Friday, September 30, 2011

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:




 Lc= 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.

Monday, September 19, 2011

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:


Best wishes on your decision!