# (updated)Electric Power Generation using Buoyancy under Automatic Circulation of a Constant Volume of Water

**Major Characteristics and Merits**

**Introductory Review: **

** Archimedes’ Principle**

*Ｔhe loss in measured weight of an object i**mmersed in a fluid is equal to the weight of the displaced fluid.*

**As illustrated by Ｆig.1, as far as buoyancy> weight holds， the object keeps rising in water. Eventually, the object emerges from the water surface. And the water volume displaced by the object decreases. As a result, buoyancy decreases. During this development, there exists a water depth where balancing occurs , namely, buoyancy=weight. This is referred to as water line. **

** Afterall, buoyancy is attributed to the terrestrial gravity. Thus, below the water line, buoyancy does not change regardless of the water depth. **

**Pre-Operation Framework**

**Preliminary Notes**

**Befor****e **** explanations of cyclic time series ****①～⑧****, ****there are some points important throughout.**

### **Explanation of one -cycle time series ①②③…⑧**

# Electric Power Evaluation

** T****he object repeats up-down motion between upper and lower limit stoppers. It is the ultimate goal**** to**** acquire electricity out of the mechanical up-down movements.**

**As for the mechanical-to-electrical energy conversion, a dynamo connected to a crank shaft m isay be considered. Instead of such conventional method, ****one may think about alternative approach based on electromagnetic induction illustrated below (pruning101.com/?p=2829). This is a prospect since energy loss is expected to be small. Further research remains to be made in this regard. **

** When the object arrives at the upper-limit stopper position, the object has potential energy U （the lower-limit position is the base）.**

**Thus,**

** ****Ｕ＝Ｍｄ [** kg·m^{2}·s^{−2} , **Joule]**** Eq.1**

**Here, ****M [kg] is mass of the object , ****d [m] **** is distance between upper**** and lower limit stoppers.**** U corresponds to the work (term of physics) done by the object during **** one up-down stroke. **

**Electric power is related to the time spent to attain the work mentioned above. ****That is to say, the shorter the time, the larger the electric power. **** Thus, if T [sec] denotes the time, the electric power P [Joule/sec=watt], is expressed as follows.**

**P****＝U / T**

**Now, T is estimated as follows.**

**Regarding the motion environments of the object, two types A and B are recognized, **

**A Buoyancy environment**

** The whole or a part of the object is surrounded by water, so that buoyancy works. **

**B Rain (shower) environment**

** There is no buoyancy. Basically ( aerodynamic resistance is negligible), The object motion is gravitational free fall. **

**Overlook view of Fig3 and Fig4 tells that A and B are roughly half-and-half during a single cycle. So, estimation of T in question is considered separately. And, it is assumed that the object moves between the upper and lower limit stoppers.**

**Then, the average of T of A and T of B will be calculated. This result is thought to be an approximation of the time spent for one cycle.**

**A B****uoyancy environment**

**Let us denote entities relevant to the object as follows.**

** M: mass [kg] W: weight: [kg m sec-2 ] α: acceleration [m sec-2] ****F: buoyancy [kg m sec-2 ]**

** ****Newton’s law of motion says mass x acceleration = force. Accordingly,**

**Mα＝F-W **

** ****Since W=Mg (where g stands for the gravity acceleration constant), the forgoing equation leads to**

** ****α= (F/W-1)g Eq. 2 **

**A problem is how large buoyancy should be in comparison to the weight W.**

**For example, if F>=2W (buoyancy is twice greater than weight) , then α**** > = g. ****This means that the object in water would move upward with a speed much faster than free fall. This is not realistic for hydrodynamic reasons. That is, high ****speed not only increases water resistance, but also induces turbulence and eddy.**

**Such troublesome flows would invite instability of the object’ movements, and ****in the first place, there is no guarantee that Archimedes’ principle will work.**

** .**

**So, for the condition F>W to be moderate, **

**F=1.1 W Eq. 3**

**.is set for the rest of present analysis. Then, fr****om Eq.2 , upward acceleration is **

**α＝0. 1g . Eq. 4 **

**This is 10% of the free fall acceleration and is thought to cause no hydrodynamic problem. **

**In general, distance=acceleration x time ^{2} /2.**

**In the present case, distance is between upper and lower limit stoppers****、thus**

**distance=d [m]****．Thereby, d=αT ^{2}/2 , and **

**Since ****α＝0. 1g from Eq. 4, **

** T= Eq.5**

**B Rain ****environment**

** There is no buoyancy. Acceleration ****α ****is downward with gravitational constant g, ie ****α=g****. Thus, the time for the object to spend from the upper to bottom limit stopper is**

** T= Eq.6.**

** As mentioned before, for the present approximation analysis, T **** to complete one cycle is average of T of Eq. 5 **** and T of Eq.6.**

**Accordingly, by using Eq.1 the electric power is estimated as follows.**

**Implementation Example 1 **

**Home power supply site for everybody of a family— illumination, air conditioner,**

**refrigerator, washing machine, television sets, computers, etc**

**The object itself (the hollow part) is a cube of 50 cm x 50cm x 50 cm****[liter]. Since the specific gravity of water is 1, the mass M= 125 [kg]**

**The distance d between the upper and lower limit stopper is 1 [m].**

**As for the object’s weight W****，W is adjusted such that F=1.1 W for stability reasons discussed earlier. This adjustment is made appropriately by choosing materials and thick ness of the framework and walls. **

**Next, the time T for the object to spend one cycle is estimated.**

**For the buoyancy environment A (upward movements), the time is 1.43 sec from Eq. 5.**** For the rain environment B (downward movement), the time is 0.43 sec from Eq.6. Thus, using Eq. 7,**

**T= ( 1.43 + 0.45 )/2=0.94 sec.**

**In practice, however, the time will be somewhat longer than this, due to hydrodynamic drag and friction as well as air resistance. ****Accordingly, **

**T=2 sec **

**is assumed here. This is thought a moderate setting from stand point of the power generating performance.**

**Now, since M=125 [kg] and m=1 [m], potential energy U that the object possesses at the upper limit stopper position is**

** .U=Md=125 [ Joule] and the electric power is P= U/T= 62.5 2 [Joule sec ^{-1} = watt ]. **

**Thus, electrical quantity gained by 1 hour operation is 62.5 x 3,600 =225kwh.**

**If this electricity is accumulated by storage battery, average electricity needed by four – people family is covered for 12 days.**

__Average electric quantity par day used by four-people family __

__Average electric quantity par day used by four-people family__**air conditioner 10.8 kwh**

**refrigerator 3.4 kwh**

**illumination 2.1 kwh **

**washing machine, television sets, etc. **

**Total 18.5 kwh**

**Implementation Example 2**

**The water tank is 30 m high, the cross-sectional area is 6 x 6=36 m ^{2}. **

**The object is a hollow cub of 2 x 2 x 2 = 8 m**

^{3 }. Volume of the replaced water is**2 x 2 x 2 = 8 x 100**

^{3}cm^{3}= 8,000 [liter]. So, M=8,000 [kg] =8 [ton].**The distance d between upper and lower limit stoppers is 15 [m].**

**Time T is ,**

**５．５ sec for the upward motion (Eq. 5), and 1.7 sec for the downward**

**motion (Eq. 6).**

**T for one cycle is estimated from Eq.7. as follows.**** **

**T = (5.5 + 1.7)/2 = 3.6 sec.**

**On the other hand, when the object is at its highest position, the potential energy is U=8000 kg x 15 m =120,000 [ Joule]. Thus, the power is**

**P = 120000/3.6=33000 [Joule**** sec ^{-1} ].**

**Accordingly, the electric quantity par hour is about 200 kwh.**

**Implementation Example ****３**

**Unlike Implementation Example 1, this is a cocept that aimes at intra-house **

**individual rooms.**

**For small implementations like this, the present Archimedean device can be ****fabricated by using a 3D printer.**

# Closing Notes

**In case of hydraulic power generation, water falls from a high site such as dam, and forms rivers’ water, and ultimately the water is released to the sea. **

**If the sea temperature is high there, water vapor is created. And the vapor is diffused into the sky with ascending air current. Then, rain clouds are formed for ****rain to fall. This way, endless circulation of water ****is acomplished by nature. Namely**

**high↓low , low↑high 、high↓low , low↑high,・・・・**

**An analogy may be recognized from correspondences such as**

** rain ⇔ shower dam ⇔ reservoir sea ⇔ ****saucer**** ascending air vapor ⇔ buoyancy**

**Water plays the leading role for both conventional hydro-electric and the present Archimedean methods. Like here, the hydro-electric power stations work also irrespective of night and day. ****However, such stations depend on weather conditions. **

**Moreover, wheras various environmental protection issues always exist, the present Archimedean method ****affects very little natural, social and economic environments.**

**Finally, ****constraction/meintenance cost is low in** **parison to other kinds of plant of similar performance for electric power generation. This point seems worthy to consider when construction of new plants or re-operation of old plants (eg., nuclear power station) are being planned.**