected.

### 4-cell spaceships

Lightest spaceships have 4 cells and there are exactly 6 different spaceships of such size (I don't have a proof, but it should be easily provable by enumeration). Lightest spaceships have some interesting properties, mentioned in the previous post.

### 5-cell spaceships

Only 6 spaceships made of 5 cells were discovered during the search, all with significant hit counts. This suggests that there are no other 5-cell spaceships. From them, 2 are diagonal, and 4 are orthogonal. Curious coincidence: the most common diagonal 5-celler have a phase, that looks exactly as the Conway's Life glider.

It moves by 1 cell every 15 generations, and its intermediate steps don't resemble glider at all, so this resemblance is nothing more than a coincidence. The other diagonal 5-celler is the R-pentomino.

### Heavier spaceships

Number of different spaceships increases with weight, the experiment discovered 57 different 6-cell spaceships and 69 spaceships of 7 cells. Some of them are extremely rare, having probability less than 3.31e-07, which suggests that there are more spaceships, too rare to be discovered. The heaviest spaceships discovered are 10 different 9-cellers. Of them the most common one, detected 26 times, is $b2o$b3o$3bo$obo$2bo, fast diagonal spaceship with period 31 and velocity c/31. No heavier spaceships were detected. ### Periods #### Smalest periods Among the discovered spaceships, the smallest period are: • orthogonal: 12 generatons, achieved by the 4-cell spaceship b2o2$b2o,
• diagonal: 11 generations, 6-cell spaceship obo$obo$b2o.

Interestingly, record for the diagonal spaceships is achieved by the relatively heavy (6 cells) and rare (probability 0.08%) pattern. Are there any heavier moving patterns with smaller periods? I don't know (though intuition says no).

#### Highest period

While in the Conway's Game of Life naturally produced spaceships and oscillators commonly have small periods, in the "Single Rotation" rule there are numerous patterns taking extremely long time to start repeating.

Obviously, there is no upper limit for the spaceship period. But among known (to me) patterns the one with highest period is orthogonal 8-cell spaceship b3o$o3b2o2$bo$2bo. It takes whopping 9046 generations to move horizontally by 2 cells (thus its speed is c/4523). ### Velocities #### Fastest spaceships As one may expect, the pair of the spaceships with the smallest periods also hold the speed record. Their movement speeds are: Whether there are any faster patterns is an open question. It is easy to prove that in the "Single Rotation" rule spaceship velocity can not exceed c/2, but this estimation is very rough. #### Slowest spaceships The slowest spaceship found is the 8-cell pattern 2bo$b2o$2bo4$bo$bo$2b2o. Its speed is c/4755.

The period of this spaceship is not the largest, but it moves only by 1 cell, where the P9046 moves by 2 cells.

## Open questions

• Smallest period. Current record is 11 generations, held by the 6-cell pattern: obo$obo$b2o. Are there spaceships with faster period?
• Fastest spaceship. It is easy to prove that spaceship velocity can't be more than c/2. Currently, the speed record is c/6 for orthogonal and c/11 for diagonal spaceships. Are there any faster ones?
• Oblique spaceships (knightships). So far, no oblique (neither diagonal nor orthogonal) spaceships were found. Are there any?
• Spaceship families. Some spaceships are clearly forming families. For example, family of period 31 spaceships with 7, 8 and 9 cells. How to find them? Are there infinite families?
• 10-cellspaceships. None found yet.


text: From the above figures, it can be observed that, the pressure is maximum at the convergent section of the C. Circular pipe diameter change, Non-circular duct area change Venturi flow meter (C=0. , low flow rate) and about 0. The mass flow rate, liquid density, and nozzle diameter determine the mean velocity Umean- By using tabulated inlet loss coefficients (Kinļet) (as shown in Fig. Air enters a nozzle at 180 kPa, 67 °C, and 48 m/s. The total inflow feed stream flow rate is the same for all four configurations (2. nozzle for the job. For operation with equal mass flow rates, the square-edged inlet nozzle requires a greater pressure drop and produces higher turbulence intensities than a nozzle with a well rounded inlet. r c = critical pressure ratio. 356 after the combustion chamber. Hunter Pro-Spray sprinkler heads are engineered to ensure long-lasting, reliable performance in residential applications. The convergent parts of the nozzle are sharp and frictionless. If the inlet area of the nozzle is 90 cm2, determine (a) the mass flow rate through the nozzle, and (b) the exit area of the nozzle. The nozzle is horizontal and there is a negligible heat loss from it. Consider a converging-diverging nozzle with a throat area of 100 cm 2 and an exit area of 240 cm exiting into an altitude simulating test chamber at 50 kPa. A converging-diverging nozzle has an exit to throat area ratio of 2. where Ao is the area of the orifice. So now we obtain a simplified expression for nozzles and diffusers: q + (h in +o. Parameter Dimensions. Nozzles are profiled ducts for speeding up a liquid or a gas to a specified velocity in a preset direction. 1/8"-27 NPT lower nozzle thread. The present study investigates the effects of the orifice nozzle number and the inlet pressure experimentally on the cooling performance of the counter flow-type vortex tube. Hussaini 2013-2014 We see that if we are given a physical converging–diverging nozzle (area ratio is known) and an operating pressure ratio between the first and second critical points, it is a simple matter to. That is, the flow area in a subsonic nozzle decreases in the flow direction. Air enters an adiabatic nozzle steadily at 300 kPa, 200°C, and 30 m/s and leaves at 100 kPa and 180 m/s. The inlet velocity is 4 m/s. Atmospheric pressure acting through the atmospheric vent (C) depresses the metering diaphragm (D) forcing fuel from the main nozzle (B) and simultaneously opening the inlet needle valve. 1 bar Must operate in a stable, narrow range. Evaporating water into the inlet air can provide critical cooling of heat sensitive areas such as electronic server rooms or animal enclosures. 21 kg/m3 and 40 m/s and leaves at 0. 2 kg/s when the inlet conditions are 1 MPa and 250C, and the exit pressure is 200 kPa. 2- Yellow Nozzle 15˚ is for brick, concrete and other hard surfaces with stubborn stains. Optional British Pipe Thread available. But the flow velocity for DME fuel is more than the diesel fuel due to decrease in pressure. g GE Aviation Technical History GE Aircraft Engines U. At this point, gas velocity is maximized, and the mass flow rate is a function of the inlet pressure, inlet temperature, and the type of gas. Since thrust is based largely on momentum, no expansion downstream of the throat is necessary (just look at commercial jet engine nozzles) and the gas is expelled to the atmosphere. Order Wiper-Windshield Washer Nozzle for your vehicle and pick it up in store—make your purchase, find a store near you, and get directions. The coefficient of discharge for nozzle is 0. • Nozzle inlet vortex can change nozzle restrictiveness on the two-phase flow. In order for the flow to accelerate, the pressure gradient through the nozzle must be such that the fluid feels a force towards the exit, requiring a lower exit pressure than inlet pressure. While searching online for minimum Area ratio required, most of them are related to CD nozzle. 41×104 Pa Fig. 50 m2, M 1=0. Since the nozzle is axisymmetric, it was modeled as an angular slice of the complete geometry, with a wedge angle of 18 degrees (see Fig. The throat diameter and exit diameter is same for all nozzles. Nozzle Type. Compressible Flow with Area Change and Friction as a Tool for Inlet Performance Evaluation Compressible Flow with Area Change and Friction as a Tool for Inlet Performance Evaluation Wilson , , Erich A. is a convergent nozzle with a circular-arc wall contour of 5. This flow is supersonic in a portion of the nozzle, terminating in a normal shock inside the nozzle. A blast nozzle apparatus for directing abrasive particles against a surface to remove therefrom adherent materials includes a flow straightener intermediate the blast nozzle and flexible supply hose so as to rehomogenize the abrasive particles throughout the total area of the blast nozzle inlet and counteract the nonuniform concentrating effect on the blast media as the abrasive particles are. Isentropic Flow with Area Change -9 AE3450 • For nozzle shown, inlet area A1=0. The converging-diverging nozzle has an exit area of 0. Some petrochemicals companies recommend to use tangential flow when there is deep vacuum. 6) This selection is useful for determining the change in static pressure in a pipe due to a diameter change, determining flowrate, or designing a flow meter. Answers: (a) 0. Combine with Teejet Threaded Caps CP1325 Brass, Teejet Nozzles, and TeeJet Strainers for a complete nozzle assembly. Combined Friction and Heat Transfer in a constant area pipe. Chapter Eleven/Normal shock in converging-diverging nozzles ----- 5-6 ch. Larger upstream pipe on the suction side joins with a smaller suction nozzle using a reducer. 0254 Inlet to exit Pressure ratio 40. A de Laval nozzle using hot air at a pressure of 1,000 psi (6. v1 + ( p2 - p1 ) a2Where, m is the mass flow rate, v is the velocity, p is the pressure of gas and a is the area of the nozzle, 1 and 2, namely the inlet and outlet of nozzle. If the inlet area of the nozzle is 90 cm 2, determine (a) the mass flow rate through the nozzle, and (b) the exit area of the nozzle. These variable area nozzles on an F-14 fighter accelerate the hot gases exiting the engine to provide propulsive thrust. velocity and the ratio of the exit area to the nozzle inlet area A ex /A in in case of a reversible adiabatic flow. de Laval nozzle I just need throat area to finish sizing. Referring to our station numbering, the nozzle entrance is station 5 and the nozzle exit is station 8. The higher turbulence level which we observed with the square-edged inlet nozzle correlates with wider spray dispersion angles observed by Reitz and Bracco. 15 of the isentropic enthalpy drop. to 140 Deg. Cylindrical nozzles each of which has a lateral orthogonal inlet of fluid into it we recommend to apply in pressure distributive pipelines. versus mass flow rate at Ar =5 Inlet Temperature of 435. Inlets and Nozzles: Design Considerations EGR 4347 Analysis and Design of Propulsion Systems Subsonic Inlets: Major Design Variables/Choices Inlet total pressure ratio and drag at cruise Engine location on wing or fuselage Aircraft attitude envelope Inlet total pressure ratio and distortion envelope Engine out windmilling airflow and drag Integration of diffuser and fan flow path contour. This creates a depression in the area of the main nozzle (B). the inlet area to the exit area. 15 14:54 to air-entraining surface vortices and air/gas-filled submerged vortices. It indicates that nozzle to throat area ratio has the important aspect on characterization of Jet Pumps than the nominal diameter. 005 m 2 and it discharges into the atmosphere. A converging-diverging nozzle has an exit to throat area ratio of 2. Journal of Engineering and Development, Vol. 1 m2 and the specific volume at the inlet is 0. nozzle area = (nozzle diameter) 2 x 0. Chapter Eleven/Normal shock in converging–diverging nozzles ----- 5-6 ch. 1 bar Must operate in a stable, narrow range. The fume extraction device plays a fundamental role because, if it cannot follow the innumerable shapes and positions of the exhaust pipes of vehicles, the entire system will be pointless. 1 Answer to Carbon dioxide enters an adiabatic nozzle steadily at 1 MPa and 500°C with a mass flow rate of 6000 kg/h and FIGURE P5-37E leaves at 100 kPa and 450 m/s. Test range of inlet capture area ratios 43 26. Show transcribed image text A supersonic nozzle with an inlet area of 0. , low flow rate) and about 0. 4a and 4b show the impact of inlet momentum (rV 2, actually inlet energy in this case, where ρ is the bulk fluid density and V is bulk velocity) in a scrubber with a plain inlet nozzle. (To achieve supersonic velocities, the area must first decrease, then increase, a so-called converging-diverging nozzle. Take vol flow rate/inner area (length/time). Your order may be eligible for Ship to Home, and shipping is free on all online orders of $35. FV = 0 Using gauge pressures, the pressure force at exit is zero. r c = critical pressure ratio. A horizontal nozzle discharging water into the atmosphere. 356 after the combustion chamber. amount of air. 762 kg/m 3 and 180 m/s. materials used it. 00 atm and the exit static pressure is 1. Steam enters a nozzle at 400°C and 800 kPa with a velocity of 10 m/s, and leaves at 300°C and 200 kPa while losing heat at a rate of 25 kW. The graph on the left shows the shape of the nozzle, chamber on the left, exit on the right. Steam expands in a set of nozzle from 10 bar, 300 0C to 1 bar. and British 2½" instantanous coupling as standard. The Sonic Nozzle (Critical Flow Nozzle, Critical Flow Venturi or Sonic Venturi) consists of a smooth rounded inlet section converging to a minimum throat area and then diverging along a pressure recovery section or exit cone. the ratio of outlet pressure to inlet pressure only when mass flow rate per unit area is min-imum D. We will use this information to determine the exit Mach number and pressure at the inlet and outlet. 35 m^2 is designed to be choked when the incoming Mach number is 2. g GE Aviation Technical History GE Aircraft Engines U. After you have entered the values, click OK to close the window. Determine. Nozzles and diffusers are ducts of varying area which are intended to increase or decrease the velocity of the flow. 1 + x 2 ; -0. The length of the nozzle will depend upon the divergent angle which is mostly ~ 15. Available at:. his value was deemed suicient to enable stagnation conditions to be assumed at the nozzle inlet and quarter circle of radius 2. It is highly recommended a high pressure filter be attached to the inlet of the rotating nozzle. I'm going to assume the flow here is adiabatic and that there is no mass flow added from the side walls through the nozzle. But the flow velocity for DME fuel is more than the diesel fuel due to decrease in pressure. 64 at 100 MPa. The velocity of the steam at the inlet is 5 m/s. This greatly restricted airflow through the boiler, requiring a restricted exhaust nozzle opening to proivde sufficient draft. It indicates that nozzle to throat area ratio has the important aspect on characterization of Jet Pumps than the nominal diameter. Heat losses from the nozzle per unit mass of the steam are estimated to be 1. There is a normal shocks in the converging section at an area of A, =0. Fanno Flow – flow with friction in an adiabatic constant area pipe. 00 atm and the exit static pressure is 1. nozzle: Shouldn't all hydrants be equipped with hoses and nozzles?, BIB VALVE, PNEUMATIC WEIGHING SYSTEM, NOZZLE-MIX GAS BURNER, NOZZLE-EXPANSION RATIO, NOZZLE EXIT AREA. v1 + ( p2 - p1 ) a2Where, m is the mass flow rate, v is the velocity, p is the pressure of gas and a is the area of the nozzle, 1 and 2, namely the inlet and outlet of nozzle. For such a nozzle, the area of the exit, throat, or both is varied so that the optimum exit pressure can be obtained. 1 m2 and specific volume at inlet is 0. After you have entered the values, click OK to close the window. It is highly recommended a high pressure filter be attached to the inlet of the rotating nozzle. 5–7 Air enters a nozzle steadily at 2. If the inlet pipe is the same size sa the PSV inlet flange, the ratio of orifice flow area to pipe flow area varies significantly. nozzle area e. The inlet stagnation pressure, p 0, and the inlet stagnation temperature, T 0, may be. 013 mm^2 Nozzle Exit Area 0. inlet nozzle into the combustor. Determine the exit area of the nozzle, in m2. Because the area of the vane inlet is several times larger than the inlet nozzle, the fluid velocities are much smaller, allowing for good gas/liquid separation as well as smooth entry into the vessel. Atomizing Air Volume ± 5 cfm/ nozzle Reflects reproducibility of nozzle set-up. This initial guess value can be calculated from the 1D analysis since we know the area ratio at the inlet. The inlet has a geometric capture area of 893 in 2. 0254 Inlet to exit Pressure ratio 40. 50 m2, M 1=0. a) The mass flow rate through the nozzle, b) The exit temperature of the air, and c) the exit area of the nozzle. reactor vessel inlet pipe splits or ruptures at a paint near the inlet nozzle at the bottom of the presssure vessel. It is frequently used to control the flow rate, speed, direction, mass, shape and the pressure of the stream that emerges from them. 34 Mass Flow Rates 0. Ideal to fit nozzles with inlet G1/4" F on lances fitting Kärcher with connection M 18x1,5 M. The nozzle exit area, A e, corresponding to the exit Mach number is given by. The nozzle. Since the nozzle is axisymmetric, it was modeled as an angular slice of the complete geometry, with a wedge angle of 18 degrees (see Fig. If the mass flow rate and inlet pressure are high enough, the fluid will reach Mach Number = 1 (speed of sound) at the throat. PARAMET NOZZL NOZZL NOZZL NOZZL ER E1 E2 E3 E4 Throat 8 8 6 6 RaidusRth(m m) Exit area, 236. is a convergent nozzle with a circular-arc wall contour of 5. g GE Aviation Technical History GE Aircraft Engines U. Journal of Engineering and Development, Vol. Discharge coefficient. effective nozzle area ratio of the e ngine to increase. In this example, the baseline case with no rounding of the nozzle inlet shows a C d of about 0. 005 m2 and it discharges into the atmosphere. d Surface Area of the Droplet (m 2) AD or D21 Absorption Diameter (m) inlet air fogging nozzles to understand the terminology used to characterize nozzle performance. The fuel nozzle inlet guides employ primary and secondary swirlers where the primary swirlers continuously deliver compressor air around the fuel nozzles and the secondary swirlers deliver compressor air solely when the pressure drop across the front end of the burner reaches a predetermined level in one embodiment or the inlet temperature goes. 15 of the isentropic enthalpy drop. Consider air flowing at high-speed through a convergent-divergent nozzle having a circular cross-sectional area, A, that varies with axial distance from the throat, x, according to the formula A = 0. If the steam floe rate is 1 kg/s and the initial velocity of steam is negligible, find the minimum area of the nozzle. Agricultural and Industrial applications. Attach the Gun/Extension Nozzle holder to the side of the body. The RPS™ 50 is a gear-driven, rotary sprinkler, capable of covering an area of 18' to 36' (5,5 to 11 M) radius at nozzle pressure of 30 to 50 PSI (2,1 to 3,4 bar) with a. • The control range of inlet pressure and mass flow rate is large enough for real applications. the ratio of outlet pressure to inlet pressure of nozzle B. 28 and at 6. The area ratio is double valued; for the same area ratio, there is a subsonic and a supersonic solution. cross-sectional area at least sixteen times the area of the fan outlet or outlet duct for fans with axis of rotation parallel to the discharge flow. inlet pipes are only available in standard sizes. At the nozzle inlet the velocity is 30 m/s, calculate values of the ratio A/A 1 (where A 1 is the cross-sectional area of the nozzle inlet) for the sections where the pressure is 600, 500, 400, 300, and 200 kPa. Air Inlet Cooling—-Air Atomizing Nozzles. The Turbo Nozzle rotates in a zero degree spray pattern in a circular motion to break down tough dirt and grime. Static pressure p at the exit = 2. Solid brass nozzle. Some suppliers recommend a tangential nozzle inlet to help the separation. the nozzle whose inlet dia is greater and outlet dia is smaller is known as convergent while the nozzle who has low dia at inlet and large dia at outlet is known as divergent type nozzle. Area A Spray Dryer: Dimensions 20 ft. If the flow is traveling at 12 ft/s and you determine the density to be 0. Inital conditions: The nozzle is placed on a pump which has a volume flow rate of 0. inlet throttling and exhaust-nozzle-area modulation. Angle (degree) Mach number. the ratio of outlet pressure to inlet pressure only when mass flow rate = c. At the center of the spiral is mathematically defined as an area of infinite density. For a fixed area ratio, defined as the exit area to the throat area, Ae/At of 1. For an inlet area of 800 cm2, determine the velocity and the volume flow rate of the steam f at the nozzle exit. 76 and know the diameter of the motor case. If the steam floe rate is 1 kg/s and the initial velocity of steam is negligible, find the minimum area of the nozzle. Atomizing Air Volume ± 5 cfm/ nozzle Reflects reproducibility of nozzle set-up. Ratio of specific heats (. 015m² as shown in the figure and the flow leave. To reduce the usual 20 hour setup time to an hour or less and also eliminate the arbitrariness of the subdivision, a computer. Because the area of the vane inlet is several times larger than the inlet nozzle, the fluid velocities are much smaller, allowing for good gas/liquid separation as well as smooth entry into the vessel. Type in '4' and press the 'Set' button. Change in potential energy from inlet to exit is negligible, ΔPE = 0. Your order may be eligible for Ship to Home, and shipping is free on all online orders of$35. Converging Diverging Nozzle - Background Hour glass shaped nozzle Inlet of decreasing cross sectional area Outlet of increasing cross sectional area Used to accelerate pressurized fluid to supersonic speeds Useful in many turbine and jet engine applications. 0, is conducted to verify the shock-capturing capability of non-Monotonic Upstream Centered Schemes for Conservation Laws (non-MUSCL) symmetric Total Variation Diminishing (TVD) scheme and to investigate the effect of inlet/exit pressure ratio on the starting performance of supersonic nozzle. 5 and the amount of inlet air pressure is 153580. Other Pro Spray Head Options. Assume the nozzle operates isentropically. While an axial full cone nozzle's vane typically has a smaller free passage than the nozzle's orifice diameter, the series. Note that this value will be updated by the code. the inlet area to the exit area. All nozzles have radius shaped convergent inlet with the cylindrical throat, and Venturi nozzle also has a divergent part as an outlet. However many factors can affect the noise level such as walls, buildings, trees or any object in the noise path. nozzle area e. Atomizing Air Volume ± 5 cfm/ nozzle Reflects reproducibility of nozzle set-up. Steady state operation of the nozzle 2. Design and Optimization of Aerospike nozzle using CFD. For proper flame, only a tiny bit of gas is mixed with a large. nozzle for the job. To evaluate the effects of inlet pressure on the ignition process of spray combustion, the images of the ignition process were recorded and the outlet temperatures were measured under inlet pressure of 0. Ratio of the cross-sectional area for gas flow at the nozzle inlet to that at the throat Explanation of nozzle-contraction-area ratio. 0 in2 and an outlet area that is 15 times larger than the inlet. g GE Aviation Technical History GE Aircraft Engines U. The choice button at the right top selects the solution that is presented. system is strongly influenced by the inlet and the nozzle. 2[mg/s] –2[mg/s] Parameters of numerical model. Spray nozzles can be categorized based on the. 1 m2 and specific volume at inlet is 0. Double brass 3/16 in. Determine (a) the mass flow rate through the nozzle, (b) the exit temperature of the air, and (c) the exit area of the nozzle. κ [-] heat capacity ratio; r [J·kg-1 ·K-1] individual gas constant; T [K] absolute temperature of gas; p [Pa] pressure of gas. Consider a Laval Nozzle with inlet area $A_{inlet}$ in a subsonic flow with velocity $V$ and density $\rho$ (and total pressure $p_0$, total temperature $T_0$ and. Inlet is given to the pressure of nozzle. The exit of the nozzle has a diameter of 2. At the inlet, the stagnation pressure is 400 kPa and the stagnation temperature is 800K. iii) If the specific volume at the nozzle exit is 0. For example, if you had a nozzle with a circular inlet with a radius of 3 feet, A = pi * r^2 = 3. his value was deemed suicient to enable stagnation conditions to be assumed at the nozzle inlet and quarter circle of radius 2. For unique sizes, we have the capability and experience to make the fabrication according to the needs of the customer. It indicates that nozzle to throat area ratio has the important aspect on characterization of Jet Pumps than the nominal diameter. v² value than for example a plain nozzle, half-open pipe or V-baffle, is that it combines several positive effects: Prevents liquid shatter. Determine (a) The mass flow rate through the nozzle, (b) The exit temperature of the air, and (c) The exit area of the nozzle. Determine (a) the mass flow rate through the nozzle, (b) the exit temperature of the air, and (c) the exit area of the nozzle. Drain Cleaning Nozzles, Sewer Cleaning Nozzles, Tube Cleaning Nozzles Drain Cleaning Nozzles perform the tough job of dislodging blockages in a drain or sewer pipe. 0135m² and exit area of Az = 0. After you have entered the values, click OK to close the window. It should be used for heavy duty/industrial strength work such as removing tar, resin, glues and rust. 04–0. Nozzles are used for three purposes: to distribute a liquid over an area, to increase liquid surface area, and create impact force on a solid surface. Search the Engineering ToolBox. Check the Inlet Strainer for any blockage, including packaging debris. For example with 1 000 000 units of inlet airspeed to obtain 1 000 050 units of outlet airspeed, the outlet area must be 1 000 000 / 1 000 050 of the inlet area. The present study investigates the effects of the orifice nozzle number and the inlet pressure experimentally on the cooling performance of the counter flow-type vortex tube. c d r inlet nozzle hot air or steam inlet yarn path texturizing nozzle outlet nozzle inlet slotted chamber area of crimp formation exhaust area of crimp setting slotted chamber outlet slotted chamber exit pipe 3D texturized yarn flat yarn from draw godes exhaust by negative pressure plug is formed and controlled by. The length of the nozzle will depend upon the divergent angle which is mostly ~ 15. Test range of inlet capture area ratios 43 26. The program assumes you are dealing with an axisymmetric nozzle so, for example, your nozzle (with an area ratio of 4) will appear as having an exit with a diameter of twice that at the throat. b) A nozzle is a device for increasing the velocity of a steadily flowing fluid. 1/8"-27 NPT lower nozzle thread. 5 bar, 7 bar, 11 bar and 15 bar (for both cold and hot flow). 6) This selection is useful for determining the change in static pressure in a pipe due to a diameter change, determining flowrate, or designing a flow meter. The nozzle inlet has a cross sectional area (Ain) of 240mm2; the nozzle outlet has a cross sectional area (Aout) of 80mm2. This suggests that for any inlet airspeed, if we want to accelerate the air flowing through a tube, the outlet area must always be less than the inlet area. All nozzles have radius shaped convergent inlet with the cylindrical throat, and Venturi nozzle also has a divergent part as an outlet. The energy generation has been conducted using a stream-tek generator (model GNMD-KIT) with different numbers of nozzles (2, 3, and 6), an aspect ratio of 1:6, and an. 4a—Simple scrubber with an unbaffled inlet. A horizontal nozzle discharging water into the atmosphere. A nozzle is used to give the direction to the gases coming out of the combustion chamber. The nozzle is supplied with clean dry air (γ=1. The Turbo Nozzle rotates in a zero degree spray pattern in a circular motion to break down tough dirt and grime. If you are an experienced user of this calculator, you can use a sleek version of the program which loads faster on your computer and does not include these instructions. 11 Prepared by A. Although an orifice is commonly used to describe the minimum flow area constricted in the valve, the geometry resembles a nozzle and area is determined by applying the equation for flow in an isentropic nozzle. Determine the exit area of the nozzle, in m2. Larger upstream pipe on the suction side joins with a smaller suction nozzle using a reducer. In this project we designed a new Tri-nozzle to increase the velocity of fluids flowing through it. Your order may be eligible for Ship to Home, and shipping is free on all online orders of $35. The higher turbulence level which we observed with the square-edged inlet nozzle correlates with wider spray dispersion angles observed by Reitz and Bracco. s e _ B a r m a g _ 0 1. 2 kg/s when the inlet conditions are 1 MPa and 250C, and the exit pressure is 200 kPa. Right: Machine nozzle-tip orifice is the same size as the inlet orifice, eliminating any stagnant material at the wall of the inlet orifice. Nozzle seat sectional area and total nozzle sectional area c. The diameter of the inlet nozzle The diameter of the inlet device depends on the inlet flow and on the type of inlet device. This is discussed in Section 3. 0, is conducted to verify the shock-capturing capability of non-Monotonic Upstream Centered Schemes for Conservation Laws (non-MUSCL) symmetric Total Variation Diminishing (TVD) scheme and to investigate the effect of inlet/exit pressure ratio on the starting performance of supersonic nozzle. It consists of a smooth rounded inlet section converging to a minimum throat area and diverging along a pressure recovery section or exit cone. The higher turbulence level which we observed with the square-edged inlet nozzle correlates with wider spray dispersion angles observed by Reitz and Bracco. Determine (a) the mass flow rate through the nozzle, (b) the exit temperature of the air, and (c) the exit area of the nozzle. The inlet isentropic stagnation air pressure is 2. At the inlet, the stagnation pressure is 400 kPa and the stagnation temperature is 800K. Operation A. Nozzles are used in steam and gas turbines, in rocket motors, in jet engines and in many other applications. How to use inlet in a sentence. FV = 0 Using gauge pressures, the pressure force at exit is zero. Air Inlet Cooling—-Air Atomizing Nozzles. The inlet area of nozzle is 110 cm{eq}^2{/eq}. The nozzle turns the static high pressure high temperature gas into rapidly moving gas at near-ambient pressure. The process operating practices include inlet velocity profile. Circular pipe diameter change, Non-circular duct area change Venturi flow meter (C=0. The Area Of The Nozzle Is Given By A = A+0(1 - Bx) And The Inlet Velocity Varies According To U = 1/2 U_0(1 + Cos W T) Where A_0 = 5 Ft^3, L = Ft, B = 0. Hydraulic atomizing ln, lnd, n and m - 1/4. Steady state operation of the nozzle 2. If the flow is traveling at 12 ft/s and you determine the density to be 0. It is used to accelerate a hot, pressurized gas passing through it to a higher supersonic speed in the axial (thrust) direction, by converting the heat energy of the flow into kinetic energy. The diameter of the inlet nozzle The diameter of the inlet device depends on the inlet flow and on the type of inlet device. 3 times the vessel diameter, with a minimum of 0. Divergent nozzle: The crossectional area of the duct increases from inlet to the outlet then it is called as divergent nozzle. At the inlet, the stagnation pressure is 400 kPa and the stagnation temperature is 800K. Gravitational acceleration. 005 m2 and it discharges into the atmosphere. The inlet has a bore cross sectional area of 600mm^2 and exit has a bore cross sectional area of 200mm^2. But, higher flow through the system results in reduced pump efficiency. this high precision of distribution is due to internal vanes which create swirl chambers inside the nozzle. The inlet stagnation pressure, p 0, and the inlet stagnation temperature, T 0, may be. However many factors can affect the noise level such as walls, buildings, trees or any object in the noise path. All three types differ from each other based on its shape. even inside the manifold plenum area, so they can. v1 + ( p2 - p1 ) a2Where, m is the mass flow rate, v is the velocity, p is the pressure of gas and a is the area of the nozzle, 1 and 2, namely the inlet and outlet of nozzle. Steam leaves the nozzle at 250oC and 0. To what pressure must the inlet stagnation pressure be raised so that: (a) A shock appears at the exit plane; (b) A shock of an area equal to 0. Is there an optimum ratio for a nozzle inlet area/throat area. MV NOZZLE WRENCH NOZZLE INLET PRESURE - MIN 3. q=heat transferred per unit mass h in = specific enthalpy of inlet fluid h out = specific enthalpy of outlet fluid c in = velocity of inlet fluid c out. radius - Full of part circle, Stainless steel arm spring, trip spring, trip collars, and trip pin, Anti-backsplash tube. Calculate the local area ratio at which the shock is located inside the nozzle…. Converging Diverging Nozzle - Background Hour glass shaped nozzle Inlet of decreasing cross sectional area Outlet of increasing cross sectional area Used to accelerate pressurized fluid to supersonic speeds Useful in many turbine and jet engine applications. 2) and the Blasius or the laminar equation for wall. A pitot tube measures the stagnation pressure at the exit of the nozzle. 015m² as shown in the figure and the flow leave. Inlet nozzles should be reduced by 25 percent of the inlet supply pipeline or the nearest standard pipe size. The convergent parts of the nozzle are sharp and frictionless. 76 and know the diameter of the motor case. The diameter of the inlet nozzle The diameter of the inlet device depends on the inlet flow and on the type of inlet device. 4 and R=287 J/kg K) at a fixed stagnation temperature of 300 K and a supply pressure that can be varied. EXAMPLE 5-5 Acceleration of Steam in a Nozzle Steam at 250 psia and 700°F steadily enters a nozzle whose inlet area is 0. Chapter Eleven/Normal shock in converging–diverging nozzles ----- 5-6 ch. The cross sectional area of inlet and outlet are same, throat area and nozzles length are exactly same. 90ms ASOI 2. Hi I'm preparing a singe 2d nozzle model for analysis but there appeared a problem which I cannot solve. The surfaces which the inlet stream strikes are gently curved and the open area available for it increases as the flow passes through it. Design and Optimization of Aerospike nozzle using CFD. To evaluate the effects of inlet pressure on the ignition process of spray combustion, the images of the ignition process were recorded and the outlet temperatures were measured under inlet pressure of 0. CONVERGENT 2. If so how would I find that? I found my optimum exit/throat ratio to be 8. Your order may be eligible for Ship to Home, and shipping is free on all online orders of$35. (Report) by "SAE International Journal of Passenger Cars - Mechanical Systems"; Transportation industry Air conditioning Equipment and supplies Air conditioning equipment Automotive air conditioners Nozzles Vacuum pumps Design and construction Vacuum-pumps. This flow is supersonic in a portion of the nozzle, terminating in a normal shock inside the nozzle. Ensure inlet line loss (non-recoverable loss) is maximum 3% of set pressure (as per API RP 521) except permitted by API RP 521 Part II, section 4. 15 of the isentropic enthalpy drop. (Shipping charge of $8. If you need to adapt the Conejet to something other than a Teejet brass wand or nozzle body, order either the Female or Male inlet nozzle body. There are 2 more seals on the riser making leaks a thing of the past. Figure 6 presents the velocity distribution w related to the jet inlet velocity wo for the jet axis for the first and. Agid: 6110043. 025mm) and a highly polished surface ensures incomparable spray performance and. MAXON offers a wide variety of premixing. The optimized x-style swirl insert ensures a high operating safety due to its large free cross-sections. Assume the nozzle operates isentropically. The dual waterway design, lightweight stainless steel construction and integral bronze nozzle make the High-Flow Series well suited for marine applications and other corrosive. Air Inlet Cooling—-Air Atomizing Nozzles. The diameter of the inlet nozzle The diameter of the inlet device depends on the inlet flow and on the type of inlet device. Specific features include a dedicated lift fan inlet, an auxiliary engine inlet, a variable‐area lift fan nozzle, and a three bearing swivel duct (3BSD) feature on the engine. Nozzles are devices used to accelerate the fluid velocity at the cost of pressure. 1 1/4in and 1 1/2in models have an aluminum discharge tube with a knurled area for hand assembly; the 1in model has a brass tube standard. 8 bar and 450°C to a back pressure of 1 bar at the rate of 16 kg/s. In the recent years significant studies have been conducted in the area of micro-propulsion systems. Referring to our station numbering, the nozzle entrance is station 5 and the nozzle exit is station 8. Denote the inlet, throat, wall, and outlet. A de Laval nozzle using hot air at a pressure of 1,000 psi (6. Show transcribed image text A supersonic nozzle with an inlet area of 0. "nozzle hole" in the Bunsen burner IS the "gas. the inlet area to the exit area. Attached to the end of a hose, these jetting nozzles are fed into the drain where they blow the debris down the pipe where it can be removed. Because the area of the vane inlet is several times larger than the inlet nozzle, the fluid velocities are much smaller, allowing for good gas/liquid separation as well as smooth entry into the vessel. The causes of the nozzle inlet flow nonuniformities could be- as in the area of the arc which is along the axis of a heating chamber. 90Â° Airflow Blowguns Air flows out of the side of the nozzle to clean air filters, radiators, and inside recesses. 307 in 2), the PSV inlet arrangement is considered acceptable. 11 Prepared by A. Double brass 3/16 in. If I do a 30/15 deg. The nozzle operates at steady-state. Moreover, the Inlet 2 and Inlet 3 nozzle inlet micro tube obtain a higher mean outer wall temperature and a lower exergy destruction, which are more appropriate for the application of the micro thermophotovoltaic (TPV) system. the area between inlet nozzle and up to one third of impeller width there is flow separation, high turbulent flow and reversed flow [2, 3]. Hi I'm preparing a singe 2d nozzle model for analysis but there appeared a problem which I cannot solve. The engine intake on this commercial airliner is configured as a diffuser to slow the incoming air and provide a uniform flow into the compressor. 21 kg/m 3 and 40 m/s and leaves at 0. This temperature drops off to approximately 1000K close to the wall of the heating chamber. The nozzle section is a two foot long converging diverging nozzle to choke and accelerate the. Type in '4' and press the 'Set' button. Angle (degree) Mach number. But with the increase of nozzle to throat area ratio there is a significant increase in Suction Pressure for same nominal diameter jet pump at different Inlet Pressure. i) Find the velocity at the nozzle exit. 0 Introduction to Air Cooled Heat Exchanger. At transonic and supersonic flight speeds, the air upstream of the nozzle is not able to move out of the way quickly enough, and is compressed within the nozzle before being diffused into the combustor. 307 in 2), the PSV inlet arrangement is considered acceptable. The inlet area of the nozzle is 80 cm 2. To what pressure must the inlet stagnation pressure be raised so that: (a) A shock appears at the exit plane; (b) A shock of an area equal to 0. The results reveal that area enlargement and contraction affect the peak differently, depending on the inlet pressure, nozzle geometry, and standoff distance; while area contraction always improves the amplitude regardless of these factors. Double brass 3/16 in. If the inlet pipe is the same size sa the PSV inlet flange, the ratio of orifice flow area to pipe flow area varies significantly. Consider a converging-diverging nozzle with a throat area of 100 cm 2 and an exit area of 240 cm exiting into an altitude simulating test chamber at 50 kPa. PRDC Enrico Fermi Reactor The "worst reasonable accident" consists of the liberation of the pressurized. By Thomas R. pop-up height. Cylindrical nozzles each of which has a lateral orthogonal inlet of fluid into it we recommend to apply in pressure distributive pipelines. Assume that there are no energy losses and that the pressure (Pout) at the point of exit from the outlet of the nozzle is 100KPa. The flow area at the inlet is 800 cm^2. The nozzle is horizontal and there is a negligible heat loss from it. Please keep in mind that the nozzle exit area of the WTA was reduced by the inlet area of the boundary layer scoop - 144 [10. Vapours formed near the inlet of the orifice reduce the available flow area and hence increasing the flow velocity. (local area) to send us a copy of the firearms. It was found, that the highest stresses correspond to the center of nozzle fillet. Those fields were calculated using full WWER-1000 RPV model without crack embedded into the mesh (Fig. The throat area of the Rocket Nozzle in m^2. 1 Fog Nozzles Located Upstream of Inlet Filters. mized (for the given reservoir conditions and nozzle geometry) and the ﬂow is subsonic on both sides of the throat. Steady state operation of the nozzle 2. Evaporating water into the inlet air can provide critical cooling of heat sensitive areas such as electronic server rooms or animal enclosures. It indicates that nozzle to throat area ratio has the important aspect on characterization of Jet Pumps than the nominal diameter. Thermal gradients are caused by differences and fluctuations in temperatures between the inlet steam on the inside of the header and the outside surface on the fire-side and boiler room-side. nozzle definition: 1. To get a basic feel for the behavior of the nozzle imagine performing the simple experiment shown in figure 2. Design and Optimization of Aerospike nozzle using CFD. The image above is of an Euler spiral. 36 m^3/s, so I transfered this to a velocity of 3,75 m/s according the continuity equation Q=V*A. By applying this equation, thrust can be calculated, F = m2. To evaluate the effects of inlet pressure on the ignition process of spray combustion, the images of the ignition process were recorded and the outlet temperatures were measured under inlet pressure of 0. The impact of nozzle hole inlet rounding on nozzle performance is illustrated in Figure 16 where the effect of different nozzle hole inlet chamfer radius (r c) is simulated [2208]. 6) This selection is useful for determining the change in static pressure in a pipe due to a diameter change, determining flowrate, or designing a flow meter. 0024 slugs/ft^3, then the mass flux is:. Nozzle Type. Referring to our station numbering, the nozzle entrance is station 5 and the nozzle exit is station 8. Work and heat transfer are negligible, Q° = W°= 0. Standard features include a 100 mesh tube strainer, a dust cap assembly, and a 6-1/2 ft. Design Optimization of Nozzle Shapes for Maximum Uniformity of Exit Flow Karla K. Test range of inlet capture area ratios 43 26. Inlet Chambers Inlet chambers (Figure 13, 14, or 15) shall have a cross- sectional area at least five times the fan inlet area. At the exit from the nozzle the specific enthalpy is. 4, December (2008) ISSN 1813-7822 159 Flow Distribution in Manifolds Abstract A Numerical study of two-dimensional, turbulent, incompressible fluid flow through. 1 MPa or 1 atm), temperature of 502 K at the nozzle exit. 1 m2 and specific volume at inlet is 0. Determine (a) the mass flow rate through the nozzle, (b) the exit temperature of the air, and (c) the exit area of the nozzle. For an inlet area of 800 cm2, determine the velocity and the volume flow rate of the steam f at the nozzle exit. Engine thrust can be high only if the inlet duct supplies the engine with the required airflow at the highest posible pressure. Fume extraction nozzles The nozzles are the expression of the professionalism and completeness of WORKY's fume extraction product range. Consider a converging-diverging nozzle with a throat area of 100 cm 2 and an exit area of 240 cm exiting into an altitude simulating test chamber at 50 kPa. The initial flame formation and flame propagation and the effects of the inlet pressure on the initial flame formation were observed. It indicates that nozzle to throat area ratio has the important aspect on characterization of Jet Pumps than the nominal diameter. They tend to have a weak coverage area on either side of the nozzle (the yellow area in the drawing above). 1 Answer to Carbon dioxide enters an adiabatic nozzle steadily at 1 MPa and 500°C with a mass flow rate of 6000 kg/h and FIGURE P5-37E leaves at 100 kPa and 450 m/s. So the conventional impingement plate (Saunders, 1988), is a small flat or curved plate placed inside the shell facing the shell inlet nozzle. Larger upstream pipe on the suction side joins with a smaller suction nozzle using a reducer. Converging Diverging Nozzle - Background Hour glass shaped nozzle Inlet of decreasing cross sectional area Outlet of increasing cross sectional area Used to accelerate pressurized fluid to supersonic speeds Useful in many turbine and jet engine applications. The Area Of The Nozzle Is Given By A = A+0(1 - Bx) And The Inlet Velocity Varies According To U = 1/2 U_0(1 + Cos W T) Where A_0 = 5 Ft^3, L = Ft, B = 0. The clearance between the HLL and the inlet nozzle shall be 0. At the exit from the nozzle the specific enthalpy is. Simply: the rocket generates high pressure, a few hundred atmospheres. • Use the tester in a well ventilated area. The flow area at the inlet is 800 cm^2. Agricultural and Industrial applications. Inlet size and pressure recovery characteristics are important in determining the engine exhaust flow conditions. 1 Total nozzle length (mm) 13 2 Inlet diameter (mm) 32 3 Outlet diameter (mm) 18 Fig. The 15° Nozzle is optimized for concentrated cleaning power and time. We are most familiar with the shape of a subsonic nozzle. 44 m (8 ft) away from the shell. 14159 * 3^2 = 28. For example with 1 000 000 units of inlet airspeed to obtain 1 000 050 units of outlet airspeed, the outlet area must be 1 000 000 / 1 000 050 of the inlet area. At transonic and supersonic flight speeds, the air upstream of the nozzle is not able to move out of the way quickly enough, and is compressed within the nozzle before being diffused into the combustor. The larger the ratio, the higher the Mach number of the flow that your nozzle will produce (if you set this number very high, say >10) it may be difficult to see all the results clearly. Material Code. If you know what is the schedule of the 3/4" pipe you can calculate the flow area and if not smaller than 0. The nozzle design parameters studied include the shape, angle, height, width, and thickness of the ports and the bottom geometry. ASOI), the intrusion of bubbles into the nozzle sac and hole was observed with the MS nozzle. b) A nozzle is a device for increasing the velocity of a steadily flowing fluid. As a gas accelerates through the Nozzle, its velocity increases and its density decreases. Apex2 Inlet Nozzles (pkt 5) JavaScript seems to be disabled in your browser. Air Inlet Cooling—-Air Atomizing Nozzles. 5 and therefore P = 1. (cone bottom) Wheel Atomization Capability Yes - 14,000 RPM max, 8in/8 hole wheel Nozzle Atomization Capability Yes - 5 max. It is frequently used to control the flow rate, speed, direction, mass, shape and the pressure of the stream that emerges from them. The air leaves the nozzle at 15 m/s, 100 kPa, and 60°C. Brass construction with stainless steel trim. The pump has a head pressure of 127530Pa. 005 m 2 and it discharges into the atmosphere. Problem Specification. To avoid this the air in the nozzle inlet is heated to a temperature T 0 that depends on the required value of the Mach number. 6°C, (c) 38. A converging-diverging nozzle has an exit to throat area ratio of 2. The shock wave occurs at the exit plane. Nozzles are the basic components of wind tunnels (see Wind. Air enters a nozzle at 180 kPa, 67 °C, and 48 m/s. 20 and po, To fixed 1 2 • How far can A2 be reduced from. A nozzle increases the flow velocity. It is frequently used to control the flow rate, speed, direction, mass, shape and the pressure of the stream that emerges from them. Determine the forces and moments the left flange exerts on the right flange. Determine. This Item: 32SA/4PK - 32SA Simple Adjust Series Closed-Case ½” Inlet Gear Drive Rotor Sprinkler - 4 Pack 32SANZLPK - 32SA Rotor Nozzle Tree Regular Price:$2. Inlet Chambers Inlet chambers (Figure 13, 14, or 15) shall have a cross- sectional area at least five times the fan inlet area. 2 SOLUTION Since the areas are only in the vertical plane, there is no vertical force. Common Area Dust Control Nozzle Uses and Industries:. Combine with Teejet Threaded Caps CP1325 Brass, Teejet Nozzles, and TeeJet Strainers for a complete nozzle assembly. By means of rotating these nozzles about their longitudinal axes, we can change the values of the angles $$\beta$$ between the direction of the flow of fluid inside the distribution pipeline and the. Stagnation temperature T o at the inlet = 300 K. even inside the manifold plenum area, so they can. 1 m2 and specific volume at inlet is 0. Mach number N m is the ratio of the gas velocity to the local speed of sound. The inlet area of the nozzle is 80 cm2. Determine. The inlet area of the nozzle is 110 cm 2. To evaluate the effects of inlet pressure on the ignition process of spray combustion, the images of the ignition process were recorded and the outlet temperatures were measured under inlet pressure of 0. 5 degree bent tube (X1 is the spray pattern, X2 is the length from inlet to bend, X3 is length from bend to tip) SP1045-X1-X2-X3: Spray Nozzle with 45 degree bent tube. 3 mm^2 Throat Area 0. This greatly restricted airflow through the boiler, requiring a restricted exhaust nozzle opening to proivde sufficient draft. Over the past several decades, high-pressure fogging has evolved into a mature technology with a wide range of. Equations / Data / Solve : Let's begin by writing the steady-state form of the 1st Law for open systems. Choose a Rotator Nozzle Model: MP Rotator & MP Rotator SR Series. Steady state operation of the nozzle 2. Find (a) the velocity at the nozzle exit, and (b) the volume flow rate at the nozzle exit. Determine (a) the mass flow rate through the nozzle, (b) the exit temperature of the air, and (c) the exit area of the nozzle. iii) If the specific volume at the nozzle exit is 0. 02 ft^-1, w = 0. But in my case I'm looking for Convergent nozzle only so that I can get an estimate how wide combustion chamber should be to the exhaust area. A that the entry speed can be neglected, only modifies the flow inside the nozzle, but not the exit conditions. 4, December (2008) ISSN 1813-7822 159 Flow Distribution in Manifolds Abstract A Numerical study of two-dimensional, turbulent, incompressible fluid flow through. All atomizing nozzles produce a range of droplet sizes so, in order to characterize a nozzle with a. The dual waterway design, lightweight stainless steel construction and integral bronze nozzle make the High-Flow Series well suited for marine applications and other corrosive. This flow is supersonic in a portion of the nozzle, terminating in a normal shock inside the nozzle. If the inlet area of the nozzle is 90 cm2, determine the mass flow rate through the nozzle, and the exit area of the nozzle. Free Online Library: Influence of Nozzle Divergent Part Length and Throat Diameter on Vortex Control of Initially Subcooled Flashing Flow. Two nozzle shapes are usually used: a simple circular (iris) nozzle or a plug. nozzle for the job. - The steamhas a very high velocity at the end of the expansion, and the enthalpy decreases as expansion takes. It is important that the outer edges of inlet 48 and outlet 52 of flow straightener 44 butt against the edges of outlet 50 of supply hose 12 and inlet 34 of nozzle 10, respectively, such that substantially no gaps are present between the interior of supply hose 12, bore 46 of flow straightener 44 and bore 32 of blast nozzle 10. OR here is another easy way to predict the necessary thrust to lift an object which weighs 9,000lbs. Check that the micrometer dial has been correctly zeroed. The inlet area of the nozzle is 40 cm. Nozzle with controllable vortex at the nozzle inlet have been designed and manufactured for experiments, as shown in Figure 1. Consider a converging-diverging nozzle with an inlet area of A = 0. i) Find the velocity at the nozzle exit. The inlet area of the nozzle is 40 cm2. , low flow rate) and about 0. The length of the nozzle will depend upon the divergent angle which is mostly ~ 15. On a monthly basis, operate the pattern control through a full range of motion from fog to straight stream. 15×103 Pa at the inlet area and then it increased to get to a value of 7. indicate poor nozzle performance. Larger upstream pipe on the suction side joins with a smaller suction nozzle using a reducer. But the flow velocity for DME fuel is more than the diesel fuel due to decrease in pressure. The nozzle is horizontal and there is negligible heat loss from it. Nozzles are used for three purposes: to distribute a liquid over an area, to increase liquid surface area, and create impact force on a solid surface. A propelling nozzle is the component of a jet engine that operates to constrict the flow, to form an exhaust jet and to maximise the velocity of propelling gases from the engine. Orifices and nozzles are typically used to deliberately reduce pressure, restrict flow or to measure flow rate. A converging-diverging nozzle has an exit to throat area ratio of 2. PRDC Enrico Fermi Reactor The "worst reasonable accident" consists of the liberation of the pressurized. 0 Tt8 / Tt5 = 1 The total pressure pt across the nozzle is constant as well: pt8 / pt5 = 1. For each set of replicates at a specified test pressure (inlet), record means (across reps) of: accuracy (percent error), standard deviation, and CV (coefficient of variation) of outlet pressure. it is basically a tapering tube, with the outlet diameter lesser than the inlet diameter. The convergent parts of the nozzle are sharp and frictionless. Steam leaves the nozzle at 200 psia with a velocity of 900 ft/s. Location Options for the Inlet Fog Nozzles. Some turbojet aircraft including the Su-7, MiG-21, English Electric Lightning, and SR-71 also use an inlet cone. The pressure reducer adjusts from 30 psi for spray nozzles to 40 psi for rotary nozzles with the turn of a screwdriver. Assume that there are no energy losses and that the pressure (Pout) at the point of exit from the outlet of the nozzle is 100KPa. By applying this equation, thrust can be calculated, F = m2. Note that the exit area is the same for all the nozzles. nozzle are generally of two kinds 1. The graph on the left shows the shape of the nozzle, chamber on the left, exit on the right. Off-design condition is considering, the flow through the nozzle at the over-expanded condition with a shock wave inside the diverging section of the nozzle. The coefficient of discharge for nozzle is 0. The modeling has been done in ANSYS. C m is the mass flow rate, v is the specific volume, A is the cross-sectional area and C is the velocity. They are primarily used on ramjets, such as the D-21 Tagboard and Lockheed X-7. 5304 kg/s, (b) 184. If the inlet pipe is the same size sa the PSV inlet flange, the ratio of orifice flow area to pipe flow area varies significantly. It should be used for heavy duty/industrial strength work such as removing tar, resin, glues and rust. Right: Machine nozzle-tip orifice is the same size as the inlet orifice, eliminating any stagnant material at the wall of the inlet orifice. Determine (a) The mass flow rate through the nozzle, (b) The exit temperature of the air, and (c) The exit area of the nozzle. Simply: the rocket generates high pressure, a few hundred atmospheres. Here we use a converging diverging nozzle to connect two air cylinders. Isentropic Nozzles • Apply equations for isentropic flow with area change to nozzles • Nozzles - increases velocity of fluid (no work) - converts thermal energy to KE (T→u) • For conventional (wall-bounded) nozzles, two types: - converging - converging-diverging (CD) M≤1 0

format: plain


text:
# American Institute of Mathematical Sciences

July  2007, 1(3): 477-543. doi: 10.3934/jmd.2007.1.477

## Renormalization and central limit theorem for critical dynamical systems with weak external noise

 1 Department of Mathematics & Statistics, McMaster University, Hamilton, ON L9H 1X2, Canada 2 Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, United States

Received  August 2006 Revised  April 2007 Published  April 2007

We study the effect of weak noise on critical one-dimensional maps; that is, maps with a renormalization theory.
We establish a one-dimensional central limit theorem for weak noise and obtain Berry--Esseen estimates for the rate of this convergence.
We analyze in detail maps at the accumulation of period doubling and critical circle maps with golden mean rotation number. Using renormalization group methods, we derive scaling relations for several features of the effective noise after long periods. We use these scaling relations to show that the central limit theorem for weak noise holds in both examples.
We note that, for the results presented here, it is essential that the maps have parabolic behavior. They are false for hyperbolic orbits.
Citation: Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477
 [1] Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597 [2] James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167 [3] Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979 [4] Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917 [5] Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106 [6] Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223 [7] Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143 [8] Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Period doubling and reducibility in the quasi-periodically forced logistic map. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1507-1535. doi: 10.3934/dcdsb.2012.17.1507 [9] Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012 [10] Frédéric Naud. The Ruelle spectrum of generic transfer operators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2521-2531. doi: 10.3934/dcds.2012.32.2521 [11] Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933 [12] Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651 [13] Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597 [14] Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099 [15] Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098 [16] Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633 [17] Alexander Blokh, Michał Misiurewicz. Dense set of negative Schwarzian maps whose critical points have minimal limit sets. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 141-158. doi: 10.3934/dcds.1998.4.141 [18] Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555 [19] Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095 [20] Wenyu Pan. Effective equidistribution of circles in the limit sets of Kleinian groups. Journal of Modern Dynamics, 2017, 11: 189-217. doi: 10.3934/jmd.2017009

2018 Impact Factor: 0.295


text:
Endre søk
Begrens søket
1 - 5 of 5
Referera
Referensformat
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Annet format
Fler format
Språk
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Annet språk
Fler språk
Utmatningsformat
• html
• text
• asciidoc
• rtf
Treff pr side
• 5
• 10
• 20
• 50
• 100
• 250
Sortering
• Standard (Relevans)
• Forfatter A-Ø
• Forfatter Ø-A
• Tittel A-Ø
• Tittel Ø-A
• Type publikasjon A-Ø
• Type publikasjon Ø-A
• Eldste først
• Nyeste først
• Disputationsdatum (tidligste først)
• Disputationsdatum (siste først)
• Standard (Relevans)
• Forfatter A-Ø
• Forfatter Ø-A
• Tittel A-Ø
• Tittel Ø-A
• Type publikasjon A-Ø
• Type publikasjon Ø-A
• Eldste først
• Nyeste først
• Disputationsdatum (tidligste først)
• Disputationsdatum (siste først)
Merk
Maxantalet träffar du kan exportera från sökgränssnittet är 250. Vid större uttag använd dig av utsökningar.
• 1. Agmon, Shmuel
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Persistence of embedded eigenvalues2011Inngår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 261, nr 2, s. 451-477Artikkel i tidsskrift (Fagfellevurdert)

We consider conditions under. which an embedded eigenvalue of a self-adjoint operator remains embedded under small perturbations. In the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m < infinity we show that in favorable situations, the set of small perturbations of a suitable Banach space which do not remove the eigenvalue form a smooth submanifold of codimension in. We also have results regarding the cases when the eigenvalue is degenerate or when the multiplicity of the continuous spectrum is infinite.

• 2. Derks, Gianne
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. matematik.
Perturbations of embedded eigenvalues for the bilaplacian on a cylinder2008Inngår i: Discrete and Continuous Dynamical Systems: Series A, ISSN 1078-0947, Vol. 21, nr 3, s. 801-821Artikkel i tidsskrift (Fagfellevurdert)
• 3. Derks, Gianne
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Perturbations of embedded eigenvalues for the planar bilaplacian2011Inngår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 260, nr 2, s. 340-398Artikkel i tidsskrift (Fagfellevurdert)

Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues are linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials.

• 4.
University of Virginia, Department of Mathematics.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. London School of Economics, Department of Mathematics.
Existence and exponential decay of solutions to a quasilinear thermoelastic plate system2008Inngår i: NoDEA. Nonlinear differential equations and applications (Printed ed.), ISSN 1021-9722, E-ISSN 1420-9004, Vol. 15, nr 6, s. 689-715Artikkel i tidsskrift (Fagfellevurdert)

We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in , n ≤ 3. Existence of finite energy solutions describing the dynamics of a nonlinear thermoelastic plate is established. In addition asymptotic long time behavior of weak solutions is discussed. It is shown that finite energy solutions decay exponentially to zero with the rate depending only on the (finite energy) size of initial conditions. The proofs are based on methods of weak compactness along with nonlocal partial differential operator multipliers which supply the sought after “recovery” inequalities. Regularity of solutions is also discussed by exploiting the underlying analyticity of the linearized semigroup along with a related maximal parabolic regularity.

• 5.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Generators for rings of compactly supported distributions2011Inngår i: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 69, nr 1, s. 63-71Artikkel i tidsskrift (Fagfellevurdert)

Let CUnknown control sequence '\tt' denote a closed convex cone in \mathbb RdRd with apex at 0. We denote by E¢(C)Unknown control sequence '\tt' the set of distributions on \mathbb RdRd having compact support contained in CUnknown control sequence '\tt'. Then E¢(C)Unknown control sequence '\tt' is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on [^(f)]1,..., [^(f)]nf1fn for f1,... ,fn Î E¢(C)Unknown control sequence '\tt' to generate the ring E¢(C)Unknown control sequence '\tt'. (Here [^(  ·  )] denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Lars Hörmander. En route we answer an open question posed by Yutaka Yamamoto.

1 - 5 of 5
Referera
Referensformat
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Annet format
Fler format
Språk
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Annet språk
Fler språk
Utmatningsformat
• html
• text
• asciidoc
• rtf


text:
# Two combinatorial geometry problems

1) Prove that a (convex or concave) polygon having the perimeter of 2 can be placed inside a circle which has the diameter of 1 pefectly

Hint from teacher:using symmetry of point.

2) With each point X inside convex polygon:A(1)A(2)...A(2009),we set S(X)=XA(1)+XA(2)+...+XA(2009)

Do 2 point P,Q exist inside the polygon that they satisfy:

PQ=1 and |S(P)-S(Q)|=2007

Note by Hunter Killer
4 years, 10 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

It is best to separate out your questions into 2 discussion posts, since they are very different questions.

For your second question, you will need to specify extra conditions. For example, if the polygon is too small, then we might not be able to find 2 points such that $$PQ = 1$$, or allow for $$S(P) \geq 2007$$. I have a slight suspicion that the vertices of the polygon lie on the unit circle.

Here's another hint for question 1, with an approach that is very different from what you teacher suggests. This is the first approach that I thought of, which is influenced by how I approached similar problems in the past.

Hint: Kelly's Theorem.

Staff - 4 years, 10 months ago

Thank Sir.I have just examined Helly Theorem and surprisingly,i prove problem 1 immediately.The most beautiful theorem that I have ever seen :))

- 4 years, 10 months ago

Can you write up your solution so that others can learn from it? You have to be careful with some details.

Staff - 4 years, 10 months ago


text:
# How do you find the asymptotes for 1. f(x) = (3x) / (x+4)?

Mar 6, 2016

vertical asymptote x = -4
horizontal asymptote y = 3

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.

solve: x + 4 = 0 → x = -4 is the equation

Horizontal asymptotes occur as lim_(x→±∞) f(x) → 0

If the degree of the numerator and denominator are equal , as is the case here , both of degree 1. The equation can be found by taking the ratio of the leading coefficients.

$\Rightarrow y = \frac{3}{1} = 3 \Rightarrow y = 3 \text{ is the equation }$

Here is the graph of the function.
graph{3x/(x+4) [-20, 20, -10, 10]}

Mar 6, 2016

Just another way of looking at the same thing

#### Explanation:

Given:$\text{ } f \left(x\right) = \frac{3 x}{x + 4}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Lets 'get rid' if the $x$ in the numerator!

f(x)=(cancel(x)(3))/(cancel(x)(1+4/x)

Straight away you can observe that the denominator is 0 when $x = - 4$. Thus the expression is undefined. The result is an asymptote at $x = - 4$

As absolute $x \to \infty$ then $\frac{4}{x} \to 0$ so we end up with

${\lim}_{x \to \pm \infty} f \left(x\right) = {\lim}_{x \to \pm \infty} \frac{3}{1 + \frac{4}{x}} = + 3$