DOE: Resolution III Plackett-Burman design

In Design of Experiments DOE, Plackett-Burman design is one of the screening designs which is used to detect the most significant factors out of large number of factors. The idea behind this design is to make number of level combinations of any two factors is the same through the whole design. To achieve this criteria, massive iterations are required; for this reason statisticians decided to calculate this design offline and save it instead of calculating it each time. 

Plackett-Burman design is based on 2-level factors and has a resolution of III. 

To construct Plackett-Burman design we first calculate number of runs. Number of runs is a multiplier of 4: e.g. 4,8,12,16,.... where number of runs should be larger than number of factors. For example, 9 factors will require 12-runs Plackett-Burman design. Simply, number of runs of this design can be calculated by the simple following equation:

Number of runs=ceiling((Number of factors+1)/4)*4

Since the treatments of this design are pre-calculated, there are two ways to get these treatments. In this article we will go through an example of 12-runs Plackett-Burman design.

The first method is to use what is called the "generating vector". The generating vector is a pre-calculated vector of all levels of the first factor (factor A) except for the last level. For example, the generating vector of 12-runs design will has the length of 11. In this method, treatments will be created by writing all levels of each factor column one by one.

To create treatments, we start with the first factor and assign the following vector to its levels:

++-+++---+-

Where "+" denotes the high level and "-" denotes the low level. Reading this vector from left to right represent factor levels from top to bottom in the design matrix.

After that, each vector of each factor will be derived from the previous vector. For example, vector B will be derived from vector A, vector C will be derived from vector B and so on. Simply, the last element of vector A will be the first element of vector B. This is better imagined by assuming that the generating vector forms a closed circle and each factor vector will start with the last value of the previous vector. The following image can explain how this may look like.

At the end, after generating all vectors, the last level of each factor will be set to low level "-". Or in other words, the last treatment in the table will be with all factors set at low level.

The following .GIF animation shows how the generation of each vector is derived from the previous vector.

Alternatively, the second method, some books prefer to give the entire table of factor levels for all treatments. Programming wise, the single generating vector has the advantage of small memory consumption and the disadvantage of extra processing time. The following table shows all treatments for 11 factors. It is very noticeable how diagonal the matrix looks like because of shifting explained before.

References:

https://www.itl.nist.gov/div898/handbook/pri/section3/pri335.htm

https://en.wikipedia.org/wiki/Plackett%E2%80%93Burman_design


Key words:

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Plackett-Burman design step by step

How automatic water feeder works

Few days ago, I was thinking how the automatic water feeder (dispenser) of pets and birds works. I searched over the internet and the answers seemed to be very traditional "air has to replace the water going out"!!!!.

The automatic water feeder is a simple mechanical device that is intended to continuously and automatically feed water for pets and birds without the need to refill it frequently. The innovation in this device that water feeding is done without flooding or over flow.

Simply, the automatic water feeder consists mainly of two elements: tank and sump. Referring to the schematic below, the tank has two properly sized holes at two different heights (extremely small holes will not allow air flow or water flow because of surface tension force).  The bottom hole is the water feeding (bleeding) hole and the top one is the air hole. The ratio between the volume of the tank and the volume of the sump decides how frequent the user has to refill the tank. The top hole level also decides the maximum water level in the sump. The lower hole level decides the wasted water volume inside the tank that can not be fed to the sump.

The same concept also exist in the domestic cold and hot water dispenser.

The tank used in most common automatic water feeder for birds is a lid which is filled with water, the sump is placed over it, and then the whole setup is flipped vertically to start working.

So, how this automatic water feeder works?

To answer this question I will explain this on the general physical model of this automatic water feeder as shown in the image above.

After filling the tank and flipping it over, air pressure inside the tank is atmospheric pressure. Water pressure at top and bottom holes is higher than the atmospheric pressure outside, so water flows from both holes but with higher rate from the lower hole (assuming that both holes have the same diameter). As a result, air vacuum increases inside the tank.

Modelling water as a rigid body, there will be a force acting on it because of pressure difference between its sides: a vacuum pressure from one side and atmospheric pressure from the other side. Outside air will push water surface (at hole 1)  inside till the applied force can overcome the surface tension of water, penetrate water, trapped in, and then move up because of buoyancy force until it joins the trapped air at the top of the tank resulting in decreasing the vacuum effect.

This surface tension describes why air is getting inside in an intermittent way in a form of large bubbles. As a result, water will flow out and so on. This will repeat over and over until water level is above the top hole center where a tiny area of the hole is not covered with water. At this instant, air pressure can not exert the enough force on this very tiny area to overcome surface tension and penetrate water and air and water flow will stop.

When outside water level is just covering the top hole, and although the top layer of the outside water still has atmospheric pressure, air can not get inside the tank because air pressure is applied on the whole water surface and water is an incompressible fluid. 

By consuming outside water by pets or birds or due to evaporation, and as long as the water level inside the tank is going down, the flow rate of water from the bottom hole is getting slower because the pressure difference across its sides is reduced.

In this post I tried to proof why water stops at the level of the top hole using mathematical method by studying different general water levels in the sump and assuming a static steady state where the water level is static and this -as a result- means that there is no water flow through both holes. By concept, water will not flow through hole when the pressures on its sides are the same (ignoring pressure drop across the hole). In this model, vapor pressure of water will be ignored.

First case: Outside water level below the bottom hole:
By applying the following equations and solving them simultaneously it has two physical solutions:

[A] Top and bottom holes are coincident (one hole). This is not our case because we have two holes.
[B] Impossible to happen if there are two holes at different levels.

Second case: Outside water level above the bottom hole:

By applying the following equations and solving them simultaneously it has two physical solutions:

[C] Water is at the level of the top hole.
[D] Impossible to happen

based on solutions [B] and [C], water will flow as long as it is below the top hole and it will stop flowing only when the water level reaches the top hole.

Third case: Outside water level above the top hole:

This case will only happen when extra water is added by the user above the top hole level. By applying the following equations and solving them it has only one physical solution:

[E] It is possible for all water levels above the top hole level as long as the water level is lower than the tip of the sump.

Fourth case: Inside water level below the top hole:

In this case water levels inside and outside will be same.

This theory of operation is only valid for relatively short tanks which are used in domestic and household applications and is impossible to apply for any tank with water level higher than approximately 10 meters (assuming zero absolute pressure above water inside the tank).

References and useful links:

https://physics.stackexchange.com/questions/88669/physics-of-the-inverted-bottle-dispenser

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Kitchen door frame aluminum profile

The following cross-section drawing shows one of the commonly used aluminum profiles in kitchen cabinets in Egypt. It is usually used for cabinet door frames. I drew this profile myself using a scrap piece (it was painted) from a nearby aluminum workshop. The real dimensions may slightly vary because of painting thickness. The drawing shows only the governing dimensions of the profile in millimeters. You can also download the CAD model of the profile from this link.

Orthogonal bars of this profile are connected at door frame corners using the following plastic connector or joint (in the render below). You can download the CAD model from this link.

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Desktop FDM 3D printer bed leveling

In this post I propose a simple and smart method to level the bed or build plate of most desktop FDM 3D printers (like CReality CR-10S 3D printer). It comes as an alternative to the traditional A4 paper tightness method.

The method simply uses a digital multi-meter (a cheap one can work fine) and aluminum tape of known thickness. First of all, the build plate is cleaned and four pieces of aluminum tape will be placed at the corners of the build plate with no wrinkles or trapped air beneath. Then, the digital multi-meter will be set on electric continuity tester (buzzer) which will beep when the resistance is lower than certain small value. One probe -let's say the red one- of this multi-meter will be connected to the hot end heater block (which is electrically conductive) and the probe can be placed in the hole of the NTC sensor to be clamped using the set screw.  After that, the user will configure the printer to enter the step-by-step bed leveling mode and the extruder moves to the first point. At this instant, the other probe -the black one- will be connected to the piece of aluminum tape at the corner at which adjustment is being done. Bed level adjustment screw will be tuned until the buzzer sound is heard and so this point is calibrated. The same process will be done for the rest of points and the whole build plate is leveled.

If the entire area of the build plate will not be used in the future prints, you can leave aluminum tape pieces in their places for future bed leveling.

 

Electrical cable tray oblong punching dimensions

The picture below shows two standard metric sizes for oblong punching used in electric cable trays (specifically in Egypt).

External gear pump swept volume

The following picture is a visual representation for the swept volume of one gear in external gear pump. The total swept volume of the pump is twice this value.

Radio pallet shuttle system selection and design

Few days ago, I came across what is called "Pallet shuttle system" which is used in pallet rack system as an alternative to "drive-in" and "drive through" of fork lifts to save time and increase safety levels.

The following table shows some useful selection criteria. It is also useful for people who are willing to design this piece of equipment. Based on the table below, and by some reversed engineering of the top brands manufacturing this equipment; I found that "Mecalux" is really the best design from all aspects.

Radio pallet shuttle system selection
All pallets are of the same item	Yes
Mixed items: Pick one item from one end and the other from the other end
Handling equipment	Fork lift and stacker
Shuttle dimensions	_
Minimum pallet size	_
Maximum pallet size	_
Maximum pallet overhang distance	_
Maximum pallet deflection	_
Maximum track distance	_
Lifting distance	_
Underneath ground clearance	_
Operating conditions
Operating temperature: Negative operating temperatures for refrigerators
Humidity
Ingress protection IP
Dynamics
Four wheel drive	Preferred
Maximum load [kg]	_
Full-load speed	_
No-load speed	_
Acceleration and deceleration profiles	_
Lifting speed	_
Weight-dependent acceleration
Number of wheels to extend life of wheels
Lift the whole upper plate
Energy and efficiency
Energy consumption to fully charge the battery	_
Regenerative breaking system	Preferred
Standby mode	Yes
Monitoring and control
Monitor battery charge remotely	Yes
Monitor battery remaining time remotely	Yes
Switching between manual and automatic modes	Yes
Remote control	Yes
Remote control communication technology requires no infrastructure	Yes
Handheld remote control	Yes
Rechargeable remote control	Yes
Remote control battery level indicator
Light display in remote control	Yes
Remote control drop down distance	3 meters
Magnetic mount remote control	Yes
Large buttons and icons in remote control	Yes
Iconic color-coded buttons	Yes
Single-hand operated remote control	Yes
Single remote controls multiple shuttles	Yes
No. of controlled shuttles using single remote	Unlimited
Remote control range [m]	_
No interference between multiple remote controls	Yes
User accounts and authentication	Yes
Configuration
PC programmable	Preferred
Programmable maximum speed	Preferred
Programmable maximum acceleration	Preferred
Programmable loading and unloading positions	Yes
First In Last Out Loading and unloading in the same location	Yes
First In First Out Loading and unloading are in opposite locations	Yes
Programmable home (for FIFO system)	Yes
Connectivity
RS232	Yes
RS485	Yes
Ethernet	Preferred
Wi-Fi	Preferred
Features
Shuttle power permanent push button	Yes
Power indicator light	Yes
Forward and backward blinking indication lights	Yes
Auto-homing	Yes
Auto-home and stop when battery is very low	Yes
Partial loading	Yes
Partial unloading (unload certain amount of pallets)	Yes
Automatic loading (LIFO)	Yes
Automatic unloading (LIFO)	Yes
Unload one-by-one, to reduce number of reallocations. This saves energy and increase the life time of the shuttle. In this mode, the default home position will be the unloading side and when the shuttle detects that the pallet is removed it will go back and bring the next pallet	Yes
Partial compacting	Yes
Full compacting	Yes
Self-align shuttle with pallet	Yes
Weight measurement	Preferred
Odometer	Preferred
On-deck pallet detection	Yes
Detect any pallet regardless the material: [1] Wood [2] Plastic [3] Steel	Yes
Detect any pallet regardless the color	Yes
Detect any pallet regardless the pattern	Yes
Pallet size detection	Preferred
Variable pallet size	Yes
Programmable pallet spacing	Yes
Counting number of pallets	Yes
Calculate remaining pallets to fully-load the track	Preferred
Wide-angle navigation camera on both sides	Optional
Flash light for camera	Yes
Speed monitoring	Preferred
Audible alarm	Yes
Shuttle number (ID) label on: [1] Front [2] Back [3] Bottom (large) [4] Top (large)	Yes
Bottom lights to easily recognize the shuttle specially when the shuttle is at elevated heights	Preferred
Possibility of two shuttles in the same lane (for FIFO)	Preferred
Forced operation function	Yes
Location signal To know where the shuttle cart is	Yes
Alarm when there is a place left for only one pallet and the labor place a pallet above it	Yes
Auto-seek remote coverage	Yes
Battery
Rechargeable battery	Yes
Battery technology	Lithium polymer Lithium ion
Sealed battery	Yes
Fire retardant casing	Yes
Battery life (years)	5
Battery life (number of charges)	1000
Battery price (cheaper battery is preferred)
Battery charger	Yes
Battery charger rating	220 V - 50 Hz
Battery charger stop after charging is complete	Yes
Battery full-charge alarm	Preferred
Battery charging time	< 8 hours
Battery charge percent indicator in the charging station	Yes
Battery discharge time at maximum load	> 8 hours
Battery location is close as possible to the side edge of the shuttle, so it can be removed ergonomically	Yes
Battery weight (for ergonomics)
Battery number label	Yes
Onsite charging	Preferred
Charging station for single battery	Yes
Spare battery	Yes
Standby operating time	_
Remove battery without any hand tools	Yes
Safety
Rail end stoppers	Yes
Pallet centralizers	Yes
Emergency push button on both sides (front and back)	Yes
Human hit safety (specially for ground racks)	Yes
Auto-home at enough distance from the rack end, so the fork lift will not hit it during unloading.	Yes
End of rail (track) detection (using two sensors)	Yes
End of rail (track) detection sensors can detect track end of any color	Yes
End of rail (track) detection sensors can detect track end full of dust	Yes
Open rail (track) detection	Yes
Anti collision with other pallet	Yes
Obstacle detection	Yes
Rail detection	Yes
Rail bump detection	Preferred
On-floor detection	Preferred
Bottom anti-slip surface to avoid slipping from fork lift	Yes
Electromagnetic locking system: Bottom very strong electromagnets to hold firmly with fork lift	Preferred
Electromagnetic locking system indictor light in the bottom	Preferred
Detect if it is lifted on fork lift, so it will not operate even if it is in automatic mode	Yes
Stop laser beams when it is lifted on fork lift	Yes
Rubber bumper at the front and end	Yes
Rubber stoppers at both sides in the front and back to avoid hitting track stoppers	Yes
Wheels are almost at the end of the shuttle to prevent flipping back	Yes
Pallet slip detection: Stop in case of pallet slip.	Yes
Pallet flip detection: Stop in case of pallet flip.	Yes
Side rollers to make sure that the shuttle is always on track	Yes
Anti-slip surface	Yes
Braking system	Yes
Sealing plugs (caps) for all communication interface ports on the shuttle	Yes
No wires, shrink wraps, clamps, frames can tangle with wheels	Yes
Ratchet lifting	Preferred
Human cart accessory (with chair inside)	Yes
Shuttle can not move if placed on normal floor	Yes
Reliability
Electromechanical lifting system	Yes
Overload protection: [1] Over current [2] Over temperature	Yes
Overload indication	Yes
Battery over-discharge protection	Yes
Spring loaded rail wiper to clean rails and avoid wheel slippage and wheel pitting	Yes
On-shuttle battery level indicator	Preferred
Wheel slip detection	Yes
Normally free to move when battery is dead	Yes
Normally not-elevated when battery is dead	Yes
Rust free materials	Yes
Dust proof sensors	Yes
Moisture proof sensors	Yes
Another shuttle can push or pull faulty shuttle	Yes
Shuttle can be lifted using fork lift if it is placed on flat floor	Yes
Shuttle have underneath pocket for forklift	Preferred
Front and back hooks for pulling it in case of getting faulty	Yes
Easy replaceable wheels	Yes
Protective guard for buttons and user interface when hit another object	Yes
Maximum load label on the top (readable from both sides)	Yes
Weight overload audible alarm	Yes