Piped Water Cooling of Concrete: An Exercise in Scaling
Fowkes, Neville D., Bassom, Andrew P., Australasian Journal of Engineering Education
During dam construction, concrete is poured in large slabs that typically have dimensions of the order of 3 x 10 x 10 m. Hydration heat is released as the concrete sets and temperature rises in excess of 50 [degrees] are quite typical. The resulting temperature build-up can lead to thermal-stress-induced cracking and structural weakening, which limits the size of the slab, see Springenschmid (1994). In addition, a second slab cannot be poured on top of the first until sufficient cooling has occurred, which causes construction delays. To hasten the cooling and avoid cracking, water is passed through pipes set within the concrete; these pipes are later in-filled with concrete. The water network is made up of a number of individual pipes, each supplied with cold water that warms as it is forced through the pipes and often exits at temperatures close to boiling. This method of cooling was first used during the construction of the massive Hoover dam; the size was such that construction could not have proceeded without piped water cooling (US Bureau of Reclamation, n. d.). The task to be addressed here is: how should the pipes be configured within the concrete so as to remove the heat in an optimal way? To answer this we need to determine the water fluxes, pipe radius and lengths, spacing between pipes, and the overall network design so as to maintain temperatures within acceptable levels with minimum cost.
The above problem arose out of the South African Mathematics in Industry Study Group (MISG) meeting held in Johannesburg in February 2004 (Charpin et al, 2004), and was subsequently presented as a problem for a Canadian Pacific Institute for the Mathematical Sciences (PIMS) graduate workshop held in Regina, Saskatchewan during June 2008, see Fowkes & Broadbridge (2008). An extended account of the results obtained can be found in Myers et al (2009), but our simplified discussion here illustrates the modelling and scaling principles that were used in this work, and that also underlie the successful study of most continuum mechanics problems arising out of industrial applications. As such, it is an excellent example to present to students in their later academic years, or could be set as a project for group study. The problem is "non-standard" (as almost all industrial problems are) in that the methodology required cannot be easily obtained from a textbook. Our task is sufficiently open-ended that it is not even clear how one might start; for example, what water piping network should one investigate?
The problem is also ideal for recent graduate students in that the technical background required to address it should be in place for almost all students of applied mathematics or engineering by the end of their third year. It is unfortunate that many students of applied mathematics can complete a full undergraduate degree without experiencing even an introductory course on solid or fluid mechanics. Engineering students at the University of Western Australia would normally have attended such courses, but are unlikely to know the mathematical techniques (especially asymptotics) required to handle most industrial problems. They will, however, have seen separation of variables and have some vague notion of diffusion, but probably only within very standard settings. The students at the PIMS graduate workshop were a cut above those normally present at such meetings as they were carefully-selected bright students from internationally-recognised institutions. Most were either confident about the physics or about the mathematics required in this Australasian journal of Engineering Education, Vol 15 No 2 problem, but few understood the need to combine the two, and the need for intuition as a driving force. This of course requires experience, and early experience is invaluable.
Another unfortunate fact of life is that too often students immediately and mindlessly resort to numerics due to the ready availability of powerful programs. …