Introduction
The purpose of this experimental task was to investigate the possibility of using circular rods of different materials made from different materials and having different diameters as heat fins. Extend surfaces of heat fins allow for additional heating and cooling capacity in a cheap and passive way. The fins are intended for use in cooling a reactor by use of the surrounding air. Models were developed based on several assumptions for the simplification of the derivation of the steady and unsteady state temperature distribution state in a solid rod heated at one of the edges.
The models derived shall then be compared to the experimental data and the assumptions analyzed critically to determine the credibility of the models for use in fin design. The results of this experiment will also be used in the models improvement. It will also help in recommending the appropriate diameter lengths and material for the fins
Theory
In heat transfer problems, for relatively simple geometry prediction of system temperature distribution is fairly accurate. It’s simply done by solving the energy conservation equations and that of thermal energy flux examples include the Stefan Boltzmann law of radiation, newton’s law of cooling for convection and Fourier’s law for conduction (Bruins, 1984).
Consider a given solid rod, shown in Fig 1 below, with radius R, initially at the ambient air temperature T1. The left end of the rod (x = 0) is raised to a high steam temperature Tst at time t = 0 by filling the chamber with steam. Heat is conducted along the solid rod from left to right. The rod loses heat to the surroundings (air) by radiation and convection.
Solid rod at ambient temperature
ASSUMPTION 1: heat conduction is one dimensional; meaning that there are no radial temperature gradients and temperature at all points of element dx is constant. ASSUMPTION 2: Assume that radiation negligible. These two assumptions lead to the unsteady state heat balance on the element of length dx
ASSUMPTION 3 Bi is the dimensionless Biot Number, defined as hR/k, and represents the ratio of the conductive (internal) resistance to heat transfer to the convective resistance to heat transfer
The boundary conditions for this problem are:
(a) At t = 0 ( = 0), then T = Ta ( = 0) for x, > 0 (b) At x = 0 ( = 0), then T = Tst ( = 1) for t, ≥0
(c) ASSUMPTION 4: Assume that the rod is infinitely long as far as heat can tell.
Data analysis
ay
Gradient =√Bi = 2.9722-4.426560.72-0.72 = -0.0374
Bi=-0.0374²
Bi=0.001401
Y intercept = ln (Tst-Ta )=4.4559
Tst-Ta= 1.4949, Ta=0
Tst=1.4949
Gradient =√Bi = 0.69315-4.40672121.44-1.44 = -0.0312
Bi=-0.0312²
Bi=0.00097344
Y intercept = ln (Tst-Ta) = 4.4362
Tst-Ta= 1.4897, Ta=0
Tst=1. 1.4897
Measured and theoretical values of temperature for the 1” aluminum rod at each thermal couple
Are the assumptions in the theory valid?
There is a radial cross sectional temperature gradient at temperature T2, T5 and T8 of +3, -7 and -5 respectively. The temperature gradient is because heat has to move by conduction radially before dissipation. The assumption is thus not valid
The thermal diffusivity for aluminum is 8.418 × 10−5
Where is thermal conductivity (W/ (m·K))
Is density (kg/m³)
Is specific heat capacity (J/(kg·K))
The variation in thermal conductivity which results to variation in thermal diffusivity varies with body temperature and not the surrounding temperature. For aluminum, thermal conductivity is mainly dependent on lattice vibrations. Increased vibration causes an increase in the mean free path of molecules that obstructs the free flowing electrons thus reducing conductivity (Carslaw, 1948).
The three rods are mathematically infinite.
The value of Bi decreases along the length of the aluminum rod the percentage variation in B I is 20.48%
The ½” rod dissipates heat much more faster as compared to the 1” rod as demonstrated by the Biot values below
Bi=0.001401
Bi=0.00097344
Recommendation
References
Bruins, A. P. (1984). F. W. McLafferty (ED). Tandem mass spectrometry. John Wiley & Sons Ltd., Chichester, UK, 1983. £9.95. ISBN 471 865 974. Biol. Mass Spectrom. Biological Mass Spectrometry, 11(9), 493-493.
Carslaw, H. S., Jaeger, J. C., Ingersoll, L. R., Zobel, O. J., Ingersoll, A. C., & Vleck, J. H. (1948). Conduction of Heat in Solids and Heat Conduction. Phys. Today Physics Today, 1(7), 24.
