Ultimate Engineering Study Guide - Questions & Answers
EXAMPLE 3.6 The possible existence of an optimum insulation thickness for radial systems is suggested by the presence of competing effects associated with an increase in this thickness. In partic- ular, although the conduction resistance increases with the addition of insulation, the con- vection resistance decreases due to increasing outer surface area. Hence there may exist an insulation thickness that minimizes heat loss by maximizing the total resistance to heat transfer. Resolve this issue by considering the following system. 3.3. Radial Systems 139 1. A thin-walled copper tube of radius r; is used to transport a low-temperature refrigerant and is at a temperature T; that is less than that of the ambient air at T. around the tube. Is there an optimum thickness associated with application of insulation to the tube? 2. Confirm the above result by computing the total thermal resistance per unit length of tube for a 10-mm-diameter tube having the following insulation thicknesses: 0, 2, 5, 10, 20, and 40 mm. The insulation is composed of cellular glass, and the outer surface convection coefficient is 5 W/mK SOLUTION Knoun: Radius r; and temperature T; of a thin-walled copper tube to be insulated from the ambient air. Find: 1. Whether there exists an optimum insulation thickness that minimizes the heat transfer rate. 2. Thermal resistance associated with using cellular glass insulation of varying thickness. Schematic: = 5 W/m2K Alr -Insulation, Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer in the radial (cylindrical) direction. 3. Negligible tube wall thermal resistance. 4. Constant properties for insulation. 5. Negligible radiation exchange between insulation outer surface and surroundings. Properties: Table A.3, cellular glass (285 K, assumed): k = 0.055 W/m-K. Analysis: 1. The resistance to heat transfer between the refrigerant and the air is dominated by con- duction in the insulation and convection in the air. The thermal circuit is therefore 7. 21 Inlaira 2KR where the conduction and convection resistances per unit length follow from Equations 3.33 and 3.9, respectively. The total thermal resistance per unit length of tube is then In(rin) 2& 2/h RO where the rate of heat transfer per unit length of tube T.-T RE An optimum insulation thickness would be associated with the value of r that minimized q' or maximized Rice Such a value could be obtained from the requirement that DR = 0 dr Hence 1=0 2 mkr 2wrh or To determine whether the foregoing result maximizes or minimizes the total resis- tance, the second derivative must be evaluated. Hence dR 1 2 krah or, at r = k/h. + dr RO! 1 1 >0 dr? (k/h2k 2k) 2nk/h2 Since this result is always positive, it follows that r= klh is the insulation radius for which the total resistance is a minimum, not a maximum. Hence an optimum insulation thickness does not exist. From the above result it makes more sense to think in terms of a critical insulation radius Por which maximizes heat transfer, that is, below which q' increases with increasing r and above which a' decreases with increasing r. 2. With h = 5 W/mK and k = 0.055 W/m-K, the critical radius is 0.055 W/m = 0.011 m 5 W/mK Hence r>r; and heat transfer will increase with the addition of insulation up to a thickness of For - - r;= 0.011 -0.005) m = 0.006 m The thermal resistances corresponding to the prescribed insulation thicknesses may be calculated and are plotted as follows: cond Ri(m/w) Com 2 1 0 0 6 10 20 30 40 50 - (mm) Comments: 1. The effect of the critical radius is revealed by the fact that, even for 20 mm of insula- tion, the total resistance is not as large as the value for no insulation. 2. Ifr: re, any addition of insulation would increase the total resistance and therefore decrease the heat loss. This behavior would be desirable for steam flow through a pipe, where insu- lation is added to reduce heat loss to the surroundings. 3. For radial systems, the problem of reducing the total resistance through the application of insulation exists only for small diameter wires or tubes and for small convection coeffi- cients, such that >1,For a typical insulation (k = 0.03 W/m-K) and free convection in air (h 10 W/mK), ra = (k/h) 0.003 m. Such a small value tells us that, normally, :>rand we need not be concerned with the effects of a critical radius. 4. The existence of a critical radius requires that the heat transfer area change in the direction of transfer, as for radial conduction in a cylinder (or a sphere). In a plane wall the area per- pendicular to the direction of heat flow is constant and there is no critical insulation thick- ness (the total resistance always increases with increasing insulation thickness).
Write a recursive function named stutter_list that accepts a list of integers as a parameter and replaces every value in the list with two occurrences of that value. For example, suppose a list namedsstores these values, from bottomtop:[13,27,1,4,0,9]Then the call of stutter__ist (s) should change the list to store the following values:[13,13,27,27,1,1,4,4,,,9,9]Notice that you must preserve the original order. In the original list the 9 was at the top and would have been popped first. In the new list the two 9 s would be the first values popped from the list. If the original list is empty, the result should be empty as well. Constraints: Your solution must be recursive. Do not use any loops. Do not use any auxiliary collections or data structures to solve this problem.