Evaluation of Nonisothermal Semi-Infinite Cylinder withSpecularly Reflecting Walls as a Blackbody Source

W. Richard Powell

The fractional deviation from blackbody radiation due to wall emissivity modestly differing from unityand due to the presence of a temperature gradient (dT/dx) at the open end of a specularly reflectingsemi-infinite cylinder of diameter D and end temperature T1 is shown to be (16\ 2 - ' d(T/T 1 )

\3/ \ e d(x/D) -

I. Introduction

The extent to which the flux emitted from the endof a heated tube approximates blackbody radiationhas been the subject of numerous investigations.1-6Reference 3 is a general review of the isothermalcavity emission problem with an -extensive bibliogra-phy and raises some questions about the validity ofRef. 2 results. References 4 and 5 consider noniso-thermal tubular cavities with diffusely reflectingwalls. Both Refs. 3 and 6 observe that cavities withspecularly reflecting walls provide better approxima-tions of blackbody radiation. The present work iscomplementary to Refs. 4 and 5 in that it is also con-cerned with the emission of nonisothermal tubularcavities but with specularly reflecting walls. A sim-ple expression for the deviation from blackbody ra-diation due to temperature gradients at the cavityexit and nonunity wall emissivity is derived.

I 1. Analysis

It is shown in Ref. 7 that the net flux traveling inthe negative x direction deep inside a long hollowtube with specularly reflecting walls is

(1)

where a is the Stefan-Boltzmann constant, D is thetube diameter, T is the wall temperature, and iswall emissivity. This expression was obtained bymaking- a Taylor series expansion and retaining firstand second order terms. It is not valid for highly re-flecting walls (e - 0) as then the radiation from re-mote parts of the tube that were not accuratelytreated in Ref. 7 would be significant.

The author is with the Applied Physics Laboratory, Johns Hop-kins University, Silver Spring, Maryland 20910.

Received 6 August 1973.

Consider an infinitely long specularly reflectingtube with a wall emissivity differing modestly fromunity whose negative x section is isothermal, i.e.,T(x) = T for x < 0. The flux into the positive xsection from the negative x section at the origin is

f_,(0) = (7rD'/4)aT14 (2)

as the negative section of the tube is a perfect black-body radiator. This flux attenuates as it specularlyreflects into the positive x section of the tube. Theportion of it remaining at x 2 0 is

f-+(x) = f-+(O)tb (X), (3)

where tb(X) is the fraction the incident blackbodyflux transmitted by a specularly reflecting circulartube of length x whose absorption coefficient is E.The procedure for calculating tb(X) is given in Ref.8, but this calculation is not necessary unless one isinterested in the flux distribution inside the tube.In addition to f + (x) there is a flux f + (x) also travel-ing in the positive x direction at x due to the radia-tion originating on the walls between x = 0 and xand arriving at the plane of x before being absorbed.Thus the flux directed in the negative x direction atx is

f(X) = '1 (x) + f(X) + f+ (x) . (4)

Clearly f-(xl) can be interpreted as the total nega-tively directed flux reaching xl from those sections ofthe tube with x > x just as f+(x,) is the total posi-tively directed flux reaching xl from that section ofthe tube between x = 0 and x. Neither of thesefluxes change if the negative x section of the tube isremoved provided that the wall temperature is heldconstant. Thus the net flux in the negative x direc-tion for a semi-infinite tube is

f, (X) = f (X) - f+ (X) = (X) + f+(x) - (5)

That is, the net flux directed toward the open end ofa semi-infinite tube is

March 1974 / Vol. 13, No. 3 / APPLIED OPTICS 593

2 - E dT,I,,, (x) -(4/3)7rY(DT)3 �� --,( E ) �x

f,(X) (rD2/4)aT4 {t (X) +

(16)(D)[T(x)](2 e )dT} (6)

and the apparent emissivity at the open end isE. = f(O)[(wD 2/4)crT 4]-l, (7a)

or,

= [1+ () (2 ) d(x/D)xo1 (7b)

The error or deviation from blackbody flux due tononunity wall emissivity and temperature gradientsis simply the second term on the right side of Eq.(7b). The fact that temperature gradients are moretroublesome as E departs from unity is reasonable asthe more remote sections of the wall (not at T) con-

tribute more significantly as the wall reflectivity in-creases.

References1. H. Buckley, Philos. Mag. 17, 576 (1934).2. J. C. DeVos, Physica XX, 669 (1964).3. E. M. Sparrow, Symposium on Thermal Radiation of Solids

(Fed. Sci. Tech. Information Clearing House, Springfield, Va.,1964), Report AD 629 980, p. 103.

4. E. M. Sparrow, Appl. Opt. 4, 41 (1965).5. B. A. Peavy, J. Res. Nat. Bur. Stand. U.S. 70C, 139 (1966).6. K. S. Krishnan, Nature 188, 652 (1960).7. K. S. Krishnan, Proc. Roy. Soc. (London) A257, 302 (1960).8. W. R. Powell, "Transmission Characteristics of Specularly Re-

flecting Light Pipes, Uniformly Irradiated by Obliquely In-clined Rays," submitted to Appl. Opt. (1973).

A photograph taken at an Eppley Conference in October 1963 includes John Strong (then at JHU), Jule Charney (then atMIT), A. Karoli (Eppley Laboratory), and A. J. Drummond. John Beckman of Beckman and Whitley is speaking.

594 APPLIED OPTICS / Vol. 13, No. 3 / March 1974