Friday, January 26, 2018

solar energy utilization

Innovative idea
                                                                                                                       M.Meyyappan
Ingenious design of a device for the industrial effluent treatment with solar energy
There are many popular methods for purifying waste water from industries and salt water from sea by using solar energy. In general all these processes are based on boiling the contaminated water to produce steam and condensing the steam with cool surface. It is not only the oldest but also most effective, since it removes bacteria far below the acceptable level, but it is inefficient in eliminating the chemical toxins there in particularly with low boiling point. However complete sterilization of water is not required. Solutes not dissolved in water , contaminates of high boiling point will remain in the boiling water itself. In general  99.9 % pure water can be obtained by distillation.
Solar disinfection is a low-cost method of disinfecting water that can often be implemented with locally available cheap materials. No complicated technical knowledge is required so that  it can be well used by all kind of people even uneducated. One of the major advantages of this method is it has low impact on the environment. It does not require any fire-wood or electricity for heating and it is well suitable to all countries where solar flux is available  almost through the year. It helps to solve the problem of energy crisis by saving the energy used for water purification and water crisis by distributing drinking water to all.
An ingenious device for the industrial effluent treatment with solar energy is designed. It is based on the vaporization at reduced pressure.  The rate of vapourization can be increased  without heating by  reducing the pressure lower than the normal atmospheric pressure. This is because lower the applied pressure , lower will be the boiling point. The normal boiling point of pure water is 1000C at NTP. Its boiling point decrease exponentially with the reduction in the applied pressure. When the applied pressure is reduced by half water boils at 810C and at one fourth of the atmospheric pressure it is 520C. Since the distillation is carried out at lower temperature, it is suitable only for effluent with contaminate of high boiling point.
It works under the same principle as that of the 20 lts mineral water can used for tapping drinking water in homes. It has two cans instead of one and are connected together with a slantwise bent tube made up of glass. Initially the bent tube is completely filled with pure water and kept inverted by holding water inside. It is carefully placed such that one end of which is kept immersed in the wider metallic vessel containing salt  or contaminated water while the other end of the tube is kept immersed  in the pure water taken in a glass jar with an outlet. The narrow neck will not allow fast drainage of water and a column of water will stand in both the limbs of the bent tube. A pressure lower than the atmospheric pressure is developed in the bent tube above the water level. It promotes the process of vaporization. The  as the boiling point is lowered due to low pressure..The wider bottom of the tube  helps the heated water to flow into the bend glass tube effectively. The metallic vessel is properly housed in a flat plate collector exclusively designed for it so that solar heat collected is transferred to water by conduction Care must be taken to reduce the energy loss by convection and radiation from the system. The outer surface at the bottom of metallic vessel is also blackened to absorb the solar flux directly to improve its efficiency. With the help of a reservoir with an outlet  attached with the metallic vessel waste water is fed  to the system for continuous working. This system is best useful for waste water having contaminates of high boiling point. During continuous usage   residue will be deposited at the bottom of the vessel which must be removed then and there, otherwise the heat conduction from the flat plate collector will be reduced gradually.





Dr.M.Meyappan is a retired professor of Physics worked in Alagappa Government Arts College, Karaikudi, Tamilnadu. 

Sunday, January 7, 2018

short story

ஆன்மீகச் சிறு கதை
கடவுளைக் கண்டேன்                                                                                                                        "கடவுள் இருக்கின்றாரா "                                                                                                              "இருக்கின்றார்"                                                                                                                                  " எங்கே இருக்கின்றார் ?"
"எங்கும் இருக்கின்றார் "
" அப்படியென்றால் கடவுள் ஏன் ஒருவர் கண்களுக்கும் தென்படவில்லை ?"
"எல்லோருக்கும் தென்படவேண்டும் என்பதற்காக ஒருவருக்கும் தென்படவில்லை "
"புரியவில்லை. இன்னும் கொஞ்சம் விளக்கமாகக் கூறமுடியுமா ?
மின் விசிறியைச் சுட்டிக்காட்டி  " இந்த மின் விசிறியில் எத்துணை இறக்கைகள் இருக்கின்றன ?"
"மூன்று "
"உனக்குத் தெரிகின்றதா ? "
"தெரிகின்றது "
மின் விசிறியை ஓட விட்டுவிட்டு  " இப்பொழுது அந்த விசிறிகள் தெரிகின்றதா ?
"தெரியவில்லை '
"ஏன் ? "
"அவை எங்கும் சுற்றிக் கொண்டிருப்பதால்  கண்களால் பார்க்கமுடியவில்லை ஆனால் தெரிந்து கொள்ள முடிகின்றது "
" சரியாகச் சொன்னாய் .. கடவுள் உனக்கும் எனக்கும் மட்டுமல்ல . எல்லோருக்கும் பொதுவானவர். மின் விசிறி போல உலகைச் சுற்றி ஓயாது வளம் வரும் கடவுளைக் காணமுடியாது , உணரத்தான் முடியும் "
 கடவுளைக் கண்டு விட்ட மகிழ்ச்சியால்  மனம் ஆனந்தக் கூத்தாடியது. 

Tuesday, November 14, 2017

Fun with numbers

Power of a number

Any power number can be expressed as the difference of two squares.

22  =  22  -   02      ;  32  = 52  - 42
23 =  32  -  12       ; 3 = 62    - 32
24    =  52  -  32    ;  34  = 15-  122
25  =  92  -  72      ; 35  = 42- 392
26  = 172  - 152    ; 36  = 1232 - 1202
27  = 332  - 311 ;   3= 3662  -  3612

Any odd square cannot be expressed as the difference of two squares with root numbers whose difference cannot be even. Any even squares cannot be expressed as the difference of two squares with root numbers whose difference cannot be odd..

Wednesday, October 11, 2017

Beal conjecture - 4

Beal conjecture-4
In the three members like or unlike power relations both the Fermat’s assertion and Beal conjecture have some similarities and differences.
Similarities
1. All the members can be even, then the relation will be reducible.. All the member cannot be odd due to non-conservation of oddness.
2. One or two members but not all the three may be square or higher power but all the members cannot be square or with the same higher power.
3. FLT and Beal conjecture are mathematically true with fractional and rational or complex numbers.
4 . In irreducible form, FLT  and the Beal conjecture do not have a common prime factor among the members.   
Differences
1.According to FLT in ax  + by  = cz , a,b,c the base numbers cannot be positive integers when the exponents x,y,z are same and greater than 2.
 (an – 1)n  + (an  - 1)n+1   = [a(an  - 1)]n , where a and n are any two independent variables which gives a set of multi-power relations obeying Beal conjecture with exponents  (n,n+1,n).
[a(an + bn)]n  + [b(an  + bn)]n    = (an + bn) n+1  gives a set of multi-power relations with exponents (n,n,n+1)

[2n an+1]n+2  +  [2n an+1]n+2   = [2n+1 a n+2] n+1  gives  a set of such relatins with exponents  (n+1,n+1,n) 

Tuesday, October 10, 2017

Beal conjecture-3

Beal conjecture-3 : Methods  of getting  three member  multi-power relations
Method-1 
It is suitable to obtain multi-power  relation  of type  an  + bn  = c n+1
The sum of any two like powers when raised to same power will be a common multiplier.
 an  + bn  =  (an  + bn
Multiplying with the common multiplier (an  + bn )n , we get [a (an  + bn )]n  + [b (an  + bn )]n  =
(an  + b)n+1  , where a and b are any positive integers.

    a       b       n      irreducible relation                  multiplier         multi-power relation
    1       2       3           1 +   23   =  9                                  93                         93 + 183  =  94
    1       2        4           1 +   2 =  17                              174                       174 + 344  = 175
      1       2       5           1  +  25   =  33                              335                          335  + 665  = 336
    1      3        3           1  + 33   = 28                                283                           283  + 843  =  284
     2      3        3           23  + 33   = 35                               353                        703  + 1053  = 354
     3       3        3           33   + 33 =  54                              543                        1623  + 1623  = 544
     2       3        4           24  +  34  = 97                              974                    1944 +  1944   =  975

Method-2
It is suitable to get multi-power relation of type  an  + an  = 2m where a and b  are equal to some power of 2 , since sum of  (2m)n + (2m)is always expressible as a power of 2 and is equal to 2mn + 1          

                                               2n  +   2n    =  2n + 1
                                                      4n    +  4n    =  22n + 1
                                                     8n    +  8n   =  23n + 1
                                                    16n  +  16= 24n + 1

                                                   32 + 32 =  25n + 1

cubical relations

One cube is equal to sum of three cubes
According to Fermat, a3  +  b=  c is not possible with a,b,c are all positive integers. However, one cube can be expressed as a sum of three cubes.
63  = 53  + 43  + 33
93   = 83  + 63  +  13
20= 173  + 143 + 73
253  =  223 +  173  + 43
283  = 213  + 19+ 183
583  = 493  + 423 + 153
703 = 57+ 543 + 73
2103  = 1713  + 1623  + 213
2143  =  2133 + 513  + 163
2563  = 2553  + 573  + 223
2563  = 255+ 273  + 223
298= 2973 + 643  + 153
682= 6753  + 2133  + 163
10623  = 1059+ 2133  + 1083
14083  = 741+ 6753  + 3463

21763 = 21693  + 4473  + 2143

Sunday, October 8, 2017

Beal Conjecture -2

Irreducible form of  three member multi-power relation (Beal conjecture)
According to Beal all three member multi-power relation a+  by   = c, where a,b,c and the exponents x,y,z  are all positive integers such that x,y,z > 2, the base numbers a,b,c  will invariably have one more common factors.
If  all the members are expressed in the same power, any number can be used as common multiplier. But if the powers are different , only certain selective common multipliers will make reducible form of multi-power relations. If one of the members in the irreducible form of a multi-power relation is 1, it can be multiplied with any multiplier of any form..It is exemplified with a typical example.
1+ 8 = 9   =   1 + 23  =  32
    X 2 à     22  +  25  = 62
   X 33   à     33  +  62  =  35
  X 43   à    4+  8 =  242  
 X 36    à   36  +  183  =  38   (  X 93  à  93  +  183  = 812)
X 46  à 4+  32 = 1922 (  x 163  -à 163  + 215  = 1922 )
One of the properties of the multi-power relation in the form a +  bn  = c is existence of common multiplier  knm
a +  bn  = cm   x   k mn   -à  (km a)n  + (km b)n  =  (kn c)m
 There are many irreducible form of multi-power relation. One of the general forms is
[ 1 + nm  = ( nm  + 1) ] x  (nm  + 1)m  -à (nm  + 1)m  + [n(nm  + 1)m ] = ( nm  + 1) m+1

Where m and n may have all possible values.