Free Surface Profile of a Circular Hydraulic Jump

Impinging jets have excellent heat transfer characteristics and are widely used where a lot of heat has to be moved or removed rapidly.  In the case of a liquid jet directed downward against a horizontal surface, a hydraulic jump forms when the free surface makes a sudden transition from supercritical to subcritical flow.  The jump formed by the radial spread of a circular jet is sometimes called a “circular” hydraulic jump.

This work measured the hydraulic jump surface profile and unsteady fluctuations for several configurations of jet sizes, flow rates and downstream depths.  The downstream depth was controlled by an adjustable height weir shown in the figure as height “w”.

The instantaneous position of the free surface was determined by using a fine wire probe.  A small dc voltage was applied to the water, and the probe completed the circuit between the water and the power supply ground.  The fluctuating surface of the hydraulic jump made it necessary to approximate the jump profile as a statistical representation of a large number of individual measurements.

The figure below shows data for a nozzle diameter of 7.8 mm.  Each symbol represents the dimensionless depth (y/d)  where the probe was in the water 50% of the time.  The vertical error bars on each plotted point represent the 0% and 100% submersion locations.  This  figure provides a representation of the approximate mean location of the free surface, along with the approximate vertical extent of the free surface fluctuations.  As would be expected, the fluctuations are largest near the front edge of the hydraulic jump, and are greater for higher Reynolds number.  The progression from a single jump structure to a double jump structure with increasing downstream depth described by Liu and Lienhard (1993) is clearly visible.

The next figure shows the extent of the vertical fluctuations for several sets of data.  The fluctuations are largest at the front of the hydraulic jump, then drop rapidly and decrease more slowly toward the back of the jump.

 Some measurements were taken in the thin flow layer upstream of the hydraulic jump.  The figure below demonstrates a comparison of some measured layer depths with a correlation of layer depths calculated from velocity measurements in a separate work.  Overall, the comparison shows the same general trends and similar layer depths. 

 

  

References

Stevens, J.W., 1995, “Free Surface Flow Profile and Fluctuations of a Circular Hydraulic Jump Formed by an Impinging Jet,” ASME Journal of Fluids Engineering, Vol. 117, pp. 677-682.

Stevens, J., and Webb, B.W., 1992, “Measurements of the Free Surface Flow Structure Under an Impinging Free Liquid Jet,” ASME Journal of Heat Transfer, Vol. 114, pp. 79-84.

Liu, X., and Lienhard, V.J.H., 1993, “The Hydraulic Jump in Circular Jet Impingement and in Other Thin Liquid Films,” Experiments in Fluids, Vol. 15, pp. 108-116.

Transient Heat Transfer Overview

I first saw a figure like this in Professor Adrian Bejan’s book Heat Transfer  and I really liked the succinct way that it illustrates the ballpark relationship between various transient conduction heat transfer approximations that we commonly use.

In log-log coordinates, we have the Biot number (Bi), which is a dimensionless heat transfer coefficient plotted on the vertical axis, and the Fourier number (Fo), which is dimensionless time, plotted on the horizontal axis.

In a solid which is initially at a uniform temperature and is subjected to a sudden change in the boundary condition, the temperature changes will initially occur (Fo <<1) mostly very close to the surface, and the temperature field can be approximated by the solution for a semi-infinite body.  Professor Bejan calls this the “early” regime, and it is represented by blue shading on the left half of the figure.  If the heat transfer coefficient is high enough (large Bi) or for some reason the surface temperature is specified,  then the solution for a fixed surface temperature (still semi-infinite) is appropriate as shown in a different shade of blue in the upper part of the left side of the figure.

For cases where the conduction within the solid is rapid relative to the heat transfer out, the temperature variation across the body can be neglected.  The entire body can be characterized by a single temperature at any given time and the temperature variation of the body with time can be approximated with an approach called “lumped capacitance”.  The Biot number serves as a measure of when the spatial temperature variation is expected to be small relative to other temperature differences.  Lumped capacitance solutions are appropriate for Bi << 1, or a common rule of thumb that is sometimes cited is Bi < 0.1.  Professor Bejan terms this the “late” regime, and it is represented by the reddish shading in the lower right side of the figure.

For cases where neither the Fo nor the Bi number is very small, the exact solutions must be applied.  Actually, the exact solutions could be applied anywhere, but since they are in the form of infinite series, it is most convenient to use one of the simpler approximations, when appropriate.  These infinite series converge rapidly for large values of time, so a for Fo > 1, it is an excellent approximation to use only the first term of the series.  A common rule of thumb for using only the first term of these series is Fo > 0.2.  A graphical presentation of the temperature distribution based on the first term of these series has become known as the “Heisler Charts”.  The area covered by the first term of the exact solutions, or by the Heisler Charts, is shown in the upper right quadrant of the figure shaded green.

Definitions:

 L_c= a characteristic length of the solid

h = convective heat transfer coefficient at the surface of the solid

k = thermal conductivity of the solid

a = thermal diffusivity of the solid

t = time

Reference
Bejan, Adrian.  Heat Transfer.  New York:  Wiley, 1993.

Considering (or not) Graduate School?

I seem to wind up talking relatively often to senior-level students about the pros and cons of graduate school.   Sometimes these conversations are initiated by students who are considering graduate school, other times I initiate the conversation with very good students who don’t seem to have considered the possibility at all.  Since this topic might be of general interest, I thought that today I would share my thoughts and opinions about graduate school in engineering:

(1)    Pros and Cons

If you are a good academic student (roughly GPA >3.4, although there is a lot of play in this), you should definitely consider graduate school as one option.  It isn’t necessarily right for everybody, but it ought to be something that you look at—if you had 3 job offers, you would weigh the pros and cons of salary, location, potential for growth, quality of company, etc. between the different jobs, then decide on which is best for you at the present time.  Graduate school ought to be one of the options to consider for academically talented students.  If your GPA is less than 3.4, you might still want to consider graduate school if it appeals to you.

a.       Pros:

                 i.      A master’s degree pushes you toward the funner end of the job spectrum  Whether this is even true, and the definition of “funner” are both matters of opinion.  My own opinion is that there is a spectrum of engineering jobs ranging from jobs with very little technical content to jobs with a lot of it.  Usually employers who want/need very little technical content are not interested in paying a slight premium for a master’s degree holder.  Conversely, those who pursued a master’s degree are less likely to want a job with little technical content.  So, it becomes a self-regulating thing.  It is also a general tendency, not an unbreakable rule.  My opinion on “funner” is probably obvious.

                  ii.      More of the same: classes, learning 

Your last four (or five or six) years have probably made you pretty good at succeeding in coursework and learning technical subjects.  Presumably you enjoy that, to some extent, or you wouldn’t be reading this at all.  If you plan your program and choose your advisor well, graduate school should be like the best parts of your undergraduate education in terms of the academic things that you enjoy.

                  iii.      Different things 

Graduate school will broaden and deepen your academic experience considerably.  You might attend a different institution and become acquainted with new professors, new classmates and a somewhat (or extremely) different  academic culture.  You will certainly have a different relationship with your major advisor than you typically had with professors as an undergraduate.  You will learn about and engage in research in a different way and/or to a different extent than you might have as an undergraduate.  In general, your classes might be more rigorous, more focused, and more interesting than your undergraduate classes.

                 iv.      Possibly no more debt 

See #5 Assistantships and Funding.  As a general rule you won’t get rich in graduate school, but depending on your circumstances you might be able to get through it without too much additional debt.

b.      Cons:

                 i.      Delaying your career 

After four years of rigorous study, you are undoubtedly eager to get out and earn a real salary and engage in real engineering.  Graduate school will delay some parts of that for a while.

                 ii.      More of the same 

Graduate school still entails taking classes, doing homework, and reading textbooks.  If you can’t stand any more of that, then graduate school may not be the best choice right now.

                 iii.      Long time to catch up $

A fair comparison between earning potentials of BS and MS degree holders has to measure both from the time of receipt of the BS degree.  Most of the comparisons that I see show that the average cumulative earnings of MS degree holders eventually catch up and pass.  However, this normally takes a long time (10-20 years) and is only an average (may not hold true for individuals).  The potential for making more money is not a logical reason to attend graduate school, in my opinion.

(2)    A Big Reason NOT to Go

Don’t consider graduate school only because you don’t have a job, in general.  This is one of the weakest reasons to attend graduate school, and will probably (not necessarily) wind up as an unpleasant experience for you and for your advisor.

(3)    What You Will Do and How Long It Will Take

Your graduate school classes may not be too much different from the courses that you have taken as a senior.  Many schools allow graduate classes to be used for senior-level technical electives.  In addition, if you do a thesis, (see #7 Thesis vs Non-thesis Options) you will work very closely with your major advisor to plan, carry out, and write up a major research project which will constitute your master’s thesis.  Typically, a master’s degree will take somewhere between 18 and 30 months beyond the BS degree if you are going full-time. 

(4)    Master’s vs. PhD

Some schools will allow top students to enroll in a PhD program with only a BS.  Other schools require a completed master’s degree before acceptance into a PhD program.  Normally PhD requirements are expressed in terms of the number of classes since the BS degree.  If you do get a master’s degree first, definitely choose a thesis option.  Almost everything about completing the thesis will be good preparation for working on a PhD.  If you do have a choice, there are good arguments for doing it either way.  On the one hand, the thesis gives you good experience and preparation for working on your dissertation, and the master’s degree gives you a tangible intermediate achievement if the PhD somehow doesn’t work out.  On the other hand, the time spent on the thesis doesn’t add much to your resume once you have the completed PhD degree.

(5)    Assistantships and Funding

Many times, especially if your GPA is above about 3.80, you will be able to get some kind of assistantship to attend graduate school.  The amount tends to be far less than you would make with a regular job, but for a single person with modest tastes, it is usually enough to make it through school without taking additional debt.  Many schools offer a tuition waiver for students on an assistantship.  A few do not, and in those cases sometimes the assistantship will be configured to cover tuition, as well.  There are a wide variety of assistantships and corresponding expectations of the students.  I’ll describe a few aspects and a couple common configurations, but bear in mind that you may find some hybrids or something completely different.

 Teaching Assistantships usually involve grading for a class, running a laboratory, offering help sessions, or even teaching a full class.  In effect, you are getting a part-time job, somewhat related to your academic area, while you work on your degree. 

Research Assistantships are normally given for work in the research laboratory.  Most often, most of the research in which you are involved relates directly or closely to your thesis, so in effect, you are getting paid to work on your thesis.  Research Assistantships might be funded by the department, which would generally give you a little more freedom in terms of the specifics of your project, or they might be funded by an external sponsor (government grant, industry contract, etc.) in which case you would almost surely be directed quite narrowly in the research subject and direction.

(6)    Finding an Advisor

If you are completing a thesis option, it is my opinion that selecting an advisor is the single most important decision that you will make about graduate school.  It is more important than the school that you attend, and it is more important than the specific topic of your thesis.  Throughout your time in graduate school, you will work closely with your advisor, he or she will direct your research and teach you about doing research, and will declare when you have done enough, and will be extremely influential in the shape of your graduate school experience.  

You should almost always sit down and talk with a professor before signing on as their graduate student.   Talk about assistantships, potential research projects, and their philosophy of graduate school.  Most professors are eager to talk about their research (it is usually more of a problem to get us turned off than to get us started).  If someone is too busy to visit with you, a potential graduate student, that ought to be one alarm for you already.  Pay attention to how enthused they are about their research, whether they are interested in you, how they respond to your questions, and, if you can find out, how they treat their current graduate students.  Finally, it would be a rare occurrence if you could compile enough quantitative  information to be completely sure that this is the right professor, so pay attention to your gut feel and your instincts.  I think that it is a necessary (but not sufficient) condition for you to be very comfortable with your advisor in order to have a great graduate school experience.

(7)    Thesis vs Non-thesis Options

Many programs offer the option of completing a master’s degree with either “course-work only” or with a thesis.  Typically, both options require 30 credit hours (usually 10, 3-credit hour courses).  In the thesis option, six credit hours are usually completed as “thesis credits” leaving you with 8 regular classes.  In “thesis credit” classes, you normally just work on your thesis, with your advisor, as you are doing anyway.  In non-thesis option programs, you usually just take 10 classes and leave with your diploma.

From a strict accounting perspective, it would probably be more efficient to do a non-thesis option.  That is, in terms of total effort, your thesis will probably require a lot more work than completing two regular classes.  However, in return for that effort, you engage in research, focus on a complex problem, review pertinent research that others have done, and present the results of your work orally and in writing in a coherent way.   While the effort is higher than two regular classes, the corresponding benefits are much higher.

So, the bottom line is that if you intend to sometime pursue a Ph.D., completing a thesis is probably a wise choice for the preparation that it will give you.  If you are a full time student, and have the option, completing a thesis is probably a wise choice for the benefits that you’ll receive.  If you are working full time and getting your master’s degree one or two classes at a time, it may not be possible to engage in the concentrated time and effort that a thesis will require and a non-thesis option may make sense for you.

(8)    Other Opinions

These are just a few of my thoughts about graduate school.  You can find a lot of other opinions and advice.  Here are a few sources:

http://www.gradschooltips.com/

http://www.quintcareers.com/considering_graduate_school.html

http://gradschool.about.com/

http://www.graduateschool4you.com/

Best wishes on your decision!

Blog Home

Design of a Low-cost, Active Evaporation System

We wanted a preliminary conceptual design for a low energy, low cost, high volume fresh water evaporation system. 

Figure 1 illustrates the proposed system.  A high pressure pump forces water through an atomizing nozzle.  At the same time, a low-speed fan induces a stream of ambient air into the tank. The air exits the tank through a duct configured to form a vortex inertial separator.  A portion of the atomized water evaporates into the ambient air stream through the tank, while the remainder falls back to the reservoir or is collected in the inertial separator and directed back to the reservoir.

Figure 1.  Diagram of conceptual design

Performance

   The liquid water storage capacity depends only on the design choice of reservoir size.  The evaporation rate is also a design variable dependent on energy usage.  The table below shows two preliminary estimates of evaporation performance and energy usage detailed by fan and pump power for just one possible system configuration.  These are only very rough estimates and have not been experimentally validated.

evaporation capacity	pump power (est.)	fan power (est.)
3 liter/hr	1 W	21 W
1 liter/hr	0.37 W	8.2 W

Advantages

    The proposed system offers a number of significant advantages over conventional evaporation approaches: 

           Design freedom

The combination of a high pressure atomizer with an inertial separator allows great flexibility in system design by permitting trade-offs in individual component specifications.  In general, higher pressure water supply will result in smaller droplet size distributions from atomizing nozzles, however this costs more in pumping power and pump cost.  More expensive nozzles will produce smaller droplets and narrower droplet size distributions. Larger droplets can be produced less expensively in terms of nozzle and pumping costs, but they will require greater residence time for evaporation.  Higher air flow rates and higher air stream turbulence levels increase evaporation rates but cost more in fan power.  Trade-offs between various component costs (nozzle, pump, fan, inertial separator) and between capital and operating (energy) costs can be optimized to give excellent performance at minimum overall cost.  The inertial separator at the exit will insure that liquid water is contained so that wide variations in performance of individual upstream components can be tolerated while still resulting in an acceptable design.

Off–design operation

The same flexibility that allows for tremendous design freedom also permits the design of a very robust overall system.  Many factors including nozzle wear, off-design environmental conditions, water contamination, operator interference, control system failure, etc., can result in system operation at conditions outside of the original design parameters.  Ideally, a system will continue to give acceptable performance away from a single operating specification.  The flexibility inherent in this concept will easily allow for such a robust design. 

Evaporative cooling

In general, the evaporative cooling effect will result in a cool and moist air stream exiting the system.   This may be a primary aim or a desirable side effect.  This system maintains the principal advantage of a conventional evaporative cooling system—low operating cost--while avoiding the disadvantages inherent in maintaining a wetted media for evaporation – mold/fungus growth, higher pressure drop for the air system, and maintenance/replacement of the media.  

Ground-source Heat Engine Prototype Design and Test

Figure 1. Generalized Schematic

A company wished to explore feasibility and performance of ground-source heat engines designed to generate a tiny amount of electrical power from the daily fluctuating temperature difference between the air and the ground using solid-state thermoelectric generators. 

 Figure 1 shows the general structure of the device. Multiple thermoelectric modules were sandwiched between air-side and ground-side heat exchangers. The thermoelectric modules provided the electrical power generation from heat that flowed as a result of the air-ground temperature difference. The design consisted of appropriate geometric design for both heat exchangers, selection and interface configuration of the thermoelectric modules, and thermal resistance matching between the heat exchangers and the thermoelectric modules. 

Figure 2. Instrumented Prototype

 Prototype devices were designed and built. Figure 2 shows one of the prototypes situated in the test area and instrumented for performance evaluation. This prototype included fins on the air-side heat exchanger visible in the figure. Extended test results indicated peak power of 5 milliwatts and average power of 1 milliwatt could be obtained from the prototype configurations. It was determined that approximately 50% of the power generated in extended tests could be attributed to direct solar insolation. 

 Figure 3 shows a sample of measured power generation rates over a short part of the test period, and Figure 4 shows the total energy generated by hour of the day for the entire test period.

Figure 3. Measured Power

Figure 4. Total Energy

Parameter Simplification

It is often desirable to simplify heat transfer problems by working in dimensionless quantities. In some situations it is desirable to apply formal non-dimensionalization approaches such as the Buckingham Pi theorem. In other cases, there are obvious non-dimensional groups. Sometimes it is most convenient to only partly non-dimensionalize a problem. Normally, this process yields three significant benefits:

(1) it often simplifies the governing differential equation or the boundary conditions
(2) it results in a general solution (creates a fixed scale for the problem)
(3) it identifies significant groups

As an example, we can examine a simple textbook case.

      Consider 1-D, transient conduction in a plane wall with constant properties and a convective boundary condition at the front surface and a symmetry boundary condition at the rear surface.  The governing differential equation is:

and the boundary conditions are:

Note that there are 5 parameters in this problem (including boundary conditions): k, rho, c, L, and h.

We can non-dimensionalize using the following definitions:

in preparation for substituting into the original problem, these definitions can be re-arranged as follows:

substituting into the original differential equation:

cancelling terms leaves:

for the boundary conditions:

The group hL/k is commonly referred to as the Biot Number, Bi. From the definition of T*, it can be seen that the rest of the right hand side is simply T*(x*=1).

The final problem with boundary and initial conditions is now:

–Note that the parameter count has shrunk from five to one (Bi)

–The group that was termed t* is often called the Fourier Number, Fo.

Transient Heat Transfer from a Buried Pipe

Figure 1. Computational domain and parameters

An accurate estimate of the heat transfer from a buried pipe to the surrounding ground is essential for the design of the ground loop portion of a ground-source heat pump. Exact analytical solutions to this problem are complicated by the fact that heat pump systems rarely operate continuously. Complete numerical simulations of system designs can be carried out, but these are unwieldy and difficult to justify for initial scoping calculations, or for preliminary performance estimates.  It was desirable to  develop  simple algebraic correlations that could be used to approximate the intermittent overall heat transfer between a fluid flowing in an isolated buried pipe and the surrounding ground.

Figure 2. Sample heat flow results

A finite difference model of this transient problem was developed in a SINDA-like numerical package.  Figure 1 shows the computational domain and system and material parameters that were used in the numerical model.  The model was exercised over a wide range of ground and fluid properties and operating conditions.  Figure 2 presents one sample set of results showing  dimensionless heat flow as a function of time at the pipe wall.  Algebraic correlations of the results were developed in order to provide readily accessible and simple design equations.  Figure 3 demonstrates the agreement between the dimensionless average heat transfer and the algebraic correlation as a function of dimensionless time.

Figure 3. Sample average heat flow results with algebraic correlation

References

J.W. Stevens, 2002, “Coupled Conduction and Intermittent Convective Heat Transfer From a Buried Pipe,”  Heat Transfer Engineering, Vol 23, n. 4, pp. 34-43.

J. W. Stevens, 2000, “Intermittent Convective Heat Transfer for Ground-source Heat Pump Design,”  Proceedings of the ASME Advanced Energy Systems Division – 2000, AES-Vol. 40, pp. 147-152.

J.W. Stevens, 1998, "Transient Heat Transfer Approximations for Ground-source Heat Pump Design,"  Proceedings of the ASME Advanced Energy Systems Division – 1998, AES-Vol. 38, pp. 415-424.

The Second Law of Thermodynamics (part 3)

A Third Statement of the Second Law

            A third way of stating the second law is to say that it is impossible to create a refrigerator that uses no power. This is equivalent to saying that, by itself, heat always flows from a warmer place to a cooler place. Recall that the purpose of a refrigerator (or air conditioner) is to take heat out of a cool place and move it to a warm place. According to this statement of the second law, this won’t happen spontaneously. Therefore, a common way for a refrigerator to function is to establish a region that is even colder than the space to be cooled, and a separate region that is even hotter than the spot where the heat is to go. However, in doing this, we have added something to our original setup, i.e. the refrigerator along with its associated input of power. 
Now, if it were possible to build a refrigerator that did its job without any input of power, we could take such a refrigerator, enclose it in a box, and use it to make heat flow spontaneously from a cooler place to a warmer place. Such a device has never been demonstrated and would violate the second law. Thus, saying that heat always flows down a temperature gradient is equivalent to saying that it is impossible to build a refrigerator that requires no power. 
            In a similar way, this statement of the second law implies, and is implied by the first statement that we used, i.e. that it is impossible to convert all heat into work. Recall that a heat engine is a device that operates between a hot reservoir and a cold reservoir and produces work. But if it were possible to build a refrigerator that did not require any work to operate, we could imagine a composite device consisting of a normal heat engine combined with a refrigerator that uses no work. 
The net result of the composite device would be a heat engine that changed all heat into work which would violate the second law. 
Alternately, we could imagine a composite device where a normal refrigerator was connected to a device that converts all heat into work. The combined device would violate our third statement of the second law since it would move heat from a cool place to a warm place without any input of work.








Summary
          The Second Law of Thermodynamics is based on the experience of many years of observations and is as solidly grounded as the First Law of Thermodynamics. It can be expresed in many different ways but all the expressions imply one another. The second law establishes values for different forms of energy, allowable directions for processes, and theoretical limits on all heat engine efficiencies.

Second Law, part 1

Second Law, part 2

The Second Law of Thermodynamics (part 2)

A Second Statement of the Second Law
            A second way of stating the second law is sometimes phrased as something along the lines of “the entropy of the universe always increases.” For the purposes of thermodynamic calculations a property can be defined called “entropy.” This property is very exact and rigorous in a quantitative way, and can be used for both practical and theoretical calculations. However, for the purposes of this discussion, it will serve to think a little more loosely of entropy as being representative of the amount of “disorder” in a system. This expression of the second law signifies that for any real process the disorder of the universe will increase. It is important to note the part about “the universe”. That is, for any particular system going through a process the entropy may increase or decrease. However, if it decreases, then we can be sure that the increase in the entropy of the surroundings (i.e. the rest of the universe) caused by the process is bigger than the decrease in entropy was for the system. 
            This statement of the second law leads to the thermodynamic equivalent of the “frictionless pulley”, i.e. it establishes a hypothetical standard against which real processes can be measured. This standard is the very best that could theoretically be achieved by any process. For some processes (those in which no heat is transferred) this theoretical limit is for the process to occur with zero increase in entropy of the universe. That is, actual processes will have an increase in entropy, theoretically ideal processes will have no increase in entropy and no process is even theoretically possible in which the entropy of the universe decreases. 
            This limit, along with similar limits for processes that do include heat transfer, is often utilized as a check by people who are tasked with evaluating proposed inventions involving energy transformations. Before investing the effort to understand and evaluate all the details of what might be a very complicated machine, the evaluator can simply check whether the overall operation of the device as described would result in a net decrease in entropy for the device and the surroundings. If it does, the device would violate the second law and cannot possibly function as described.

            A second implication of this expression of the second law is that all processes have directions in which they will proceed and directions in which they will not. The allowable directions will be those that result in an increase in entropy for the universe. Perhaps the easiest example of this to visualize is the case of a hot bowl of soup cooling off in a cool room. The room will never spontaneously cool off while the hot soup gets hotter even though such a process could be imagined that would not violate the first law. The total amounts of energy flowing are governed by the first law, and the direction is governed by the second law.
            This behavior of heat flow is sometimes expressed by saying “heat always flows downhill” where “downhill” is established by the temperature gradient. We’ll come back to this later as another statement of the second law, however, it is immediately clear that such an observation for heat is not stranger than the corresponding observation that “water always flows down hill” where “down hill” is determined by gravity, or a potential energy gradient.
            Many chemical processes have a particular direction in which they will proceed for a given set of conditions (temperature, pressure, concentration, etc.) and those directions are always in accord with the second law. Further, in some cases where it is not otherwise clear, it is possible to predict the direction of a reaction from second law considerations.
            Finally, it is worth noting that this statement of the second law is a preferred starting point for many of the philosophical discussions that are connected to thermodynamics. Some of the simplest and most profound of these deal with the beginning and end of the universe. For example, if the total entropy of the universe as a whole always increases, what will happen when a state of “maximum entropy” is reached? Will the universe ultimately “wind down” to a final state of complete uniformity? How did the universe get “wound up” in the first place? These and other questions flow directly from an understanding of the second law, and constitute what may be one of the more prominent and popular contact points between engineering and philosophy.

Second Law, part 1

Second Law, part 3