Theory and Design of An Incremental Loading Machine

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THEORY ÂHD DESIGH OF AM IMCEEMEHTAL LOADIMG MACHIME

THESIS Submitted in Partial Fulfilment of the requirements for the degree of MASTER OF MECHANICAL ENGINEERING at the POLYTECHNIC INSTITUTE OF BROOKLYN

JOSEPH LEVITT AND PAUL SOROS May 1950

Approved:

sis Adviser

lead

tmen-

ProQuest Number: 27591416

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VITA JOSEPH LEVITT * ie * Date of Birth:

November 17, 1921

Place of Birth: Binghamton, New York Undergraduate Education: Cooper Union School of Engineering; Bachelor of Mechanical Engineering, June, 1944» Professional Experience; September 194# - Present: Instructor in Physics and Mechanical Engineering, Pratt Institute, Brooklyn, N.Y. September 194*7 - June 194&: Instructor in Physics and Mathematics, University of Illinois, Chicago, 111. January 1947 - July 1947: Research Assistant (Diesel Engines) at Engineering Experiment Station, Pennsylvania State College, Pa. August 1945 - December 194&: Mechanical Engineer, Schwarz Laboratories Inc., N.Y.C. (Consulting Engineers to the Food and Brewing Industries). August 1944 - August 1945: Mechanical Engineer, N.Â.C.A., Langley Field, Va. (Aerodynamic Research). Publications; ^Entropy - The Thermodynamic Dilemma,” Engineer, April, 1947.

Penn State

"Wind Tunnel Investigation of Effect of Canopies on Directional-Stability." N.A.C.A. TH# 1052, May, 194&» ^ The work connected with the present, Thesis was carried out in the Physics Laboratories of Pratt Institute, Brooklyn, New York. Construction of the Incremental Loading Machine was undertaken in February 1950 and completed in May 1950. The Total amount of time consumed for the entire project was a little more than 1000 man-hours.

VITA PAUL SOROS * * Date of Birth;

June 5, 1926

Place of Birth; Budapest, Hungary Undergraduate Education; Budapesti Gepeszeti Ipari Kizepiskola (High School of Metal Technology of Budapest). Grad­ uated in 1944 • Studied Mechanical Engineering at the Jozsef lador Muegyetem (Jozsef Nador Polytechnic Institute) in Budapest until 194&' Publications: ”Uj Konnyusulyu Automotor,” (A New Light-Weight Internal Combustion Engine), Elet Es Tudomany (Life & Science), Budapest, March 194#» ”Uj Motorgyatasi Eljarasor,” (New Methods in Motor Car Engine Manufacturing), Technikus, Budapest, April 194#* *

*

# *

The work connected with the present Thesis was carried out in the Physics Laboratories of Pratt Institute, Brooklyn, New York. Construction of the Incremental Loading Machine was undertaken in February 1950 and completed in May 1950. The total amount of time consumed for the entire project was a little more than 1000 man-hours.

ACKNOWLEDGEMENT The authors are greatly indebted to Prof. M. J, Steinberg of Brooklyn Polytechnic Institute and the Consolidated Edison Company of New York, who graciously extended his valuable time in giving assistance, advice, and encouragement, through­ out the entire investigation.

ABSTRACT # # # The aim of this investigation has been to design and construct a machine which will automatically compute, from the input-output data, the optimum division of load among a group of power stations comprising any power generating system. Various methods of load distribution are described in­ cluding that of the Incremental method which is the most important as its application results in the attainment of maximum operating efficiency.

The present method of cal­

culating Incremental load distribution is presented. The mathematical proof of the principle on which the machine operates is developed; also included, are descrip­ tions of its construction and operation. Results of tests on the machine are presented and com­ pared with values obtained by straightforward calculation. While this comparison proves to be quite favorable, the need for improvement in the existing machine is noted and a revised design is presented.

TABLE OF CONTENTS

TOPIC

PAGE

I n t r o d u c t i o n ....................

1

Mathematical Proof of the Principle of Equal Incremental Rates . . . . . . . . .

6

Calculation of Incremental R a t e s ..................

8

Discussion of a Mechanical Device for Establishing Incremental Loading

....

10

Mathematical Proof of Theory of Incrementometer . . .

18

Experimental Verification of T h e o r y ..............

21

Experimental Procedure and Results

26

................

C o n c l u s i o n s ...........................•..........

33

Bibliography

35

......................................

TABLE OF FIGURES AND GRAPHS

TITLE

PAGE

Fig. 1 - Solution of Typical Load Division Problem by the Incremental M e t h o d ..............

4

Fig. 2 - The Incrementometer....................

12

Fig. 3 - Bottom Side of Disc Illustrating Method of Tightening Steel B a n d ................

14

Fig. 4 - Upper Side of Disc Showing Radial Lever and Rider ................................ 14 Fig. 5.- Relative Rotation Between Discs ........

17

Fig. 6 - Free Body Diagram of D i s c ................ 19 Fig. 7 - Performance Curves of FourPower Plants . . 22 Table I - Incremental Load Division Among Four Power S t a t i o n s .................................. 24 Fig. 8 - Effect of Loading on SystemInput • • . •

25

Table II - Load Division Among Four Power Stations as Furnished by Incrementometer........ 31 Fig. 9 - Working Drawing Showing Details of Present Design of Incrementometer.................. VInside of Lpocket. Fig. 10 - Working Drawing Showing Details of Cback cover ............ / Proposed Design of Incrementometer

1 - Introduction The general problem of power-station operating econ­ omies may broadly be considered to consist of three phases which are: 1.

The provision of sufficient capacity in operation to insure continuity of service throughout the system.

2.

The selection of the necessary equipment to supply the load from the standpoint of its effect on the best over-all system economy.

3.

The correct division of load on the equipment which has been placed in operation.

This Thesis is concerned with the last of these phases. Even without a detailed explanation it is apparent that random loading of boilers or turbines in any power plant, or random loading of individual power plants in a power generat­ ing system would not result in the attainment of the greatest possible efficiency.

This of course has long been recognized

and through the years, many methods of load division have been developed.

In a number of cases the different methods,

for a certain range of operation, result in identical loading. A few of the methods in use are; 1.

Base loading

2.

Proportional loading

3.

Equal-efficiency loading

4.

Incremental loading

To arrive at the correct combination of equipment and then the correct division of load for the combination placed in operation, the performance characteristics of each piece of equipment must be known.

With proper equipment available

and station characteristics known, the appropriate solution of any momentary load-division problem may be "reached. The different methods of load division will now be dis­ cussed in the same order as stated above. The method of base loading gives load to the best sta­ tion or machine up to its maximum capacity before any load beyond its capacity is assigned to the next best station or machine. The method of proportional loading consists merely of plotting the sum of the turbine loads at their full load points, and connecting these points through zero with straight lines; from these, unit loads can be scaled off for any given station load. In the equal-efficiency method, the ratio of output to input of the various turbines or boilers must be equal.

Where

this scheme is used, a schedule may be made up by maintain­ ing equal efficiencies on a variety of machines, progressively as the load changes. It is by means of incremental loading that the minimum input for any output with any given combination of equipment in operation is achieved.

This method will not of itself

show what equipment should be placed in operation, but with

the aid of input-output data it will assist materially in determining the proper combination of equipment for various system or station loading characteristics. In order to appreciate fully the method of incremental loading it may be desirable to define "incremental rate.” For a given item of equipment or power plant station, there will exist a definite relation between the output and the input.

In other words, if output is designated by the symbol

L, and input by the symbol I,' it may be said that I=-F(L) . The incremental rate is then defined as ^

.

Mathematically

it is the derivative of input with respect to output.

Physic­

ally it represents the change in input associated with a corresponding change in output. With incremental loading, an attempt is made to assign load to individual machines or stations so that they oper­ ate at equal incremental rates.

Where thé performance char­

acteristics are such as to make this impossible, load is assigned to that machine or station which has the lowest incremental rate. An illustration of the basic principle of the incre­ mental method of loading is indicated in Fig. 1.

Heat rates

are shown for the two stations having 9,000 BTÜ difference between their best points.

Assume that the stations are so

loaded that station A is operating at a heat rate of 30,000 BTU/kwh and station B at a heat rate of 18,000 BTU/kwh as shown.

From the heat-rate curves alone it would appear

«it

o

in ri

uU Qj UJ-

O UL

000

# p ii3

that if an additional 1,000 kw load were required to he furnished by the system comprising stations A and B, then station B should be the one to supply it.

An inspection

of the incremental heat rate curves reveals, however, that the added load can be supplied to the system for 20,000 BTU by station A, whereas station B would require 27,000 BTU to produce the same load increment.

At the points selec­

ted, then, it is clear that the additional load ought to be furnished by station A. While this example served to demonstrate that it was

more economical to assign load to the station having the smaller incremental rate, it can be proved mathematically that the most economical operation is achieved when the loads are divided so that their incremental rates are equal.

2- MATHEMATICAL PROOF OF THE PRINCIPLE OF EQUAL IHCREMEHTAL RATES Statement of Principle:

When two or more machines are

operating in parallel to supply a common load, the maximum overall efficiency is obtained when the machines are operat­ ing at outputs which correspond to the same incremental rate value. Applied to two machines, the proof is as follows: Let

L^ = total output of the 2 units L^-output of unit Mo. 1 L2 - output of unit Mo. 2 I^ = total input to the 2 units t I ^ = input to unit Mo. 1 I^- input to unit Mo. 2

It is assumed that Iq^F^CL^)

and I2=F2^^2^

represent two

continuous input-output curves, and have continuous deriv­ atives, iL and

, respectively, which always increase as

L^ and L_ increase.

and

Then

If the first derivatives

, 4 L always increase, the second «•-» clti derivatives will be positive and It will be a minimum when ^ 4L,

= 0.

But

K7= t: Furthermore

--

— = ^clL, dt,

=

Hence

= ■3lT

S

-I

4L- — 4r^ ~ ^ ^

t = i t This means that the minimum input •

for a given com­

bined output is obtained when the incremental rates of the two units are equal.

Furthermore, there is only one pair of

loads for any given output at which the incremental rates will be equal. The preceding proof may be extended to show that if the number of machines operating in parallel is greater than two, then optimum efficiency will still be obtained when the in­ cremental rates are equal. It is not unreasonable to expect that cases will arise where, at least for a certain range of operation, it will not be possible to load machines so that their incremental rates are equal.

Under such conditions the machine which has the

lowest incremental rate is loaded progressively until its incremental rate has reached that of the machine having the second lowest incremental rate.

Thereafter, the two machines

are loaded progressively so that their incremental rates are equal. The justification for this procedure is easily demon­ strated.

Letdj equal the small increment of input associated

8

with a small increment of output, AL. are taken reasonably small, then

If these increments and dL=AL.

While the incremental rate is mathematically defined as IR=^, in practice it would have to be established from AL

But,

dl 81 = ( ^ )

Hence, and

d l = (IE) dL

or

AI=j(IR) dL But, y"(lR)dL

represents the area under the IR curve.

Hence the incremental input will be the least when the area under the curve is least.

Clearly the machine having the

smaller incremental rate will be the one of choice.

3 - CALCULATlOM OF IMCREMEMTAL HATES The incremental rate may be obtained by one of the following methods; (a)

When the input-output curve can be expressed by an algebraic equation, the incremental rate can be determined by differentiation, since it.is the first derivative of input with respect to output.

(b)

The incremental rate may be graphically determined by drawing a tangent to the input-output curve at the point corresponding to the output.

The slope

of the tangent is equal to the incremental rate at the given point.

(c)

From the input-output curve, a series of output values are chosen and the corresponding input values are read.

The difference between succesive values

of the output are usually made constant and are made small enough so that the characteristic shape of the incremental rate curve can be determined with reasonable accuracy.

The incremental rate is

merely the ratio of the input difference to the output difference, or the incremental input divid­ ed by the incremental output, and is assumed to be a function of the midpoint. Bf the three methods, the third is the most practical. Having obtained the incremental rates, a schedule of loading is then prepared in which the individual machines are loaded progressively at equal incremental rates to fur­ nish the total load required.

10

4 - DISCÜSSIOM OF A MECHAMICÂL DEVICE FOR ESTABLISHING IMCREMENTAL LOADING Of the several methods of loading that have so far been discussed, the incremental method results in minimum input for a given total output and hence no other method can sur­ pass it.

This fact is readily admissible from mathematical

considerations and has, moreover, been demonstrated in actual operation. The search for an optimum method of load distribution has thus been closed.

All methods for improvement in power

station loading must now be directed toward the attainment of rapid or perhaps even automatic calculation and application of incremental rates. An excellent method of applying the method of incremental rates is presently utilized by The Consolidated Edison Company of Mew York, Inc. through the device of a loading slide-rule on which are marked off the incremental rates of the various power stations that comprise a system.

The application of

this method involves the calculation of incremental rates from the input-output data for each station.

Once these

incremental rates are obtained, the loading schedule is readily established.

For further details the reader is re­

ferred to Reference #1 in the Bibliography. If a loading schedule could be obtained which provides incremental loading without actually having to compute the individual incremental rates, a reduction of time and money would be effected, since the preparation of a schedule for a

11

power generating system involves from 4 to 8 man-hours of work for each power plant. One method of accomplishing this desired end result is achieved by the use of a machine which the authors call an ^Incrementometer.^

This device consists of a series of

equally spaced discs mounted on a vertical shaft by means of ball bearings, so that the discs are free to rotate about their common center with a negligible amount of friction. The assembled machine is shown in Fig. 2. On each of the discs is plotted, in polar coordinates, the performance data of a given power generating station. Output is plotted as angular displacement (O) and input, as radial displacement (r).

Each disc has a series of holes

equally spaced on radial lines.

The radial lines are also

spaced equally at intervals of 12 degrees.

The reasons for

the adoption of this design will be made clear when the oper­ ation of the machine is discussed. On any radial line, the distance between consecutive holes represents a unit of input.

Starting from zero, each

radial line represents a unit of output. To plot a performance curve on any disc the procedure is as follows:

Starting with zero output (6=0) the corres­

ponding input is recorded by inserting a proper hole on the first radial line.

bolt in the

For the next value

of output, the corresponding input is entered by inserting a bolt on the next radial line in the hole corresponding to

12

FIG. 2 - THE INCREMENTOMETER,

13

the value for the input.

This is continued progressively

until the entire performance data have been entered.

A

steel band is now tightened around the outside of the bolts on the bottom side of the disc and thus represents the com­ plete performance curve of the power generating station. This is illustrated in Fig. 3. The same procedure is repeated for each of the other stations.

The final result is that the performance curves

have now been entered on the Incrementometer and it is ready for use. To understand its operation it is necessary to explain the mechanical connections between discs. Each disc has a radial, horizontal lever connected to its upper side

(Fig. 4)y as an integral part of the disc.

The lever cannot, therefore, rotate with respect to the disc upon which it is mounted.

On each lever is mounted a fric-

tionless rider which is free to move radially in and out along the lever.

The rider maintains rolling contact along

the outside surface of the steel band of the disc above it. The rider cannot move away from the steel band with which it is in contact due to a constant radial tension which is pro­ vided by means of a continuous string which seeks to draw the rider to the center of the disc.

The continuous string

exerts the same radial tension on all riders no matter what their positions may be but never comes in contact with the steel bands.

u

FIG. 3 - BOTTOM SIDE OF DISC ILLUSTRATING METHOD OF TIGHTENING STEEL BAND,

Radial

Lever

,rFIG. 4 '

UPPER SIDE OF DISC SHOWING RADIAL LEVER AND RIDER

15

Along the periphery of each disc is a uniform scale of output. If the upper disc is held fixed and the lowest lever is rotated through any angle 6^, the various discs will rotate through respective values of

6^, 0^, ...

where

Gg^ is the relative rotation between the top disc and the disc below it, G^ is the relative rotation between the second disc and the one below it, etc., and G^ is the relative rotation between the lowest lever and the bottom disc which is direct­ ly above it.

(The lowest lever is not fixed to any disc).

Clearly, 8% + 8% + &C ^

G^= G^.

Assume, for the moment, that there are only three discs in the Incrementometer, on which the steel bands correspond respectively to the performance curves of three power gener­ ating stations. The top disc is, as always, fixed with respect to the shaft.

Keeping in mind the fact that angular displacement

is equivalent to output, suppose that a total output of 10 units is to be supplied by the three generating stations. The levers of all discs are initially at zero.

To find the

load division, the master (lowest) lever is rotated to 10 on a fixed bottom scale (which is therefore also the total­ izing scale).

The following possibilities arise:

(1) The bottom disc will rotate, and the middle

and

top discs will remain stationary. (2) The bottom and middle discs will rotate and

the

16

top disc will remain stationary. In either case, however, the sum of the relative ang­ ular displacements between the three discs will be 10 units. Furthermore, each lever indicates the load on the scale of the disc above it.

In the particular case illustrated in

Fig. 5, the load on A is 3 units, that on B is 5 units, and that on C is 2 units, making a total of 10 units of output. Even with more than three discs the operation of the machine remains the same.

The master lever is rotated to

the required system outppt on the totalizing scale and each lever then indicates the load on the disc above it.

The

division of load among the various power stations is thus obtained directly without resorting to any calculations other than those necessary to establish the relations between input and output. It remains, now, to prove that the division of load as furnished by the Incrementometer is truly Incremental loading.

17

r a o f Dtisc

F

5

R e ia liv e Ro+aiioA Be+vwcerv Discs

18

5 - MATHEMATICAL PROOF OF THEORY OF IKCREMEKTOMETER Consider two consecutive discs on the machine.

Let the

top one be fixed and let the one below it be free to rotate. (See Fig. 6).

For the sake of simplicity, only the more

important elements have been included. Let reaction between periphery of curve and its rider (acts normal to curve) p=reaction between lever and its rider (acts perpendicular to radius) T - tension in string attached to the rider (acts along the radius) 0 = angle between radius and normal to curve at any point on curve 0 - output r c input = distance of rider from center of rotation If the rider on the upper curve is taken as a free body, then T =Bc os0

(1)

p = Hsin0

(2)

and combining (2) and (l) p -T tan0

(3)

If the rider on the lower curve is taken as a free body the same treatment exactly as above yields pT = T^ tan0^

(4)

19

BOTTOM VIEW OF TOP DISC

\ FREE BODY DIAGRAM OF RIDER FREE TO ROTATE

Fig 6

FREE BODY DIAGRAM OF LOWER DISC

20

Let us now take the lower disc as a free body.

Applying

the equation 2 M=0 pr-î3^sin0^ r *

(5) «

substituting yields But,

p*r^K* sin0* prnp*r*

p ^ T tan0

(6)

and p*= T^tan0*

Making these substitutions, we get Tr tan0 = T*r* tan0*

(7)

In polar coordinates, r tan0 and

=^

r*tan0*-^^

(8) (9)

If (8) and (9) are substituted in (?) we have T dr- T* ^ 3© d©' If the tension T = T *

(10)

then

^ - dr^ (ll) a©" a©' Thus it is established that if the radial tension on each rider is the same, then the relative motion of the discs will take place in such fashion that the incremental rates will be equal and results in optimum loading. It is quite easy to extend the proof for three or more curves by taking two consecutive curves at a time and con­ tinuing progressively in this fashion until all the curves have been included.

21

6 - EXPERIMENTAL VERIFICATION OF

THEORY

Although the correctness of the theory upon which the Incrementometer is based has been demonstrated by mathematical considerations, it is still desirable to provide experimental verification. Toward the attainment of this objective, the actual machine was constructed. Four power generating stations were selected whose re­ spective performance characteristics are given by the follow­ ing equations: (a)

1 = 100 + 51 + .251

(b)

I = 75 +5L + .751^

(c)

1= 20 +

lOL + .07l 3

(d)

I = 50 +

15L + .03P

+ .03L^

In these equations I represents input in million BTU/HR, and L represents load or output in MW.

These curves are

plotted in Fig. 7* The incremental rates for the four stations are obtained by differentiating each of the preceding four equation to yield: (a')

IR c5

+ ,5L + .091^

(b»)

IE = 5

+ 1.5L

(o')

IE = 1 0 +

.211^

(d>)

IE = 15 + .091^ To obtain a loading schedule for these four stations,

the loads onthe individualstations were values of incremental rateby solving

obtained forcommon

the aboveequations

for

22

400

t

iCV R V t S +

— jÈliiË Z5_IAL

%

23

each value of incremental rate selected. Setting IR-6, for instance, equation (a*) yields 6 = 5 + .51» + .09L^

from which

La=1.55 Similarly, equation (b*) yields 6 = 5 + 1.5L

and L^=.87

Substitution of IR= 6 in equations (c*) and (d*) yield imaginary values for L.

Since the load on any station must

be real, the minimum value of L is assigned. Hence Lg=0, and For a value of IR= 16, the various individual loads are computed to be 8.57 L^=: 7.34 L = 5.35 L°=3.33 These computations were made for incremental rates ranging from 5 to 38 and are presented in increasing order in Table I.

This table also indicates the loading schedule

for optimum load division. curves shown in Fig.

From these values

7, total

and from the

system input is plotted against

total system output in Fig. 6 . It was felt that it would be desirable also to include in this graph a plot of input against output as obtained from a schedule prepared by the method of base loading to capacity. An inspection of Fig. Ô reveals that only for zero output and maximum output are the inputs the same for the two different

24

TABLE I TWCBEMKMTAL LOAD DIVISION AMONG FOUR POWER STATIONS IR

5 :6 7 8 9 10 12 13 15 16 18 21 24 27 29 31 34 38

Load on Station A B C 0.00 1.55 2.68 3.62 4.45 5.17 6.47 7.04 8.12 8.57 9.52 10.00 10.00 10.00 10.00 10.00 10.00 10.00

m D

0.00 0.00 0.00 0.87 0.00 0.00 1.33 0.00 0.00 2.00 0.00 0.00 2.67 0.00 0.00 3.33 0.00 0.00 4.66 3.09 0.00 5.33 3.78 0.00 6.66 4.88 0.00 7.34 5.35 3.33 8.66 6.17 5.76 10.66 7.24 8.15 12.66 8.15 10.00 13.00 9.00 11.52 13.00 9.50 12.46 13.00 10.00 13.31 13.00 10.00 14.55 13.00 10.00 16.00

Total Output MW 0.00 2.42 4.01 5.62 7.12 8.50 14.22 16.15 19.66 24.59 30.11 36.05 40.81 43.52 44.96 46.31 47.55 49.00

25

_

l i i i l l i p

p

m

5 ¥ S 1 » W W [^

26

methods of loading.

For all system loads intermediate be­

tween zero and maximum, the input obtained by base loading to .capacity is higher than that resulting from incremental loading as, indeed, it should be.

7 -KPERIMEHTAL PROCEDURE ABD RESULTS Having computed the inputs corresponding to the re­ spective outputs or loads for the full range of each of the four power generating stations by means of equations a, b, c, and d, the data were ready to be entered in the Incremen­ tometer. The scale selected for output was 1 M M - 12 degrees and that for input was 1 million BTU/HR- 1 mm. Because the Incrementometer operates on the basis of incremental rates it is permissible in plotting the inputs, to subtract a constant amount from each value as the incre­ mental rates will remain unaffected, i.e. A[ f (L)] - ^[F(L)-k] 4L

4L

This procedure was decided upon in order to include the entire range of input for the various individual stationsDifferent amounts were subtracted from the input values of each station, but for any one station the amount subtracted from each value of input was, of course, the same. Station D is of special interest.

Its range of oper­

ation was deliberately made much larger than any of the other three stations in order to demonstrate the flexibility

27

of the machine. The range of operation of station D is from 0 to 16 MW output and from $0 to 413 million BTU/HR input.

Even if

the maximum amount of 50 is subtracted from each value of input, it would still be impossible to include the entire range of performance on one disc. used for this station.

Hence two discs were

The first disc covered the range

from 0 to 10 MW and the second covered the range from 10 to 16 MW.

Since the incremental rates increase progressively

with load, it follows that there is no danger of the second disc coming into operation before the first has completed its full range of 10 MW. As discussed above, the input-output data would normally be entered in the machine by inserting riate holes of each of the discs.

bolts in the approp­

This would ordinarily

require each disc to have many holes on each radial line so that all possible values of input could be included with­ in a certain range. To save construction time, however, only those holes were drilled which actually were needed.

A steel band was

tightened around the bolts of each disc thus producing the desired performance curves. After the data had been entered, the next step was to assemble the machine by mounting the discs on the vertical shaft.

The continuous string,

which provides the constant

radial tension on each rider had to be inserted simultaneously

28

with the assembling of the discs. most difficult job of all.

This proved to be the

The close spacing between discs

and the many turns over tiny pulleys which the string takes in its travel from the bottom rider to the top one made 'the task a most exacting and time-consuming operation.

After

assembly, a 15 lb tensile force was applied to the free end of the string and the machine was ready for its first test. The master lever was rotated to a value of 4 MW total output on the totalizing scale.

For this position the read­

ings on the individual discs were as follows : Station Station Station Station

A B 6 D

-

2.65 MW 1.35 0.00 0.00

Referring to the loading schedule, it is seen that for a total output of 4*01 MW, the incremental distribution of load should theoretically be: Station Station Station Station

A B C D

-

2.68 1.33 0.00 0.00

The situation looked very promising and it was decided to continue the test and compare the load distribution as computed by the machine with the theoretically correct values for the entire range of load up to the maximum value of 49 The master lever was then rotated to indicate a system load of 8.5 MW on the totalizing scale.

The load on Station

A was read off to be 5*3 MW (compared to the theoretically

29

correct value of 5.17 MW;.

Before the readings could he

obtained from the rest of the discs, the continuous string which provides the tensile force on each rider snapped. An inspection of the string revealed the fact that it had broken at a point where it had become frayed by continuous rubbing. The machine was dismantled, a stronger string was sub­ stituted, and the machine re-assembled.

A tensile force of

only 3 lb was now applied to the free end of the string.

The

test of the machine was then continued by rotating the master lever to various total outputs on the totalizing scale and recording the loads on the individual stations. One set of readings obtained from the machine are com­ pared with the theoretically correct ones below; Station

Incrementometer Reading

A B C D

10.0 MW 6.9 8.9 4.2

9.5 MW 8.6 6.2 5.7

30.0

30.0

Total

Theoretical Value

At first glance it appears that these readings do not compare too favorably with the theoretical ones but it is instructive to examine more fully the consequence of these departures. For a system output of 30 MW, the input that would be required in incremental load distribution is 610 million BTU/HR (obtained from Fig. 8).

If the inputs corresponding

to the load distribution as furnished by the Incrementometer are summed up they equal 624 million BTU/HR. (obtained from

30

Fig. 7).

This represents an error in input of 2.29^ and

can not he regarded as very serious. Continuing with the test of the machine, the tension was next increased to 6 Ihs and readings were taken for values of output ranging from 2.4 MW to 46.2 MW. Finally the tension was increased to 9 lbs and several more readings were obtained. The complete set of readings for the entire test is presented in Table II.

An inspection of this Table reveals

that the greatest error made by the machine during the test was 2.29^ and the least error was 0.37

Moreover, while

the error varies with the total output, it generally decreases as the tension on the continuous string is increased.

This

would indicate that the maximum precision would be achieved if the tension on the string were allowed to increase to a much larger value than that used in this investigation. Unfortunately, this could not be done with the existing machine A thicker string could not be used because of the narrow grooves in the small pulleys and the slender string which did fit had an ultimate strength of about 15 lbs. Other experimental difficulties also presented them­ selves during the test of the machine.

Besides the frequent

breaking of the string, the principal difficulty which arose came from the instability of the individual riders.

Whenever

the string tension was relaxed the riders would fall out of position and the continuous string would immediately jump off

31

TABLE II LOAD DIVISION AMONG FOUR POWER STATIONS AS FURHI8HED BY IT(ICREMF.HTnMETF.R

String Tension

A

Load on Station ^ 0 B D

Total Total Outputs Input

Theoretical Error Mimimum % Input

3 lb

8.9 10.0

3.5 6.9

7.1 8.9

0.0 4.2

19.5 30.0

454 624

445 610

2.02 2.29

6 lb

1.5 2.6 8 *4 9.2 10.0 10.0

0.9 1.4 4.8 12.5 13.0 13.0

0.9 0.0 5.9 5.6 7.3 8.3

0.0 0.0 0.3 8.2 13.5 14.9

2.4 4.0 19.4 35.5 43.8 46.2

259 269 442 723 915 987

257 268 440 712 910 980

0.78 0.37 0.46 1.54 0.55 0.72

9 lb

1.5 3.5 5.1

0.8 1.6 1.9

0.0 0.1 1.1

0.0 0.0 0.0

2.3 5.2 8.1

258 280 306

256 278 303

0.78 0.72 0.99

**

Load on Station and Total Output given in MW

'

Total Input and Theoretical Minimum Input expressed in million BTU/HR

32

the pulleys.

To get the string back into position it was

necessary to dismantle the machine, insert the string correct­ ly, and then re-assemble the machine.

Because of this diffi­

culty a more complete set of data was not obtained. Sufficient data were, however, collected to verify the mathematical theory of the Incrementometer and also to give support to the supposition that a loading schedule may be prepared directly from the input-output data without resort­ ing to calculation of incremental rates.

33

8 - CONCLUSIONS The design and construction of a machine which automat­ ically indicates the optimum division of load among a group of power generating stations has been completed in this study. It is believed that the machine is the first of its kind in that it eliminates the need for calculation of incremental rates but utilizes, instead, the input-output data directly. The machine showed some weakness and a need for improve­ ment particularly with respect to the quality of the work­ manship.

$he authors are not professional machinists, however,

and this together with the lack of suitable facilities pre­ vented the construction of a machine entirely free from mechanical defects. Based on the experience gained during this investigation, a new design was developed and it is believed that it will prove to be entirely satisfactory.

The important changes

that have taken place are: 1.

The use of a continuous string has been abandoned and in its place are individual strings which provide the necessary radial force on each rider.

2.

To each of the riders has been added an additional ball bearing which will provide the proper support at all times.

Detailed drawings of the existing as well as the proposed new design are furnished in Fig. 9 and Fig. 10 respectively.

34

In closing, it should be pointed out that the use of this machine need not be limited merely to calculating distribution of load.

It can, indeed, through servo­

mechanisms, be adapted to control automatically the loading of individual power generating stations in any given power generating system. The authors hope that further studies will be under­ taken in this direction by those who are presently engaged in this field of power plant supply economy.

35

BIBLIOGRAPHY (1)

Steinberg, M.J., and Smith, T.Hà

"Economy Loading of

Power Plants and Electric Systems," 1943.

Pub; John

Wiley and Sons, Inc. (2)

Johnson, H.H., and Umbenhauer, M.S: Slide Rule,"

(3)

"Station Loading

Power, November, 1938.

Steinberg, M.J. and Smith, T.H:

"The Theory of Incre­

mental Rates and Their Practical Application to Load Division — Part I," (4)

Stahl, E.C.M:

Electrical Engineering, March, 1934.

"Economic Loading of Generating Stations,"

Electrical Engineering, September, 1931.

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AMENDED DESIGN OF INCREMENTAL LOADING MACHINE F is -

10