Critical thickness of insulation

Let us consider a layer of insulation which might be installed around a circular pipe, as shown in Fig. 2-7. The inner temperature of the insulation is fixed at T and the outer surface is exposed to a convection environment at T.. From the thermal network the heat transfer is 

Now let us manipulate this expression to determine the outer radius of insulation ra which will maximize the heat transfer. The maximization condition is 

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For small diametor pipes there is a critical thickness of insulation that produces the minimum thermal resistance. To be effective insulation thicknesses must be greater than this value. In this problem the critical thickness of insulatitm is O.OOSm as shown on the figure. This can be verified by differentiating the sum of the conductive and convective resistance with respect to the outside radius and setting the result equal to uao. Thus... 

Equation (2-18) expresses the critical-radius-of-insulation concept. If the outer radius is less than the value given by this equation, then the heat transfer will be increased by adding more insulation. For outer radii greater than the critical value an increase in insulation thickness will cause a decrease in heat transfer. The central concept is that for sufficiently small values of h the convection heat loss may actually increase with the addition of insulation because of increased surface area. 

By varying input parameters it is found that the critical thickness of the lagging decreases with increasing temperature of the pipe, i.e. the lagging must be thinner in order to avoid self-ignition. This is a requirement counteracting the quality of heat insulation. 

The electronic insulation of these electrodeposited polymer layers must hold to a two-terminal voltage of 4 V if lithium (or lithium ion) anodes are to be used in the 3-D nanobattery. Because the polymers must also be thin, high dielectric strengths are critical. As seen in Table 2, diminishing the thickness of the dielectric to the nanoscale exacts a higher standard in terms of the quality of the dielectric. For example. 

A 1.0-mm-diameter wire is maintained at a temperature of 400°C and exposed to a convection environment at 40°C with h = 120 W/m2 °C. Calculate the thermal conductivity which will just cause an insulation thickness of 0.2 mm to produce a "critical radius." How much of this insulation must be added to reduce the heat transfer by 75 percent from that which would be experienced by the bare wire ... 

The pipe in Prob. 2-51 is covered with a layer of asbestos [k = 0.18 W/m °C] while still surrounded by a convection environment with h = 12 W/m2 °C. Calculate the critical insulation radius. Will the heat transfer be increased or decreased by adding an insulation thickness of (a) 0.5 mm, (b) 10 mjn ... 

In 4.2.6. (see above), we illustrate the t5T)ical setup for the crucible and melt in the Czochralski apparatus. We place the Ir crucible containing the mixture of oxides on a circular Zr02 platform. This acts as a thermal barrier for the bottom of the apparatus. We may then place a larger Zr02 cylinder around the crucible for further insulation (Note that the choice of a thermal insulator is not critical except that it must be able to withstand the anticipated temperatures to be used). An R.F. coil is placed around the outside of the insulation in a position where it can electrically couple with the metal crucible. Finally, an outside wall of insulation is put into place and atop cover plate is put into position. At 2000 °C., the outer wall thickness of Zr02 needs to be at least 2.5-3.0 cm. The whole is then... 

In certain instances, the heat loss from an insulated pipe may exceed that from an uninsulated one. In these systems, the insulator has a relatively high thermal conductivity and its resistance to heat flow is insufficient to compensate for the additional heat loss resulting from the increased exposed area. The heat loss increases to a maximum, then it decreases with an increase in thickness. The thickness of the insulator which corresponds to the maximum heat loss is called the critical insulation thickness Xcr, ft.). It is verified by the following equation ... 

There is a critical current value for each electrolytic dressing system when the current is smaller the thickness of the insulating oxide layer increases with the value of the current otherwise, it decreases. 

The Computer Sciences Division at the Oak Ridge National Laboratory has undertaken a program to evaluate the criticality criteria for transport packaging of fissionable materials as expressed in Title 10 of the Code of Federal Regulations Part 71. The study is focused presently on the definition of criticality, for, water-reflected arrays of loaded cbiitainers as a function of container volume, type of insulating material and its thickness, and the number of packages., ... 

Initial calculations are of U(93.2) metal cylinders having a height-to-diameter ratio, h/d, of unity. The calculated critical arrays of air-spaced units are sliown in Fig. 1. The introduction of Cclotex as the insulating material placed at the cell boundaries results in a reduction of the mass needed to maintain criticality i.e., reactivity of the arrays would increase were the mass not reduced. The results for three thicknesses of Celotex at a density of 225 kg/m (15 Ib/ft ) are presented in the figure. The calculations were of 216 unit arrays having cubic cell volumes of 56.8, 113.6, and 208.2 litres (15, 30, and 55 gal, respectively). 

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