The Numerical Method of Lines facilitates the instruction of unsteady heat conduction in simple solid bodies with convective surfaces

Abstract

The present study on engineering education addresses the Method of Lines and its variant the Numerical Method of Lines as a reliable avenue for the numerical analysis of one-dimensional unsteady heat conduction in walls, cylinders, and spheres involving surface convection interaction with a nearby fluid. The Method of Lines transforms the one-dimensional unsteady heat conduction equation in the spatial and time variables x, t into an adjoint system of first-order ordinary differential equations in the time variable t. Subsequently, the adjoint system of first-order ordinary differential equations is channeled through the Numerical Method of Lines and the powerful fourth-order Runge–Kutta algorithm. The numerical solution of the adjoint system of first-order ordinary differential equations can be carried out by heat transfer students employing appropriate routines embedded in the computer codes Maple, Mathematica, Matlab, and Polymath. For comparison, the baseline solutions used are the exact, analytical temperature distributions that are available in the heat conduction literature.

Keywords

One-dimensional unsteady heat conduction simple solid bodies (wall cylinder and sphere)Numerical Method of Lines fourth-order Runge–Kutta algorithm time-dependent center and mean temperatures total heat transfer

Get full access to this article

View all access options for this article.

References

Pinder

GF.

Numerical methods for solving partial differential equations: a comprehensive introduction for scientists and engineers. New York, NY: Wiley and Sons, 2018.

Holman

JP.

Heat transfer. 10th ed. New York, NY: McGraw-Hill, 2010.

Kreith

Manglik

Principles of heat transfer. 7th ed. Boston, MA: Cengage, 2010.

Bergman

Lavine

Incropera

, et al. Fundamentals of heat transfer. 7th ed. New York, NY: Wiley and Sons, 2011.

Ҫengel

Heat transfer, fundamentals and applications. 5th ed. New York, NY: McGraw-Hill, 2014.

Moffat

RJ.

Contributions to the theory of single-sample uncertainty analysis. J Fluids Eng 1982; 104: 250–260.

Liskovets

OA.

The method of lines, review (in Russian), Differenzial’nie Uravneniya 1965; 1: 1662–1668. English Translation: Differential Equations 1965; 1: 1308–1323.

Chapra

Canale

RP.

Numerical methods for engineers. 7th ed. New York, NY: McGraw–Hill, 2014.

Verwer

Sanz-Serna

JM.

Convergence of method of lines approximations to partial differential equations. Computing 1984; 33: 297–313.

10.

Reddy

Trefethen

LN.

Stability of the method of lines. Numer Math 1992; 62: 235–267.

11.

Solving a system of differential equation by finding eigenvalues and eigenvectors, https://yutsumura.com/solving-a-system-of-differential-equation-by-finding-eigenvalues-and-eigenvectors/ (accessed 20 February 2020).

12.

Schiesser

WE.

The numerical method of lines: integration of partial differential equations. Math Comput 1993; 60: 433–437.

13.

Schiesser

WE.

The numerical method of lines. New York, NY: Academic Press, 2012.

14.

Campo

Sieres

Masip–Macía

Plain correlation equations to predict the center and mean temperatures and total heat transfer in simple solid objects with prescribed surface temperature at limiting small-time conditions. Int J Heat Mass Transfer 2019; 130: 733–739.

15.

Arpaci

Conduction heat transfer. Reading, MA: Addison-Wesley, 1966.

16.

Schneider

PJ.

Conduction heat transfer. Reading, MA: Addison-Wesley, 1955.

17.

Liukov

AV.

Analytical heat diffusion theory. New York, NY: Academic Press, 1968.

18.

Özişik

MN.

Heat conduction. 2nd ed. New York, NY: John Wiley & Sons, 1993.