# Heat Transfer Lab Report

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• Published : November 25, 2012

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UNSTEADY STATE HEAT TRANSFER
Heat transfer processes are prominent in engineering due to several applications in industry and environment. Heat transfer is central to the performance of propulsion systems, design of conventional space and water heating systems, cooling of electronic equipment, and many manufacturing processes (Campos 3). Unsteady state conduction is the class of heat transfer in which the temperature of the conducting medium varies with time and position. This occurs frequently in industrial processes, especially food preservation and sterilization, where the temperature of the food or of the heating or cooling medium constantly changes (Farid2).

The work reported here involves the investigation of unsteady state heat transfer in two cylindrical rods and the conformity of experimental results to different methods of theoretical analysis. Aluminum and Plexiglas cylinders were used. Thermocouples were placed at different radial and axial positions, and the cylinders, which were in thermal equilibrium with an ice bath, were placed in a warm water bath at 370C. Temperature profiles were obtained using a data acquisition system on a computer.

Theory
The applicable form of the heat transfer equation for conduction in solids is given by (Welty1):

ρcp∂T∂t=∇∙k∇T+q (1)

If the thermal conductivity is constant and the conducting medium contains no heat sources, Equation 1 reduces to Fourier’s second law of heat conduction (Welty1):

∂T∂t=α∇2T (2)

Where α = (k/ρcp). Equation 2 can be written in cylindrical coordinates as (Welty1):

∂T∂t=α∂2T∂r2+1r∂T∂r+1r2∂2T∂θ2+∂2T∂z2 (3)

Assuming that no heat transfer occurs in the axial position, and temperature varies with radial position and time only,

∂2T∂θ2=∂2T∂z2=0 (4)

Equation 3 therefore becomes (Welty1):

∂T∂t=α∂2T∂r2+1r∂T∂r (5)

Nomenclature for all equations is shown in the appendices.

For a cylindrical rod immersed in a higher temperature fluid, heat transfer occurs by convection from the body of fluid to the surface of the rod, and by conduction from the rod’s surface to its center.

If conduction through the rod occurs much faster than convection from the fluid, convection is the rate-limiting heat transfer mechanism, and the temperature within the solid will vary with time only. This condition, in which the external resistance is large relative to the overall resistance, is the primary characteristic of a “lumped” system. The Biot number, (Bi = hV/kA), is a ratio of the internal (conductive) resistance to heat transfer, to the external (convective) resistance to heat transfer. A general rule of thumb is that a body can be assumed to be lumped if Bi < 0.1 (Welty1).

For lumped bodies, the temperature variation with time is described by Equation 6 (Welty1):

T=T0-T∞e-Bi∙Fo+T∞ (6)

Where Fo = αt/[(V/A)2]

For cases in which the internal and external resistances are significant, Equation 5 must be solved numerically or graphically to determine the temperature variation with position and time. Graphical solutions (Heisler charts) are shown in Welty1 for different shapes and geometries. To use the Heisler charts, three dimensionless ratios must be known, and a fourth will be read on the appropriate axis. These dimensionless ratios are:

Y, unaccomplished temperature change=T∞-TT∞-T0 (7)

X, relative time=αtx12 (8)

n, relative position=xx1 (9)

m, relative resistance=khx1...