RETRACTED: Critical radius for radial heat conduction: a necessary criterion but not always sufficient

Abstract

Use of the critical radius for radial heat conduction in thermal insulation systems has been widely reported in the literature. When it is desirable to increase heat dissipation in these systems, the critical radius can be used in a definitive manner to maximize the heat dissipation. However, if it is desirable to decrease heat gain or heat loss, the critical radius only serves as a necessary condition, but it is not sufficient. To address design issues of such thermal systems, the crossover radius is utilized. The crossover radius is defined as a radius greater than the critical radius such that the heat transfer with the corresponding amount of insulating material is equal to that of the bare thermal system. Both cylindrical and spherical systems are considered. Justification: It will be pointed out later in this paper that the concept of the crossover insulation radius is applicable when the Biot number is less than 1 in a cylindrical system or when it is less than 2 for the spherical system. Once the Biot number is larger than these respective values, the heat transfer reduction is immediate with application of any amount of insulation. However, the justification for the present work is threefold. First, all the available heat transfer textbooks discuss only the concept of the critical insulation radius and the subsequent increase in the relative heat transfer as compared to the bare radial system. The present work extends the discussion beyond the critical insulation radius along with the impact on the relative heat transfer and thus completing the discussion. Secondly, even today there are smaller, cylindrical thermal systems that are employed in the refrigeration and cryogenics industry which have rather small value of the Biot numbers. Third, with the current trends in the compact heat exchangers, as thermal systems get smaller and smaller, it is natural that the smaller values of Biot number will be encountered more frequently where the concept of the crossover insulation radius is applicable. At times such systems are improperly insulated resulting actually in an energy penalty due to increased heat transfer. With adequate knowledge about the crossover insulation radius, these systems can be properly insulated while improving their energy effectiveness. Such present-day smaller thermal systems are discussed next. In this work the Biot number is defined as the ratio of convective heat transfer coefficient times the inner radius of the insulation divided by the thermal conductivity of the insulating material. Thus all the three parameters will be addressed first before discussing the smaller Biot numbers. As reported in one cryogenic heat transfer study, the convective heat transfer coefficient varies almost linearly with the prevailing temperature difference [Adv. Cryog. Eng. 6 (1960) 499]. The value varies from 8.1 to 11.6 W/m2 K over the temperature difference of 100–300 K. According to the most recent developments, the smallest pipe or tubing diameter that is employed in the industry is 2/16 in. for aluminum and 3/16 in. for copper [[12]; [13] ; [14], http://www.johnstonesupply.com/main/]. The capillary tubing is available even in smaller sizes. The thermal conductivity values of foam insulation at cryogenic temperatures range from 0.033 to 0.055 W/m K [R.H. Kropschot, Low-Temperature Insulation, Applied Cryogenic Engineering, John Wiley, 1962, pp. 152–169]. The interstitial gas contained in the cells plus internal radiation and a contribution due to solid conduction determines the thermal conductivity of foam. Evacuation of foam provides the initial apparent low thermal conductivity. But many Freon-blown types of foam will be penetrated by air over a period of time, which may increase their conductivity by as much as 30%. Based on the available data for refrigeration tube diameters (2/16 in. for aluminum and 3/16 in. for copper) and the insulation thermal conductivity values, the current range of smaller Biot numbers for the refrigeration systems turns out to be 0.23–0.84, which makes the present work relevant. It should be noted that as the thermal conductivity increases with age, the Biot number will become even smaller. Further, with the current trends in miniaturization, it is expected that the smaller Biot numbers will be encountered more frequently in the future. Representative energy penalties are analyzed next.