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UBC Theses and Dissertations

Turbulent natural convection coupled with thermal radiation in a boundary layer Hood, Innes William 1989

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TURBULENT NATURAL CONVECTION COUPLED WITH T H E R M A L RADIATION IN A BOUNDARY L A Y E R By INNES WILLIAM HOOD B.A.Sc, The University of British Columbia, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF MASTER OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA September 1989 © Innes William Hood, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of A A E C H A A ^ M i^/E^tLNC The University of British Columbia Vancouver, Canada Date O C T . \1_ , /9ft<7  DE-6 (2/88) ABSTRACT Tu r b u l e n t n a t u r a l c o n v e c t i o n coupled with thermal r a d i a t i o n i n an absorbing gas, adjacent t o a v e r t i c a l heated p l a t e has been s t u d i e d . The r e s u l t s o"f a s i m p l i f i e d t h e o r e t i c a l model are compared with experimental d a t a obtained u s i n g a heated v e r t i c a l p l a t e i n a carbon d i o x i d e and n i t r o g e n gas mixture. The t h e o r e t i c a l model assumes a gray, o p t i c a l l y t h i n , n o n s c a t t e r i n g and i n f i n i t e gas ad j a c e n t t o a b l a c k , isothermal p l a t e . In a d d i t i o n , the Boussinesq approximation was a p p l i e d , u s i n g the v e l o c i t y and temperature p r o f i l e s of E c k e r t and Jackson. The momentum and energy equations were s o l v e d by an i n t e g r a l t e c h n i q u e , with the r a d i a t i o n term modelled by two d i f f e r e n t methods: the mean beam length method; and an e x p o n e n t i a l k e r n a l approximation. The experimental work c o n s i s t e d of o b t a i n i n g the temperature and v e l o c i t y p r o f i l e s o f a t u r b u l e n t n a t u r a l boundary l a y e r at one l o c a t i o n over a range of carbon d i o x i d e and n i t r o g e n gas compositions, and c a l c u l a t i n g and comparing the e n t h a l p y f l u x f o r the d i f f e r e n t cases. R e s u l t s are g i v e n i n terms of a combined c o n v e c t i v e and r a d i a t i v e N u s s e l t number as a f u n c t i o n of gas a b s o r p t i v i t y . I t was found t h a t E c k e r t and Jackson's v e l o c i t y p r o f i l e d i d not p r e d i c t the f l o w a c c u r a t e l y . By m o d i f y i n g the p r o f i l e t o f i t the data, good ageement between the t h e o r y and experiment was o b t a i n e d . i i i Table of Contents A b s t r a c t i i L i s t of F i g u r e s v i Acknowledgements v i i Nomenclature v i i i Chapter I I n t r o d u c t i o n 1 P r e l i m i n a r y Remarks 1 Terminology 3 1.3 Review of the l i t e r a t u r e 4 1.3.1 T u r b u l e n t N a t u r a l Convection 5 1.3.2 Gas R a d i a t i o n 7 1.3.3 R a d i a t i o n Laminar N a t u r a l Convection 7 1.3.4 Numerical S o l u t i o n s of N a t u r a l Convection, R a d i a t i o n Problems 12 1.3.5 M i s c e l l a n e o u s 13 1.4 Scope of the Present I n v e s t i g a t i o n 14 Chapter II T h e o r e t i c a l Development 16 2 . 1 I n t r o d u c t i o n 16 2 .2 Governing Equations 18 2 .2 . 1 The Boundary Layer 18 2 .2. 2 The Outer Region 21 2 .3 Order of Magnitude A n a l y s i s 22 2 .4 Equations with T u r b u l e n t Q u a n t i t i e s 27 2 .5 The R a d i a t i o n Term 30 2 .5. 1 Mean Beam Length Model 31 2 .5. 2 O p t i c a l l y T h i n Model 32 2 .6 S o l u t i o n of Governing Equations 37 2 .7 D i s c u s s i o n of T h e o r e t i c a l R e s u l t s 45 i v Chapter I I I Experimental Apparatus and Procedure 51 3.1 General 51 3.2 Experimental Apparatus 51 3.3 Instrumentation 56 3.4 Experimental Procedure 62 3.4.1 C a l i b r a t i o n of Instrumentation 62 3.4.2 Routine Experimental Procedure 64 Chapter IV R e s u l t s and D i s c u s s i o n 66 4.1 General 66 4.2 E r r o r A n a l y s i s 62 4.3 Experimental Observations 70 4.4 Experimental R e s u l t s and D i s c u s s i o n 73 Chapter V C o n c l u s i o n s 87 Chapter VI Recommendations 89 References 90 Appendix A Apparatus Drawings 97 Appendix B C a l i b r a t i o n Curves 101 Appendix C Sample C a l c u l a t i o n s 103 v L i s t o f F i g u r e s F i g u r e 1 Problem Geometry 19 F i g u r e 2 T h e o r e t i c a l Temperature P r o f i l e 46 F i g u r e 3 T h e o r e t i c a l V e l o c i t y P r o f i l e s 47 F i g u r e 4 T h e o r e t i c a l Heat T r a n s f e r 50 F i g u r e 5 Comparisson of V e l o c i t y P r o f i l e s 75 F i g u r e 6 Comparisson of Temperature P r o f i l e s 78 F i g u r e 7 Experimental Temperature P r o f i l e s 80 F i g u r e 8 Experimental V e l o c i t y P r o f i l e s 81 F i g u r e 9 Heat T r a n s f e r R e s u l t s 83 F i g u r e 10 P l a t e C o n s t r u c t i o n 97 F i g u r e 11 E n c l o s u r e C o n s t r u c t i o n 98 F i g u r e 12 D e t a i l s of E n c l o s u r e C o n s t r u c t i o n 99 F i g u r e 13 D e t a i l o f Hot Wire Holder 100 F i g u r e 14 V e l o c i t y C a l i b r a t i o n Curve 101 F i g u r e 15 Temperature C a l i b r a t i o n Curve 102 v i Acknowledgements I wish t o thank Dr. M. Iqbal and Dr. E. G. Hauptmann f o r t h e i r guidance and h e l p through a l l stages of t h i s work. F i n a n c i a l support f o r t h i s work by the N a t u r a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l of Canada i s g r a t e f u l l y acknowledged. v i i Nomenclature La/t i n a constant. b c o n s t a n t [m] B o p t i c a l t h i c k n e s s = K6 c c o n s t a n t 0 . 5 Cj c o n s t a n t [ g e c ] C 2 c o n s t a n t [m°' 3] C^ c o e f f i c i e n t of f r i c t i o n C P s p e c i f i c heat Cj^g^rjl e 4 b l a c k body power [-^ ] m E n n=l,2,3,... i n t e g r o - e x p o n e n t i a l f u n c t i o n g g r a v i t a t i o n a l constant [ j ^ ] h heat t r a n s f e r c o e f f i c i e n t [ ] m °C W n k thermal c o n d u c t i v i t y T-^ Ul Lm C J L p l a t e l e n g t h [m] L e mean beam length [m] m c o n s t a n t n c o n s t a n t P p r e s s u r e [-^1 m •_W-m2 q u w a l l heat f l u x [-^ ] m q heat f l u x [-^ ] t dummy v a r i a b l e T temperature [K] u streamwise v e l o c i t y [ s m c ] v i i i u* s c a l e v e l o c i t y [ s m c ] U j c h a r a c t e r i s t i c v e l o c i t y [ s m c ] v t r a n s v e r s e v e l o c i t y [ s m c ] x streamwise c o o r d i n a t e [m] y t r a n s v e r s e c o o r d i n a t e [m] n n u -P u g / ? ( T « , - T o o ) x 3 Gr Grashox number = — 5 -— Nu N u s s e l t number= k Pr P r a n d t l numbe r= g Greek a thermal d i f f u s i v i t y [ s m ; c] A ag gas a b s o r p t i v i t y /? c o e f f i c i e n t of thermal expansion [K 7 «y 6 boundary l a y e r t h i c k n e s s [m] tg gas e m i s s i v i t y 2 eh t u r b u l e n t thermal d i f f u s i v i t y [ g ^ ] em eddy v i s c o s i t y [ g ^ ] s e c 2 sec-a b s o r p t i o n c o e f f i c i e n t [m x] T o p TUJ — T o o H dynamic v i s c o s i t y [m.^s^c] 2 k i n e m a t i c v i s c o s i t y [ S e C ] 1 B i x p d e n s i t y [—#] m <T S t e f a n - B o l t z r a a n n c o n s t a n t [ )f m2K TW w a l l s h e a r s t r e s s [-^ >] m v d i f f u s e r e f l e c t i v i t y $ v i s c o u s d i s s i p a t i o n f u n c t i o n T — - — s i LK g-m 2 j S u b s c r i p t s black b l a c k c o n d i t i o n s g g a s t e m p e r a t u r e gray g r a y c o n d i t i o n s in/ e v a l u a t e a t y = oo I e v a l u a t e a t x = L r r a d i a t i o n component ref r e f e r e n c e t e m p e r a t u r e w a t t h e wal1 x i n x d i r e c t i o n y i n y d i r e c t i o n oo e v a l u a t e a t y = oo x I . INTRODUCTION 1.1 P r e l i m i n a r y Remarks Heat; " t r a n s f e r by c o n v e c t i o n combined w i t h i n f r a - r e d r a d i a t i o n o f a g a s a r i s e s i n many e n g i n e e r i n g a p p l i c a t i o n s . T h i s mode o f h e a t t r a n s f e r w i l l be most i m p o r t a n t when a h e t e r o p o l a r g a s s u c h a s w a t e r v a p o u r o r c a r b o n d i o x i d e i s h e a t e d by i t s s u r r o u n d i n g s . T h e s e g a s e s have s t r o n g a b s o r p t i o n bands i n t h e i n f r a - r e d r e g i o n . The e f f e c t o f s u c h s i t u a t i o n s i s t o add a n o t h e r mechanism f o r h e a t t r a n s f e r i n t o t h e s y s t e m , r e s u l t i n g i n an i n c r e a s e i n t h e o v e r a l l r a t e o f h e a t t r a n s f e r . A p r a c t i c a l example o f t h i s mode o f h e a t t r a n s f e r i s f o u n d i n f u r n a c e s , where t h e h o t f l u e g a s e s c o n v e c t h e a t t o t h e c o l d f u r n a c e w a l l s , a s w e l l a s r a d i a t i n g e n e r g y t o t h e b o u n d i n g s u r f a c e . A n o t h e r example o f t h i s phenomenon a r i s e s i n t h e h e a t t r a n s f e r f r o m f l a m e s . An i m p o r t a n t a p p l i c a t i o n o f t h i s i s t h e p r e d i c t i o n o f t h e s p r e a d o f f i r e , smoke, and t o x i c g a s e s i n a room f i r e [ 1 ] . In t h i s s i t u a t i o n , t h e g a s f l o w i s b u o y a n c y d r i v e n , and b e c a u s e o f t h e 1 presence of water vapour and carbon d i o x i d e , t h e r e w i l l be an exchange of thermal r a d i a t i o n between the gas and the e n c l o s u r e . F i n a l l y , an i n t e r e s t i n g and unusual a p p l i c a t i o n i s found i n the case of the proposed M a r t i a n b a l l o o n . The atmosphere of Mars i s composed mainly o f carbon d i o x i d e , so a b a l l o o n open at the bottom would be heated by n a t u r a l c o n v e c t i o n as a r e s u l t of the warming of the b a l l o o n s u r f a c e by the sun, and a l s o by the a b s o r p t i o n of thermal r a d i a t i o n by the carbon d i o x i d e w i t h i n the b a l l o o n . The examples i l l u s t r a t e the p r a c t i c a l importance of unde r s t a n d i n g the i n t e r a c t i o n between thermal r a d i a t i o n and c o n v e c t i o n , and i t s e f f e c t on the o v e r a l l r a t e o f heat t r a n s f e r . I t i s necessary a t t h i s stage t o d i s t i n g u i s h between f o r c e d and n a t u r a l convect i on. The p r e s e n t work w i l l d e al mainly with the e f f e c t of a r a d i a t i n g gas on n a t u r a l c o n v e c t i o n , although the methods used t o s o l v e the problem are analogous f o r f o r c e d f l o w . Before proceeding, i t i s important t o d e f i n e some of the terms f r e q u e n t l y encountered when d e a l i n g with a r a d i a t i n g f l u i d . 2 1.2 Terminology The terms gray, o p t i c a l l y t h i n , o p t i c a l l y t h i c k and s c a t t e r i n g are f r e q u e n t l y encountered i n d i s c u s s i o n s on thermal r a d i a t i o n p rocesses of p a r t i c i p a t i n g gases. The purpose of t h i s s e c t i o n i s t o d e f i n e these terms i n an e f f o r t t o a v o i d ambiguity and c o n f u s i o n when d e a l i n g with the s u b j e c t . The term gray i m p l i e s a s u r f a c e o r gas which emits o r absorbs r a d i a t i o n independently o f the wavelength of the r a d i a t i o n . T h i s i s a c o n s i d e r a b l e s i m p l i f i c a t i o n , as most s u r f a c e s and gases e x h i b i t a s t r o n g dependence of r a d i a t i v e p r o p e r t i e s with wavelength. The j u s t i f i c a t i o n f o r making such an assumption i s one of convenience; without the assumption, i t would be v e r y d i f f i c u l t t o s o l v e r a d i a t i o n problems without the use of l a r g e computers. The term o p t i c a l l y t h i n i m p l i e s a f l u i d i n which the photon mean f r e e path i s much l a r g e r than the c h a r a c t e r i s t i c p h y s i c a l dimension. R a d i a t i o n exchange occ u r s between elements o f gas volume and the bounding s u r f a c e o n l y . Any r a d i a n t energy emitted by a molecule 3 of gas passes "through "the gas unattenuated. C o n v e r s e l y , the term o p t i c a l l y t h i c k i m p l i e s a f l u i d i n which the photon mean f r e e path i s much s m a l l e r than the c h a r a c t e r i s t i c p h y s i c a l dimension. T h e r e f o r e , r a d i a n t exchange occurs between adjacent molecules and the i n t e r a c t i o n i s a d i f f u s i o n p r o c e s s . S c a t t e r i n g i s d e f i n e d as any i n t e r a c t i o n between a photon and another p a r t i c l e , r e s u l t i n g i n the l o s s of some of the energy of the photon. R a y l e i g h s c a t t e r i n g i s c h a r a c t e r i z e d by the c r i t e r i o n t h a t the dimension of the s c a t t e r i n g p a r t i c l e s be much s m a l l e r than the wavelength of the i n c i d e n t r a d i a t i o n . The i n t e n s i t y o f R a y l e i g h s c a t t e r i n g i s i n v e r s e l y p r o p o r t i o n a l t o the f o u r t h power of the wavelength. For thermal r a d i a t i o n , the wavelength i s s u f f i c i e n t l y l a r g e so as t o make R a y l e i g h s c a t t e r i n g n e g l i g a b l e . 1.3 Review o f the L i t e r a t u r e Numerous papers have been p u b l i s h e d on n a t u r a l c o n v e c t i o n and on gas r a d i a t i o n . These works were s t u d i e d i n an attempt t o g a i n i n s i g h t i n t o each mode of heat t r a n s f e r , and the methods employed t o q u a n t i f y them. In a d d i t i o n , a computer search was conducted, and m a t e r i a l r e l a t i n g s p e c i f i c a l l y t o coupled r a d i a t i o n - n a t u r a l c o n v e c t i o n was found t o be l i m i t e d t o the laminar regime and numerical s o l u t i o n s o f the t u r b u l e n t case i n e n c l o s u r e s and between p a r a l l e l p i a t e s . 1.3.1 T u r b u l e n t N a t u r a l Convection E c k e r t and Jackson [2] were one of the f i r s t t o develop equations f o r heat t r a n s f e r r a t e s from an i s o t h e r m a l , v e r t i c a l p l a t e . By analogy with the near w a l l r e g i o n of the f o r c e d c o n v e c t i o n case, they assumed p r o f i l e s f o r the temperature and v e l o c i t y d i s t r i b u t i o n w i t h i n the boundary l a y e r , and shear s t r e s s a t the w a l l . In a d d i t i o n they assumed the v a l i d i t y of Reynold's analogy, and were t h e r e f o r e a b l e t o s o l v e the momentum and energy i n t e g r a l e q u a t i o n s . Cheesewright [3], and Cheesewright and I e r o k i o p i t i s [ 4 ] conducted experiments on t u r b u l e n t n a t u r a l c o n v e c t i o n i n order t o v e r i f y E c k e r t and Jackson's work. In a d d i t i o n , Cheesewright and Doan [ 5 ] have made measurements of space-time c o r r e l a t i o n s i n the t u r b u l e n t n a t u r a l c o n v e c t i o n boundary l a y e r i n an 5 attempt t,o understand the s t r u c t u r e of the problem. Numerous o t h e r papers e x i s t i n c l u d i n g works by F u j i l et a l [ 6 3, Miyamoto e t a l [7] and Hoogendorn and Euser [8] . An e x t e n s i v e review i s g i v e n by George and Capp [10] . Of t h e o r e t i c a l i n t e r e s t are the works of K u t a t e l a d z e [11] , Kato e t a l . [12] , Mason [13] , and George and Capp [9] , who have developed the concept of a two l a y e r model i n d e s c r i b i n g t u r b u l e n t n a t u r a l c o n v e c t i o n . The l a s t paper i s p a r t i c u l a r l y noteworthy. The t u r b u l e n t n a t u r a l boundary l a y e r next t o a v e r t i c a l heated p l a t e i s analyzed by s c a l i n g arguments. I t i s shown t h a t a f u l l y developed boundary l a y e r must be t r e a t e d i n two p a r t s with two c o r r e s p o n d i n g l e n g t h s c a l e s . The o u t e r r e g i o n c o n s i s t s of most of the boundary l a y e r and corresponds t o a r e g i o n of n e g l i g a b l e v i s c o u s and c o n d u c t i o n terms i n the governing e q u a t i o n s . The inner l a y e r , which i s shown t o be a constant heat f l u x l a y e r , i s found t o c o n s i s t of two s u b l a y e r s : the innermost being a "conductive s u b l a y e r " and the o u t e r a "buoyant s u b l a y e r " . U n i v e r s a l p r o f i l e s are g i v e n f o r v e l o c i t y and temperature p r o f i l e s i n the two r e g i o n s as w e l l as asymptotic heat t r a n s f e r and f r i c t i o n laws. 6 R e s u l t s are g i v e n i n terms of u n i d e n t i f i e d u n i v e r s a l c o n s t a n t s and f u n c t i o n s of P r a n d t l number. 1.3.2 Gas R a d i a t i o n Gas r a d i a t i o n i s a v a s t s u b j e c t and no attempt i s made t o l i s t a l l the volumes p u b l i s h e d . Of i n t e r e s t t o the t h e o r e t i c a l development of the r a d i a t i o n term are the works of Sparrow and Cess [13] and S i e g e l and Howell [14]. The work of H o t t e l [15], [16], and [17] i s of importance f o r i t s development of the mean beam len g t h method f o r e v a l u a t i n g r a d i a t i o n exchange and the e m p i r i c a l d a t a i t i s based upon. Howell [18] and V i s k a n t a [19] have w r i t t e n comprehensive reviews on the p r e s e n t l e v e l of understanding and the methods used f o r s o l v i n g r a d i a t i v e t r a n s f e r i n p a r t i c i p a t i n g media. 1.3.3 R adiation-Laminar N a t u r a l Convection Although the work of L i c k [20] d e a l s s t r i c t l y with coupled c o n d u c t i o n - r a d i a t i o n between p a r a l l e l p l a t e s , h i s treatment of the r a d i a t i o n term f o r an o p t i c a l l y t h i n gas i s the b a s i s of s o l u t i o n of many ot h e r a u t h o r s . In i t s most ge n e r a l form, the r a d i a t i o n term i s an i n t e g r o - d i f f e r e n t i a l e q u a t i o n . By assuming an o p t i c a l l y t h i n , gray gas, L i c k a p p l i e d an approximate k e r n e l technique by matching areas and f i r s t moments, and was able t o s i m p l i f y the problem t o a d i f f e r e n t i a l e q u a t i o n . Cess [13,21,22] has made a number of c o n t r i b u t i o n s t o the study of the e f f e c t s of r a d i a t i o n on the boundary l a y e r f l o w of a gas. In 1964, he presented the f i r s t i n v e s t i g a t i o n of the i n t e r a c t i o n of thermal r a d i a t i o n with n a t u r a l c o n v e c t i o n f o r the p a r t i c u l a r case of a v e r t i c a l p l a t e i n a nonabsorbing gas [21] . From t h i s work, he concluded t h a t i t i s p o s s i b l e t o t r e a t the c o n v e c t i v e and r a d i a t i v e components as independent and superimposed. Sparrow and Cess s o l v e d the problem of r a d i a t i o n i n t e r a c t i o n w ith f o r c e d c o n v e c t i o n i n a gray, a b s o r b i n g gas by l i n e a r i z i n g temperature i n the energy e q u a t i o n and s o l v i n g f o r the o p t i c a l l y t h i c k and o p t i c a l l y t h i n l i m i t s . Cess [22] a l s o t r e a t e d the case of an o p t i c a l l y t h i n gas c o n s i d e r i n g the e f f e c t s of the n o n l i n e a r i t y of temperature dependance and a nongray gas. From t h i s l a s t paper, he found t h a t the f i r s t o r d e r r e s u l t s of the nongray case d i f f e r s u b s t a n t i a l l y from those f o r the gray gas. A r p a c i [23] has s t u d i e d " t h e o r e t i c a l l y the e f f e c t of the i n t e r a c t i o n between r a d i a t i o n and laminar n a t u r a l c o n v e c t i o n i n an a b s o r b i n g , gray, stagnant gas a d j a c e n t t o a v e r t i c a l p l a t e . Both o p t i c a l l y t h i c k and t h i n c o n d i t i o n s are t r e a t e d . The problem was s o l v e d by a p p l y i n g a p e r t u r b a t i o n technique t o the momentum and energy i n t e g r a l e q u a t i o n s . England and Emery [24] were the f i r s t t o attempt experimental o b s e r v a t i o n of the e f f e c t s of thermal r a d i a t i o n upon the laminar f r e e c o n v e c t i o n from a v e r t i c a l p l a t e . V e l o c i t y and temperature p r o f i l e s were o b t a i n e d i n both a b s o r b i n g and nonabsorbing gases. In a d d i t i o n , they a p p l i e d a f i n i t e d i f f e r e n c e procedure t o the problem assuming an o p t i c a l l y t h i n , gray gas. From t h e i r work, they concluded t h a t the e f f e c t of r a d i a t i o n i s n e g l i g a b l e — c o n t r a r y t o the f i n d i n g s of o t h e r papers. A p o s s i b l e cause of the d i s c r e p a n c y may have been due t o the experimental procedure employed. For example, the v e l o c i t y p r o f i l e s were ob t a i n e d by i n j e c t i n g p a r t i c l e s i n t o the flow and a p p l y i n g a photographic t e c h n i q u e . The i n j e c t e d p a r t i c l e s were z i n c s t e a r a t e with an average diameter of 6.6 (im. In n a t u r a l c o n v e c t i o n experiments where l a s e r d o p p l e r anemometry i s a p p l i e d , i t has been found necessary t o have a seed of the order of 0.2 ftm diameter f o r the seed t o f o l l o w the flow. T h e r e f o r e the r e s u l t s of England and Emery i s r a t h e r q u e s t i o n a b l e . Cheng and O z i s i k [25] c o n s i d e r e d the more gen e r a l case of an a b s o r b i n g , e m i t t i n g , and i s o t r o p i c a l l y s c a t t e r i n g gray gas. They s o l v e d the problem by a p p l y i n g a normal mode expansion t e c h n i q u e . In a d d i t i o n , they performed a parameter survey t o study the e f f e c t s of s i n g l e s c a t t e r i n g albedo, a b s o r p t i o n c o e f f i c i e n t , and conduction t o r a d i a t i o n parameter on the temperature and v e l o c i t y d i s t r i b u t i o n i n the boundary l a y e r . E l s a y e d and F a t h a l a h [26] have s t u d i e d n a t u r a l c o n v e c t i o n adjacent t o a v e r t i c a l t r a n s p a r e n t p l a t e which i s heated and t r a n s m i t s r a d i a n t energy. The working f l u i d i s water and i s modelled as an o p t i c a l l y t h i c k , c o l d f l u i d . By c o l d , i t i s i m p l i e d t h a t the water absorbs but does not emit r a d i a n t energy. The assumption of an o p t i c a l l y t h i c k f l u i d i s d i f f i c u l t t o j u s t i f y and i s d i s c u s s e d i n [27] . 10 The work of Audunson and Gebhart [27] i s important;, f o r i t seems t o c o n t a i n the o n l y r e l i a b l e experimental v e r i f i c a t i o n o f n a t u r a l c o n v e c t i o n from a v e r t i c a l p l a t e with a p p r e c i a b l e r a d i a t i o n e f f e c t s . The experiment c o n s i s t e d of measuring power i n p u t , w a l l temperature and temperature p r o f i l e s w i t h i n the laminar boundary l a y e r of a v e r t i c a l heated p l a t e with uniform heat i n p u t . The gases used were a i r , argon, and ammonia, and r e s u l t s were obtained over a range of p r e s s u r e s . T h e o r e t i c a l r e s u l t s were obtained assuming a gray gas and a l i n e a r i z e d r a d i a t i o n term. The s o l u t i o n was obtained u s i n g a p e r t u r b a t i o n technique and i s g i v e n i n terms of an inn e r and o u t e r r e g i o n . The i n n e r r e g i o n corresponds t o the boundary l a y e r , with s t r o n g temperature and v e l o c i t y g r a d i e n t s . The ou t e r r e g i o n e x i s t s o u t s i d e the boundary l a y e r where g r a d i e n t s are small and heat t r a n s f e r o c c u r s mainly through conduction and r a d i a t i o n . From t h e i r work, Audunson and Gebhart found t h a t a r a d i a t i n g gas caused an i n c r e a s e i n heat t r a n s f e r r a t e s o f as much as 40% f o r the c o n d i t i o n s i n t h e i r experiments. T h i s value was found t o be s t r o n g l y dependent upon the emissive c h a r a c t e r i s t i c s o f the heated p l a t e . I t was observed t h a t the presence of a 11 r a d i a t i n g gas decreased "the non-dimensional "temperature by the order of 20%. For non-absorbing gases, they v e r i f i e d t h a t i t i s p o s s i b l e t o t r e a t the r a d i a t i o n and c o n v e c t i v e proccesses as independent and superimposed. F i n a l l y , they d i s c u s s the a p p l i c a b i l i t y of the o p t i c a l l y t h i c k approximation a p p l i e d i n numerous ot h e r s t u d i e s , and conclude t h a t i t i s not an a p p r o p r i a t e assumption. 1.3 .4 Numerical S o l u t i o n s o f N a t u r a l Convection R a d i a t i o n Problems. In the case of m u l t i - d i m e n s i o n a l problems, i n c l u d i n g the e f f e c t s of a non-gray gas and p r o p e r t y v a r i a t i o n , numerical modeling i s of g r e a t importance. The p r e s e n t work w i l l not deal with numerical methods, so the reader i s r e f e r r e d t o the review a r t i c l e by Yang [28] f o r a d e t a i l e d d i s c u s s i o n of c u r r e n t works i n t h i s f i e l d . 1.3.5 M i s c e l l a n e o u s Very l i t t l e has been done on the i n t e r a c t i o n of a 12 r a d i a t i n g gas with t u r b u l e n t •forced c o n v e c t i o n f l o w s . The work by Naidenov and S h i n d i n [29] c o n s i d e r s the e f f e c t on t u r b u l e n t q u a n t i t i e s i n a gas l a y e r f l o w i n g over a h o r i z o n t a l s u r f a c e , and ab s o r b i n g r a d i a t i o n from an e x t e r n a l source. From t h e i r work, they concluded t h a t the temperature p r o f i l e i n the thermal s u b l a y e r ceases t o be l i n e a r . In a d d i t i o n , they found an i n c r e a s e i n the temperature f l u c t u a t i o n s under the i n f l u e n c e o f r a d i a t i o n . The t h e o r e t i c a l work by Cheung et a l . [30] on r a d i a t i v e heat t r a n s f e r i n a t u r b u l e n t h o r i z o n t a l l a y e r i s of note. The authors s t u d i e d the beh a v i o r of the thermal boundary l a y e r over a range of o p t i c a l t h i c k n e s s , conduction t o r a d i a t i o n parameter, and R a y l e i g h number. They found t h a t an i n c r e a s e i n the r a d i a t i v e component of the heat t r a n s f e r produced an i n c r e a s e i n the t h i c k n e s s of the boundary l a y e r , e v e n t u a l l y r e s u l t i n g i n the boundary l a y e r approximation c e a s i n g t o be v a l i d . 1.4 Scope of the Present I n v e s t i g a t i o n In r e v i e w i n g the p u b l i s h e d m a t e r i a l on n a t u r a l 13 c o n v e c t i o n coupled with r a d i a t i o n , t h e r e seems t o be an absence of m a t e r i a l d e a l i n g with t u r b u l e n t f l o w s , and a l a c k of experimental v e r i f i c a t i o n i n g e n e r a l . A c c o r d i n g l y , the purpose of t h i s i n v e s t i g a t i o n i s t o t h e o r e t i c a l l y and e x p e r i m e n t a l l y study the e f f e c t of a b s o r p t i o n of thermal r a d i a t i o n by a r a d i a t i v e l y p a r t i c i p a t i n g gas i n a t u r b u l e n t n a t u r a l c o n v e c t i o n boundary l a y e r a djacent t o a v e r t i c a l heated p l a t e . The j u s t i f i c a t i o n f o r s t u d y i n g r a d i a t i o n -t u r b u l e n t n a t u r a l c o n v e c t i o n heat t r a n s f e r i s t h a t i t combines two fundamental modes of heat t r a n s f e r i n ways t h a t have not been w e l l documented or f u l l y e x p l o r e d e x p e r i m e n t a l l y . In a d d i t i o n , because of the complexity of the r a d i a t i o n term and the c o u p l i n g of the momentum and energy equations, t h e o r e t i c a l s o l u t i o n s are l i m i t e d i n scope by the assumptions needed i n o r d e r t o s o l v e the e q u a t i o n s . F i n a l l y , although e x t e n s i v e numerical s o l u t i o n s do e x i s t , they tend t o be l i m i t e d t o flow i n e n c l o s u r e s or between i n f i n i t e p a r a l l e l p l a t e s , and t h e r e seems t o be l i t t l e e xperimental v e r i f i c a t i o n of these works, so i t i s d i f f i c u l t t o assess t h e i r accuracy. F i n a l l y , the presence of carbon d i o x i d e and water 14 vapour i n "the atmosphere means t h a t many n a t u r a l c o n v e c t i o n heat t r a n s f e r problems w i l l be i n f l u e n c e d by thermal r a d i a t i o n . T h i s i n f l u e n c e i s not u s u a l l y c o n s i d e r e d , even though i t may have a c o n s i d e r a b l e e f f e c t on the r a t e of heat t r a n s f e r . 15 II THEORETICAL DEVELOPMENT 2.1 I n t r o d u c t i o n A mathematical model of the net heat t r a n s f e r t o a t u r b u l e n t n a t u r a l c o n v e c t i o n boundary l a y e r i n an ab s o r b i n g gas adjacent t o a v e r t i c a l heated p l a t e i s developed and s o l v e d i n t h i s chapter. In a d d i t i o n , the assumptions and s i m p l i f i c a t i o n s i n t r o d u c e d i n t o the model are l i s t e d and j u s t i f i e d . The momentum and energy equations are developed with the i n c l u s i o n of a buoyancy term, t u r b u l e n t f l u c t u a t i o n s , and a r a d i a n t f l u x term. In order t o s o l v e t h i s problem without r e s o r t i n g t o l a r g e computers, a number of s i m p l i f y i n g assumptions are invoked. I t i s assumed t h a t the fl o w i s two-dimensional and can be expressed u s i n g the usual boundary l a y e r approximations. The Boussinesq approximation i s a p p l i e d and the v e l o c i t y and temperature p r o f i l e s of E c k e r t and Jackson are assumed t o p r e d i c t the fl o w . In a d d i t i o n , the gas i s assumed t o be o p t i c a l l y t h i n , gray, and n o n s c a t t e r i n g . F i n a l l y , the r e g i o n o u t s i d e the boundary l a y e r i s assumed i s o t h e r m a l , stagnant, c o l d and i n f i n i t e i n ex t e n t . By m o d e l l i n g the out e r r e g i o n i n t h i s manner, 16 "the heat t r a n s f e r o u t s i d e the r e s t r i c t e d t o the a b s o r p t i o n of the boundary l a y e r and the p l a t e . boundary l a y e r i s r a d i a n t energy from The g o v e r n i n g equations are i n t r o d u c e d i n §2.2, f o l l o w e d by an order of magnitude a n a l y s i s i n §2.3. The e f f e c t o f t u r b u l e n c e i s i n c o r p o r a t e d i n t o the model i n §2.4 by assuming t h a t the v e l o c i t y , temperature and pr e s s u r e can be expressed i n terms of an average component and a f l u c t u a t i n g component. Although n o t h i n g new i s presented i n these t h r e e s e c t i o n s , they are i n c l u d e d t o review the assumptions and l i m i t a t i o n s of the boundary l a y e r approximation. The r a d i a t i o n term i s developed i n §2.5 by the a p p l i c a t i o n o f two models. The f i r s t model i s the mean beam le n g t h f o r m u l a t i o n due t o H o t t e l [15,16,17], and the second method c o n s i s t s of d e v e l o p i n g the r a d i a t i o n t r a n s p o r t e quation based upon an o p t i c a l l y t h i n , gray gas approximation. In §2.6, the i n t e g r a l method i s a p p l i e d t o the momentum and energy e q u a t i o n s , r e s u l t i n g i n two n o n l i n e a r a l g e b r a i c equations; the r e s u l t s o f which are presented i n §2.7. In summary, the problem s o l v e d i n t h i s chapter i s the heat t r a n s f e r due t o thermal r a d i a t i o n coupled 17 with t u r b u l e n t n a t u r a l c o n v e c t i o n i n the boundary l a y e r of an a b s o r b i n g gas adjacent t o a heated v e r t i c a l s u r f a c e . I t i s assumed t h a t the Boussinesq and t h i n l a y e r approximations can be a p p l i e d t o the g o v e r n i n g e q u a t i o n s . In a d d i t i o n , t o s i m p l i f y the r a d i a t i o n term, the gas i s assumed t o be gray, o p t i c a l l y t h i n , n o n s c a t t e r i n g , and c o l d and stagnant o u t s i d e the boundary l a y e r . 2.2 Governing Equations 2.2 .1 The Boundary Layer The problem s o l v e d i s assumed t o be of a two-dimensional nature. The c o o r d i n a t e s used i n the development of the governing equations are shown i n F i g u r e 1, with x r e p r e s e n t i n g the streamwise d i r e c t i o n , and y the d i r e c t i o n p e r p e n d i c u l a r t o the p l a t e . I t i s p o s s i b l e t h a t the a d d i t i o n of the r a d i a t i o n term w i l l cause a l a r g e i n c r e a s e i n the t h i c k n e s s of the boundary l a y e r , which would i n v a l i d a t e the boundary l a y e r approximations. However, the pr e s e n t a n a l y s i s w i l l be based upon the assumption 18 Figure 1.: Problem Geometry. of the v a l i d i t y of the boundary l a y e r approximation, for the experiment performed by Audunson and Gebhart [27] i n d i c a t e s t h a t t h i s regime does e x i s t under cir c u m s t a n c e s s i m i l a r t o the present work. In o r d e r t o s i m p i f y the s o l u t i o n of the g o v e r n i n g e q u a t i o n s , the Boussinesq approximation i s a p p l i e d t o the problem. I t i s known t h a t the assumption of c o n s t a n t p r o p e r t i e s , when a p p l i e d t o t u r b u l e n t f l o w s can i n t r o d u c e l a r g e e r r o r s i n heat t r a n s f e r c a l c u l a t i o n s . A f u r t h e r c o m p l i c a t i o n i s the number of c o n f l i c t i n g c o r r e l a t i o n s found i n the l i t e r a t u r e f o r d e f i n i n g the r e f e r e n c e temperature T r e^ a t which p r o p e r t i e s should be e v a l u a t e d , and the temperature at which the buoyancy term, /? should be d e f i n e d . T h i s problem has r e c e i v e d a good d e a l of a t t e n t i o n , and the 19 work of S i e b e r s , Moffat, and Schwind [31] review r e c e n t works. They suggest a c o r r e l a t i o n of the form T r e / = T o o + 0 . 2 ( T , „ - T o o ) and e v a l u a t i n g /? at T m . The assumption of constant p r o p e r t i e s i s of q u e s t i o n a b l e v a l i d i t y , and the j u s t i f i c a t i o n f o r a p p l y i n g i t t o t u r b u l e n t f l o w s i s one of convenience. A p p l y i n g the assumptions of a two d i m e n s i o n a l , steady, i n c o m p r e s s i b l e flow with constant p r o p e r t i e s , the mass c o n s e r v a t i o n , and momentum equations i n the x and y d i r e c t i o n s may be expressed as 1^  + ^  = 0 2.1 ox dy u Ju + v 5u = _ g 1 ffi + v{ p,\ + p| ) 2.2 ox dy & P dx. v ax2 dy2 and u p. + v 2* = _ 1 fiE + V( fi!v + fi!v ) 2 . 3 <9x dy P 9y v dx2 dy2 and from [13], the energy equation as u a T + v 5 T = a r a i T + a 2 T ) , U 9x + V dy * { dx2 + dy2 > + PC* The term j/ i s the kinematic v i s c o s i t y , a i s the thermal 20 d i f f us i v i t y d e f i n e d as - T S T » $ i s the v i s c o u s d i s s i p a t i o n term, and q r i s the r a d i a t i o n term. 2.2.2 The Outer Region The r e g i o n o u t s i d e the boundary l a y e r i s assumed i s o t h e r m a l , stagnant, c o l d , and i n f i n i t e i n e x t e n t . The assumptions of an i s o t h e r m a l , stagnant gas are based on the r e s t r i c t i o n t h a t the f l o w i s induced s o l e l y as a r e s u l t of the buoyancy f o r c e s produced by a temperature g r a d i e n t adjacent t o the w a l l . The term " c o l d gas" i m p l i e s the gas absorbs thermal r a d i a t i o n , but i t s emittance i s n e g l i g a b l e . T h i s approximation i s v a l i d p r o v i d e d the temperature of the gas i n the outer l a y e r i s much lower than the temperature of the p l a t e or the bulk temperature w i t h i n the boundary l a y e r . By m o d e l l i n g the o u t e r r e g i o n i n t h i s way, i t s o v e r a l l e f f e c t on the i n t e r a c t i o n of n a t u r a l c o n v e c t i o n with thermal r a d i a t i o n i s t o i n c r e a s e the temperature at the o u t e r edge of the boundary l a y e r . Because the ou t e r l a y e r i s modeled as i n f i n i t e , t h i s temperature i n c r e a s e i s assumed s m a l l . 21 2.3 Order of Magnitude A n a l y s i s . To g a i n insight. i n t o the terms dominating the problem, an order of magnitude a n a l y s i s i s performed. I t i s assumed t h a t : u ~ 0 ( u 1 ) , where u 1 = ^ g /?(T w — Too)x; x ~ 0 ( L ) , where L i s a c h a r a c t e r i s t i c dimension such as the l e n g t h of the p l a t e ; y~0(5) , where 6 i s the boundary l a y e r t h i c k n e s s ; and from c o n s e r v a t i o n of mass, v ~ 0 ( ^ j ^ ) . I n s e r t i n g these v a l u e s i n t o E q n . ( 2 . 2 ) , and comparing magnitudes, u J u + v du= 1 9P „ ( g u | ! u ) 2 > 5 flx dy & P dx y dx2 dy2 0 ( j j ) O ( ^ ) ? ? 0 ( £ ) 0(p i t can be concluded t h a t the f i r s t term of the v i s c o u s f o r c e can be n e g l e c t e d . Although the o r d e r s of magnitude of the body and pressure f o r c e s are not known, i t may be argued t h a t i f the e q a t i o n s are t o balance, these terms w i l l at most be of equal magnitude t o the other terms in the e q u a t i o n . In a d d i t i o n , as the body f o r c e i s the d r i v i n g mechanism of t h i s f l o w , i t cannot be n e g l e c t e d . T h e r e f o r e , the momentum equation i n the x d i r e c t i o n may be r e w r i t t e n as 22 I f the same procedure i s a p p l i e d t o the momentum equation i n the y d i r e c t i o n , u ax + v a y ~ P a y + ^ a x 2 + a y 2 j O ( ^ ) 0(Hj*) ? 0 ( * x £ ) O ( ^ ) from which i t i s concluded t h a t the momentum equation i n the y d i r e c t i o n i s O(^) l e s s than the momentum equation i n the x d i r e c t i o n and may be approximated as P = 0, 2.8 ay T h e r e f o r e , the momentum equation i n the x d i r e c t i o n can be expressed with the pre s s u r e term as a f u l l d e r i v a t i ve It may be argued t h a t ^ i s imposed on the boundary l a y e r by the stagnant r e g i o n f a r from the p l a t e . A p p l y i n g E u l e r ' s equation, 23 Dt 8 P o o v r ' 2. 10 •to t h i s r e g i o n , i"t i s found t h a t 3^ x = ^°°S" I n s e r t i n g t h i s i n t o the momentum equation, and a p p l y i n g the r e l a t i o n s h i p f o r a p e r f e c t gas g - _1 (®£-\ - I 2 11 p ~ P \dTjp~ T' where p can be approximated by i t s T a y l o r s e r i e s expansion, D _ . , dp , d2p (T-Too) 2 /> — Poo + (,1-looJ + 21 +••• 2 . 1 2 the momentum equation can be expressed as + v |H = Sfi (T-Too) + "( g^S )• 2.13 C o n s i d e r i n g next the energy e q u a t i o n , where T i s assumed of 0 ( T w — T o o ) = AT. Here T t u and T o o are the w a l l and ambient temperatures r e s p e c t i v e l y . The 24 v i s c o u s d i s s i p a t i o n term i s assumed n e g l i g a b l e f o r the p r e s e n t case. For n a t u r a l c o n v e c t i o n f l o w s i n g e n e r a l , the v i s c o u s d i s s i p a t i o n term i s important o n l y i f the k i n e t i c energy of the f l u i d i s comparable t o the heat t r a n s f e r e d , or i f the body f o r c e i s l a r g e , f o r i n s t a n c e i n r o t a t i n g machines. A review of the s u b j e c t i s g i v e n by Gebhart [32] . Looking now at the magnitude of the terms i n the energy equation i t i s found t h a t ; u d T d T _ a ( d ^ T + d^T) 1_ , aqr , dqr . U dx + V dy ~ a ( 5x 2 + dy2 ) PCP ( dx + dy ) 2 " 1 5 O ( ^ ) , 0 ( M ] ) , 0 ( «£E ) , 0 ( ^ E ) , ? , From the o r d e r s of magnitude, the conduction term i n the x - d i r e c t i o n i s seen t o be much s m a l l e r t h a t the conduction term i n the y - d i r e c t i o n , and may be n e c l e c t e d . The r a d i a t i v e term i n the x d i r e c t i o n can be n e g l e c t e d based on the f o l l o w i n g argument. By comparing the c o n v e c t i o n term i n the x d i r e c t i o n t o the component of r a d i a t i v e t r a n s f e r i n the x d i r e c t i o n , q r,a; i s n e g l i g a b l e i f 2 5 u SL » 1 * b f . 2.16 ox /)CP ox Assuming agai n -that u ~ 0 ( u 1 ) , and "that ^ ~0^"' y^ 0 0) > U l . i—!fLj-—2Lij. To p r e d i c t the magnitude of the r a d i a t i v e term, the Rosseland approximation f o r an o p t i c a l l y t h i c k gas i s a p p l i e d t o q r , r . By a p p l y i n g the o p t i c a l l y t h i c k l i m i t t o the p r e s e n t case, i t i s understood t h a t the r a d i a t i v e term w i l l be overestimated. Consequently, the order of magnitude of q r , r may be estimated from 4 deb _ 16<xT3 QT 16<rT3 ( T « , - T « x , )  qr>* - 3 flx " 3K O X 3K L 2 z ' x < where e 6 i s the blackbody emissive power, a i s the Stefan-Boltzmann constant, and K i s the a b s o r p t i o n c o e f f i c i e n t of the gas. Comparing the c o n v e c t i o n and r a d i a t i v e terms, i t i s observed t h a t q r,jr i s n e g l i g a b l e i f 16<rT 3/3«pCp > 1" 2 , 1 8 For the present a n a l y s i s , t h i s e x p r e s s i o n i s ~0(100). A p p l y i n g the above s i m p l i f i c a t i o n s t o the energy equ a t i o n , the r e s u l t i n g e x p r e s s i o n i s , 26 u 21 + v 2 T _ a ( 2fT ) 1_ ( dQl ) 2 19 U 3x + fly" a ^ ay 2 ' PC P <- dy }m An e x p r e s s i o n f o r w i l l be d e r i v e d i n §2.5 2.4 Equ a t i o n s i n Terms of T u r b u l e n t Q u a n t i t i e s The next step i n the development of the governing e q u a t i o n s i s the i n t r o d u c t i o n of t u r b u l e n c e i n t o the mathematical model. T h e r e f o r e , the a n a l y s i s which f o l l o w s i s v a l i d f o r Grashof numbers, Gr > 0(10 9). T h i s i s accomplished by t r e a t i n g the v e l o c i t y , temperature and p r e s s u r e as a q u a n t i t y c o n t a i n i n g average and f l u c t u a t i n g p a r t s : u = u + u', v=v+v', T=T+T', p = p + p'. 2.20 If the the momentum and energy equations are put i n c o n s e r v a t i v e form, the r e s u l t i s fe2 + ^ = SP (T-Too) + v( 0 )• 2.21 and duT , gvT_ n. ( d2T \ _ _1_ ( a_qr . „ 9 9 dx + dy ~ a ^ dy2 ) PCP ( dy >' Z ' Z Z Next, the e x p r e s s i o n s f o r v e l o c i t y , temperature, and pres s u r e are i n s e r t e d and the time mean taken. The 2 7 result; i s and N 21 + v 2I_ a a!T _ 9 u ¥ _ o / r 1 i _ , aqr u 5x + V dy" a ay2 5x a y pCp < a y ^ 4 In d e r i v i n g "these equation, any "term containing a s ing le f l u c t u a t i n g quanti ty i s zero , so "for instance, u'v = 0. By applying a dimensional ana lys i s to these equations, i t i s noted that dV2 „ auV ax * a y and ax ay ' so they may be neglected. E l i m i n a t i n g the overbar f o r convenience, the momentum and energy equations s i m p l i f y to and du ax + au dy v jj!= dy g/? ( T - T o o ) + v p \ - ^uV a 2!l 3y 2 ay />CP 5y 2.25 2.26 It i s important to note here the form of the governing equations, f o r t h i s w i l l determine the 28 method employed f o r o b t a i n i n g a s o l u t i o n . The momentum and energy equations are n o n - l i n e a r , and coupled through the appearance of the temperature i n the momentum equ a t i o n . In a d d i t i o n , they c o n t a i n f l u c t u a t i n g q u a n t i t i e s which must be modelled. F i n a l l y , the r a d i a t i o n term i s a n o n - l i n e a r i n t e g r o -d i f f e r e n t i a l equation which depends on p h y s i c a l q u a n t i t i e s t h a t are themselves s t r o n g l y dependent on temperature and wavelength. As a r e s u l t of the complexity of t h i s system of equations, a gen e r a l s o l u t i o n f o r the problem i s not f e a s i b l e . 2.5 The R a d i a t i o n Term Two methods are presented i n t h i s s e c t i o n f o r the e v a l u a t i o n of the r a d i a t i o n term i n the energy e q u a t i o n . The f i r s t method i s due t o H o t t e l and i s known as the mean beam length method. For the development of t h i s t h e o r y , the reader i s r e f e r e d t o S i e g e l l and Howell [14] , McAdams [15] or Holman [33] . The method employs e m p i r i c a l d a t a of gas e m i s s i v i t y v e r s u s temperature over a range of the product of gas pres s u r e and t h i c k n e s s of the gas l a y e r . The second method employed i s s i m i l a r t o the model used by A r p a c i [23]. In i t , the r a d i a t i o n term i s s i m p l i f i e d by 29 assuming a gray, o p t i c a l l y "thin, n o n s c a t t e r i n g gas. With these assumptions, an e x p o n e n t i a l k e r n a l t e chnique may be a p p l i e d t o the r a d i a t i o n t r a n s p o r t e q u a t i o n . A p p l y i n g these approximations, and an assumed temperature p r o - f i l e i n the boundary l a y e r , the r a d i a t i o n term may be s o l v e d . 2.5.1 Mean Beam Length Model The mean beam length model i s an e m p i r i c a l method developed by H o t t e l [15,16,17] . Charts are gi v e n "for the e m i s s i v i t y of blackbody r a d i a t i o n f o r carbon d i o x i d e , eg, i n terms of the temperature of the gas, Tg, and the product of the p a r t i a l p r e s s u r e of the gas with the mean beam le n g t h , P-Le. The a b s o r p t i v i t y of the gas, a g w , i s c a l c u l a t e d from the same c h a r t and i s approximated by e v a l u a t i n g the e m i s s i v i t y at the w a l l temperature and the e x p r e s s i o n P-Le-Q^P^j, and T 0 . 6 5 3 m u l t i p l y i n g t h i s r e s u l t by yf^J • The r a d i a n t exchange can be eva l u a t e d from the e x p r e s s i o n Qr.black . . . i i — — — energy emitted by gas — energy from e n c l o s u r e absorbed by gas z<r{tgTgt-ag,wrY%), 30 2.27 where a i s "the Stefan-Boltzmann constant;. To apply "the mean beam length method t o a v e r t i c a l p l a t e i n an i n f i n i t e gas, the f o l l o w i n g approximations are made. The beam len g t h L e i s approximated by 1.8-5, where 6 i s the boundary l a y e r t h i c k n e s s . The value of the f a c t o r 1.8 i s based on a d e r i v a t i o n t o equate the beam length t o the geometric c o n f i g u r a t i o n o f the problem. For f u r t h e r d e t a i l s , see Holman [33]. Because of the temperature g r a d i e n t w i t h i n the boundary l a y e r , an average temperature i s taken f o r e v a l u a t i n g the gas e m i s s i v i t y and a b s o r p t i v i t y . F i n a l l y , the gas o u t s i d e the boundary l a y e r i s assumed c o l d , so i t absorbs r a d i a t i o n but i t s emittance i s n e g l i g i b l e . The e f f e c t of t h i s w i l l be t o i n c r e a s e s l i g h t l y the temperature o u t s i d e the boudary l a y e r , but i t s e f f e c t on the boundary l a y e r i s a n t i c i p a t e d t o be s m a l l . For the case of heat t r a n s f e r t o a gray s u r f a c e , H o t t e l [16] found the r e l a t i o n Cjgray _ew+l <y 28 Qblacck 2 t o be v a l i d f o r ew>0.&. 31 2.5.2 O p t i c a l l y T h i n , Gray Gas Model The model developed i n t h i s s e c t i o n i s s i m i l a r t o the d e r i v a t i o n g i v e n by A r p a c i [23] f o r development of the laminar case of r a d i a t i o n - n a t u r a l c o n v e c t i o n . The gas i s assumed gray, o p t i c a l l y t h i n and n o n s c a t t e r i n g . The model i s d e r i v e d from the r a d i a t i o n t r a n s f e r e q uation and s o l v e d based upon an assumed boundary l a y e r temperature p r o f i l e . The r a d i a t i o n t r a n s f e r equation w i l l not be developed here, and the reader i s r e f e r e d t o Sparrow and Cess [13] f o r the d e r i v a t i o n . For the present purposes, the r a d i a n t f l u x equation used by A r p a c i [23] i s a p p l i e d , q r =2*[ewTt E 3( 7) + JJ T 4( 7') E 2( 7- 7') d T ' - J ~ T 4( 7') E 2( 7'- 7) d 7' + 2vw E 3( 7) J ~ T 4( 7') E 2( 7') d 7' ]. 2.29 The term ew i s the d i f f u s e w a l l e m i s s i v i t y , 7 = /cy where K i s the a b o r p t i o n c o e f f i c i e n t of the gas and y i s the d i s t a n c e normal t o the p l a t e , vw i s the d i f f u s e r e f l e c t i v i t y of the w a l l , and E 2, and E 3 are i n t e g r o -e x p o n e n t i a l f u n c t i o n s of n d e f i n e d by, 32 E n( y)=.f 0t"- 2exp(-y/t)dt. 2.30 If "the radiant, f l u x equation i s i n t e g r a t e d from the w a l l t o the outer edge of the boundary l a y e r , the r e s u l t i s , qr|S,=8<r { Jg« T 3 [ E 3 ( T ' ) - E 3 (KS-y') } dy' - vw[l-2E3 (««)] x Jo* T 3 ( J T ) E S (T ' ) dy' }. 2.31 ay D e t a i l s of the i n t e g r a t i o n are presented by V i n c e n t i and Baldwin [34]. In order t o s o l v e t h i s e q u a t i o n , the approximation due t o L i c k [20] i s a p p l i e d . The i n t e g r o - e x p o n e n t i a l f u n c t i o n i s approximated by an ex p o n e n t i a l of the form E„(y)= a exp (-by), 2.32 where a and b are chosen i n order t o match the areas and f i r s t moments of the two f u n c t i o n s . N oting t h a t o n l y E 3 i s r e q u i r e d , the f u n c t i o n can be approximated as E 3(y)= J exp (-§y). 2.33 If Eqn. (2.33) i s i n s e r t e d i n t o Eqn. (2.31), and i n t r o d u c i n g the d i m e n s i o n l e s s q u a n t i t i e s B = K<5 = opt i c a l 33 •thickness, £' = y/B, and 1— £', "the r a d i a n t f l u x may be r e w r i t t e n as ^ = 4^{JiT 3 exp(-lL') dC' - { l - ^ [ l - e x p ( - ^ ) ] } e x p ( - ^ ) x ft T 3 ( g ) exp( + ^ ! ) dC' }• 2.34 For the case of an o p t i c a l l y t h i n gas, the o p t i c a l t h i c k n e s s , B<Sl• With t h i s l i m i t a t i o n , the e x p o n e n t i a l f u n c t i o n s can be r e p l a c e d with the f i r s t two terms of t h e i r M a c l a u r i n s e r i e s t o g i v e , qr|&= (^) [ ( i+tM Jo T3 ( g j ) dC' •2 Jo T3 (S)C'dC' + 0(B) ]. 2.35 The s o l u t i o n t o t h i s equation may be obt a i n e d i f the temperature d i s t r i b u t i o n w i t h i n the boundary l a y e r i s known. C o n s i d e r i n g the p r e l i m i n a r y nature of the assumptions employed i n t h i s approximation, the f i r s t o r d e r temperature p r o f i l e , (T-TQQ) _ . y 1 (T w -Too) ~ 1 K6> J ^-Jb i s a p p l i e d t o s o l v e Eqn . (2.35). D i s c u s s i o n of the 34 c h o i c e of the temperature p r o f i l e i s gi v e n i n §2.5. C a r r y i n g out the s u b s t i t u t i o n and i n t e g r a t i n g , produces a f t e r some m a n i p u l a t i o n , q r = 4<r AT 4 { (l+vw) [ (1 + A) 3-|(1 +A)2 + (1 + A ) - J ] "[ g(l+A) 3-?(l+A) 2 + X(l+A)-^ ] } 2.37 where AT=T u,-T O 0, and A = r p r^°T-i— • 2 .38 l iv — 1 oo To o t b a i n the a b s o r p t i v i t y ,a s, of carbon d i o x i d e , the a l g e b r a i c models of Steward and Kocaefe [35] , and Farag [36] were c o n s i d e r e d . There were a number of i n c o n s i s t e n c i e s i n the former, so the method of Farag was employed. No attempt i s made here t o develop the method, and the reader i s r e f e r e d t o the o r i g i n a l p u b l i c a t i o n . To r e l a t e the a b s o r p t i v i t y t o the a b s o r p t i o n c o e f f i c e n t , the equation a«, = l-exp(-«6) 2.39 i s used, where S i s the boundary l a y e r t h i c k n e s s , and oig i s ev a l u a t e d at the bulk temperature of the boundary 1aye r. 3 5 2.6 S o l u t i o n o f Governing Equations T h i s s e c t i o n d e a l s with the s o l u t i o n of the momentum and energy equations f o r t u r b u l e n t n a t u r a l c o n v e c t i o n i n a r a d i a t i v e l y p a r t i c i p a t i n g gas. The s o l u t i o n procedure used i s as f o l l o w s : a pply the i n t e g r a l technique t o the momentum and energy equations; assume an e x p r e s s i o n f o r the w a l l shear s t r e s s and the v a l i d i t y of Reynolds analogy; use the v e l o c i t y and temperature p r o f i l e s of E c k e r t and Jackson; and f i n a l l y , express the r a d i a n t f l u x term u s i n g e i t h e r the mean beam l e n g t h , or the o p t i c a l l y t h i n approximation developed by A r p a c i [23]. The procedure reduces t o the s o l u t i o n of two n o n - l i n e a r a l g e b r a i c e x p r e s s i o n s by an i t e r a t i v e method. The f i r s t s tep i n the s o l u t i o n of the problem i s to express the momentum and energy equations as i n t e g r a l e q u a t i o n s . By f o l l o w i n g t h i s procedure, the govern i n g equations are not s a t i s f i e d at every p o i n t w i t h i n the boundary l a y e r , b u t o n l y at the w a l l and the outer edge of the boundary l a y e r . I t i s assumed i n t h i s d e r i v a t i o n t h a t the f l u i d and thermal boundary l a y e r t h i c k n e s s e s are e q u i v a l e n t . T h i s i s not s t r i c t l y 36 •true, but as "the P r a n d t l number f o r carbon d i o x i d e i s of the order of 0.7 over the temperature range of i n t e r e s t , and the t u r b u l e n t P r a n d t l number i s c l o s e t o u n i t y , the assumption i s not unreasonable. R e w r i t i n g the momentum and energy equations i n c o n s e r v a t i v e form g i v e s , Jo d y + Jo $7 d y =Jo E0 ( T - T o o ) dy + S° (" P " ^ ) ^ 2 ' 4 ° and , J° 9x a y + J o dy d y _ J o a dy2 dy d y -J« A ( w ) d y ' 2 ' 4 1 I n t e g r a t i n g the momentum equation through the boundary l a y e r r e s u l t s i n Ar* u 2 dy =r0 g/? ( T - T o o ) dy + [(„ + c m) ^ ] ^ . 2.42 The f i r s t c o n v e c t i o n term i s ob t a i n e d through the a p p l i c a t i o n of L e i b n i t z ' formula, •fofx-'dy = 5y-J>ady - «(y=«) 2.43 37 where u(y = 5) = 0 . I"fc has been assumed t h a t the t u r b u l e n t shear s t r e s s , —u'v' can be expressed as em §^ with e m oy c o r r e s p o n d i n g t o the eddy v i s c o s i t y . The momentum equation may be r e w r i t t e n as d - f g u 2 dy = Jg g/? (T-Too) dy - rw, 2.44 dx where the w a l l shear s t r e s s i s gi v e n by Tw = (v + em)(§j^ The boundary c o n d i t i o n s a r e , y = 0 : u = 0; g/? (T^-Too) = -v dy v = <5- 2 u-n O - o 2 _ U - o y ~ 6 - S y - U ' dy2 ' dy3 ' A p p l y i n g the same technique t o the energy e q u a t i o n , and making the s u b s t i t u t i o n ( « g - - ' T ' ) = ( * + <* ) ( g ) 2.45 where eh i s the thermal d i f f u s i v i t y , the energy equation can be expressed as, 3 x / § u T d ^ + w . = ( * + <* i!3)w - wP 2 - 4 6 R e w r i t i n g v as v =— J dy from c o n t i n u i t y , and 38 s u b s t i t u t i n g t h i s -for the second c o n v e c t i v e term e v a l u a t e d at y=5 g i v e s £ j * uCT-Tco) dy = jSg. _ _1_ ( q r ) , 2.47 where the w a l l heat f l u x has been s u b s t i t u t e d f o r (a + eh)(j^-j . The boundary c o n d i t i o n s f o r the energy equation a r e , and , y = 0 : T=Tu,; g/? (T„-Too) = -v gl"; | ^ = 0 dy dy' y=6 : T — T o o ; g = 0 ; 0 = 0 , To o b t a i n a s o l u t i o n t o Eqs. (2.44) and (2.47), the assumptions and procedures used by E c k e r t and Jackson f o r t h e i r s o l u t i o n of t u r b u l e n t n a t u r a l c o n v e c t i o n are a p p l i e d . I t i s r e a l i z e d t h a t the a d d i t i o n of the r a d i a n t term w i l l a l t e r the boundary l a y e r t h i c k n e s s , and temperature and v e l o c i t y p r o f i l e s , but as the gas i s assumed t o be o p t i c a l l y t h i n , the two s o l u t i o n s w i l l converge as the a b s o r p t i o n of the gas goes t o z e r o . I t should be noted t h a t although t h e r e i s s t r o n g evidence i n f a v o u r of a p p l y i n g a two l a y e r model t o the v e l o c i t y and temperature p r o f i l e s , no attempt i s made t o t h i s end. 39 The j u s t i f c a t i o n of a p p l y i n g a l e s s a c c u r a t e model comes -from the f i r s t o rder nature of the t h e o r e t i c a l development, and the d e s i r e t o have a s o l u t i o n t h a t does not r e q u i r e the a p p l i c a t i o n of l a r g e s c a l e numerical methods. The assumptions used t o s o l v e the problem are: ( i ) The v e l o c i t y and temperature p r o f i l e s o f E c k e r t and Jackson p r e d i c t the flow. The p r o f i l e s a r e , 1 u = u*(l-r?)V 2.48 and T—T 0 1 T i o — T o o (l-»77), 2.49 where u* i s a s c a l e v e l o c i t y d e f i n e d by u*= C^x"1, and 77=^. The v e l o c i t y p r o f i l e i s deduced from the s o l u t i o n of the f o u r t h - o r d e r polynomial l u = u*(a+br? + c»72+dr?3 + er;4)777 2 . 50 and the boundary c o n d i t i o n s . The l a s t term a r i s e s from the assumption of s i m i l a r i t y of v e l o c i t y p r o f i l e s i n the near w a l l r e g i o n of f o r c e d and n a t u r a l f l o w s . The temperature p r o f i l e i s obtained from the a p p l i c a t i o n of the boundary c o n d i t i o n s , and the argument t h a t f o r 40 • f l u i d s with P r a n d t l numbers c l o s e "to u n i t y , the v e l o c i t y and temperature p r o f i l e s are s i m i l a r i n shape. The choice of the v e l o c i t y and temperature p r o f i l e s of E c k e r t and Jackson over one of the more s o p h i s t i c a t e d models l i s t e d i n the i n t r o d u c t i o n may be j u s t i f i e d by the s i m p l i c i t y of the model combined with i t s reasonable accuracy. I t i s aknowledged t h a t the presence of a p a r t i c i p a t i n g gas i s l i k e l y t o change the shape of the v e l o c i t y and temperature p r o f i l e s somewhat. However, c o n s i d e r i n g the s i m p l i f i e d nature of the present model, the s e l e c t i o n of the temperature and v e l o c i t y p r o f i l e s may be r e s t r i c t e d t o f i r s t o rder approximat i o n s . ( i i ) The w a l l shear s t r e s s depends on the v e l o c i t y s c a l e u* and the boundary l a y e r t h i c k n e s s 6 i n the same manner as f o r f o r c e d c o n v e c t i o n , r„, = 0.0225 /m*2(y^) 3 . 2.51 ( i i i ) Reynolds analogy can be a p p l i e d t o r e l a t e the w a l l shear s t r e s s and the c o n v e c t i v e heat t r a n s f e r c o e f f i c ent, • ^ i — - 9l - 2 . 5 2 Re (Pr) 1 2 _ *2' nl 3 P u 41 The just, i f i c a t i o n f o r the l a s t assumption i s t h a t the P r a n d t l number i s c l o s e t o 0.7 over the range of i n t e r e s t , and t h i s corresponds t o the r e g i o n over which the analogy i s most a c c u r a t e . I t i s aknowledged t h a t the t u r b u l e n t P r a n d t l number P r ^ l 1 2 , w i l l not have the constant value of u n i t y a c r o s s the boundary l a y e r , and t h a t t h e r e i s l i k e l y t o be s i g n i f i c a n t p r o p e r t y v a r i a t i o n s through the boundary l a y e r . However, experimental v e r i f i c a t i o n of the heat t r a n s f e r e q u a t i o n based on the above assumptions suggest t h a t the assumptions are reasonable. I f Nuj i s w r i t t e n as N u ' = i ? S W 2 - 5 3 then combining assumptions ( i i ) and ( i i i ) g i v e s _1 _2 q„, = 0.0225 (a^) 4 u* C P ( T ^ - T o o ) Pr 3. 2.54 I n s e r t i n g the equations f o r the w a l l heat f l u x (2.54), and v e l o c i t y and temperature p r o f i l e s , (2.48), and (2.49) i n t o Eqs. (2.44) and (2.47), and c a r r y i n g out the i n t e g r a t i o n y i e l d s 42 0.0523 ^ (Su*2) = 0.125g/? ( T ^ - T o o ) 6 - 0.0225pu*2 ("j^)"5 2.55 and 2 - i 0.03663 ( T „ , - T o o ) ^ ( . 5 u * ) = 0.0225(Pr)~3 ( T „ , - T o o ) u* (^) 4 - -3^- 2.56 pCp In the i n t e g r a l s o l u t i o n f o r t u r b u l e n t n a t u r a l c o n v e c t i o n from a v e r t i c a l p l a t e with no r a d i a t i o n , i t i s assumed t h a t the c h a r a c t e r i s t i c v e l o c i t y can be p r e d i c t e d by the equation u*=C 1x m, and the boundary l a y e r t h i c k n e s s by 6 = C 2x n. For the p a r t i c u l a r case of a nonabsorbing gas, i f these e x p r e s s i o n s are s u b s t i t u t e d i n t o Eqs. (2.55) and (2.56), and c a r r y i n g out the d i f f e r e n t a t i o n , i t i s found t h a t the o n l y way Eqs. (2.55) and (2.56) can be s a t i s f i e d f o r a l l x i s i f m = 0.5 and n = 0.7. With the presence of an o p t i c a l l y t h i n r a d i a t i o n term, a s e l f s i m i l a r s o l u t i o n can no longer be obtained; s i n c e the v a r i a t i o n of q r with x i s not known i n general , the cons t a n t s and C 2 would become dependent on the p o s i t i o n along the p l a t e , x. Sinc e the presen t problem d e a l s with an o p t i c a l l y t h i n gas, i t i s assumed t h a t the constants G± and C 2 are 43 n e a r l y constant: over the s t u d i e d , and the exponents the nonabsorbing case. (2.56) become range of gas absorpt i v i t i e s m and n are the same as f o r T h e r e f o r e , Eqs. (2.55) and 0 . 08894C 2C 2x° • 7 = 0. 125g/?(Ttu-T00) C 2x° • 7 - 0.0225C 2x 2 . 5 7 and 0.0440^3 (T u,-T 0 0)x 0- 2 = _i 0.0225(Pr ) "3 (T^-Too) C^x0-5- (9l£g—} - ^ _ , 2.58 where Cj and C 2 are f u n c t i o n s of p o s i t i o n x. To s o l v e these equations f o r Cj and C 2 at any p o i n t x, the r a d i a t i o n term i s d e f i n e d u s i n g e i t h e r the mean beam le n g t h f o r m u l a t i o n , or the o p t i c a l l y t h i n approximation. In e i t h e r case, the momentum and energy i n t e g r a l equations can be s o l v e d , g i v e n the length and temperature of the p l a t e , the ambient temperature and the composition of the gas. For the present work, the equations and parameters were loaded i n t o an equation s o l v i n g software package. Because the a b s o r p t i o n c o e f f i c i e n t of the carbon d i o x i d e i s s p e c i f i e d i n terms of the t h i c k n e s s of the r a d i a t i n g l a y e r , an 44 i t e r a t i v e process was r e q u i r e d t o o b t a i n a s o l u t i o n . By s o l v i n g Eqs. (2.57), and (2.58), the c h a r a c t e r i s t i c v e l o c i t y , boundary l a y e r t h i c k n e s s , and the l o c a l r a d i a t i v e component of heat t r a n s f e r can be e v a l u a t e d . 2.7 D i s c u s s i o n of T h e o r e t i c a l R e s u l t s The r e s u l t s of a p p l y i n g the i n t e g r a l s o l u t i o n t o the momentum and energy equations are d e s c r i b e d i n t h i s s e c t i on. F i g u r e s 2 and 3 p r e d i c t the temperature and v e l o c i t y p r o f i l e s i n the boudary l a y e r over the range of gas compositions examined i n the experiment. The r e s u l t s shown were obtained u s i n g the o p t i c a l l y t h i n approximation t o the r a d i a t i o n term. The p r o f i l e s p r e d i c t e d u s i n g the mean beam length method are s i m i l a r , and are omitted f o r c l a r i t y . There i s o n l y one temperature p r o f i l e because the r a d i a t i o n term was modelled u s i n g the temperature p r o f i l e . In m o d e l l i n g the r a d i a t i o n term i n t h i s way, the temperature p r o f i l e i s independent of the a b s o r p t i v i t y of the gas. A d i s c u s s i o n of the e f f e c t of c o n s t r a i n i n g the temperature p r o f i l e on the p r e d i c t i o n of the heat t r a n s f e r i s presented i n chapter 4. The v e l o c i t y 45 F i g u r e 2.: T h e o r e t i c a l Temperature P r o f i l e . P l a t e length=0.76m (the length of the p l a t e used i n the experiment). F i g u r e 3. : (the l e n g t h T h e o r e t i c a l V e l o c i t y P r o f i l e s . P l a t e of the p l a t e used i n the experiment). l e n g t h =0.76m p r o f i l e i s s c a l e d by -the f a c t o r u 1 = ^ g/?(Tu,-T0O)x. 2.59 The e n thalpy f l u x from the p l a t e s u r f a c e may be ob t a i n e d by i n t e g r a t i o n of the product of the v e l o c i t y and temperature p r o f i l e s i n the boundary l a y e r , I=bJ* pC pu(T -Tco) dy. 2.60 The e n t h a l p y f l u x corresponds t o the net heat t r a n s f e r e d t o the boundary l a y e r by c o n v e c t i o n and r a d i a t i o n f o r the gi v e n p l a t e l e n g t h . By n o n d i m e n s i o n a l i z i n g t h i s q u a n t i t y as N u = blYT-k' 2 ' 6 1 where b i s the p l a t e width, A T = T u , — T o o > and k i s the thermal c o n d u c t i v i t y , the t o t a l heat t r a n s f e r as a f u n c t i o n o f gas a b s o r p t i v i t y i s o b t a i n e d . Curves of the N u s s e l t number ve r s u s the gas a b s o r p t i v i t y are presented i n F i g u r e 4. In p r e s e n t i n g the d a t a here, / x o . 33 the N u s s e l t number i s s c a l e d by (Gr-Pr) . To s c a l e the a b s o r p t i o n c o e f f i c i e n t of the gas, the a b s c i s s a i s p l o t t e d as the o p t i c a l t h i c k n e s s , c o r r e s p o n d i n g t o 48 B = KS. The r e s u l t s are giv e n f o r "the mean beam le n g t h method and the o p t i c a l l y t h i n method, and the agreement i s seen t o be good. I t must be emphasized t h a t the presence of the r a d i a t i o n term i m p l i e s the govern i n g equations are no longer s e l f s i m i l a r . Because the con s t a n t s C1 and C 2 are now dependent on x, the heat t r a n s f e r r e s u l t s obtained may not be a p p l i e d t o a p l a t e of a r b i t r a r y l e n g t h . T h e r e f o r e i n order t o s o l v e a g e n e r a l problem, F i g u r e 4 may not be used, but the procedure f o l l o w e d i n order t o o b t a i n F i g u r e 4 must be a p p l i e d . 49 0.05 legend optically thin mean beam length 0.02 0.04 0.06 0.08 0.1 O p t i c a l T h i c k n e s s F i g u r e 4.: T h e o r e t i c a l Heat T r a n s f e r R e s u l t s . P l a t e length=0.76m (the length of the p l a t e used i n the experiment). I I I . EXPERIMENTAL APPARATUS AND PROCEDURE 3.1 General In o r d e r "to v e r i f y "the " t h e o r e t i c a l a n a l y s i s of "the p r e v i o u s chapter, an experimental apparatus was c o n s t r u c t e d , and a s e r i e s of t e s t s performed. A d i s c u s s i o n of the apparatus and i n s t r u m e n t a t i o n i s gi v e n i n t h i s chapter, f o l l o w e d by a d e s c r i p t i o n of the procedure used t o o b t a i n the data. 3.2 Experimental Apparatus Drawings of the apparatus are presented i n appendix A. E s s e n t i a l l y , the apparatus c o n s i s t s of a heated v e r t i c a l p l a t e e n c l o s e d i n a s e a l e d , t r a n s p a r e n t box. The working f l u i d i s a carbon d i o x i d e , n i t r o g e n gas mixture. In order t o change the a b s o r p t i o n c o e f f i c i e n t of the gas, the gas composition i s c o n t r o l l e d . Measurements are taken at one l o c a t i o n a l o n g the p l a t e , through a range of gas compositions, and a t one s u r f a c e temperature. The q u a n t i t i e s measured are the s u r f a c e temperature of the p l a t e , v e l o c i t y and temperature p r o f i l e s through the boundary 51 l a y e r , the ambient temperature i n the e n c l o s u r e and the composition of the gas. By changing the composition of the gas, the component of r a d i a t i o n absorbed i n the boundary l a y e r w i l l change, and t h i s w i l l a f f e c t the heat t r a n s f e r r a t e . In order t o o b t a i n the net heat t r a n s f e r t o the boundary l a y e r of the p l a t e , the v e l o c i t y and temperature p r o f i l e s are combined and i n t e g r a t e d , r e s u l t i n g i n an e x p r e s s i o n f o r the en t h a l p y f l u x . R e s u l t s are giv e n i n terms of a combined c o n v e c t i v e and r a d i a t i v e N u s s e l t number as a f u n c t i o n of the carbon d i o x i d e content. The h e a t e r p l a t e i s c o n s t r u c t e d of 2.54 cm t h i c k , 304 s t a i n l e s s s t e e l p l a t e with dimensions of 76 cm x 38 cm. The p l a t e i s heated by ten 1500 watt stove h e a t e r elements, f a s t e n e d t o the back with studs and nuts. Power t o the hea t e r s i s s u p p l i e d from a 3 phase, v a r i a b l e output source. A 2.54 cm t h i c k p l a t e was choosen on the b a s i s of c a l c u l a t i o n s t o estimate the d e v i a t i o n of s u r f a c e temperature as the r e s u l t of the uneven d i s t r i b u t i o n of heat by the stove elements. I t was estimated t h a t the maximum temperature d e v i a t i o n would be of the order of 2% of the mean s u r f a c e temperature. In order t o o b t a i n a t u r b u l e n t boundary l a y e r , the p l a t e has a shrp l e a d i n g edge exposed t o 52 "the f l o w , and a " t r i p p i n g wire l o c a t e d about ^ of the way up the p l a t e . The back and edges of the p l a t e are i n s u l a t e d with a 7 cm t h i c k l a y e r of r e f r a c t o r y i n s u l a t i o n . The j o i n t s i n the i n s u l a t i o n are s e a l e d with c a l c i u m s i l i c a t e . The f r o n t of the p l a t e i s coated with a high temperature b l a c k p a i n t t o maximize the e m i s s i v i t y of the s u r f a c e . From Holman [33] , the e m i s s i v i t y i s a pproximately 0.95. No attempt was made t o v e r i f y t h i s v a l u e . To o b t a i n the s u r f a c e temperature of the p l a t e without d i s t u r b i n g the flow, small h o l e s were d r i l l e d i n t o the p l a t e through which 24 gage, type-k thermocouple wire was i n s e r t e d . The wire was then welded onto the f r o n t s u r f a c e and the a r e a was ground f l a t ; see d e t a i l A of appendix 1. Because the thermocouples conduct heat from the p l a t e t o the environment, t h e r e i s a d i s t o r t i o n of the s u r f a c e temperature adjacent t o the sensor. For a d i s c u s s i o n of the e r r o r s a s s o c i a t e d with the heat t r a n s f e r between the p l a t e and the thermocouple wire, the reader i s r e f e r r e d t o the work by Sparrow [37]. A p p l y i n g t h a t a n a l y s i s , i t was concluded t h a t the 53 "temperature d i s t o r t i o n i n the v i c i n i t y of the sensor i s s m a l l f o r the present work. To v e r i f y the accuracy of the thermocouples mounted t o the p l a t e , a type-k thermocouple was t e m p o r a r i l y p l a c e d on the f r o n t s u r f a c e of the p l a t e and i n s u l a t e d from the environment. The r e a d i n g o b tained from t h i s thermocouple was compared t o the readings from the thermocouples welded t o the p l a t e and the agreement was found t o be w i t h i n 5%. The box e n c l o s i n g the p l a t e i s c o n s t r u c t e d with an aluminum back and base, and 3 mm a c r y l i c sheet s i d e s and t o p . The dimensions of the e n c l o s u r e are 1.6 meters h i g h , 1.2 meters wide, and 0.8 meters deep. A c r y l i c was choosen because of i t s high t r a n s m i s s i v i t y and a v a i l a b i l i t y . The frame t o support the a c r y l i c i s c o n s t r u c t e d of angle aluminum, and the box i s f a s t e n e d with machine screws and s e a l e d with s i l i c o n e . The temperature w i t h i n the box i s monitored u s i n g 5 t y p e - j thermocouples which are s h i e l d e d from the p l a t e with aluminum f o i l . The heater p l a t e i s supported on a s h e l f c o n s t r u c t e d of angle i r o n , l o c a t e d a t the back of the e n c l o s u r e . The p l a t e i s h e l d v e r t i c a l by an a d j u s t a b l e 54 screw mechanism which i s t h e r m a l l y i s o l a t e d from the p l a t e . The p l a t e was checked f o r plumb u s i n g a s p i r i t l e v e l d u r i n g i n s t a l l a t i o n , and throughout the t e s t i n g procedure i t was v e r i f i e d u s i n g a d i a l gage. The buoyant plume from the p l a t e i s allowed t o r i s e i n t o a 6 inch duct where i t i s c o o l e d by a heat exchanger. The reason f o r c o o l i n g the f l o w i s the l i m i t a t i o n i n the temperature of the a c r y l i c e n c l o s u r e . The heat exchanger c o n s i s t s of a 30 meter l e n g t h of 1.27 cm diameter s o f t copper pipe wound i n t o a c o i l of 7.5 cm diameter, with water p a s s i n g through the i n s i d e of the p i p e . A f t e r the gas i s c o o l e d , i t i s r e c i r c u l a t e d i n t o the e n c l o s u r e . C a l c u l a t i o n s were performed t o determine the o v e r a l l l o s s e s a s s o c i a t e d with the c o o l i n g loop, and i t was concluded t h a t a small f a n was r e q u i r e d t o m a i n t a i n the c i r c u l a t i o n . The f a n was c a l i b r a t e d so t h a t i t s mass flow r a t e c o u l d be s e t t o the f l o w r a t e p r e d i c t e d by i n t e g r a t i n g the t h e o r e t i c a l v e l o c i t y p r o f i l e through the boundary l a y e r . The f a u l t of t h i s method i s t h a t the o n l y d r i v i n g mechanism i n d u c i n g the flow should be the buoyancy f o r c e as a r e s u l t of the temperature g r a d i e n t i n the boundary l a y e r . The presence of the f a n c o u l d be d r i v i n g the flow, r e s u l t i n g i n a mixed c o n v e c t i o n 55 regime. However, r e f e r i n g "to Holman [33] , "the c r i t e r i o n f o r "the onset of the mixed regime i s G r / ( R e ) 2 > l . C a l c u l a t i o n s f o r the present case suggest t h a t the heat t r a n s f e r i s buoyancy d r i v e n . The f l o w e x i t i n g from the c o o l i n g loop r e - e n t e r the e n c l o s u r e below the heater p l a t e . Because the f l o w v e l o c i t y i n the heat exchange i s low, and the volume of the e n c l o s u r e i s l a r g e , no d i f f u s e r was p l a c e d at the e x i t t o the heat exchange. The e f f e c t on the r a t e of heat t r a n s f e r due t o a small v e l o c i t y o u t s i d e the boundary l a y e r i s s m a l l . The gases used i n the experiment, carbon d i o x i d e and n i t r o g e n , are s u p p l i e d through tubes from a b o t t l e d supply. A constant f l o w of gas t o the apparatus i s maintained d u r i n g a t e s t t o ensure the p r e s s u r e w i t h i n the system i s s l i g h t l y g r e a t e r than atmospheric. By m a i n t a i n i n g t h i s flow, leakage i s from the e n c l o s u r e t o the environment. 3.3 Instrumentation In o r d e r t o c a l c u l a t e the o v e r a l l heat t r a n s f e r t o the boundary l a y e r , the v e l o c i t y and temperature p r o f i l e s are r e q u i r e d . From these p r o f i l e s , the t o t a l 56 heat: f l u x , components, c o n s i s t i n g of c o n v e c t i v e and r a d i a t i v e may be deduced from the e x p r e s s i o n I =!i bpC pu(T-Too) dy, 3.1 where b i s the width of the p l a t e . T h i s e x p r e s s i o n corresponds t o the net heat t r a n s f e r t o the boundary l a y e r over the l e n g t h of the p l a t e from the l e a d i n g edge t o the l o c a t i o n of measurement. The v e l o c i t y and temperature p r o f i l e s are ob t a i n e d with a constant temperature hot wire anemometer. The choice of the hot wire anemometer over a thermocouple and l a s e r doppler anemometer t o p r o f i l e the boundary l a y e r i s the a b i l i t y t o measure temperature and v e l o c i t y with one instrument. In a d d i t i o n , the q u e s t i o n arose of the s u i t a b l e c h o i c e of a seed f o r the l a s e r doppler; the combination of high temperature and low v e l o c i t y made c o n v e n t i o n a l seeds i n a p p r o p r i a t e . I t must be recognized t h a t the b e n e f i t s of the hot wire are o f f s e t by the l i m i t a t i o n s of the anemometer at low v e l o c i t i e s and the u n c e r t a i n t y i n what i s a c t u a l l y being measured. The l i m i t a t i o n s of the hot wire are d i s c u s s e d l a t e r . 57 To o b t a i n the temperature p r o f i l e , the anemometer i s used as a r e s i s t a n c e thermometer. A p o t e n t i a l of about 0.5 v o l t s i s maintained a c r o s s the b r i d g e and the r e s i s t a n c e of the sensor measured through the boundary l a y e r . The temperature may be o b t a i n e d from the e q u a t i o n R = Rc[ 1+w ( T-T c )], 3.2 where Rc i s the r e s i s t a n c e at the temperature T c, and u i s the c o e f f i c i e n t of r e s i s t i v i t y of the wire. I t i s acknowledged t h a t the v a r i a t i o n of r e s i s t i v i t y of the wire i s not e x a c t l y l i n e a r with temperature, but the d i s c r e p a n c y over the temperature range of i n t e r e s t i s s m a l l . Another problem a s s o c i a t e d with u s i n g the hot wire as a temperature measuring d e v i c e i s t h a t by p a s s i n g a c u r r e n t through the wire, the temperature of the wire w i l l be r a i s e d above the ambient temperature. In o r d e r t o study the magnitude of t h i s e r r o r , the v o l t a g e passed through the wire was v a r i e d over the range f r o n 0.2 t o 2.5 v o l t s , and the c o r r e s p o n d i n g sensor r e s i s t a n c e measured. I t was found t h a t the v a r i a t i o n i n wire r e s i s t a n c e was small over t h i s range. The wire used i s 6.3 fim diameter, 80% platinum, 20% i r i d i u m m a t e r i a l with a c o e f f i c i e n t of r e s i s t i v i t y 58 of 0.0009 c r 1 . In a p p l y i n g the hot wire anemometer t o v e l o c i t y measurements, e x t e n s i v e r e f e r e n c e t o the work of C o l l i s and W i l l i a m s [38] was made. The v e l o c i t y p r o f i l e i s obtained through the use of the di m e n s i o n l e s s e x p r e s s i o n , rp 0.17 Nu ( ^ J = A + B (Re)", 3.3 where Nu = / n , „ >,^8R/m =r-?, 3.4 and "(R+R3)lk7r(T«;—T^) _TU,+T/ T m — — o — 5 3-5 Re = ^ . 3.6 The v a r i a b l e s E and R are the sensor v o l t a g e and r e s i s t a n c e , R3 has a value of 40fi and i s the r e s i s t a n c e of the f i x e d l e g of the b r i d g e , / i s the len g t h of the sensor, k i s the thermal c o n d u c t i v i t y of the f l u i d , and Tw and T^ are the temperatures of the wire and f l u i d r e s p e c t i v e l y . The length of the sensor was measured with a t r a v e l l i n g microscope, and the r e s i s t a n c e of the probe and c a b l e s were checked with a d i g i t a l multimeter. The v a l u e s of A, B, and n are 59 determined from a c a l i b r a t i o n curve obtained i n a i r , and the p h y s i c a l p r o p e r t i e s are ev a l u a t e d at T m . In r e l a t i n g the c a l i b r a t i o n curve t o the experiment, the v a l u e s of the thermal c o n d u c t i v i t y and kinematic v i s c o s i t y f o r each gas are used, and i t i s assumed t h a t the v a l u e s of A, B, and n are independent of the gas composition. Equn. (3.4) i s v a l i d i n the Reynolds number range 0.01 t o 140, pro v i d e d t h a t buoyancy, r a d i a t i o n and non-continuum e f f e c t s are n e g l i g a b l e . C o l l i s and W i l l i a m s [38] p r e d i c t t h a t buoyancy e f f e c t s are n e g l i g a b l e i f Reoo > Grgo, 3.7 where the s u f f i x e s denote the e v a l u a t i o n of the p h y s i c a l p r o p e r t i e s at the ambient temperature. Davies and F i s h e r [39] have performed e x t e n s i v e t e s t i n g on hot w i r e s and determined t h a t f o r wires o p e r a t i n g at a maximum temperature of 1000 K, r a d i a t i o n l o s s e s from the wire may be n e g l e c t e d f o r a l l but the lowest v a l u e s of Reynolds number. From Reynolds [40] , non-continuum e f f e c t s may be n e g l e c t e d f o r a Knudsen number Kn< 0.015. The e f f e c t of heat l o s s t o the wire supports i s d i s c u s s e d by C o l l i s and W i l l i a m s and by Davies and F i s h e r . I t i s concluded t h a t f o r 60 p l a t inum-i r i d ium w i r e s , "the v a r i a t i o n i n wire temperature i s n e g l i g a b l e f o r a length t o diameter r a t i o g r e a t e r than 200. The e f f e c t of t u r b u l e n c e i n t e n s i t y must be c o n s i d e r e d next. A hot wire i s l i m i t e d t o a p p l i c a t i o n s where the t u r b u l e n c e i n t e n s i t y does not exceed about 20%. In the o u t e r r e g i o n of the n a t u r a l c o n v e c t i o n boundary l a y e r , however, the i n t e n s i t y approaches 100%, so i t i s not c l e a r what i s a c t u a l l y b e i n g measured i n t h i s r e g i o n . I t should be emphasized t h a t the hot wire i s l i m i t e d t o f l o w s with v e l o c i t i e s g r e a t e r than about 0.1 m/sec, so i t cannot be expected t h a t the outer r e g i o n of the boundary l a y e r w i l l be measured with g r e a t p r e c i s i o n . A d d i t i o n a l i n f o r m a t i o n on the use of hot wire anemometers i s given i n Hinze [41] , Bradshaw [42] ; and an e x t e n s i v e b i b l i o g r a p h y on the s u b j e c t i s presented by Freymuth [43]. The hot wire anemometer i s mounted on the t r a v e r s i n g mechanism shown i n d e t a i l D of appendix 1. The anemometer i s connected t o a d i a l gage so t h a t displacement may be measured t o w i t h i n 0.0254 mm. The t r a v e r s i n g mechanism i s f a s t e n e d t o the frame of the box t o ensure r i g i d i t y . 61 R e c a l l i n g "that "the carbon d i o x i d e content i s c o n t r o l l e d i n order t o a d j u s t the a b s o r p t i v i t y of the gas, an a c c u r a t e means i s r e q u i r e d of a d j u s t i n g the gas f l o w r a t e and m o n i t o r i n g the composition of gas i n the e n c l o s u r e . The flow of each gas i s c o n t r o l l e d u s i n g a r e g u l a t o r and monitored with a rotameter. The composition i s obtained by t a k i n g a sample of gas from the e n c l o s u r e and performing a gas chromatograph t e s t . 3.4 Experimental Procedure 3.4.1 C a l i b r a t i o n o f Instrumentation A c a l i b r a t i o n of the f a n i n the c o o l i n g loop and of the hot wire anemometer were performed. To c a l i b r a t e the f a n , a Davis vane anemometer was p l a c e d downstream of the f a n and the average f l o w r a t e measured over a range of f a n speeds. The hot wire anemometer was c a l i b r a t e d f o r temperature and v e l o c i t y measurement; t y p i c a l c a l i b r a t i o n curves are l o c a t e d i n appendix B. The temperature c a l i b r a t i o n c o n s i s t e d of measuring the c o l d r e s i s t a n c e of the wire over a range of temperatures, and comparing the r e l a t i o n 62 R = R e [ l + a ( T - T c ) ] with "the "temperature p r e d i c t e d by a thermocouple l o c a t e d c l o s e t o the hot wire. Agreement was found t o be w i t h i n 2%, and t h i s d i s c r e p a n c y may have been p a r t l y due t o the s p a c i a l and temporal s e p a r a t i o n of the two s e n s o r s . To c a l i b r a t e the hot wire f o r v e l o c i t y measurement, the b r i d g e v o l t a g e of the anemometer was measured a g a i n s t the v e l o c i t y p r e d i c t e d by a p i t o t tube i n a wind t u n n e l . I t was not p o s s i b l e t o c a l i b r a t e the anemometer i n v e l o c i t i e s l e s s than 0 . 5 m/sec. due t o the l i m i t a t i o n s of the wind t u n n e l , so i n the range from 0 . 5 m/sec t o 0 . 1 m/sec, i t was necessary t o e x t r a p o l a t e the c a l i b r a t i o n curve. For v e l o c i t i e s l e s s than 0 . 1 m/sec, the c a l i b r a t i o n i s no longer v a l i d because of the onset of buoyancy d r i v e n flow . I t i s important t o emphasize t h a t the c a l i b r a t i o n was performed i n a i r at room temperature, whereas the experiment was performed i n a carbon d i o x i d e , n i t r o g e n gas mixture over a range of temperatures. As a r e s u l t , i t i s not p o s s i b l e t o d i r e c t l y r e l a t e the b r i d g e 63 v o l t a g e t o the v e l o c i t y of the "flow. Instead, Equn (3.3) i s u t i l i z e d , where the v a l u e s of the c o n s t a n t s A, B, and n are determined from the c a l i b r a t i o n curve. I t was not necessary t o c a l i b r a t e the gas chromatograph, as i t was a l r e a d y s e t up t o measure carbon d i o x i d e and n i t r o g e n . However, t e s t s u s i n g c o n t r o l l e d samples of gas were done t o v e r i f y the accuracy of the apparatus. 3.4.2 Routine Experimental Procedure A d e s c r i p t i o n of the experimental procedure i s presented i n order t o i d e n t i f y the s t e p s taken t o o b t a i n d a t a t h a t r e p r e s e n t s the t h e o r e t i c a l model with the g r e a t e s t p o s s i b l e accuracy and without ambiguity. Each t e s t took approximately t h r e e t o f o u r hours t o perform. In p r e p a r a t i o n f o r t a k i n g measurements, the h e a t e r was turned on and allowed t o come t o steady s t a t e . In a d d i t i o n , gas was f l u s h e d through the system, and the gas composition i n the e n c l o s u r e checked r e g u l a r l y with the gas chromatograph. The c r i t e r i a used t o determine steady s t a t e were; the 64 s u r f a c e "temperature of "the p l a t e and the ambient temperature i n the e n c l o s u r e d i d not change by more than 1°C. over a t e n minute i n t e r v a l ; the composition of gas was s t a b l e t o w i t h i n 5% over a one hour i n t e r v a l . I t proved d i f f i c u l t t o maintain a constant gas composition i n the e n c l o s u r e , mainly as a r e s u l t of the d r i f t of one of the gas r e g u l a t o r s . The time r e q u i r e d t o reach steady s t a t e was u s u a l l y about two t o t h r e e hours. A gas chromatograph was taken i m e d i a t e l y before and a f t e r each t e s t . For the purpose of a n a l y s i s , the average of the two r e s u l t s was used. To o b t a i n the temperature p r o f i l e , the c o l d r e s i s t a n c e of the sensor was measured through the boundary l a y e r . The d i s t a n c e between measurements v a r i e d from 5.08 mm. o u t s i d e the boundary l a y e r t o 0.254 mm. i n t e r v a l s near the w a l l . U s i n g the same i n t e r v a l s , the bri d g e v o l t a g e was measured t o o b t a i n the v e l o c i t y p r o f i l e . The d a t a was recorded by hand as r a p i d d a t a a q u i s i t i o n was not r e q u i r e d . Each t e s t was done an average of two times t o check f o r random e r r o r s . 65 IV. RESULTS AND DISCUSSION 4.1 General The r e s u l t s of the experiment are presented and d i s c u s s e d i n t h i s chapter. To i n f e r i n f o r m a t i o n from the data, i t i s necessary t o f i r s t know the l i m i t s of accuracy of the measuring d e v i c e s . In a d d i t i o n , some o b s e r v a t i o n s about the apparatus are presented, and t h e i r i n f l u e n c e s on the experiment are d i s c u s s e d p r i o r t o the p r e s e n t a t i o n of the data. 4.2 E r r o r A n a l y s i s As s t a t e d i n the l a s t chapter, the u s e f u l n e s s of the hot wire anemometer i s o f f s e t somewhat by the u n c e r t a i n t y i n what i s a c t u a l l y being measured. T h e r e f o r e , some l e v e l of confidence must be atta c h e d t o the d a t a before any c o n c l u s i o n s may be reached. Because the experiment was c a r r i e d out u s i n g moderately high temperatures, an i n i t i a l check on the r a d i a t i v e heat t r a n s f e r c o e f f i c i e n t between the hot wire and the w a l l was made. T h i s i s expressed as 66 _ecr(T 4-T 4 J) nrod — T —Tu, ' 4 . 1 where e i s "the e m i s s i v i t y of the hot wire sensor m a t e r i a l . Comparing the r a d i a t i v e heat t r a n s f e r c o e f f i c i e n t t o the c o n v e c t i v e heat t r a n s f e r c o e f f i c i e n t p r e d i c t e d from King's law, i t was concluded t h a t the temperature and v e l o c i t y measurement e r r o r due t o r a d i a t i o n exchange between the hot wire and the w a l l was n e g l i g i b l e . In a p p l y i n g an e r r o r a n a l y s i s t o the temperature and v e l o c i t y measuements, the equations r e l a t i n g the output of the anemometer t o the q u a n t i t i e s of i n t e r e s t , namely, R=R e[l-w(T-T e)], 4.2 and T 0.17 0 . 45 (T£) = a + b W • 4- 3 E 2R \2 (R+R3) l k T r (Tw-Tf) are expanded i n a T a y l o r s e r i e s . R e t a i n i n g the f i r s t o r d e r terms o n l y , the measurement u n c e r t a i n t y of the temperature can be approximated as; 4.4 where e r , (r , Cu, and et are the u n c e r t a i n t i e s i n 67 measurement; of the wire r e s i s t a n c e , c o l d r e s i s t a n c e , wire r e s i s t i v i t y and the temperature a t which the c o l d r e s i s t a n c e i s measured. When, -for example, i s OR. e v a l u a t e d , the o t h e r q u a n t i t i e s i n the e x p r e s s i o n are h e l d c o n s t a n t . The e x p r e s s i o n -for the measurement e r r o r o f the v e l o c i t y i s obt a i n e d i n the same manner. To o b t a i n the u n c e r t a i n t y o f the en t h a l p y f l u x , the same tec h n i q u e i s a p p l i e d t o the e x p r e s s i o n I=/o PCpbu (T-Too) dy, 4.5 where t h e u n c e r t a i n t y a s s o c i a t e d with u and T have been p r e v i o u s l y e v a l u a t e d . The r e s u l t o f t h i s a n a l y s i s i s g i v e n i n Tab l e 1. Table 1 U n c e r t a i n t y Due t o Measurement E r r o r u n c e r t a i n t y temperature p r o f i l e 1.5% v e l o c i t y p r o f i l e 15% e n t h a l p y f l u x 25% s u r f a c e temperature 0.1'C ambient temperature r c gas chromatograph 5% 68 Included i n t h i s t a b l e i s the measurement u n c e r t a i n t y of the oth e r i n s t r u m e n t a t i o n used i n the experiment.The u n c e r t a i n t y i n the v e l o c i t y measurement appears u n u s u a l l y high. However, L a n c a s t e r [42] , suggests the u n c e r t a i n t y t o be g e n e r a l l y of the order of 10 t o 20% f o r v e l o c i t y measurement employing a hot wire anemometer. The o v e r a l l accuracy of the v e l o c i t y measurements i s based on the assumption of the v a l i d i t y of King's law. The l i m i t a t i o n s of the f o r m u l a t i o n are d i s c u s s e d i n § 3.3. These c r i t e r i a were a p p l i e d t o the data, and i t was concluded t h a t the v e l o c i t y i n the out e r r e g i o n of the boundary l a y e r would not be a c c u r a t e l y p r e d i c t e d as the r e s u l t of buoyancy e f f e c t s and the l a r g e t u r b u l e n c e i n t e n s i t y . However, because the v e l o c i t y i s low and the temperature d i f f e r e n c e small i n the out e r p a r t of the boundary l a y e r , the c o n t r i b u t i o n t o the en t h a l p y f l u x from t h i s r e g i o n i s s m a l l . T h e r e f o r e the e f f e c t of measurement e r r o r i n t h i s r e g i o n on the p r e d i c t i o n of the o v e r a l l heat t r a n s f e r r a t e i s not expected t o be too s i g n i f i c a n t . I t should be noted t h a t the u n c e r t a i n t y a s s o c i a t e d with the composition of the gas i s the r e s u l t of the i n s t a b i l i t y of one of the gas r e g u l a t o r s r a t h e r than i m p r e c i s i o n of the gas chromatograph. 69 4.3 Experimental Observations Some o b s e r v a t i o n s about the experimental apparatus are made i n t h i s s e c t i o n . In p r e s e n t i n g t h i s i n f o r m a t i o n , the problems and anomalies encountered d u r i n g e x p e r i m e n t a t i o n , and the p o s s i b l e e f f e c t s on the d a t a are ex p l o r e d and d i s c u s s e d . With t h i s i n f o r m a t i o n , some l e v e l of conf i d e n c e on the o v e r a l l experiment may be i n t e r p r e t e d . The most s i g n i f i c a n t problems encountered d u r i n g the experiment were the h e a t i n g of the e n c l o s i n g box, the s t r a t i f i c a t i o n of the temperature i n the e n c l o s u r e , and the i n s t a b i l i t y of one of the gas r e g u l a t o r s . I t was found t h a t the a c r y l i c e n c l o s u r e d i d not t r a n s m i t thermal r a d i a t i o n as w e l l as p r e d i c t e d . T h e r e f o r e , the m a t e r i a l heated up, r e s u l t i n g i n the l i m i t a t i o n o f the temperature at which the experiment c o u l d be conducted, and secondly, c r e a t i n g the p o s s i b i l i t y of heat t r a n s f e r between the e n c l o s u r e and the gas, or the e n c l o s u r e and the heated p l a t e . To reduce t h i s probem, fa n s were p l a c e d around the apparatus i n order t o c o o l the outer s u r f a c e of the a c r y l i c . The l i m i t a t i o n i n the o p e r a t i n g temperature of the apparatus r e s t r i c t e d the range of experimental 7 0 i n v e s t i g a t i o n , but d i d not add any u n c e r t a i n t y t o the r e s u l t s . The second p o i n t , however, i s the source o"f some concern. The e n c l o s u r e reached a temperature o"f about 60°C at some areas on i t s s u r f a c e . T h i s may have caused a n a t u r a l c o n v e c t i o n boundary l a y e r t o form a d j a c e n t t o the e n c l o s i n g w a l l s , r e s u l t i n g i n a nonzero v e l o c i t y o u t s i d e the boundary l a y e r of the p l a t e . I t i s d i f f i c u l t t o assess the magnitude of t h i s problem s i n c e the v e l o c i t y was p r o f i l e d o n l y a t one l o c a t i o n up the p l a t e . At t h a t l o c a t i o n , the v e l o c i t y was found t o be zero o u t s i d e the boundary l a y e r , but i t i s not known i f t h i s i s t r u e f o r a l l l o c a t i o n s . Because the a c r y l i c i s not 100% t r a n s m i s s i v e , i t absorbed and emitted thermal r a d i a t i o n , r e s u l t i n g i n heat exchange with the p l a t e and boundary l a y e r . However, s i n c e the e n c l o s u r e was at a much lower temperature than the p l a t e or the bulk temperature of the boundary l a y e r , and the r a t i o of the s u r f a c e a r e a of the p l a t e t o the s u r f a c e a r e a of the e n c l o s u r e i s about 0 . 0 8 , the e f f e c t of r a d i a t i o n heat t r a n s f e r with the e n c l o s u r e was assumed t o be s m a l l . I t was observed t h a t the temperature v a r i e d by about 15°C from the bottom of the e n c l o s u r e t o the top. The e f f e c t of the s t r a t i f i c a t i o n i s t o decrease the 71 o v e r a l l heat f l u x . The magnitude of the decrease can be approximated by Q = - / Q C P g ( T o o(x)-T r) dx 4.6 where rh i s the mass fl o w r a t e i n the boundary l a y e r at any streamwise l o c a t i o n x, and T r i s a r e f e r e n c e temperature. To estimate t h i s q u a n t i t y , the t h e o r e t i c a l v e l o c i t y p r o f i l e of E c k e r t and Jackson was i n t e g r a t e d through the boundary l a y e r t o o b t a i n the mass f l u x at any l o c a t i o n up the p l a t e . E q u a t i o n (4.6) was n u m e r i c a l l y i n t e g r a t e d , and i t was concluded t h a t the s t r a t i f i c a t i o n of the temperature d u r i n g the experiment reduced the net heat f l u x by about 2.3% compared t o a n o n s t r a t i f i e d environment e v a l u a t e d at the mean ambient temperature i n the e n c l o s u r e . T h i s i s co n s i d e r e d small compared t o the u n c e r t a i n t y i n the measurement of the enthalpy f l u x , and i s t h e r e f o r e n e g l e c t e d . The i n s t a b i l i t y of one of the r e g u l a t o r s s u p p l y i n g gas t o the apparatus r e s u l t e d i n a v a r i a t i o n i n the gas composition of about 5% over any one t e s t . The f l o w was checked and ad j u s t e d f r e q u e n t l y d u r i n g a t e s t , and the e r r o r a s s o c i a t e d with the v a r i a t i o n i n 72 gas composition i s r e f l e c t e d i n the r e l a t i v e l y l a r g e u n c e r t a i n t y a p p l i e d t o the p h y s i c a l p r o p e r t i e s i n the e r r o r a n a l y s i s of the v e l o c i t y p r o f i l e and heat f l u x . D u r i n g e x p e r i m e n t a t i o n , a check on the frequency and i n t e n s i t y of v i b r a t i o n of the p l a t e was performed. The p l a t e v i b r a t i n g c ould produce e x c e s s i v e motion of the gas i n the e n c l o s u r e . To check f o r t h i s problem, an acce1erometer was f a s t e n e d t o the p l a t e support. The output of the accelerometer was put t o a charge a m p l i f i e r and measured with an o s c i 1 1 i s c o p e . I t was found t h a t the s t r u c t u r e v i b r a t e d with a n a t u r a l f r e q u e n c y of about 120 h e r t z and with an a c c e l e r a t i o n of the o r d e r of approximately 0.02 m/sec 2. An a c c e l e r a t i o n of t h i s magnitude i s not expected t o a f f e c t the r e s u l t s s i g n i f i c a n t l y . 4.4 Experimental R e s u l t s and D i s c u s s i o n The r e s u l t s of the experiment are presented and d i s c u s s e d now. Graphs of the experimental v e l o c i t y , and temperature p r o f i l e s are g i v e n , from which a N u s s e l t number versus gas a b s o r p t i v i t y curve was o b t a i n e d by numerical i n t e g r a t i o n . Sample c a l c u l a t i o n s 73 of the d a t a m a n i p u l a t i o n are presented i n appendix C. F i g u r e 5 p r e s e n t s the experimental v e l o c i t y p r o f i l e o b t ained f o r a nonabsorptive gas, compared with the p r o f i l e p r e d i c t e d by the model of E c k e r t and Jackson. In a d d i t i o n , a s c a l e d v e r s i o n of the E c k e r t and Jackson p r o f i l e i s presented here, and the v e l o c i t y p r o f i l e s of v a r i o u s experimental i n v e s t i g a t i o n s of t u r b u l e n t n a t u r a l c o n v e c t i o n from a v e r t i c a l p l a t e are i n c l u d e d f o r the purpose of comparison. The s c a l i n g terms of the graph are The p o s i t i o n i s s c a l e d a c c o r d i n g to Eqn. (4.7) i n order t o conform t o the s c a l i n g used by other i n v e s t i g a t o r s . Although t h e r e i s a l o t of s c a t t e r i n the present data, i t i s c l e a r t h a t the t h e o r e t i c a l v e l o c i t y p r o f i l e of E c k e r t and Jackson does not adequately p r e d i c t the t r e n d suggested by the r e s u l t s o b t a i n e d i n t h i s experiment, nor the d a t a of Mason, Hoogendoorn and Euser, or Cheesewright. In a d d i t i o n , 4.7 and 4.8 74 data theory -*~ Cheesewright Moson *"*"" modified theory Hoogendoorn F i g u r e 5 . : Comparison of V e l o c i t y P r o f i l e s . P i ate Length =0.76m. "the present d a t a v a r i e s from these t h r e e works, although the agreement i s f a i r . The v e l o c i t y i n the o u t e r p a r t of the boundary l a y e r of the present work i s seen t o be lower, and the boundary l a y e r appears t o be t h i n n e r than p r e d i c t e d by the o t h e r works. The d i f f i c u l t y i n l o c a t i n g the o u t e r edge of the boundary l a y e r can be a t t r i b u t e d t o the l i m i t a t i o n of the hot wire anemometer at low v e l o c i t i e s . As e x p l a i n e d i n §4.2, i t i s not expected t h a t an i n a c c u r a c y of the p r o f i l e i n t h i s p a r t of the boundary l a y e r w i l l a f f e c t the r e s u l t s too much. The m o d i f i e d E c k e r t and Jackson p r o f i l e i s o b t a i n e d by m u l t i p l y i n g the o r i g i n a l p r o f i l e by a c o n s t a n t , i n t h i s case, the v a l u e of 0.75 was a p p l i e d . The value of the constant i s chosen by the c r i t e r i a t h a t the m o d i f i e d p r o f i l e f o l l o w the d a t a as c l o s e l y as p o s s i b l e . The j u s t i f i c a t i o n f o r a p p l y i n g such a technique t o the t h e o r e t i c a l p r o f i l e i s t h a t t h e r e has been a l o t of c r i t i c i s m t h a t i t does not a c c u r a t e l y p r e d i c t the v e l o c i t y i n the boundary l a y e r . The p r e s e n t i n v e s t i g a t i o n has found the agreement between the t h e o r e t i c a l and experimental p r o f i l e s t o be poor, while the agreement between the s c a l e d t h e o r e t i c a l p r o f i l e and the experiment t o be much b e t t e r . Exact agreement between the s e t s of d a t a cannot be expected and i n g e n e r a l , the d a t a i s seen 76 "to agree reasonably w e l l with the s c a l e d t h e o r e t i c a l p r o f i l e of E c k e r t and Jackson, and the f i n d i n g s of Hoogendoorn and Euser, Cheeswright, and Mason. The p r o f i l e of E c k e r t and Jackson i s based on the h y p o t h e s i s of a s i n g l e c h a r a c t e r i s t i c l e n g t h s c a l e i n the boundary l a y e r . T h i s i s c o n t r a d i c t o r y t o more rece n t models which suggest the need f o r two length s c a l e s , so the d i f f e r e n c e i n shape between E c k e r t and Jackson and the experimental f i n d i n g s i s not s u r p r i s i ng. The temperature p r o f i l e f o r the case of a nonabsorbing gas i s compared t o the c o r r e s p o n d i n g t h e o r e t i c a l p r o f i l e i n F i g u r e 6, where ^ rjT r i : . 4 . 9 1 w A co The agreement i s seen t o be very good, and i s c o n s i s t e n t with the f i n d i n g s of o t h e r i n v e s t i g a t i o n s . F i g u r e 7 shows the temperature p r o f i l e s f o r t h r e e of the cases s t u d i e d e x p e r i m e n t a l l y . There appears t o be a decrease i n the temperature g r a d i e n t with i n c r e a s i n g a b s o r p t i v i t y of the gas, although the change i s s l i g h t over the range s t u d i e d . Because the change i n the shape of the temperature p r o f i l e s i s s m a l l , the e r r o r 77 a s s o c i a t e d with c o n s t r a i n i n g i t t o be constant i n the o p t i c a l l y - t h i n t h e o r e t i c a l model i s s m a l l . The other cases s t u d i e d have s i m i l a r temperature p r o f i l e s , and are omitted f o r c l a r i t y . F i g u r e 8 pre s e n t s the experimental v e l o c i t y p r o f i l e s f o r the cases of 0%, 50% and 61% carbon d i o x i d e content i n the e n c l o s u r e . Again, the l a r g e s c a t t e r i n the v e l o c i t y d a t a i s apparent. From the p r o f i l e s , i t i s p o s s i b l e t o d i s c e r n an i n c r e a s e i n the peak v e l o c i t y and a r e a under the curve with i n c r e a s i n g gas a b s o r p t i v i t y , although t h i s d i d not occur m o n o t o n i c a l l y with i n c r e a s i n g carbon d i o x i d e content. T h i s i s i n c o n t r a s t t o the t h e o r e t i c a l model, which p r e d i c t e d a monotonic i n c r e a s e i n a r e a and peak v e l o c i t y with i n c r e a s i n g a b s o r p t i v i t y . A p o s s i b l e cause f o r the c o n t r a d i c t i o n i s the magnidude of e r r o r of the v e l o c i t y p r o f i l e s . R e f e r i n g t o §4.2, the u n c e r t a i n t y i n the v e l o c i t y measurement i s about 15%, so the e f f e c t of small changes i n the composition of the gas on the v e l o c i t y p r o f i l e s may have been too s l i g h t f o r the experimental procedure t o d e t e c t . To o b t a i n the enthalpy f l u x , the e x p r e s s i o n 79 F i gu re 7.: Exper imenta l Temperature Prof lies'. P l a t e Length =0.76m * • « 8 . . E x p < ! r i m e n t a l V e I o c i t y p r o f i U 8 _ 76m I=b/g PCPu ( T - T o o ) dy, 4.10 was n u m e r i c a l l y i n t e g r a t e d u s i n g the t r a p e z o i d a l r u l e , where 6 was d e f i n e d as the l o c a t i o n where the boundary l a y e r v e l o c i t y was z e r o . T h i s s p e c i f i c a t i o n i s not ve r y a c c u r a t e , but the c o n t r i b u t i o n t o the en t h a l p y f l u x from the outer r e g i o n of the boundary l a y e r i s s m a l l . Knowing the temperature at each l o c a t i o n , the d e n s i t y p, was obtained u s i n g the p e r f e c t gas law, and the s p e c i f i c heat C P, from a t h i r d o r d e r polynomial e x p r e s s i o n . C a r r y i n g out the i n t e g r a t i o n , the N u s s e l t number was found from the e x p r e s s i o n N u = b k ( T ; - T o o ) - - 4 - 1 1 The r e s u l t s are shown i n F i g u r e 9 , where the o r d i n a t e i s s c a l e d with the f a c t o r ( G r - P r ) 0 - 3 3 . The o p t i c a l t h i c k n e s s i s an i n d i c a t i o n of the gas a b s o r p t i v i t y and i s d e f i n e d by B = K6 , where K i s the a b s o r p t i o n c o e f f i c i e n t of the gas. Again, 6 i s r a t h e r p o o r l y d e f i n e d e x p e r i m e n t a l l y , so i n d e f i n i n g the o p t i c a l t h i c k n e s s here, the t h e o r e t i c a l boundary l a y e r t h i c k n e s s was employed. To a v o i d c l u t t e r on the graph, o n l y the t h e o r e t i c a l r e s u l t s based on the o p t i c a l l y 82 legend opt. thin ~4~' modified opt. thin dota ignore radiation 0 0.02 0.04 0.06 0.08 0.1 Optical Thickness 9.s Comparison of Experimental and Theoretical Heat Transfer r e s u l t s . p l a t e Length =0.76m. t h i n approximation are i n c l u d e d . R e f e r r i n g t o chapter 2, i t was concluded t h a t the mean beam len g t h and o p t i c a l l y t h i n approximations gave s i m i l a r r e s u l t s , so t h i s o mission seems j u s t i f i e d . The heat t r a n s f e r r e s u l t s based on the m o d i f i e d v e l o c i t y p r o f i l e are seen t o agree much b e t t e r with the experimental f i n d i n g s than the N u s s e l t number p r e d i c t e d u s i n g the o r i g i n a l t h e o r y . T h i s i s h a r d l y s u r p r i s i n g s i n c e the t h e o r y i s a l t e r e d t o f i t the data. In defence of t h i s a l t e r a t i o n t o the t h e o r y , i t i s s t r e s s e d t h a t the v e l o c i t y p r o f i l e of E c k e r t and Jackson d i f f e r s c o n s i d e r a b l y from the f i n d i n g s of t h i s and o t h e r i n v e s t i g a t i o n s , and by s c a l i n g the v e l o c i t y p r o f i l e , the agreement between the two v e l o c i t y p r o f i l e s i s found t o be much b e t t e r . The ch o i c e of a p p l y i n g a s c a l e f a c t o r t o the v e l o c i t y p r o f i l e r a t h e r than imposing an a r t i f i c i a l s t a r t i n g l e n g t h i s based on the f i n d i n g s t h a t the t h e o r e t i c a l v e l o c i t y p r o f i l e does not a c c u r a t e l y p r e d i c t any experimental f i n d i n g s . By a p p l y i n g an a r t i f i c i a l s t a r t i n g p o i n t , the i m p l i c a t i o n i s t h a t the t h e o r e t i c a l p r o f i l e i s a c c u r a t e , but the experiments t o v e r i f y the v e l o c i t y p r o f i l e of E c k e r t and Jackson are not. I t i s acknowledged t h a t the ch o i c e o f the s c a l i n g f a c t o r i s 84 q u i t e a r b i t r a r y , but t h i s seems t o be the most s u i t a b l e means of s c a l i n g the v e l o c i t y p r o f i l e . The maximum d e v i a t i o n between the experimental f i n d i n g s and the m o d i f i e d t h e o r e t i c a l r e s u l t s i s about 20%, which i s w i t h i n the measurement u n c e r t a i n t y of the experiment. In c o n t r a s t , the experiment i s found t o d e v i a t e from the o r i g i n a l t h e o r y by as much as 40%. Included i n t h i s graph i s the t r e n d p r e d i c t e d i f the r a d i a t i o n component of the heat t r a n s f e r i s n e g l e c t e d . In g e n e r a l , the r e s u l t s are s a t i s f a c t o r y : the experimental r e s u l t s are found t o be w i t h i n the measurement u n c e r t a i n t y of the t h e o r e t i c a l N u s s e l t number which i s based on the modifided v e l o c i t y p r o f i l e of E c k e r t and Jackson. Exact agreement cannot be expected because of the s i m p l i f i e d nature of the t h e o r e t i c a l model. Even so, the t h e o r e t i c a l model i s q u i t e u s e f u l i n t h a t i t i s capable of p r e d i c t i n g the beha v i o r o f a complex process with moderate accuracy, u s i n g a model which i s easy t o apply and s o l v e . The agreement between the experiment and the o r i g i n a l t h e o r y proposed i s seen t o be r a t h e r i n a c c u r a t e . The cause of the ina c c u r a c y appears t o be the d i s p a r i t y 85 between the t h e o r e t i c a l and experimental v e l o c i t y p r o f i l e s . 86 V. CONCLUSIONS Turbulent, n a t u r a l c o n v e c t i o n coupled with thermal r a d i a t i o n i n the boundary l a y e r of a v e r t i c a l heated p l a t e has been s t u d i e d t h e o r e t i c a l l y and e x p e r i m e n t a l l y . From t h i s work, the f o l l o w i n g may be cone1uded: 1 . The t h e o r e t i c a l p r e d i c t i o n of t u r b u l e n t n a t u r a l c o n v e c t i o n coupled with thermal r a d i a t i o n u s i n g the mean beam length model based on the e m p i r i c a l work of H o t t e l , and the o p t i c a l l y t h i n model developed by A r p a c i and L i c k are found t o g i v e s i m i l a r r e s u l t s . 2. The e m p i r i c a l work of E c k e r t and Jackson p r e d i c t i n g the t u r b u l e n t n a t u r a l c o n v e c t i o n v e l o c i t y p r o f i l e does not c o r r e l a t e c l o s e l y with t h a t found e x p e r i m e n t a l l y . 3. The peak v e l o c i t y and ar e a under the v e l o c i t y curve were found t o i n c r e a s e with an i n c r e a s e i n the a b s o r p t i v i t y of the working gas, although t h i s d i d not occur m o n o t o n i c a l l y . 87 4. The -temperature p r o f i l e s were found "to agree c l o s e l y with the p r o f i l e p r e d i c t e d by E c k e r t and Jackson. 5. The g r a d i e n t of the temperature p r o f i l e decreased s l i g h t l y with i n c r e a s i n g gas absorpt i v i t y . 6. The t h e o r e t i c a l heat t r a n s f e r r e s u l t s based on the v e l o c i t y p r o f i l e of E c k e r t and Jackson were found t o c o r r e l a t e p o o r l y with the f i n d i n g s of the experiment. 7. By s c a l i n g the t h e o r e t i c a l v e l o c i t y p r o f i l e t o f i t the experimental p r o f i l e , the agreement between the t h e o r y and the experimental r e s u l t s were w i t h i n the experimental u n c e r t a i n t y . 88 VI. RECOMMENDATIONS The f o l l o w i n g r e c o m m e n d a t i o n s may be u s e f u l i f work on t h i s s u b j e c t i s t o c o n t i n u e . 1. I f t h e o r e t i c a l work i s t o c o n t i n u e , a more s o p h i s t i c a t e d v e l o c i t y p r o f i l e s h o u l d be u s e d . The v e l o c i t y p r o f i l e u s e d i n t h i s i n v e s t i g a t i o n seems i n a p r o p r i a t e , and t h e t e c h n i q u e o f s c a l i n g t o match t h e d a t a i s t o o a r b i t r a r y . 2. C o n t i n u e e x p e r i m e n t a l o b s e r v a t i o n s u s i n g a l a r g e r a p p a r a t u s w h i c h i s c a p a b l e o f m a i n t a i n i n g a h i g h e r t e m p e r a t u r e . By i n c r e a s i n g t h e s i z e and o p e r a t i n g t e m p e r a t u r e o f t h e a p p a r a t u s , t h e v e l o c i t y i n t h e b o u n d a r y l a y e r , and t h e r a d i a t i v e component o f t h e h e a t t r a n s f e r w o u l d i n c r e a s e . An i n c r e a s e i n t h e s c a l e o f t h e v e l o c i t y p r o f i l e w o u l d mean t h e p r o f i l e s w ould n o t be b a s e d so h e a v i l y on an e x t r a p o l a t e d c a l i b r a t i o n c u r v e . By i n c r e a s i n g t h e r a d i a t i v e component, t h e r a n g e o f t h e e x p e r i m e n t c o u l d be i n c r e a s e d , f r o m w h i c h t h e l i m i t on t h e a p p l i c a b i l i t y o f t h e o p t i c a l l y t h i n a p p r o x i m a t i o n c o u l d be o b t a i n e d . 89 REFERENCES [1] Chang, L. C , Yang, K. T., L l o y d , J . R., " R a d i a t i o n - N a t u r a l Convection i n Two Dimensional ComplexEnc1osures", J o u r n a l of Heat T r a n s f e r , V o l . 105, 1983, p. 89-95. [2] E c k e r t , E. R. G., Jackson, T. W., " A n a l y s i s of T u r b u l e n t Free-Convection Boundary Layer on F l a t P l a t e " , N.A.C.A. Report 1015, 1951, p. 255-261. [3] Cheesewright, R., "Turbulent N a t u r a l Convection From a Plane S u r f a c e " , J o u r n a l of Heat T r a n s f e r , V o l . 90, 1968, p. 1-8. [4] Cheesewright, R., I e r o k i o p i t i s , E., " V e l o c i t y Measurements i n a T u r b u l e n t N a t u r a l Convection Boundary Layer", Proc. 7th I n t . Heat T r a n s f e r Conf., ed. G r i g u l l , U . , Hahne, E., Stephan, K., V o l . 2, 1982, p. 305-309. [5] Cheesewright, R., Doan, K. S., "Space-Time C o r r e l a t i o n Measurements i n a T u r b u l e n t N a t u r a l Convection Boundary Layer", I n t . J . Heat Mass T r a n s f e r , V o l . 21, 1978, p. 911-921. [6] Fuj i i , T. , Takeuch i , M. , Fuj i i , M. , S u z a k i , K. , Uehara, H., "Experiments on N a t u r a l Convection Heat T r a n s f e r From the Outer Surface of a V e r t i c a l C y l i n d e r t o L i q u i d s " , I n t . J . Heat Mass T r a n s f e r , V o l . 13, 1970, p. 753-787. [7] Miyamoto, M., K a j i n o , H., Kurima, J . , Takanami, 90 I., "Development of Turbulence C h a r a c t e r i s t i c s i n a V e r t i c a l Free Convection Boundary Layer", Proc. 7th I n t . Heat T r a n s f e r Conf., ed. G r i g u l l , U., Hahne, E., Stephan, K., V o l . 2, 1982, p. 323 -328. [8] Hoogendoorn, C. J . , Euser, H., " V e l o c i t y P r o f i l e s i n the T u r b u l e n t Free Convection Boundary Layer", Proc. 6th I n t . Heat T r a n s f e r Conf., Hemisphere, Washington, V o l . 3, 1974, p. 193-197. [9] George, W. K., Capp, S. P., "A Theory f o r N a t u r a l Convection T u r b u l e n t Boundary Layers Next t o Heated V e r t i c a l S u r f a c e s " , I n t . J . Heat Mass T r a n s f e r , V o l . 22, 1979, p. 813-826. [10] K u t a t e l a d z e , S. S., "The Model of T u r b u l e n t Free Convection Near a V e r t i c a l Heat T r a n s f e r S u r f a c e " Heat T r a n s f e r and Turbu1ent Buoyant Convect i o n . ed. S p a l d i n g , D. B., Afgan, N., Hemisphere, New York, 1977, p. 488-495. [ l l ] K a t o , H., N i s h i w a k i , N., H i r a t a , M., "On the T u r b u l e n t Heat T r a n s f e r by Free Convection From a V e r t i c a l P l a t e " , I n t . J . Heat Mass T r a n s f e r , V o l . 11, 1968, p. 1117-1125. [12] Mason, H. B., Seban, R. A., "Numerical P r e d i c t i o n s f o r T u r b u l e n t Free Convection From V e r t i c a l S u r f a c e s " , I n t . J . Heat Mass T r a n s f e r , V o l . 17, 1974, p. 1329-1336. 91 [13] Sparrow, E. M. , Cess, R. D. , R a d i a t i o n Heat T r a n s f e r Hemisphere, Washington, 1978. [ 1 4 ] S i e g e l , R. , Howel1, J . R. , Thermal R a d i a t i o n Heat  T r a n s f e r . 2nd ed., McGraw-Hill, New York, 1981. [15] Mcadams, W. H., Heat T r a n s m i s s i o n . McGraw-Hill, New York, 1964. [ 1 6 ] H o t t e l , H. C , Mangelsdorf, H. G., "Heat T r a n s m i s s i o n From Nonluminous Gases I I . Experimental Study of Carbon D i o x i d e and Water Vapour", Trans. Am. Inst.Chem. Engrs., V o l . 38, 1942, p. 531-565 [ 1 7 ] H o t t e l , H. C , Smith, V. C , " R a d i a t i o n From Nonluminous Flames", Trans. Am. Soc. Mech. Engrs., V o l . 57, 1935, p. 463-470. [18] Howell, J . R. , "Thermal R a d i a t i o n i n P a r t i c i p a t i n g Media: The Past, The Present, and Some P o s s i b l e F u t u r e s " , ASME Jou r n a l of Heat T r a n s f e r , V o l . 110, 1988, p. 1220-1229. [19] V i s k a n t a , R., "R a d i a t i o n Heat T r a n s f e r : I n t e r a c t ion With Conduction and Convectionand Approximate Methods i n R a d i a t i o n " , Proc.7th I n t . Heat T r a n s f e r Conf.4, ed. G r i g u l l , U., Hahne, E., Stephan, K., Vo l . 1 , 1982, p. 103-122. [20] L i c k , W. , "Energy T r a n s f e r by R a d i a t i o n and Conduction", Proc. Heat T r a n s f e r F l u i d Mech. I n s t . S t a n f o r d U n i v e r s i t y P r e s s , S t a n f o r d , 1963, p. 14-26. 92 [21] Cess, R. D., "The I n t e r a c t i o n of Thermal R a d i a t i o n With Free Convection Heat T r a n s f e r " , I n t . J . Heat Mass T r a n s f e r , V o l . 9, 1966, p. 1269-1277. [22] Cess, R. D. , " R a d i a t i o n E f f e c t s Upon Boundary Layer Flow of an Absorbing Gas", ASME Jo u r n a l of Heat T r a n s f e r , V o l . 86, 1964, p. 469-475. [23] A r p a c i , V. S., " E f f e c t of Thermal R a d i a t i o n on the Laminar Free Convection From a Heated V e r t i c a l P l a t e " , I n t . J . Heat Mass T r a n s f e r , V o l . 11, 1968, p. 871-880. [24]England, W. G., Emery, A. F., "Thermal R a d i a t i o n E f f e c t s on the Laminar Free Convection Boundary Layer of an Absorbing Gas", ASME Jo u r n a l of Heat T r a n s f e r , V o l . 91, 1969, p. 37-44. [25]Cheng, E. H., O z i s i k , M. N., "R a d i a t i o n With Free Convection i n an Absorbing, E m i t t i n g and S c a t t e r i n g Medium", Int. J . Heat Mass T r a n s f e r , V o l . 15, 1972, p. 1243-1252. [2 6 ] E l s a y e d M. M., F a t h a l a h , K. A., "Natural Convection Over a V e r t i c a l F l a t P l a t e due t o Ab s o r p t i o n of Thermal R a d i a t i o n " , Warme-und S t o f f u b e r t r a g u n g , V o l . 19, 1985, p. 121-129. [27] Audunson, T., Gebhart, B., "An Experimental and A n a l y t i c a l study of N a t u r a l Convection with A p p r e c i a b l e Thermal R a d i a t i o n E f f e c t s " , J . F l u i d Mech., V o l . 52, 1972, p. 57-95. 93 [28] Yang, K. T., "Numerical Modeling of N a t u r a l C o n v e c t i o n - R a d i a t i o n I n t e r a c t i o n s i n E n c l o s u r e s Proc. 8th I n t . Heat T r a n s f e r Conf., ed. T i e n , C. L., Carey, V. P., F e r r e l , J . K., V o l . 1, 1986, p. 103-121. [29] Naidenov, V. I., S h i n d i n , S. A., " I n t e r a c t i o n s of R a d i a t i o n With T u r b u l e n t F l u c t u a t i o n s i n a Boundary Layer", High Temperature, V o l . 19, 1981, p. 106-109. [30] Cheung, F. B., Chan, S. H., Chawla, T. C , Cho, D. H. " R a d i a t i v e Heat T r a n s f e r i n a Heat Ge n e r a t i n g and T u r b u l e n t Convecting F l u i d Layer", I n t . J . Heat Mass T r a n s f e r " , V o l . 23, 1980, p. 1313-1324. [ 3 1 ] S i e b e r s , D. L., Moffat, R. F., Schwind, R. G., "Experimental, V a r i a b l e P r o p e r t i e s N a t u r a l Convection From a Large, V e r t i c a l , F l a t S u r f a c e " ASME Jo u r n a l of Heat T r a n s f e r , V o l . 107, 1985, p. 124-132. [32]Gebhart, B . , " E f f e c t s of V i s c o u s D i s s i p a t i o n i n N a t u r a l Convection", J . F l u i d Mech., V o l . 14, 1962, p. 225-232. [33]Holman, J . P., Heat T r a n s f e r 5th ed. McGraw-Hill, New York, 1981. [ 3 4 ] V i n c e t t i , W. G., Baldwin, B. S., " E f f e c t of R a d i a t i o n on the Propagation of Plane A c c o u s t i c Waves", J . F l u i d Mech. V o l . 12, 1961, p. 449-477. 94 [35] Steward, F. R. , Kocaefe, Y. S., " T o t a l E m i s s i v i t y and A b s o r p t i v i t y Models For Carbon D i o x i d e , Water Vapour and T h e i r M i x t u r e s " , Proc. 8th I n t . Heat T r a n s f e r Conf., ed. T i e n , C. L., Carey, V. P., F e r r e l , J . K., V o l . 3, 1986, p. 735-740. [36]Farag, I. H., "Nonluminous Gas R a d i a t i o n : Approximate E m i s s i v i t y Models, Proc. 7th I n t . Heat T r a n s f e r Conf., ed. G r i g u l l , U., Hahne, E., Stephan, K., Vol.2, 1982, p. 487-492. [37] Sparrow, E. M., " E r r o r E s t i m a t e s i n Temperature Measurements" , Measurements i n Heat T r a n s f e r . 2nd. ed., ed. E c k e r t , E. R. G., G o l d s t e i n , R. J . , Hemisphere, Washington, 1976, p. 6-13. [38] C o l l i s , D. C , W i l l i a m s , M. J . , "Two Dimensional Convection From Heated Wires at Low Reynolds Numbers", J . F l u i d Mech., V o l . 6, 1959, p. 357-384. [39]Davies, P. 0. A., F i s h e r , M. J . , "Heat T r a n s f e r From E l e c t r i c a l l y Heated C y l i n d e r s " , Proc. Royal S o c i e t y , London, S e r i e s A, V o l . 280, 1964, p. 486-527. [40] Reynolds, A. J . , T u r b u l e n t f l o w s i n E n g i n e e r i n g . Wiley, New York, 1974. [41]Hinze, J . •., Turbulence. 2nd. ed. McGraw-Hill New York, 1975. [42] Bradshaw, P. , Johnson, R. F. , "Turbulence Measurements With Hot Wire Anemometers, Notes 95 on A p p l i e d Science No.33, N a t i o n a l P h y s i c a l L a b o r a t o r y , 1963. [43] Freyrauth, P., A B i b l i o g r a p h y of Thermal Anemometry, TSI, St Pa u l , 1981. 96 F i g u r e 1 0 . : P l a t e C o n s t r u c t i o n . | J / I ^ r I see deto.1 D — JJ _J \ ff - 3 7 — — J c o p p e r coo l ing c o t shett c e t a l duct cooling voter out square to circle adapter sheet rwtol hood hole fo r anenoneter / — see de-tan C see detail B dlnenslons m Inches alumnun f r a n e Qlunmun base 7 2 5 F i g u r e 11.: E n c l o s u r e C o n s t r u c t i o n DETAIL A PLATE THERMOCOUPLE ASSEMBLY T.I.G. weld DETAIL B FRONT PANEL S E A L 6*32 machine screw .125 la acrylic wing nut gasket DETAIL C SIDE PANEL JOINT type-k thermocouple wire 6*32 machine screw on 6 in. cen te r r- 0.125 In. acrylic lin. alunlnun angle bracket seal with silicone dimensions in inches F i g u r e 1 2 . : D e t a i l s o f E n c l o s u r e C o n s t r u c t i o n . DETAIL D HOT WIRE MOUNTING 6*32 machine screw h o i wire anemometer dial gage dimensions in inches gure 13.: D e t a i l of Hot Wire Mounting. 110 9.7 9.75 9.8 9.85 9.9 9.95 10 10.05 10.1 10.15 10.2 10.25 10. Resistance F i g u r e 15.: Temperature C a l i b r a t i o n Curve. Appendix C; Sample c a l c u l a t i o n A r e p r e s e n t a t i v e s e t of c a l c u l a t i o n s i s presented t o i l l u s t r a t e the c o n v e r s i o n of the raw d a t a i n t o the q u a n t i t i e s of i n t e r e s t . To o b t a i n the temperature i n the boundary l a y e r from the r e s i s t a n c e of the hot wire sensor, the r e l a t i o n R=R c[l+w(T-T c)] C . l i s a p p l i e d where: Rc = 8.47ft, w = 0.0009oC-1, and T C = 24.7°C. For a b r i d g e r e s i s t a n c e , R4, the sensor r e s i s t a n c e R, i s o b t a i n e d from the equation R=Rj — R C o K e — Rproie > C.2 where the cable and probe r e s i s t a n c e s were measured, and found t o have v a l u e s of 0.62ft and 0.58ft r e s p e c t i v e l y . I f f o r example, the b r i d g e r e s i s t a n c e was 11.00ft, the sensor r e s i s t a n c e would be 11.00ft - 0.620 - 0.58ft = 9.80ft , 103 from which "the "temperature i s "found from 9.800 ± — - 4 7 Q . -i+ 24.7°C=199 0C. 0.0009°CT From "the measurement u n c e r t a i n t y a n a l y s i s of chapter 4, the u n c e r t a i n t y i n t h i s temperature i s 1.5%, which corresponds t o about 3°C. The v e l o c i t y i s obtained from the equation E?R <- -- 0- 1 7 (R+R 3) 3lirk(T i e-T /) ( f e ) = A + B(HJi) , C.3 where T m = 2—^. C.4 The r e s i s t a n c e R, was s e t t o 12.86Q f o r the v e l o c i t y measurements, and R3 i s the r e s i s t a n c e on the f i x e d l e g of the b r i d g e , having a value of 40f2. The sensor length 1 and sensor diameter d, are 0.00125 m. and 6.3 nm r e s p e c t i v e l y . The thermal c o n d u c t i v i t y k, and the kinematic v i s c o s i t y v were eva l u a t e d at T m . From the v e l o c i t y c a l i b r a t i o n curve, the c o n s t a n t s were found t o be A= 0.333, B = 0.297, and n = 0.45. For a r e s i s t a n c e R=12.86fi, the sensor temperature Tu, i s 104 12.860 ± 8.470 24. TC = 600°C = 873 K 0.0009°C F o r a f l u i d t e m p e r a t u r e o f 199°C, T m = 673 K, f r o m w h i c h , f o r example, t h e v i s c o s i t y , o f a c a r b o n d i o x i d e , n i t r o g e n g a s m i x t u r e i s f o u n d f r o m t h e e q u a t i o n y=xi/ n +(1— xWo , C.4 2 v ' 2 where x i s t h e mass p r o p o r t i o n o f t h e n i t r o g e n g a s . F o r a m i x t u r e w i t h 50% o f e a c h component, t h e p r o p e r t i e s a r e : p= 1.079 Kg/m 3, v- 49.52 m 2 / s e c , k= 0.0471 W/m-K, C P = 1102 J/Kg-°C, and P r = 0.715. F o r a v o l t a g e o f 2.78 v o l t s , e q a t i o n C.3 g i v e s (2.78 v o l t s ) 2 ( 12.860) / 6 7 3 K > (12.860 + 400) 2 (0.00125 m) TT (0.0471 W/mK) (873K-472K) \ 472 K , 0.333 + 0.297 ( <"> ( 6-3xlQ - « m ) ^ V49.52xl0- 6m 2/sec/ f r o m w h i c h t h e v e l o c i t y u i s p r e d i c t e d t o be 1.03 105 m/sec. The measurement u n c e r t a i n t y i s 15%, or 0.154 m/sec. 106 

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