Chapter 6: Work, Power, Energy, Moment of Inertia - Introduction Water powered mills were the factories up until about 1865. This was before the widespread use of steam power and much before the use of electricity. In parts of rural America water power was the cheapest and many times the only source of energy for industry until the mid to late 1930’s (Animal and human power was often used in the farming operation, but few farmers ground their own grain.). One of the reasons that many major cities developed along rivers, especially at falls in the river, was due to the power available from the falling water. The potential energy of the water was used in the mills in several different ways and we will discuss two of the more common in this chapter. In addition we will discuss power and how items rotate. The goal of this chapter is to gain a historical perspective how physics affected the development of the United States and specifically how the principles of energy, work, power and rotational motion were employed.

Objectives • To explain conservation of mechanical energy • To understand the functional of both overshot and undershot mills • To understand the concepts of potential and kinetic energy • To explain rotational motion • To understand the concepts of work and power

Gravitational Potential Energy

Gravitational Potential Energy is theenergy that is stored in an objectthat is located a height above thesurface of the earth. For example ifa pond of water is located on a hill20 feet above a house the water inthe pond has stored energy and thuscan be used to create waterpressure in the home. Energy isdefined as the ability to do work. The cycle for water on the earth is the following: the sun evaporates water from the surface of the earth, and it goes into the atmosphere and forms clouds. The cloud eventually becomes saturated and the water falls back to the earth as rain. Once on the ground the water at higher elevations begins to flow toward lower elevations and eventually may reach the ocean. As the water falls from higher to lower elevations the gravitational potential energy of the water is converted into kinetic energy, heat and sound.

Overshot Mill - Part 1

The simplest and most common type mill has an overshot waterwheel. An overshot mill is a potential energy device in which the water falls a vertical distance. This causes the waterwheel to rotate which changes the potential energy of the water into work inside the mill. The mill to the right is Mabry Mill just off the Blue Ridge Parkway in Southwestern Virginia. You can see the waterwheel and the water flume which supplies the water to the top of the waterwheel. The water is supplied by a small lake back in the trees. The water is released on the wheel just slightly forward of the vertical center of the wheel. If the water was supplied exactly at the center of the wheel it would be impossible for the wheel to rotate because part of the water would flow forward and part backward. There would be no net force, and the wheel would not rotate. The weight (a force which is a product of mass and the acceleration of gravity) of the water does cause an unbalanced torque (a force applied at a distance from a pivot point, i.e. the axle of the wheel) on the wheel. Because this torque is not balanced by any other torques it causes the wheel to rotate. This rotation carries the water from a higher level to a lower level and reduces the potential energy of the water. The greater the vertical distance that the water remains on the wheel or the larger diameter of the wheel, the more energy removed from the water. The potential energy removed from the water will be converted into work inside the mill. Not all mills were used to grind grain, but we will only discuss grain grinding mills (flour and grist). Other types of mills were carding mills, saw mills, woolen mills and others.

Overshot Mill - Part 2 How is physics applied in the operation of an overshot water powered mill? Potential Energy is defined as:

(6.1) U represents the potential energy. m is the mass of the water. g is the acceleration of gravity

which is a constant of . h is the height that the water falls while in contact with the waterwheel.

Acceleration is the speeding up or slowing down of an item (Acceleration will be discussed in more depth in the Chapter 7). Any item that falls will accelerate at a constant rate, and this is the acceleration of gravity. The water wheel has buckets that will catch a finite amount of water before the rotation of the wheel has reached a point that the next bucket is filled by the water. Note the water cannot remain on the wheel for a full half of rotation because it will pour out of the buckets before it reaches the bottom of the path.

Work is defined as a force moved through a parallel distance tothe direction the force was applied. Energy gives you the abilityto do work. The energy of the water is transferred to doingwork inside the mill.

Figure 6.3 Overshot

Waterwheel By Old Sturbridge Village

Work is defined as:

(6.2) W represents the work. F is a force. d is a distance parallel to the direction of the force. The force is the product of the mass of the water and the acceleration (of gravity). This relationship is Newton’s Second Law of Motion and is defined as:

(6.3) The acceleration, a will be the acceleration of gravity and is represented by g. The force is the weight of the water and is represented with w.

(6.4) When you drop a weight (force) vertically (height) you change the potential energy to work.

(6.5)

Overshot Mill - Part 3

Determining the mass of the water on a waterwheel is not a measurement that can be easily made in the real world (and would have been very hard 200 years ago) so we must look at what can be measured. We could determine the maximum water per bucket, but in reality the buckets do not get completely filled with water.

Power is the amount of work done in a given amount of time. It is defined by the following relationship:

(6.6)

P represents power. W is work. t is time. First of all a time period must be substituted into our relationship to utilize the concept of power. Using equation (6.5) and substituting it into equation (6.6) the following is obtained.

(6.7)

We have converted our relationship into a time dependent relationship. Finding the mass per time, a rate, is still a measurement that is not easily determined. We need to convert the mass of the water into the volume of the water. To accomplish this we will use the density of water which is a constant for fresh water and is defined to be mass per volume (refer to chapter 2):

(6.8)

(rho) is the density. m is the mass of the water. V is the volume of the water.

Overshot Mill - Part 4

To determine the volume of the water we need to understand how the water was delivered to the waterwheel. This was usually done by a wooden flume (sometimes by a metal pipe). The water might travel to the flume via of an earthen or stone mill race. The wooden flume is the most common delivery device and we will only analysis this device for our purposes. The

wooden flume is a rectangular shaped device as can be observed in the image below. The volume of the flume can be calculated by the following relationship.

(6.9)

The volume is defined for the flume as length (l) times the width (w) times the depth (d). Substituting into equation (6.8) and solving for mass gives the following.

(6.10)

If we take equation (6.10) and substitute it into equation (6.7) we get the following.

(6.11)

Velocity for the water is defined as the distance the water travels in a given amount of time (see chapter 7 for more detail). The relationship for velocity is defined as:

(6.12)

The l will represent the distance the water travels (length) and t is the time period in which the distance was measured. We will discuss velocity in more detail in Chapter 7. Velocity of the water is measured as it travels in the flume. To determine this velocity today we have devices that we can be placed in the water. In the days of the water powered mill they would observe a leaf floating in the water or determine how much water flowed down the flume in a given amount of time and knowing the size of the flume they could determine the velocity.

Overshot Mill - Part 5 If we substitute equation (6.12) into equation (6.11) we get the following relationship.

(6.13)

We will now substitute the values for the constants and simplify the equation. The density of water is 1000 kg/m3. The acceleration of gravity is 9.8 m/s2. The constants in equation (6.13) will become the weight density value of water which was discussed Chapter 2.

(6.14)

The width times the depth is the cross sectional area and will be represented with an A.

(6.15)

The height is the vertical distance the water falls while in contact with the wheel which is usually between 70% and 90% of the height of the wheel. Some highly engineered wheels this can reach nearly 100% We can use the equation in the form of equation (6.15) or (6.14).

Example: Mill Springs, KY - Part 1

We will use the overshot mill atMill Springs, KY for our exampleproblem. The water for the millcomes from 13 springs and thewater is retained by a dam. Thepresent mill building was built in1877 and the present waterwheel was installed in 1906. Thefirst mill on this site was builtaround 1817. A famous Civil Warbattle was fought near this site in1862.

The velocity of the water inthe flume (entering thewheel) is about 2 m/s.Water remains on thewheel for approximatelythe entire half revolution ofthe wheel therefore we willuse the vertical height ofthe wheel.

The wheel is approximately 13 meters in height and thus the value of h is approximately 13 meters. The width of the wheel is approximately 62 centimeter and the depth of the water is about 10 centimeters.

If we substitute into equation (6.14) the following is obtained.

(6.16)

Thus the input power of the system is 15,798 watts, but the normal unit of power in the United States for mechanical systems is the horsepower. One horsepower is equal to 746 watts. Therefore:

(6.17)

This mill, with one of the largest waterwheels in the country, has an input power of only 21.2

hp which is about the same amount of power that your car generates when it is traveling at 55 miles per hour and not accelerating on a highway.

Example: Mill Springs, KY - Part 2

Many of the old mills were shut down as their mill dams broke or their water wheel decayed or broke. Some of the mills remained in operation longer by using a farm tractor or a Model T to power the mill through a belt. These usually were the smaller country mills. The larger commercial mills (like Mill Springs) converted from water power to steam power or possibly electricity.

Many mills were lost due to fire or floods. The mill at Mill Springs has a gasoline engine that was obtained when the steel Fitz (manufacturer of the water wheel) waterwheel was installed in 1906. This engine provided additional power for the mill. It was used to power some of the other machinery within the mill, too.

Mill Springs operated until 1949 when the Army Corp of Engineers purchased the property as part of the Lake Cumberland project. The input power is always less than the output power and many mills were approximately 10% efficient. The mill at Mill Springs has a very high efficiency because of the balance of the wheel and the speed at which the stones are rotated for demonstration purposes. The water depth on the waterwheel was much greater and the velocity of the water was much higher when the mill was operational and the dam was in good repair.

(6.18)

Mills Today If you visit an old mill they usually grind grain for only a short period of time a few times a day. This is generally for demonstration purposes. The machinery could grind more grain, but if they produced large amounts of corn meal or flour they would not have a market for their product. At many of the old mills the flour they produce often does not meet the health standards of today. There still are water powered mills that are commercial. One of these is in

Kentucky near Midway. When the mill is not grinding very little water is required to rotate the wheel. The power output of the mill can be controlled by the amount of water which is put on the wheel. The gate box which is visible in Figure 6.8 can control the amount of water released. These gates are positioned at different places on different mills. When the miller is ready to grind grain he will raise the gate and allow more water to travel onto the wheel. The wheel will begin to rotate at a greater speed. The miller will then engage either a gear or belt so that the stones or roller mills will begin to turn. The miller will continue to increase the flow of water until the stones are turning at the proper speed. If other equipment is also being used, the gate is raised even more. The process is reversed when the grinding is complete. If the large amount of water was allowed to flow over the waterwheel as when the stones and other equipment were engaged, the wheel would rotate at such a high velocity that the wheel would be damaged. The water mills were either large commercial business or small family farm related operations. Many of the name brands of flour that we use today began with small water powered mills. At these mills in their early days they would grind 24 hours per day when the crops were being harvested. At other times of the year they would grind only occasionally. During the harvest season the farmers might wait in long lines for days to get their corn and wheat ground. Of course the mill needed to have water and many mills had mill ponds to assure that they had an ample supply of water. Floods and droughts were serious problems for these early factories. Remember also that water power was used for other purposes than just grinding grain. It was used to make cloth and yarn, for power to run the assembly lines for the manufacturing of the early automobiles, for cutting lumber and many other manufacturing purposes. Many times sawmills would be built prior to the grist mill to produce the lumber for the new community. The byproducts of the grinding operation would be used for animals feed or as the grain for a distillery or just thrown into the mill pond to produce some extremely large fish.

Kinetic Energy

Not all of the mills were overshot mills as discussed above. In a few locations undershot water mills were built however, very few remain in the United States. Many of these mills were converted into water powered steel turbines. One of the few remaining is the mill pictured on the previous page, Graue Mill. Graue Mill is located outside of Chicago, IL. The stories that are told are that when Abraham Lincoln was a young lawyer he would stop and visit the mill. They would talk with the miller about how the mill worked. It is known that Lincoln made several trips between his home and Chicago during this period of time. This mill was also a stop on the Underground Railroad.

Undershot Mill

Undershot Mills use the principles of kinetic energy. Kineticmeans to be in motion. Therefore this source of energy is theenergy of moving water. An undershot mill uses the velocityof the water to turn the waterwheel. The water gains it kineticenergy as part of a fast moving stream or being stored in amill pond behind a dam as discussed in Chapter 2. When thewater does not already have sufficient kinetic energy a dam isplaced across a river. Water is taken off the bottom of thedam and flows through a millrace. The deeper the water heldby the dam the faster the water will flow down the millrace.This occurs because there is a greater amount of pressure onthe water at the bottom of the dam due to the weight of thewater. You may have experienced this effect as you swimdeeper in a pool the pressure you feel on your body isgreater. This is because more water is above you pressingdown. Review our discussion of dams in Chapter 2.

Kinetic Energy is defined as:

(6.19)

K represents the kinetic energy of the system. m is the mass. v is the velocity. The water slows down upon striking the waterwheel therefore the kinetic energy of the water is also less. The kinetic energy lost by the water is converted to work inside the mill. If the water comes nearly to rest then all of the kinetic energy of the water has been converted into work. Combining equations (6.2) and (6.19):

(6.20)

Similar to the overshot wheel discussed earlier we will convert the equations from energy to power equations by making the equations time dependent. This is done because of the problem defining the mass of the water striking the waterwheel. We will start with equations (6.6) and (6.20) and combine them together.

(6.21)

Using the same logic as done with the overshot mill we will substitute equation (6.10) and the following is obtained:

(6.22)

As before length per time is velocity and the value for density of water is a constant. Width times depth is the cross section of the flume as before. Therefore the equation simplifies to:

(6.23)

Graue Mill, IL

For our example we will use Graue Mill near Chicago, IL. Note that the waterwheel is not like the wheel discussed in the overshot mill. Overshot wheels have buckets to hold the water as the wheel turns. Also there was an elevated flume. The wheel on Graue Mill and other undershot mills is more like the paddle wheel of a steamboat. This is because we are only using the kinetic energy of the water. It is not falling; instead it is pushing the waterwheel.

As you can see in Figure 6.16 this wheel is much wider than the wheel previously discussed, but it is also much less massive and therefore has much less inertia. The wheel must be extremely well balanced. The cross sectional area of the flume is 3m x .5m or 1.5 m2. The velocity of water is approximately 3 m/s. If we substitute into equation (6.23):

(6.24)

Again the input power is much greater than the output power of the mill. Therefore the efficient of the mill can be calculated as discussed previously.

Other Types of Mills

There are other types of waterwheels these including the breastshot wheel which uses both kinetic and potential energy. There are two types of breastshot wheel. With the first type the water hits the wheel slightly below center (also known as pitchback wheel).With the other type the water hit slightly above the center (not very efficient).

Another type of wheel is the tub wheel which took many different forms including ones that look like paddles to others that look like an overshot wheel lying on its side.

In the 1840’s the first turbines were used which were made of metal and had a much higher efficiency than did the wooden waterwheels.

The newer overshot waterwheels were also made of metalwhich lasted longer and did not have the typical problemsof wood wheels such as warping and deterioration due tothe water. Therefore as wooden wheels decayed theywere replaced with metal wheels or turbines. Waterwheelswere ordered and installed in Kentucky into the earlyparts of the 20th century. There are still commercial millsin Kentucky which operate on water power and useturbines. There are other parts of the country that havecommercial mills that still use overshot waterwheels.

Energy Transfer

Many of the early water powered mills were built togrind grain (corn and wheat). To transfer the inputpower to the turning of the stones took massivegearing both to change the direction of the powerand also to increase the velocity of the stones.Similarly when you ride most bicycles you willpeddle at one speed but the rear wheel will turn at adifferent speed due to the gears.

The power take off from the wheel at Mill Springs is typical of many of the large metal wheels. The wheel has a gear ring visible in figure 6.8. The power is transferred to a smaller gear. The large ring has 36 teeth in each of the 16 sections for a total of 576 gear teeth. The smaller gear has 38 teeth on its circumference. Therefore for every rotation of the large wheel the small wheel turned 15.2 rotations.

The large wheel turns at 2 or 3 rotations per minute so the small wheel turns at about one revolution every two seconds.

Figure 6.21 Energy Transfer by Old Sturbridge Village

Rotation Kinetic Energy

To grind grain solid stones with grooves cut into them are used. These stones called burrs, do not crush the grain but instead it is sheared. The stones are large and massive and must rotate at a speed much faster then the water wheel turns.

The equations and solutions for an object that rotates is different from an object moving in a straight line. For an object that rotates we look at the angular velocity instead of the linear velocity and the Greek letter omega (ω) is used to represent the angular velocity. Measurements are made usually in radians per second. A radian is part of a circle. All circles have 2π radians and π is equal to 3.14. A radian does not depend on the radius of the circle. Angular velocity is defined as the angle the object turn through during a defined period of time.

We are only going to use solid circular shaped objects but be aware that different shaped objects with the same mass will rotate at different rates dependent on the moment of inertia of the object. Rings, solid disks, bars and spheres will all have different equations for the moment of inertia. For the millstone which is a solid disk the equation is:

(6.25)

I represents the moment of Inertia. m is the mass of the object, and r is the radius of the object. For a typical stone in a large mill the stone is usually about 1.5 m in diameter and has a mass of 1000 kg.

Therefore:

(6.26)

Note that .75 meter is used since this is the radius, i.e. radius is half of the diameter and the height was converted into metric units instead of English.

Equation for Rotational Kinetic Energy The equation for Rotational Kinetic Energy is represented by the following relationship.

(6.27)

The stones begin at rest and increase in speed until they reach operational speed. The work done on the stones is equal to the rotational kinetic energy of the stones. A stone in a typical mill will turn at 60 revolutions per minute. Revolutions per minute need to be converted to radians per second. There are 60 seconds per minute and 2π radians per revolution. We will substitute equation (6.27) into the power equation (6.6).

(6.28)

Therefore the power required to start a millstone turning to operational speed at the Mill Spring grist mill, is:

P = 7.43 hp

Note this is much less than the input power of the mill. The energy which was lost was mainly due to friction (assume that all the power was used only by the millstones). This also assumes that it takes only one second to go from rest to operational speed and at many mills this time period is much longer.

Efficiency If the output power is the amount of power required to do the work for which the mill is designed and the input power is the amount of power which is derived from the water we can determine a simple efficiency of the mill using equation, (6.18). We will use the data from equations (6.17) and (6.28) to calculate the result. These results will be for Mill Springs with the constrains previously noted.

(6.29)

Old College Book: Waterwheels, Part 1 & 2 The concepts covered in this chapter were taught in colleges and universities in the past. I have copied a few of the pages from an old textbook for you to review. These images are used with permission of the publisher at the Society for the Preservation of Old Mills

(SPOOM, http://www.spoom.org).