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= a t/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/(r Cp) 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 time t > 0.2, which is suitable for most heating and cooling projects encountered in industrial heating applications.


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.

Figure 1 is a printout of a HEATCOOL metric units program for computing time to heat 12 460-mm (18-inch) x 610-mm (24-inch) x 3660-mm (144-inch) aluminum billets, with a head temperature set at 592°C (1097°F) or 10% above the final set-point temperature of 538°C (1000°F). The billets are set on their small side with 150-mm (6-inch) space between so radiation is not viewed 100% between billets and the small bottom side is not available for convection or radiation heat transfer. At the end of the head-temperature phase (5.29 hr), the center temperature reached 479°C (894°F) and the corner reached 509°C (948°F). This corner temperature is not included in the printout but is displayed for reference. If the corner exceeded the set point during the higher head-temperature phase, this would be displayed and corrective action recommended (decrease the center within temperature and/or the head temperature). The time to heat the center to 532°C (990°F) is computed to be about 11.2 hours, and the head temperature is computed to save about 2.0 hours heating time for this particular load compared to heating without a head temperature.


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

For more information:
Joseph D. Barnes, P.E. is a consulting engineer and principal of Barnes Associates, 900 Market St., Apt. 301, Meadville, Pa. 16335; tel: 814-724-4615; e-mail:; web:

Symbols: T=Temperature, h=heat-transfer film coefficient, Cp=specific heat, V=Volume, r=density, L=length, k=conductivity, K=degree Kelvin, W=watt, m=meter, A=area, t=time, Q=blackbody emissive power.

Additional related information may be found by searching for these (and other) key words/terms via BNP Media SEARCH at thermal diffusivity, transient heating, heat transfer, thermal conductivity, convection, radiation, conduction

SIDEBAR: Running the Numbers

When asked to put the HEATCOOL program to the test with actual measured plant data, an historic study run at Westinghouse was used. In this study, aluminum coils 36.75 inch OD x 20 inch ID x 36 inch wide and 2,669 pounds in weight had been heated. All of the data from the test was entered into the HEATCOOL program, including the final coil temperature and the furnace set-point temperature.

One of the advantages of the program is the capability to enter different percentages for each surface area of the coil as well as different gas velocities over each of these surfaces. The ID surface estimation for radiation was 20% of the ID surface area of 15.708 sq. ft. A very low air velocity (100 fpm) was estimated over this ID surface, since the flow is not in a radial direction through the coil. The actual test shows the maximum air velocity (passing coil) at 32.2 fps and the minimum velocity (before coil) at 21.7 fps. The average of these velocities (1600 fpm) was used for the air flowing past the OD and end surfaces.

The program computes the heating time at 5.81 hours. The test shows the actual heating time to be 5.7 hours (after the coil reached 200°F at the end of purge). The program indicates the head temperature saved about 1.4 hours heating time vs. if the head temperature was not employed. This is a significant saving and would be an important factor to consider when scheduling furnace time.