Numerical analysis of unsteady heat conduction in regular solid bodies comprising natural convection to nearby fluids

For the analysis of unsteady heat conduction in solid bodies comprising heat exchange by forced convection to nearby fluids, the two feasible models are (1) the differential or distributed model and (2) the lumped capacitance model. In the latter model, the suited lumped heat equation is linear, separable, and solvable in exact, analytic form. The linear lumped heat equation is constrained by the lumped Biot number criterion Bil=h¯(V/S)/ks < 0.1, where the mean convective coefficient h¯ is affected by the imposed fluid velocity. Conversely, when the heat exchange happens by natural convection, the pertinent lumped heat equation turns nonlinear because the mean convective coefficient h¯ depends on the instantaneous mean temperature in the solid body. Undoubtedly, the nonlinear lumped heat equation must be solved with a numerical procedure, such as the classical Runge–Kutta method. Also, due to the variable mean convective coefficient h¯ (T), the lumped Biot number criterion Bil=h¯(V/S)/ks < 0.1 needs to be adjusted to Bil,max=h¯max(V/S)/ks < 0.1. Here, h¯max in natural convection cooling stands for the maximum mean convective coefficient at the initial temperature Tin and the initial time t = 0. Fortunately, by way of a temperature transformation, the nonlinear lumped heat equation can be homogenized and later channeled through a nonlinear Bernoulli equation, which admits an exact, analytic solution. This simple route paves the way to an exact, analytic mean temperature distribution T(t) applicable to a class of regular solid bodies: vertical plate, vertical cylinder, horizontal cylinder, and sphere; all solid bodies constricted by the modified lumped Biot number criterion Bil,max<0.1.

Download Full-text

Numerical Analysis of Natural Convection in a Two-Dimensional Square Enclosure Including the Effect of Heat Conduction in the Boundary Walls

Journal of the Visualization Society of Japan ◽

10.3154/jvs.12.1supplement_291 ◽

1992 ◽

Vol 12 (1Supplement) ◽

pp. 291-294

Author(s):

Mitsunobu AKIYAMA ◽

Hitoshi SUGIYAMA ◽

Nao NINOMIYA ◽

Isamu URAI ◽

Yasuaki OKADA

Keyword(s):

Heat Conduction ◽

Natural Convection ◽

Numerical Analysis ◽

Two Dimensional ◽

Square Enclosure

Download Full-text

Streamlining the teaching of unsteady heat conduction in regular solid bodies with heat convection to neighboring fluids

International Journal of Mechanical Engineering Education ◽

10.1177/0306419016689496 ◽

2017 ◽

Vol 45 (3) ◽

pp. 245-259

Author(s):

Antonio Campo ◽

Jane Y Chang

Keyword(s):

Heat Transfer ◽

Heat Conduction ◽

Biot Number ◽

Engineering Students ◽

Total Heat ◽

Total Heat Transfer ◽

Unsteady Heat Conduction ◽

Cylinders And Spheres ◽

Solid Bodies ◽

Long Cylinder

In the analysis of unidirectional, unsteady heat conduction for simple solid bodies (large slab, long cylinder and sphere), the modern tendency adopted by authors of heat transfer textbooks is to calculate the temperatures and total heat transfer with “one-term” series accounting for the proper eigenquantities, which are expressed in terms of the Biot number. The supporting information is available in tables for a large slab, a long cylinder and a sphere. To avoid linear and quadratic interpolation for the Biot numbers listed in the tables, the goal of the present study is to use regression analysis in order to develop compact correlation equations for the first eigenvalues, the first eigencontants and the first constants (for the total heat transfer) varying with the Biot number for large slabs, long cylinders and spheres, all in the ample range 0 < Bi ≤ 100. This direct approach will speed up the step-by-step calculations of a multitude of unsteady heat conduction problems for engineering students.

Download Full-text