Aluminum coils awaiting heat treatment

To calculate time to heat or cool a part involves many temperature-dependent variables, including thermal conductivity, heat-transfer film coefficients, part size and shape, surface environment, etc. If the part is small, the time can be solved assuming a uniform temperature distribution throughout the body. This is calculated as heat transferred to or from the material equal to the increase or decrease in heat energy of the body during a differential time period, and it is in the form of an equation relating temperature differential ratios to exponential heat-transfer values:

(T-T¥)/(Ti-T¥) = e(b t) where b = h A/(rV Cp)

The transient temperature distribution in the part can be solved by a partial differential equation using advanced mathematical techniques, but the solution normally requires infinite series that are difficult to deal with and time consuming to evaluate. If the part is large, the analysis becomes quite involved. Transient-temperature heat-transfer charts developed by Schack, Newman, Gurney-Lurie, Heisler (1947) and Grober (1961) can be used but are difficult to read and subject to reading errors[1].

Fourier’s Law is the differential equation for heat conduction relating transient temperature distribution q= (T-T¥)/(Ti-T¥) to location within the part (distance from the center X = x/L), including the Biot number (Bi = h L/k) and time tau (t=at/L²) as the dimensionless parameters. The Biot number is the ratio of external convection and radiation heat transfer to internal heat conduction in a part. It is defined as the ratio of heat conducted to the rate of heat stored in a material. Alpha (a) is the symbol for thermal diffusivity = heat conducted/heat stored = k/(rCp) and represents how fast heat diffuses through a material. The Stefan-Boltzmann law (Q = 5.67-8 W/(m2 K4) A T4) is the rate of radiation that can be emitted from a surface. Part configuration and surface emissivity are key factors for this mode of heat transfer.

Additional equations in the program determine the nondimensional transient temperature distribution in a one-term approximation of the Fourier infinite series, including the Biot number and thermal-diffusivity parameters. They result in errors less than 2% for timet> 0.2, which is suitable for most heating and cooling projects encountered in industrial heating applications.

## HEATCOOL Program

The program can be used to compute transient heating or cooling time of any load configuration in a vacuum chamber or in a chamber equipped with or without fan circulation. The load can be a slab, rectangular bars, cylinders, spheres, coiled strip or small parts in a basket. The time includes convection, radiation and conduction heat-transfer modes.

Recirculated gas flow at any velocity can be air, argon, carbon dioxide, carbon monoxide, endo-exothermic, flue gas, helium, hydrogen, methane, nitrogen, 90%N2/10%H2 or steam. Gas properties programmed and computed at any input pressure include density, specific heat, thermal conductivity, viscosity, and Reynolds and Nusselt numbers. The convection and radiation heat-transfer coefficients are computed. Plant elevation is entered, and the barometric pressure is computed. Chamber pressure can be above or below atmospheric. Cooling can also be done in a liquid (quenching in water, oil, brine or synthetic) with agitation to enhance heat transfer.

Density, specific heat and thermal conductivity of 23 metals from aluminum to zirconium and five nonmetals are programmed. Other nonprogrammed metals and nonmetals can be entered. The thermal diffusivity and Biot numbers are computed for analysis of internal/external thermal resistances. Final temperatures are computed at the center and surface for slabs, cylinders and spheres. Three equally spaced nodal temperatures from center to surface are also computed. For rectangular shapes, temperatures are computed at the center, each face, center of the long edge and extreme corner. It is important when heating materials required to meet critical temperature specifications to be sure all surfaces of the part do not exceed the set-point temperature. Higher head temperatures can also be input above the final set-point temperature to decrease heating time, and the time saved is displayed.

## Conclusion

The HEATCOOL program transforms these complicated equations and formulas into practical data and computes heat or cool time with ease. It will save many hours in solving transient heating- and cooling-time projects. Inclusion of preprogrammed values eliminates the need to consult outside references for many physical properties, thus saving tedious and time-consuming nonproductive work. Direct, straightforward selections are used.

Heating and cooling (metal and nonmetal) slabs, billets, cylinders and coils can be readily computed with the HEATCOOL program, and different temperatures and times can be quickly evaluated.IH