Critical radius of insulation

Derive the expression for the critical radius of insulation for a sphere. 

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. 

Calculate the critical radius of insulation for asbestos [A = 0.17 W/m °C] surrounding a pipe and exposed to room air at 20°C with h = 3.0 W/m2 °C. Calculate the heat loss from a 200°C, 5.0-cm-diameter pipe when covered with the critical radius of insulation and without insulation. 

Heat Conduction in Cylinders and Spheres 150 Multilayered Cylinders and Spheres 152 3-5 Critical Radius of Insulation 156... 

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

The variation of Q with the outer radius of the insulation 2 is plotted in Fig. 3-31. The value of r2 at which Q reaches a maximum is determined from the requirement tliat dQld) - 0 (zero slope). Performing the differentiation and solving for ri yields the critical radius of insulation for a cylindrical body to be... 

Note that the critical radius of insulation depends on the thermal conductivity of the insulation k and the external convection heat transfer coefficient h. The rate of heat transfer from the cylinder increases with the addition of insulation for r2 < r teaches a maximum when rj = r , and starts to decrease for 2 > Thus, insulating the pipe may actuaUy increase the rale of heat transfer from the pipe instead of decreasing it when T2 < r -... 

The irnportant question to answer at this point is whether we need to be concerned about the critical radius of insulation when insulating hot-water pipes or even hr -water tanks. Should we always check and make sure that the outer radius of insulation sufficiently exceeds the critical radius before we install any insulation Probably not, as explained here. 

The discussions above can be repeated for a sphere, and it can be shown in a similar manner that the critical radius of insulation for a spherical shell is... 

To answer the second part of the question, we need to knov the critical radius of insulation of the plastic cover. It is determined from Eq. 3-50 to be... 

C What is the critical radius of insulation How is it defined for a cylindrical layer ... 

C Consider a pipe at a constant temperature whose ra- dius is greater than the critical radius of insulation. Someone claims that the rale of heat loss from the ] pe has increased when some insulation is added to the pipe. Is this claim valid ... 

Adding insulation to a cylindrical pipe or a spherical shell increases the rate of heat transfer if the outer radius of the insulation is less than the critical radius of insulalioit, defined as... 

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... 

Convective heat loss to the environment can be much more significant in microreactors than in conventional systems because of their high surface to volume ratios. A consequence of this is that insulation generally enhances heat loss rather than mitigating it. The critical radius of a tube below which the use of an outer insulation layer will increase heat losses is given by the ratio of the thermal conductivity of the insulation layer (k) and the heat transfer coefficient at the interface between the insulation and the ambient (/lo), rcriticai = k//iQ. For most commonly used materials and ambient air, this critical radius is on the order of 1 mm. ... 

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 value of the critical radius r, is the largest when k is large and h is smalF. Noting that the lowe.st value of h encountered in practice is about 5 W/m C for the case of natural convection of gases, and that the thermal conductivity of common insulating materials is about 0.05 W/m °C, the largest value of the critical radius we are likely to encounter is... 

This value would be even smaller when the radiation effects are considered. The critical radius would be much less in forced convection, often less than 1 mm, because of much larger h values associated with forced convection. Therefore, we can insulate hot-water or steam pipes freely without worrying about the possibility of increasing the heat transfer by insulating the pipes. 

The radius of electric wires may be smaller than the critical radius. Therefore, the plastic electrical insulation may actually enhance the heat transfer... 

The minimum of total resistance is at a radius less than the outside radius of pipe So there is no physically realizable critical radius. The total resistance increases with increasing insulation and reduces the heat loss. 

EXAMPLE 43-5. Insulating an Electrical Wire and Critical Radius An electric wire having a diameter of 1.5 mm and covered with a plastic insulation (thickness = 2.5 mm) is exposed to air at 300 K and =... 

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