*In August, Leon Glicksman, an MIT professor of architecture and mechanical engineering, and John Lienhard, a professor of mechanical engineering, published “Modeling and Approximation in Heat Transfer” (Cambridge University Press).*

*The product of a nearly 20-year-long collaboration between them, the book explores the challenges faced by engineers in systems design and research. Mastery of fancy calculations is well and good, they argue, but students must also acquire a critical and often neglected skill set: the ability to think in physical terms. To this end, the authors focus on how modeling and synthesis can be carried out in practice. This is about thinking the big picture: how to get started, how to identify key physical variables in a problem, how to focus your attention toward what matters.*

*The School of Engineering recently spoke with the coauthors (who replied collectively) by email about their new text.*

**Q: **How would you describe the origins of this textbook?

**A: **Many excellent textbooks on thermal science are already available, but most of them lack systematic discussions of how modeling and synthesis can be carried out in practice. Specifically, most textbook problems have already been reduced to some kind of clear, physical model: Only analysis remains to be done by, say, calculating a heat-transfer coefficient or manipulating a Fourier series. In practice, the detailed analysis is often the least difficult step. What’s hard is how to get started, what are the key physical variables in a problem — what should be the focus of attention and what can be neglected?

Some people seem to think these skills, which are really stages of problem formulation, cannot be taught formally and that they can be acquired only through long experience with engineering problems. We happen to disagree with that.

Over more than 15 years, we have developed and co-taught a graduate-level subject at MIT on modeling and approximation of thermal systems (Course 2.52/4.424). This new book rises directly out of our experience teaching that course.

The course, and this book, help students break down complex engineering systems into simplified thermal models. They can use these models to investigate and assess essential features of a system — all of this is done through a process of tackling example problems before class, and then discussing various approaches during class.

Both the course and the book owe a lot to the structure of the MIT Department of Mechanical Engineering’s doctoral qualifying exams, which focus primarily on a candidate’s ability to think in physical terms rather than on his or her ability to do fancy calculations.

We have also found, from our research and consulting, that a simplified model can provide a critical guide for the next steps in research and development. It allows one to weigh different approaches and assess the feasibility of each. This really requires a clear understanding of the important physical processes, and it leads to a better explanation of the need for the work to be done when communicating with colleagues and sponsors.

**Q:** What modeling techniques do you cover in the book?

**A: **We identify a number of different approaches, among which the formulation of upper and lower bounds is central. For example, if a contributing phenomenon can be bounded and shown to exert a second-order effect, you can just ignore it. Very basic principles are also essential tools: conservation of energy, the second law of thermodynamics, or the use of analogies between thermal and electrical phenomena. Sometimes a suitable simplification can reduce the model of a complicated device to a rather simple one for which a known solution exists.

**Q: **What’s a good example of using these modeling techniques in an engineering context?

**A: **Let’s consider something simple. Say you want to design a more efficient photocopier. In large copying machines, toner is fused to paper by applying pressure and heat through a heated fuser roller. To avoid warm-up delays, the fuser roller is typically kept at an elevated temperature between jobs, and the heat lost from the idling fuser roller results in substantial standby energy consumption. So how do you design a copier that is more energy efficient while still offering quick startup?

Before embarking on any design innovations, you need to verify that the heat lost from the fuser roller is the really the major source of standby energy consumption. This means approximating the heat lost from the surface of the fuser roller by convection and radiation, as well as estimating the conduction heat loss through the bearings and supports of the fuser roller. These estimates will identify the most important sources of heat loss, and they can point to the most effective ways to reduce that loss.

We make an upper bound on the rate of radiation heat loss though thermal radiation by assuming the fuser roller is a black body. By comparing this rate to the total energy consumption at idle, we will see if whether radiation can be neglected. A similar order of magnitude estimate may rule out convection heat transfer from the fuser roller. This might lead us to conclude that the most important heat loss is conduction through the roller’s bearings. In that case, one design solution would be to add insulation around the bearing supports.

An alternate approach can also be considered: Keep the fuser roller at room temperature and heat it rapidly when copies are needed. This approach will only work if the fuser roller can be heated fast. Evaluating this approach requires an estimate of the thermal time constant of the fuser roller and a possible redesign of the fuser roller to minimize its mass and heat capacity.

We find that the techniques, and the mindset, of simplified modeling and approximation can carry over to many fields beyond just thermal processes. Over the years, we have gotten some very gratifying feedback from former students who attest to the importance of this approach to their subsequent work in both industry and academia.

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