1] A handful of basic definitions:

The global balance Law of Mass :

The principle states that the total mass of a medium is conserved as the medium

undergoes a deformation. The law may be stated as: In the absence of sources and

sinks, the total mass of a body before a given deformation is equal to the total mass

of the body after transformation.

This law establishes the continuity of mass of a continuous medium since the total

mass remains unchanged during a deformation. If 0 and dV denote, respectively, the

mass density and the element of volume before deformation and if and dv denote

the corresponding quantities after deformation, then we must have

which implies that the time rate of change of the total mass in

the body is zero.

Differences ‘twixt heat (Q) and temperature (T):

We’re quite often confounding heat and temperature, though, even if they are

cognate terms, they’re in fact thoroughly different one from each other

If we’re placing upon the flame of a gas burner, a pan filled with cold water, the water

temperature is gradually stepping up, while the water is receiving heat from the gas

burner.

It might seem legit to suppose that when

an object is receiving heat, his

temperature as a consequence is

increasing too. Yet, this statement is ab

initio rejected; when the water begins to

be boiling around 100°C, we’re observing

that the temperature isn’t augmenting

anymore, while heat is still received.

Work, potential & kinetic energy: From

Newtonian mechanics, we know going

from state 1 to state 2 that the work is

done by a force moving through a

distance. The word “work” was first used

in this sense by the French mechanician

Gaspard-Gustave Coriolis.

This concept was originally introduced in mechanics fields, we’ve had initially the

kinetic energy, associated with the motion state of a body, then, the potential energy

referring to the position state and the forces deriving from it, videlicet, the state

forces.

Thus, we define the mechanic energy m as the summation of the kinetic energy,

and the potential energy associated to the external and internal forces thereof.

m=W’ex+W’in ; with m= cp,ex p, in

In the aforesaid expression, the mechanical energy and the work are distinct: the first

concept is a state function, which doesn’t depend on the path, while the other hinges

around it.

So, the work is said to be, not an energy, but a transfer of energy.

Therefore, the mechanical energy isn’t a conservative unit; hence, can we introduce

a new concept that generalizes the previous one, while being conservative?

The answer is precisely given by the first law of thermodynamics, which is a case of

the law of conservation of energy, stating that the total energy of an isolated system

is constant;

W+Q=U [Joules]

With: U, the total change in internal energy of a system

Q is the heat exchanged between a system and its surroundings

W is the work done by or on the system

Internal energy:

This subject of thermodynamics is rather complicated, for there are sundry ways to

describe the same thing. When we want to describe the comportment of a gas, we

can say that the pressure depends on the temperature and the volume. Or, we can

also say that the volume depends on the temperature and the pressure. Or

concerning the internal energy U, we can say that it depends on both temperature

and volume, if these are the variables that we arbitrarily selected – but we might just

as well express it through the couples {temperature, pressure}, or {Volume, pressure}

, hence, we can build as much variable functions as we want : U-TS is a volume and

temperature function… For ease concerns, we’ll express it with the temperature and

the volume, both regarded as independent variables:

In the Joule’s experiment: Our tank being thermally isolated, U is equal to zero;

we can expound the first law of thermodynamics to the particular case of the

adiabatic process

What are the definitions at stake with the mechanical equivalent of heat?

2] A “smidgen” of History

Surprisingly, the energy is a somewhat recent concept. The word was coined in

1807 by T.Young, famous for his contribution in optics fields, it stems from the greek

word ”énergia” meaning “action force”. It had already been used by W.Leibnitz in

1678, under the form of an amount which was preserved in free fall; the sum of the

“momentum” (kinetic energy), and the “dead force” (the potential energy).

Concerning the potential energy, an approximation of the value of “G” had been

determined through the Henry Cavendish experiment in 1798, thus leading to the

expression of the acceleration of gravity (g), used by James Joules to calculate the

work he was providing to his system with weights attached to pulleys.

Gravity is stretching through whopping distances. Instead of being obliged to gaze

towards the stars, why couldn’t we take a lead ball and a bead, and then watch the

bead going straight to the lead ball? The main hindrance of the experiment, put is

such a simple way, is the extreme weakness or sensitivity of the force, so it had to be

done with a thorough minutiae. While measuring the fiber twisting, Cavendish was

then able to express the intensity of the resulting force; here’s a simplified view of the

apparatus he used:

In Cavendish's original experiment, the following values

were used:

Mass of large ball M = 158 kg (348 lbs) | Radius of

large ball rM = 30.5 cm (12 in) | Mass of small ball m =

0.73 kg (1.6 lbs) | Radius of small ball rm = 5 cm (2 in) |

Separation of large balls L = 1.86 m (73.3 in) |

Separation of small balls L = 1.86 m (73.3 in) | Distance

between large and small balls R = 0.225 m (8.85 in)

More recent experiments have used other values.

The derived equation for G is: G = 2π2LθRe2/T2M

Where :

The calculated value of G from this experiment is: G = 6.674*10−11 m3/kg-s2

Application: Mearth = 5.973 1024 Kg | Rearth=6.378 106m

g= 9,799 [m.s-2]

About the kinetic energy: In 1720, Dutch physicist Willem Gravesande conducted

an experiment in which brass balls were dropped with varying velocity onto a soft

clay surface; Gravesande found in his experiments that a ball with twice the velocity

of another would leave an indentation four times as deep, that three times the

velocity yielded nine times the depth, and so on. He shared these results with Emilie

Gabrielle du Chatelet, and with Voltaire, after which she subsequently corrected

Newton's formula E = mv to E = mv².

In her 1740 Lessons in Physics, Chatelet, supposedly, combined the theories of

Gottfried Leibniz and the practical observations of Dutch physicist Willem

Gravesande to show that the energy of a moving object is proportional not to its

velocity, as had previously been believed by Newton, Voltaire and others, but to the

square of its velocity. Hence it is said that whereas Gravesande provided the

experimental data for the formula of kinetic energy, Chatelet provided the formulaic

explanation. (See eoht.info for more details)

Historical Milestones:

In 1524, Paracelsus adopted Aristotle’s four element theory, but reasoned that they

appeared in bodies as three principles. Paracelsus saw these principles as

fundamental, and justified them by recourse to the description of how wood burns in

fire. The Parcelsus theory included the cohesive principle, so that when it left in

smoke the wood fell apart, the mercury principle represented the aerial by-products

of the combustion, the sulphur represented the fuel, and utterly the salt represented

the solid by-products

(Besides, the etymology of the word “gas” stems from Paracelsus’ use of the word

khaos, in the occult sense of "proper elements of spirits" or "ultra-rarified water".)

1593: Galileo develops a water thermometer

In Circa, 1635, Italian physicist Giovanni Baliani showed that by placing an iron pot

filled with water on a spinning metal disk, it was possible to make water boil. This

experiment is said to have been one of the first references to an experimental

determination of the equivalence between heat and work

In 1669, German physician and chemist Johann Becher updated Paracelsus’ sulphur

model of how things burn with a terra pinguis model of combustion, wherein terra

pinguis was considered as the fatty, oily material substance of bodies that gives

things the property of combustion.

1744 : The Celsius scale was poised to be used widely

1760s : Joseph Black develops calorimetry

In 1798, the cannon boring experiment conducted by Benjamin Thompson showed

that the work expended by a 2 horsepower engine, turning cannon boring drill bit at a

rate of thirty-two revolutions per minute, inside of a cannon barrel, which itself was

submerged in a tank of water, will generate heat in the cannon barrel to the effect

that it caused the water to boil at 2 hours and 30 minutes.

1824 : Nicolas Léonard Sadi Carnot discusses idealized heat engines.

1840s: James Prescott Joule relates heat and work with its Paddle Wheel

Experiment

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In 1703, German chemist and physician Georg Stahl, one of Becher’s students,

updated the terra pinguis model with a phlogiston model of combustion. The

deficiencies of this theory, as shown by experiment in later decades, led French

chemist Antoine Lavoisier in the 1780s to develop the caloric theory of combustion.

The deficiencies of this theory led German physicist Rudolf Clausius in the 1850s to

develop an entropy model replacement for caloric and heat, which in turn gave birth

to the science of thermodynamics (1865).

Thermodynamics has a chequered history, unfortunately, it was not blessed with the

crispness of development that mechanics realized with Newton. In fact, its growth is

filled with false steps, errors and debates.

It’s almost ‘till the middle of the XIXth century that we’re still considering heat as fluid

called “the caloric”. This massless fluid, capable of pouring from one object to

another one, that cannot be created neither destroyed, provides a rather complete

explanation of many experiences of that time. Julius Robert Mayer, James Prescott

Joules and Hermann Von Helmholtz are historically related to the abandonment of

the caloric theory and the generalization of the principle of energy conservation that

considers thermal phenomena.

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Mayer was the first to suggest that the various forms of energy, including heat, are

convertible into one another without loss. When a certain amount of a certain kind of

energy disappears, an energetic equivalent is popping up in some other form as a

consequence.

Alas, Mayer and his idea suffered from a lack of consideration. Having received a

medical training in Germany, he became interested in physics and published his

observations in 1843. The article, written in a metaphysical style, isn’t considered to

be particularly convincing. The willy-nilly resistance against Mayer insights is partly

due to the fact that it isn’t downright egregious that he understands the laws of

Newton and even less physical concepts intervening in a theory of energy

conservation. (entropy)

James Joule’s 19th century experiments with beer can be used to illustrate the notion

of “entropy”. The English brewer, whose name lives on in the standard unit of energy,

sealed beer in a thermally isolated tub containing a paddle wheel that was connected

to weights falling under gravity outside. The wheel’s rotation warmed the beer,

increasing the disorder of its molecules and therefore its entropy.

But hard as we might try, we simply cannot use Joule’s set-up to decrease the beer’s

temperature, even by a fraction of a millikelvin. Cooler beer is, in this instance, a

state regrettably beyond the reach of physics

1] “heat>work transformations”

A) The most well-known: The Aelopile

A 50AD Alexandrian Hero-type aeolipile rigged to

a pulley contraption so to be able to do

mechanical work: namely the raising of a weight

(mg) by a certain distance in height (h), during

which process a certain quantity of heat Q

converted into a certain quantity of work W

(mgh), in a certain proportionate ratio (J) of work

to heat.

B) insight of the “heat>work transformation” efficiency

The gist of thermodynamic hinges

essentially on considerations of that ilk:

Because a rubber strip is more “rigid” at

increased temperatures than lower ones,

it must be possible to haul a weight while

begetting work.

We can envision this machine of a quite foolish aspect to perform exactly the above-

mentioned function. It’s formed of a bicycle wheel in which all

the spokes are rubber-made. If we were to apply heat at one

side of the wheel with a pair of lamps, the rubber strips would

gain some “rigidity”, causing the gravity center to shift and

therefore entailing a spinning of the wheel. The efficiency of

such a machine would be extremely scant. 400 Watts are

dispelled by the two lamps, but that might scarcely lift a flea!

An interesting question, tough, is to know if we could obtain

more work from the heat in a more efficient way, according to

the Carnot’s formula, we put:

2] Joule’s experiment (“work>heat transformation”)

Under steady conditions:

Measurement & uncertainty:

Before performing a measuring, it’s appropriate to mull over the following elements:

The measurement principle (scientific basis of a measurement)

The method of measurement (set of practical & theoretical operations to be

implemented during the execution of a measurement)

The procedure (detailed set of the employed methods)

The measurand (particular quantity subject to measurement)

The process variables of influence (quantities influencing the measurement

results)

The equipment: -accuracy error: it will be given by an instrument calibration

certificate

-precision error: It may properly come through repetition of measurements of a

same measurand

Process: hardships due to implementation difficulties

Materials: Elastic deformation, surface attrition, global geometry…

Environment; the temperature (acting on the measurand or upon the measurement

instrument), the electromagnetic field (electro-magnetic compatibility), the input

tension fluctuations, the vibrations, the atmospheric pressure, etc…

People: -reading error (interpolation, interpretation, issues with analogic devices)

-Positioning variability

-visual reflexes (measurement taking, over time)

Uncertainty: The measurement uncertainty is the associated parameter to a

measurement, characterizing the values dispersion which could be reasonably be

attributed to the measurand.

Type-uncertainty of type A: The estimation of that uncertainty is an evaluation

method by the statistical analysis of observations series. It’s obtained through a

probability law deducted from an observed distribution.

Most often, the best available estimation of the expected value 0 , of a quantity (q)

varying haphazardly, and for which we’ve obtained n independents observations qk

in the same conditions is the arithmetic mean value of the n observations :

Type-uncertainty of type B:

The type-uncertainty of type B is used when the type-uncertainty of type A reach little

values, or when the measure wasn’t obtained through repeated observations. In the

other cases, it is just “added” to the type-uncertainty of type A.

Combined Standard uncertainty:

In the worst of all the cases, the uncertainty around the measurement of the

temperature elevation (the most fluctuating parameter) is defined this way:

(1) In order to simplify the calculus, we will consider that the following set of

uncertainties is insignificant in the above formula; {the time, the variation of the

caloric capacity through temperature augmentation, and the inherent masses

of the objects}.

(2) After a few insights weeks, I’ve realized that it was more professional to write

the formulas in Latek code, here’s a brief overview of what it may looks like in

our case :

Quite handy, isn’t it?

Conclusion: Further improvements and remarks:

While we’ve been apprised of the sliding friction this year (Coulomb law), relative to

the lateral motion of two solid surfaces in contacts, determining the precise amount of

energy delivered by the friction of the beater to the oil, it’s still a pretty fuzzy step (the

fluid viscosity isn’t given, but it ought to have the highest possible value to produce

more heat). We have the formula for the dissipated energy of the trunk, nonetheless,

the developed kinetic energy of the anchor-shaped piece remains to be calculated

(most likely with the maximaV software), and by doing so, this might lead to a pair of

further optimizations, mainly about the ideal dimensions.

Kinetic energy of a cylinder under rotation around an axis:

The kinetic energy density at point x, and at time t, of a body moving with velocity

v(x,t) is defined as ½ .v. v

Where, m is the mass [kg], R, the radius[m] and , the angular velocity.

How could we measure a viscosity? (Dynamic)

The simpler system consists to let falling a bead of a given radius and density in the

concerned fluid in order to measure its viscosity and calculate the time necessary for

the ball to cover a certain distance, once its maximal speed is reached.

It’s a simple and fast method, but it is undeniably lacking some accuracy, besides the

obtained value would have to be corrected thereafter (inter alia, with a correction of

the Reynolds number)

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Another kind of viscometer is based on the fluid motion through a capillary tube.

We’re then measuring the time taken by the flow to pour through the tube under the

action of a differential pressure.

Could we measure more accurately the temperature?

The quartz thermometer is a high-precision, high

accuracy temperature sensor.

It measures temperature by measuring the frequency

of a quartz crystal oscillator.

The oscillator contains a specially cut crystal that

results in a linear temperature coefficient of

frequency, so the measurement of the temperature is

essentially reduced to measurement of the oscillator

frequency.

Resolutions of .0001 °C, and accuracy of .02 °C

from 0-100 °C are achievable. (Wikipedia)

Remark about the critical radius:

We’re now considering a cylinder (our calorimeter), with L being the length and

r=r0+e, if we were to apply an insulation around it, with a thermal conductivity in

W/(k.m), then we would obtain the following formula :

We can demonstrate that the thermal resistance is passing by a

Minimal point called the “critical radius”

Therefore, if we want to insulate correctly our tank, we need to properly define our

variables, otherwise, we could be increasing the thermal losses.

Are there Alternative methods to an increased thermal

insulation?

It’s possible to use an infrared camera to measure the

losses (U) instead of Q, but as the cylinder cavity isn’t

isotropic, the thermal transfer would not be homogenous,

and considering a mathematical point of view, the

realization would be slightly more complicated

Utterly, we could just as well spend some time pondering

the idea of an external regulated environment, in order to

maintain equal temperatures at both sides of the cylinder

Explanatory scheme (with a resistance):

Thereby, we could ascertain that U (mCv) is really equal to

zero, in order to have W=Q.

Bibliography :

“Physique & chimie”, Magister 2000, Philippe Auzou

Chaptire XI, « PHYSIQUE » Kane/Sternheim, InterEditions, ISBN 2

7296 0098 1

« THERMODYNAMIQUE fondements et applications » J.Ph.Pérez &

A.M Romulus, Masson, ISBN 2-225-84265-5

« Lecture Notes On Thermodynamics» Joseph .M Power,

“Departement of Aerospace and Mechanical Engineering, University

of Notre Dame”

« Le cours de physique de Feynman –mécanique 2» Editions Duno

http://en.wikipedia.org/wiki/Quartz_thermometer