Critical thickness of insulation of a square cross section

Solar Energy Vol. 25, pp. 569 570 © Pergamon Press Ltd., 1980. Printed in Great Britain

0038-092X/80/1201-0569/$02.00/0

TECHNICAL NOTE

Critical thickness of insulation of a square cross section? C. K. HSIEH Mechanical Engineering Department, University of Florida, Gainesville, FL 3261 l, U.S.A.

(Received 18 January 1980; accepted 1 July 1980)

INTRODUCTION

where Ta denotes the ambient temperature; T~, the prescribed, inner boundary temperature. In the computation, the iteration process was terminated when a repeated calculation yielded temperatures converging to within 5 x 10 -5.

In insulating a heated pipe it is well known that the critical thickness which maximizes the heat dissipation is given by the simple formula [1]. re =

k

(1)

h

RESULTS AND DISCUSSION

where r c represents the critical radius; k, the thermal conductivity; h, the convective coefficient. This formula is valid when the insulation is concentric with the circular pipe and has a circular cross section. In solar energy applications, particularly in flat-plate collectors, tubes and headers are sometimes housed inside channels which are not circular. Under these conditions eqn (1) cannot be used to determine the critical radius. To the knowledge of this author, critical thicknesses for insulations of noncircular geometry have not been found in the open literature. It is therefore the purpose of this note to report the critical thickness of insulation that has a square cross section.

The computer data are plotted in Fig. 1. For all cases tested, the heat dissipation is peaked at about hW k

- 0.79

or

1,200 --

ANALYSIS The system under investigation is depicted in the inset of Fig. 1. The insulation has a circular inner surface and a square outer boundary. It is assumed that the temperature is specified at the inner boundary; the outer surface dissipates heat by convection. Both the thermal conductivity and the convective coefficient are treated as constants. The system is in steady state. The problem as given, while appearing simple, defies an exact solution. The problem lies in the irregular geometry of the system, which normally requires the use of conformal mapping to make the problem solvable [2]. However, conformal mapping is useful in the solution of Dirichlet and Neumann problems. For a combination of these problems, representing physically a convective condition, the method becomes exceedingly difficult, if not impossible. The method used in this note is a numerical iteration scheme commonly found in the solution of heat conduction problems. A computer program was developed to evaluate the nodal temperatures on the system (outside) boundary. These temperatures were succeedingly integrated to yield the surface heat dissipation. A wide range of Biot number (hW/k) was tested, with the values of inner radius (rl) and the convective coefficient-thermal conductivity ratio (h/k) taken as parameters. The nodal temperature was nondimensionalized by expressing it as

+Work supported S.E. 2 5 / 6 ~

T(x, y) - TA Z~

(3)

Because of the numerical method used, this value could not be refined without using smaller mesh sizes in the computation. Nevertheless, such an attempt would greatly add to the computer cost because of the nature of the problem [3]. In practice, since the uncertainty in the convective

I

0(x, y) -

W = 0.79(k/h).

(2)

i

~ ~ . r i = 5 ,

/

/

L h/k =0.0625 [ArbitraryUnits]

"ri=4,h/k=o.0625

~i,OOOl-/ ,¢ =5,h/k=o.0975

~ 90C

4

=

I

h/k 00~ =

I~ 0.6

P

i 0.8

hW/k

=4, h/k=lO.0975 LO 1.2 1.4

Fig. 1. Plot of criticalBiotnumbers. 569

•

--

0.4

by U.S.DOE.

I

570

Technical Note

coefficient is large [4], eqn (3) as given is sufficiently accurate for design purposes. The way of using eqn (3) follows that of eqn (1). It is only when the outer radius of the heated pipe exceeds 0.79 k/h, adding insulation has a heat protection effect. A heated pipe having radius less than this value would be located to the left of the peak points in the figure. As a result, any addition of insulation on pipes may either promote, or impede, heat dissipation, depending on the insulation thickness used. It is interesting to note that eqn (3) appears to relate to eqn (1) on a circumference basis. That is to say, using eqn (1) to evaluate the circumference of a circle, one obtains 6.28 k/h; the corresponding circumference for the square, evaluated based on eqn (3), is 6.32 k/h. Their difference appears to be small, and could be a result of the large mesh size used in the analysis. This observation strengthens the theory that adding insulation, while raising the conductive resistance inside the insulation layer, also increases the surface area to promote heat dissipation. The former is accounted for by the k term in eqns (1) and (3). The latter is controlled by the h term, while the coefficients in these

equations appear to adjust themselves to reflect the circumference changes.

Acknowledgement--This work was supported by the U.S. Dept. of Energy and was performed when the author was associated with the Solar Energy Group, Argonne National Laboratory in the summer of 1979.

REFERENCES

1. W. H. McAdams, Heat Transmission, 3rd Edn. McGraw-Hill, New York (1954). 2. M. J. Balcerzak and S. Raynor, Steady state temperature distribution and heat flow in prismatic bars with isothermal boundary conditions. Int. J. Heat Mass Transfer 3, 113-125 (1961). 3. A. M. Clausing, Numerical methods in heat transfer. In Advanced Heat Transfer (Edited by B. T. Chao), pp. 157-216. University of Illinois Press, Urbana (1969). 4. F. Kreith, Principles of Heat Transfer, 3rd Edn. Intest, New York (1973).