Modeling the Transient Two-Dimensional Temperature Response of Cylindrical Geometry for the Enhancement of Learning Heat Transfer

Student: Jackson Jaworski – Advisors: Don Mueller, Ph.D, P.E. & Hosni Abu-Mulaweh, Ph.DDepartment of Civil and Mechanical Engineering

Abstract

Engineering is a daunting subject that includes manycomplex sub-disciplines. One of the more challengingsub-disciplines for students is heat transfer whichdescribes how energy is transferred through orbetween objects due to a temperature difference.Students are typically introduced to transient, or timevarying, conduction heat transfer problems withsimple one-dimensional geometries. While theseexamples are adequate to introduce concepts, inpractice, an engineer should know when a one-dimensional simplification is applicable or when multi-dimensional analysis is required.

Background

Geometry of several cylindrical shapes subjected to convectionboundary conditions of a uniform temperature.

One-dimensional models introduced in classes simplifythe complex nature of multi-dimensional problemswhich often require the development of computersimulations to obtain solutions. Development of aninteractive, flexible, and intuitive program forengineering students to study transient, multi-dimensional conduction heat transfer problems incylindrical geometry, would allow for betterunderstanding of fundamental concepts. Specifically,to aid as a demonstration, that under certainconditions, a finite cylinder can be approximated by aone-dimensional slab or an infinite cylinder.

Model Development GUI Development Results

Modeling an entire cylinder is computationallycomplex and would require long simulation run times.This time is reduced by looking at a two-dimensionalsimplification of a short cylinder. Since heat flowsradially into a cylinder, any slice taken through thecenter of the solid will produce the same results. Thisis further simplified by only looking at the top half of acylinder since the temperature gradient is mirroredacross the mid-plane. The resulting domain is muchsimpler to simulate with greatly reduced computationtime of a few seconds compared to minutes.

The domain is then discretized into a mesh in order tosolve the temperatures at each point within thecylinder, producing a gradient of temperatures orisotherms. The solution is produced using a standardexplicit forward finite difference method. Due to theuse of an explicit method, the stability of the systemmust be considered in order to approach the correcttemperature. The resulting mathematical equationsare shown in the table above which describe the fourdifferent types of nodes in the simplified geometry.

Transient, two-dimensional finite difference equations incylindrical coordinates (∆𝑟𝑟 = ∆𝑧𝑧) and stability criteria.

A graphical user interface (GUI) was developed inorder to have the user easily navigate the program andrun simulations more efficiently and intuitively. Theuser can modify initial and boundary conditions,material type, and geometry to see how variousoutputs are affected. More advanced users can changethe nodes and roots for analytical solutions at the costof simulation speed.

References

Conclusion

Simulations allow students the ability to see theresults from changing input parameters to see howeach variable affects the outputs of the model. Thedevelopment of an interactive program allows theusers to better understand the fundamental conceptsof multi-dimensional, transient heat transfer incylindrical geometries.

As the length of cylinder is increased the two-dimensional solution approaches the infinite cylinderapproximation. Likewise, as the radius of the cylinder isincreased, the two-dimensional solution approachesthe infinite slab approximation. The error associatedbetween the two-dimensional solution and infinitecylinder approximation can be graphed for a widerange of length to radius ratios resulting in the figurebelow. For large ratios, the infinite approximation is asuitable method for a longer period of time.

Behavior of center temperature, average temperature, totalheat transfer, and error associated with slab model and infinitecylinder model approximations.

Percent error by approximating a finite cylinder as infinite. Thevolume was held constant for the various ratios.

Figure 2: Default GUI parameter configurations.

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