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A new 1-D composite lumped model facilitates the algebraic calculation of local temperatures, mean temperatures, and total heat transfer in simple bodies

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Abstract

Estimates of spatio-temporal temperatures and total heat transfer in simple bodies (large plate, long cylinder and sphere) have been done in the past with the help of charts and recently by way of algebraic correlation equations; both avenues are valid for large times mostly, i.e., dimensionless times or Fourier numbers τ ≥ 0.2. The imposing time restriction comes from the utilization of the truncated ‘one-term series’ because the Fourier infinite series diverge for very short times approaching zero. The central goal of this technical paper is to predict the mean, surface and center temperatures, as well as the total heat transfer in those simple bodies by implementing a 1-D composite lumped analysis united to a sound physics-based computational procedure. The effortless combined methodology is new. It brings forth a handful of compact algebraic equations covering the full gamma of Biot numbers (0 < Bi < 100) and all dimensionless times, 0 < τ < ∞.

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Abbreviations

A :

Surface area

Bi :

Biot number, \( \frac{{\overline{{h_{f} }} R}}{{k_{s} }} \)

C :

Geometric parameter: 1 for large plate, 2 for long cylinder and 3 for sphere

c s :

Specific heat capacity of solid

\( \overline{{h_{f} }} \) :

Mean, external convective coefficient

\( \overline{{h_{s} }} \) :

Mean, internal convective coefficient

k f :

Thermal conductivity of fluid

k s :

Thermal conductivity of solid

m :

Mass of solid

\( \overline{{Nu_{s} }} \) :

Mean, internal Nusselt number, \( \frac{{\overline{{h_{s} }} (2R) }}{{k_{s} }} \)

\( \overline{{Nu_{f} }} \) :

Mean, external Nusselt number, \( \frac{{\overline{{h_{f} }} (2R)}}{{k_{f} }} \)

\( \overline{{Nu_{ov} }} \) :

Mean, overall Nusselt number, \( \frac{{\overline{U} (2R)}}{{k_{s} }} \)

Q i :

Initial internal energy

Q t :

Total heat transfer

r :

Transverse coordinate

R :

Characteristic length: semi-thickness of large plate, radius of long cylinder, radius of sphere

t :

Time

t fi :

Final time

T :

Local temperature

T c :

Center temperature

T f :

Fluid temperature

T m :

Mean temperature

T w :

Surface temperature

\( \overline{U} \) :

Mean, overall heat transfer coefficient

V :

Volume

α(τ), β(τ), γ(τ) :

Time-dependent coefficients in Eq. (11)

α s :

Thermal diffusivity of solid

η :

Dimensionless r

θ :

Dimensionless T for convective surface

θ* :

Dimensionless T for isothermal surface

ρ s :

Density of solid

τ:

Dimensionless t or Fourier number

Ω t :

Dimensionless Q t

i :

Initial

f :

Fluid

fi :

Final

ov :

Overall

s :

Solid

w :

Surface

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Antonio Campo

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  1. Antonio Campo
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Correspondence to Antonio Campo.

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Campo, A. A new 1-D composite lumped model facilitates the algebraic calculation of local temperatures, mean temperatures, and total heat transfer in simple bodies. Heat Mass Transfer 48, 1495–1504 (2012). https://doi.org/10.1007/s00231-012-0994-x

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  • DOI: https://doi.org/10.1007/s00231-012-0994-x

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