INFORMATION TO USERS

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1 INFORMATION TO USERS This material was produced from a microfilm copy of the original docum ent. While the m ost advanced technological means to photograph and reproduce this docum ent have been used, the quality is heavily dependent upon the quality of the original subm itted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.T h e sign or "target for pages apparently lacking from the docum ent photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you com plete continuity. 2. When an image on the film is obliterated w ith a large round black mark, it is an indication th at the photographer suspected th at the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite m ethod in "sectioning" the material. It is custom ary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections w ith a small overlap. If necessary, sectioning is continued again beginning below the first row and continuing on until com plete. 4. The majority of users indicate th at the textual content is of greatest value, however, a som ew hat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the O rder D epartm ent, giving the catalog num ber, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: -Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 N orth Z eeb R oad Ann Arbor, M ichigan 48106

2 76-10,044 SHAPIRO, Howard Neal, SIMULTANEOUS HEAT AND MASS TRANSFER IN POROUS MEDIA WITH APPLICATION TO SOIL WARMING WITH POWER PLANT WASTE HEAT. The Ohio State U n iv ersity, Ph.D., 1975 Engineering, mechanical Xerox University Microfilms, Ann Arbor, Michigan C o p y rig h t by- Howard N eal S h a p iro 1975 THIS DISSERTATION HAS BEEN M ICROFILM ED EXACTLY AS RECEIVED.

3 SIMULTANEOUS HEAT AND MASS TRANSFER IN POROUS MEDIA WITH APPLICATION TO SOIL WARMING WITH POWER PLANT WASTE HEAT DISSERTATION P re s e n te d i n P a r t i a l F u lf i llm e n t o f th e R equirem ents f o r th e D egree D o cto r o f P h ilo s o p h y i n th e G raduate S chool o f th e Ohio S ta te U n iv e r s ity By Howard N eal S h a p iro, B.S», M.Sc. * * * * * The Ohio S t a t e U n iv e r s ity 1975 R ead in g C om m ittees A pproved By M ich ael J. M oran, Chairm an C h a rle s D. Jo n e s Seppo A. K o rp e la W arren L. R o lle r / I A d v is e r D ep& rlpent o f M ech an ical I J e n g in e e rin g

4 ACKNOWLEDGMENTS I would f i r s t l i k e t o acknow ledge my a d v is o r, P r o f e s s o r M ich ael J. M oran. H is p e r s o n a l s ta n d a r d s f o r e x c e lle n c e w i l l alw ays he an i n s p i r a t i o n to me. C e r t a in l y, t h i s work i s t h a t much b e t t e r f o r h i s e f f o r t s ; alw ays s t r i v i n g to b r in g o u t th e b e s t i n me. Too, he h as been my te a c h e r, my c o -w o rk e r, and my f r i e n d. I am p ro u d o f t h i s a s s o c i a t i o n. N ex t, I w ould l i k e to th a n k P r o f e s s o r W arren L. R o lle r f o r th e o p p o r tu n ity to w ork on th e p r o j e c t upon w hich t h i s d i s s e r t a t i o n i s b a s e d. I would a l s o l i k e to e x p re s s my a p p r e c ia tio n f o r th e e f f o r t s t h a t he h as made on my b e h a lf. I g r a t e f u l l y acknow ledge th e s u p p o rt o f th e Ohio A g r i c u l t u r a l R e se a rc h and D evelopm ent C e n te r. I w ish to th a n k th e s p o n s o rs, th e A m erican E l e c t r i c Power Company, f o r fu n d in g th e p r o j e c t. I n a d d i tio n, I th a n k th e f a c u l t y, s t a f f, and s tu d e n ts o f th e M ech an ical E n g in e e rin g D epartm ent a t Ohio S ta te f o r a m o st m e a n in g fu l l e a r n in g e x p e rie n c e. F i n a l l y, to my w ife, L ee, who h as s a c r i f i c e d f a r more th a n I in o rd e r t h a t I co m p lete t h i s d i s s e r t a t i o n, I g iv e my lo v e.

5 VITA May 13, 19^ Born - C le v e la n d, Ohio B.S., The Ohio S t a t e U n iv e r s ity R e se a rc h A s s o c ia te, D ep artm en t o f M ech an ical E n g in e e rin g, The Ohio S t a t e U n iv e r s ity, Colum bus, Ohio 1971 M.S c., The Ohio S t a t e U n iv e r s ity, C olum bus, Ohio T each in g A s s o c ia te, D ep artm en t o f M ech an ical E n g in e e rin g, The Ohio S ta te U n iv e r s ity, Colum bus, Ohio T e c h n ic a l A s s i s t a n t, The Ohio A g r i c u l t u r a l R e se a rc h an d D ev elopm ent C e n te r, W ooster, Ohio A s s i s t a n t P r o f e s s o r, Iowa S ta te U n iv e r s ity, Ames, Iowa PUBLICATIONS "A V a ria b le Speed V -b e lt T ra n sm issio n w ith an A sy m m etrical B e l t," T ra n s. ASME, J. E ng. I n d., A u g u st, 1973* "D esig n E q u a tio n s f o r a Speed and Torque C o n tr o lle d V a r i a b le R a tio V -b e lt D riv e," T ra n s. SAE, P a p e r No »1973- FIELDS OF STUDY M ajor F i e l d : M ech an ical E n g in e e rin g S tu d ie s in Therm al S c ie n c e s. P r o f e s s o r M ic h ael J. Moran S tu d ie s in M achine D esig n. P r o f e s s o r s K en n eth G. Hornung and Ja ck C o llin s i i i

6 TABLE OF CONTENTS ACKNOWLEDGMENTS... V IT A... LIST OF TABLES... LIST OF FIGURES NOMENCLATURE... Page i i i i i v i i i i x x i i C h a p te r 1. INTRODUCTION B ackground The T ra n s p o rt P ro b lem The D esig n P roblem P re v io u s S o il Warming S t u d i e s Time a s an In d e p e n d e n t V a r ia b le ^1* C lo s u re. c o o.. o o o.. g. o CONSTANT PROPERTY MODEL FOR SOIL TEMPERATURE I n tr o d u c tio n The C o n s ta n t P r o p e r ty M odel ^ P r e s e n ta tio n o f G ra p h ic a l R e s u l t s..... i c. t Q u a lita t iv e O b se rv a tio n s i E v a lu a tio n o f th e M odel... 33

7 C h a p te r 3. EQUATIONS FOR HEAT AND MASS TRANSPORT IN POROUS MEDIA I n t r o d u c t i o n * 1.1 N a tu re o f th e P ro b le m Volume A v e ra g in g The C o n tin u ity E q u a tio n A The E n erg y Equation C o n s t i t u t i v e R e la tio n s h ip s * 5.1 P r e l i m i n a r i e s C o n s t i t u t i v e R e la t io n s h ip s S te a d y - S ta t e Form s o f th e G o v ern in g E q u a tio n s C lo s u r e. A. EVALUATION OF THE TRANSPORT COEFFICIENTS FOR S O I L A.l I n t r o d u c t i o n A.2 E v a lu a tio n o f th e C o e f f i c i e n t s.... A.2.1 E v a lu a tio n o f th e ' s fro m T heory A.2.2 E m p ir ic a l E v a lu a tio n o f th e ' s. A.2.3 E v a lu a tio n o f th e K 's.... A.3 V a lid a tio n o f th e C o e f f i c i e n t s.... A.A Summary...

8 v i C h a p te r Page 5. VARIABLE PROPERTY MODEL FOR TEMPERATURE AND MOISTURE I n t r o d u c t i o n D e s c r ip tio n o f th e N u m erical P roblem. 8^ P r e l i m i n a r i e s... 8^ The F i n i t e D iffe re n c e F o rm u la tio n E v a lu a tio n o f S o lu tio n T e c h n iq u e s ^ B oundary C o n d itio n s f o r S o i l W arming S o lu tio n o f th e N u m erical P ro b lem T e s tin g th e S o l u t i o n P r e s e n ta tio n o f Results Q u a lita t iv e O b s e rv a tio n s ^ S ig n if ic a n c e o f th e R e s u lts I l l 6. DISCUSSION OF SOIL WARMING... Ilk 6.1 Review o f O b je c tiv e s Ilk 6.2 M eetin g th e O b je c tiv e s C lo s u r e SUMMARY AND RECOMMENDATIONS FOR FURTHER STUDY Summary The T ra n s p o rt P roblem The D esig n P r o b l e m R ecom m endations f o r F u r th e r S tu d y C lo s in g C,omment ^

9 Page FOOTNOTES APPENDIX A. SOME USEFUL THEOREMS B. DERIVATION OF FINITE DIFFERENCE EQUATIONS. lao C. COMPUTER SOLUTION OF THE FINITE DIFFERENCE EQUATIONS BIBLIOGRAPHY... 16k

10 LIST OF TABLES T able 1 B e s t Range f o r M eetin g : A g r i c u l tu r a l and Power P l a n t O b je c tiv e s 33 T able 2 C a lc u la tio n o f 72 T able 3 P a ra m e tric S tu d y o f th e M o istu re C o n te n t V a r ia tio n 106 T ab le I* M o istu re L e v e ls f o r Optimum Y ie ld o f S e v e ra l C rops i n S i l t Loam S o il 110 Page v i i i

11 LIST OF FIGURES F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re F ig u re 1.1 ) I n c r e a s e d y i e l d f o r p l a n t s i n n u t r i e n t b a th s a t d i f f e r e n t te m p e r a tu r e s. 2.1 ) L ay o u t o f a s o i l w arm ing sy ste m. 2.2 ) S o il w arm ing m odel. 2.3 ) V alu es o f th e r a d i c a l A 2-B 2 a s a f u n c tio n o f s and r. 2.4 ) D im e n sio n le ss w a te r te m p e ra tu re drop a s a f u n c tio n o f th e la y o u t p a r a m e te rs. 2.5 ) R oot sy stem s o f s e v e r a l c r o p s. 2.6 ) E x p e rim e n ta l v e r t i c a l te m p e ra tu re p r o f i l e a b o u t a b u rie d h e a t s o u rc e ) A x ia l v a r i a t i o n o f d im e n s io n le s s r o o t zone te m p e ra tu re 2.8 ) D im e n sio n le ss s o i l te m p e ra tu re a s a f u n c tio n o f th e la y o u t p a r a m e te rs. 2.9 ) D im e n sio n le ss s o i l te m p e ra tu re a s a f u n c tio n o f th e la y o u t p a r a m e te rs ) D im e n sio n le ss s o i l te m p e ra tu re a s a f u n c tio n o f th e la y o u t p a r a m e te rs ) C om parison o f c o n s ta n t p r o p e r ty m odel w ith e x p e rim e n ta l d a t a. 3.1 ) Volume e le m e n t i n a p o ro u s medium. 4.1 ) T h e o r e tic a l t r a n s p o r t c o e f f i c i e n t s f o r P a lo u s e s i l t loam. P ags IX

12 F ig u re (4.2 F ig u re (4.3 F ig u re ( 4.4 F ig u re (4.5 F ig u re (4.6 F ig u re (4.7 F ig u re (4.8 F ig u re (5»1 F ig u re (5.2 F ig u re (5-3 F ig u re (5*4 F ig u re (5-5 F ig u re (5*6 F ig u re (5-7 E x p e rim e n ta l and t h e o r e t i c a l tem p e r a t u r e and m o is tu re p r o f i l e s i n a o n e -d im e n sio n a l sam p le. T ra n s p o rt c o e f f i c i e n t s f o r m o is tu re in d u c e d flo w. T ra n s p o rt c o e f f i c i e n t s f o r te m p e ra t u r e in d u c ed flo w. Therm al c o n d u c tiv ity f o r s i l t loam s o i l a t 25 C. C a lc u la te d o n e -d im e n sio n a l te m p e ra tu r e and m o is tu re p r o f i l e s. C a lc u la te d o n e -d im e n sio n a l te m p e ra tu r e and m o is tu re p r o f i l e s. C a lc u la te d o n e -d im e n sio n a l te m p e ra tu r e and m o is tu re p r o f i l e s. S o il r e g io n f o r n u m e ric a l c a l c u l a t i o n s. V a ria b le p r o p e r ty m odel te m p e ra tu re s o l u t i o n f o r "uniform m o is tu re. V a ria b le p r o p e r ty m odel te m p e ra tu re s o l u t i o n. E x p e rim e n ta l v e r t i c a l e q u ilib r iu m m o is tu re c o n te n t a b o u t a b u r ie d p ip e sy stem. V a r ia b le p r o p e r ty m odel m o is tu re c o n te n t s o lu t io n f o r 0^= V a ria b le p r o p e r ty m odel m o is tu re c o n te n t s o l u t io n f o r 0^= V a ria b le p r o p e r ty m odel m o is tu re c o n te n t s o l u t i o n f o r 0^= Page

13 F ig u re ( 6. 1 ) Corn f i e l d w ith p ip e s i n p la c e. P age 118 F ig u re ( 6. 2 ) P o s s ib le s o lu t io n s f o r d im e n sio n - l e s s s o i l te m p e ra tu re. 119 F ig u re ( 6.3 ) P o s s ib le s o lu t i o n s f o r d im e n sio n l e s s v /a te r te m p e ra tu re d r o p F ig u re (B l) D iagram o f c a l c u l a t i o n a l g r i d. 1^7

14 NOMENCLATURE s c a l a r, v e c t o r, o r s e c o n d -o rd e r te n s o r q u a n t i t y a s s o c ia te d w ith th e i t h p h a se i n V s p e c i f i c h e a t o f w a te r (c a l/g m - C ) d e p th o f th e p ip e s (cm) c o e f f i c i e n t f o r te m p e ra tu re in d u c e d m o is tu re flow- (cm 2/day- C ) t r a n s p o r t c o e f f i c i e n t f o r te m p e ra tu re in d u c e d flo w (cm2/d a y - C ) c o e f f i c i e n t f o r m o is tu re in d u c e d m o is tu re flo w (cm2/d a y ) c o e f f i c i e n t f o r m o is tu re in d u c e d flo w (cm2/d a y ) e n th a lp y p e r u n i t m ass (c a l/g m ) d im e n s io n le s s v o lu m e tric f l u x v e r t i c a l u n i t v e c to r c o e f f i c i e n t f o r te m p e ra tu re in d u c e d e n e rg y flo w (cm2/d a y - C ) c o e f f i c i e n t f o r m o is tu re in d u c e d e n e rg y flo w (cm2/d a y - C ) le n g th o f th e p ip e s (Km) m ass flo w r a t e i n th e p ip e s (g m /s e c ) u n i t n o rm al v e c to r 2 c o n d u c tio n h e a t f l u x v e c to r (c a l/c m -d a y )

15 x i i i r R s s S Se.S 1,Sje fsw,s y t T d im e n s io n le s s p ip e r a d iu s p ip e r a d iu s (cm) d im e n s io n le s s p ip e s p a c in g pip s p a c in g (cm) s u r f a c e o f V p a r t i t i o n s o f S tim e s o i l te m p e ra tu re ( C) T j w a te r te m p e ra tu re a t sy stem i n l e t ( C ) T s s o i l s u r f a c e te m p e ra tu re ( C ) Tw w a te r te m p e ra tu re ( C) T^,Tg d im e n s io n le s s w a te r te m p e ra tu re s u i n t e r n a l e n e rg y p e r u n i t m ass (c a l/g m ) v V v e l o c i t y (cm /sec) a v e ra g in g volume volum e o f th e i t h p h ase V,J, x,y,z xfy,z e q u iv a le n t v a p o r volum e p o s i t i o n c o o r d in a te s d im e n s io n le s s p o s i t i o n c o o r d in a te s G reek Sym bols d im e n s io n le s s la y o u t p a ra m e te r v o lu m e tric m o is tu re c o n te n t

16 x iv <6> a v e ra g e r o o t zone m o is tu re c o n te n t 6 v o lu m e tric l i q u i d c o n te n t 9 e q u iv a le n t v o lu m e tric v a p o r c o n te n t * h y d r a u llic c o n d u c tiv ity (c m /sec) * th e rm a l c o n d u c tiv i ty (m c a l/c m -se c - C ) v e l o c i t y o f liq u id - v a p o r i n t e r f a c e P d e n s ity (gm/cm^) $ d im e n s io n le s s s o i l te m p e ra tu re << > d im e n s io n le s s m id f ie ld a v e ra g e r o o t zone c te m p e ra tu re S u b s c r ip ts JL v s T l i q u i d p h a se v a p o r p h a se s o l i d p h a se te m p e ra tu re in d u c ed 0 m o is tu re in d u c e d b (i»j) lo w e r b o u n d ary n o d a l p o i n t in d ic e s

17 CHAPTER 1 INTRODUCTION 1.1 B ackground W ith m odern g e n e r a tin g p l a n t s em ploying ste am pow er c y c le s, a s i g n i f i c a n t f r a c t i o n o f th e e n e rg y r e le a s e d from th e f u e l i s u l t i m a t e l y r e j e c t e d to th e en v io rn m e n t by h e a t t r a n s f e r th ro u g h th e c o n d e n s e rs, A re v ie w o f p e r t i n e n t l i t e r a t u r e r e v e a l s t h a t many schem es have b ee n p ro p o se d f o r u t i l i z i n g unused th e rm a l e n e rg y. A t p r e s e n t th e co n s e n s u s i s t h a t few, i f an y, o f th e s e id e a s a p p e a r to be e c o n o m ic a lly d e s i r a b l e. B u t, a s e n e rg y s u p p lie s d w in d le, u s e s f o r w aste h e a t m ust n e c e s s a r i l y become more a t t r a c t i v e. The a w a re n e ss t h a t w aste h e a t i t s e l f may be a v a lu a b le r e s o u r c e i s r e l a t i v e l y r e c e n t and i s b y no means w id e s p re a d. In th e p a s t th e c o n c e p t h a s b een d is c u s s e d a s, a t m o st, an i n t e r e s t i n g s i d e l i g h t when c o n s id e r in g th e p ro b lem o f how to d is p o s e o f la r g e q u a n t i t i e s o f warm c o n d e n se r w a te r.

18 2 I t i s n o t s u r p r i s i n g t h a t th e u se o f th e rm a l e f f l u e n t h a s b een g iv e n su ch c u r s o r y c o n s id e r a tio n. The te m p era t u r e r i s e th ro u g h pow er p l a n t c o n d e n se rs i s i n th e 5 "to 15 C r a n g e. T h is r e p r e s e n t s a m inim al th e rm a l p o t e n t i a l, r e q u i r i n g a la r g e s u r f a c e a r e a to a f f e c t much e n e rg y. t r a n s f e r. W ith c o o lin g w a te r flo w s on th e o rd e r o f 500,000 g a llo n s p e r m in u te f o r a la r g e n u c le a r f a c i l i t y, th e r e a p p e a rs to be a g ra v e c a p a c ity m ism atch b etw een th e e n e rg y a v a i l a b l e i n c o o lin g w a te r and sy stem s t h a t m ig h t e x p l o i t a s i g n i f i c a n t p o r t i o n o f th e e n e rg y. Many have c o n c lu d e d, t h e r e f o r e, t h a t th e r e i s l i t t l e hope o f f in d in g a p r a c t i c a l u se f o r t h i s e n e rg y. These p o i n t s n o tw ith s ta n d in g, th e p o t e n t i a l v a lu e o f an y r e a l i s t i c and e f f i c i e n t u se f o r th e e n e rg y i n c o n d e n se r c o o lin g w a te r i s to o g r e a t to r e j e c t th e id e a a l t o g e t h e r. In d e e d, th e r e a r e a num ber o f i n v e s t i g a t i o n s g o in g fo rw a rd, in v o lv in g c o n s id e r a b le in v e s tm e n ts o f tim e and money, w ith th e g o a l o f d e v e lo p in g w o rk ab le a l t e r n a t i v e s ; s e e f o r exam ple [ 2 ] and [36]. M oreover, t h i s d i s s e r t a t i o n d e s c r i b e s an i n v e s t i g a t i o n aim ed a t s tu d y in g th e s im u lta n e o u s t r a n s f e r o f e n e rg y and m ass in p o ro u s m edia w ith a p p l i c a t i o n to a b u r ie d p ip e s o i l warm ing sy stem f o r a g r i c u l t u r e and f o r pow er p l a n t c o o lin g. The p rim a ry o b je c tiv e o f th e p r e s e n t s tu d y i s to d e te rm in e i f w aste h e a t can be e f f e c

19 3 t i v e l y e x p lo ite d to enhance a g r i c u l t u r a l p r o d u c tio n by s o i l warm ing an d, i f s o, can c o n s e n s e r w a te r be s i g n i f i c a n t l y c o o le d i n th e p r o c e s s. T here a r e two im p o rta n t a s p e c ts to th e s tu d y. One in v o lv e s m o d e llin g th e s im u lta n e o u s t r a n s p o r t o f h e a t and m o is tu re i n a p o ro u s medium. The o th e r c e n te r s upon d i s c u s s in g th e p o s s i b i l i t y o f d e te rm in in g sy stem d e s ig n s t h a t can m eet a g r i c u l t u r a l a n d /o r pow er p l a n t n e e d s. Each o f th e s e a s p e c ts w i l l now be d is c u s s e d b r i e f l y The T ra n s p o rt P ro b le m. H e r e to f o r e, d is c u s s io n s o f s o i l w arm ing have c e n te r e d upon th e am ount o f h e a t d i s s i p a te d by a g iv e n sy stem. I t h a s b een a ssum ed t h a t i f a s u f f i c i e n t q u a n t i t y o f e n e rg y i s d i s s i p a t e d to s o i l, th e th e rm a l en v io rn m e n t f o r a g r i c u l t u r e m ust n e c e s s a r i l y be en h an ced. P re lim in a r y i n v e s t i g a t i o n s i n t o s o i l warm ing have shown t h a t t h i s n eed n o t be th e c a s e. H eat d i s s i p a t i o n a lo n e i s n o t an a d e q u a te m easure o f th e o v e r a l l p erfo rm a n ce o f a s o i l warm ing sy stem, b e c a u se w h e th e r s o i l warm ing i s u se d a t a l l i s l i k e l y to depend upon th e a g r i c u l t u r a l p a y o f f. F i e l d c ro p s do re sp o n d f a v o r a b ly to r e l a t i v e l y s m a ll in c r e a s e s i n r o o t tem p era t u r e [2k] [35 1» a s e v id e n c e d by th e d a ta i n F ig u re ( 1.1 ) ;

20 YIELD - mg/plant 1000 BUSH BEANS SOYBEANS CORN 500 COTTON SOIL TEMPERATURE F i g u r e d cl) I n c re s e d y i e l d f o r p l a n t s i n n u t r i e n t b a th s a t d i f f e r e n t tem p era t u r e s 0 3 -

21 5 Tout, i n a d d i t i o n, s o i l m o is tu re h a s an im p o r ta n t in f lu e n c e on c ro p g ro w th. U n fo rtu n a te ly, th e p re s e n c e o f warm b u r ie d p ip e s c a u s e s m o is tu re to m ig ra te from h ig h te m p e ra tu re to low te m p e ra tu re r e g io n s w ith th e r e s u l t t h a t a d ry c o re c a n d e v e lo p i n th e n e ig h b o rh o o d o f th e b u r ie d p ip e s t h a t i s d i f f i c u l t to re w e t [25]. I f th e d ry r e g io n i s o f su ch an e x t e n t t h a t c ro p g ro w th i s a d v e r s e ly a f f e c t e d, th e n s o i l warm ing w ould n o t be f e a s i b l e w ith o u t some fo rm o f s u b s u rfa c e i r r i g a t i o n to m a in ta in s u f f i c i e n t w a te r i n th e r o o t r e g io n. A c c o rd in g ly, to a s s e s s p r o p e r ly th e f e a s i b i l i t y o f m e e tin g a g r i c u l t u r a l re q u ire m e n ts a d e t a i l e d d e s c r i p t i o n o f th e te m p e ra tu re and m o is tu re d i s t r i b u t i o n a b o u t a sy ste m o f warm b u r ie d p ip e s i s n e e d e d. W ith t h i s o b je c tiv e i n m ind, a s i m p lif ie d c o n s ta n t p r o p e r ty h e a t c o n d u c tio n m odel i s p r e s e n te d i n C h a p te r 2, and a v a r i a b l e p r o p e r ty a n a l y s i s o f th e s im u lta n e o u s h e a t and mass t r a n s f e r i s d e v e l oped i n d e t a i l i n C h a p te rs 3» and 5» The D esig n P ro b le m. F or i l l u s t r a t i v e p u rp o s e s, a sy stem o f b u r ie d p ip e s d e s ig n e d s p e c i f i c a l l y to d i s s i p a t e h e a t from c o n d e n se r w a te r flo w in g in s i d e i s c o n s id e r e d. The o b je c tiv e o f th e sy stem i s to m a in ta in h ig h h e a t f l u x

22 6 a t th e p ip e s u r f a c e to enhance h e a t c o n d u c tio n from th e w a te r. T h is means t h a t th e p ip e s sh o u ld be w id e ly sp a c e d and th e p ip e d ia m e te r sh o u ld be s m a ll r e l a t i v e to th e s p a c in g. F u rth e rm o re, th e ru n s o f p ip e s h o u ld be r e l a t i v e l y lo n g to a llo w th e w a te r s u f f i c i e n t o p p o r tu n ity to c o o l a s i t f lo w s. A lso, th e p ip e s sh o u ld be b u r ie d n e a r th e s o i l s u r f a c e to re d u c e th e th e rm a l r e s i s t a n c e betw een th e p ip e w a ll and th e s o i l s u r f a c e. N ex t, a sy stem d e s ig n e d s p e c i f i c a l l y f o r a g r i c u l t u r a l p u rp o s e s i s c o n s id e r e d. An im p o rta n t o b je c tiv e i s t h a t th e h e a tin g e f f e c t due to th e b u r ie d h e a t s o u rc e s be f e l t th ro u g h o u t th e r e g io n i n o rd e r to b e n e f i t c ro p g ro w th. A sy ste m s p e c i f i c a l l y d e s ig n e d to a c c o m p lish t h i s w ould r e q u ir e low h e a t f l u x n e a r th e p ip e s, im p ly in g t h a t th e te m p e ra tu re w ould n o t drop o f f r a p i d l y i n th e v i c i n i t y o f th e warm p i p e s. T h is means t h a t th e p ip e s sh o u ld be c l o s e l y sp a c ed and th e p ip e d ia m e te r sh o u ld be s iz a b le r e l a t i v e to th e s p a c in g. F u rth e rm o re, i t w ould be d e s i r a b le t h a t th e c o n d e n se r w a te r flo w be n e a r l y is o th e r m a l, so th e te m p e ra tu re p o t e n t i a l f o r h e a tin g i s h ig h a lo n g th e e n t i r e le n g th o f th e f i e l d. w ith r e l a t i v e l y s h o r t r u n s. A way o f a c h ie v in g t h i s i s A lso, th e p ip e s sh o u ld be b u r ie d s u f f i c i e n t l y deep to a llo w f o r p ro p e r r o o t fo rm a t i o n and c u l t i v a t i o n.

23 7 I t i s c l e a r, th e n, t h a t th e d e s ig n c o n s t r a i n t s f o r a g r i c u l t u r e and f o r pow er p l a n t c o o lin g a r e somewhat a t o d d s, and t h a t to d e s ig n f o r b o th may n o t be p o s s i b le w ith o u t undue com prom ise. A m ajo r t h r u s t o f t h i s s tu d y w i l l be to i n v e s t i g a t e i f su ch d e s ig n s a r e p o s s i b le by a d e t a i l e d a n a l y s i s o f th e h e a t and mass t r a n s p o r t p r o c e s s e s i n th e s o i l. 1.2 P re v io u s S o i l Warming S t u d i e s. To d a te, th e r e h a s b een no r e p o r t o f a c o m p le te ly s a t i s f a c t o r y m odel o f a s o i l w arm ing sy ste m. A c o n s ta n t th e rm a l c o n d u c tiv ity, s t e a d y - s t a t e h e a t c o n d u c tio n m odel was d e v e lo p e d by K en d rick and H avens [13 1. The m odel i s r e a s o n a b ly a c c u r a te f o r p r e d i c t i n g s o i l te m p e ra tu re s a b o u t b u r ie d p ip e s i n d ry o r s a t u r a t e d s o i l and can be u sed to o b ta in u p p e r and lo w er bounds on th e h e a t d i s s i p a t i o n from a s o i l warm ing sy stem.. However, th e m odel i s in c a p a b le o f p r e d i c t i n g th e v a r i a t i o n o f m o is tu re i n th e s o i l, s in c e i t i s b ased upon th e a ssu m p tio n o f c o n s ta n t m o is tu re th ro u g h o u t. I t sh o u ld be n o te d, howe v e r, t h a t v a lu a b le i n s i g h t can be g a in e d fro m th e co n s t a n t c o n d u c tiv ity s o l u t i o n, and th e f o rm u la tio n i s s tu d ie d i n d e t a i l i n C h a p te r 2.

24 8 A t r a n s i e n t a n a ly s i s o f a s o i l warm ing sy stem h a s b een r e p o r te d by r e s e a r c h e r s a t Oregon S t a te U n iv e r s ity E 2 ]. The m odel s im u la te s a s o i l warming sy ste m t h a t i n c lu d e s s u b s u rfa c e i r r i g a t i o n to m a in ta in a d e q u a te m o is tu re i n th e s o i l. The e f f e c t s o f v a p o r t r a n s p o r t a r e n e g le c te d i n th e m odel, s in c e v a p o r flo w i s o n ly an im p o r ta n t f a c t o r when th e m o is tu re l e v e l i s low [25]. Thus, th e m odel c a n n o t be u sed to e v a lu a te th e e x t e n t to w hich th e s o i l d r i e s o u t i n th e ab sen c e o f a s u b - i r r i g a t i o n sy ste m. A n o th er sh o rtc o m in g i s t h a t o n ly r e l a t i v e l y slow s e a s o n a l t r a n s i e n t s a r e.in tro d u c e d i n t o th e s im u la tio n once th e i n i t i a l warm-up t r a n s i e n t s have d ie d o u t. Thus, th e c a p a b i l i t y 1 o f th e sy stem to re sp o n d to d iu r n a l in p u ts, w hich c o u ld be an im p o rta n t f e a t u r e o f a t r a n s i e n t a n a l y s i s, i s n o t in c lu d e d. An im p o rta n t q u e s tio n c o n s id e re d i n S e c tio n 1.3 i s w h e th e r a t r a n s i e n t a n a l y s i s i s n eed ed f o r a s s e s s in g th e f e a s i b i l i t y o f s o i l w arm ing. 1.3 Time a s an In d e p e n d e n t V a ria b le The t r a n s i e n t c h a r a c te r o f th e t r a n s p o r t p ro b lem w i l l now be d is c u s s e d. The p u rp o se o f t h i s d is c u s s io n i s to m o tiv a te th e im p o rta n c e o f th e s t e a d y - s t a t e a n a ly s e s i n C h a p te rs 2 and 5«

25 9 T r a n s ie n t phenom ena o ccu r c o n tin u o u s ly in s o i l. D i u r n a l te m p e ra tu re f l u c t u a t i o n s a f f e c t th e u p p er l a y e r s o f th e s o i l and a r e ro u g h ly p e r i o d i c. These v a r i a t i o n s a r e su p erim p o sed upon th e g r a d u a l s e a s o n a l ch an g es t h a t p ro p a g a te much d e e p e r. I t i s u n re a s o n a b le to e x p e c t t h a t a s o i l w arm ing sy stem c o u ld n e g a te th e s e e f f e c t s and m aint a i n te m p e ra tu re p r o f i l e s c o n s ta n t w ith tim e. Of w hat v a lu e th e n, i s a s t e a d y - s t a t e a n a ly s is? T here a r e c o n d itio n s u n d er w hich th e s t e a d y - s t a t e m odel g iv e s a c c u r a te p r e d i c t i o n s. F o r in s ta n c e, th e d i u r n a l te m p e ra tu re f l u c t u a t i o n s a r e c o n fin e d to th e u p p er l a y e r s o f s o i l, a s can be se e n from th e d a ta o f S kaggs, e t a l. [26]. Thus, f o r la r g e r e g io n s th e te m p e ra tu re p r o f i l e s change i n a r e l a t i v e l y slow s e a s o n a l f a s h io n. A ls o, a s r e p o r te d by R ykbost and Boersma [24- ], on a p l o t c o v e re d w ith v e g e ta t io n th e d iu r n a l te m p e ra tu re wave i s somewhat damped o u t a t th e s u r f a c e. e f f e c t o f th e gro u n d c o v e r. T h is i s due to th e sh a d in g So, f o r tim e i n t e r v a l s much s h o r t e r th a n th e p e r io d o f a n n u a l v a r i a t i o n, th e sy stem c o u ld be m o d e lle d a s b e in g a t s t e a d y - s t a t e. A lso, i n some c a s e s, i t i s a n t i c i p a t e d t h a t th e d iu r n a l wave w ould be o f s m a ll enough m agnitu d e to be n e g le c te d.

26 10 The g r e a t e s t v a lu e o f a s t e a d y - s t a t e a n a l y s i s i s f o r d e s ig n. I n c o n s id e r in g s o i l warm ing w ith b u r ie d p ip e s i t i s u n re a s o n a b le to e x p e c t to d e s ig n a sy stem c a p a b le o f c o p in g w ith d iu r n a l te m p e ra tu re c h a n g e s. i t may n o t even be n e c e s s a r y to do s o. F o r a g r i c u l t u r e A g ro w in g p l a n t i n t e g r a t e s th e e f f e c t s o f ch an g es i n i t s en v io rn m e n t w ith tim e. I n f a c t, some p l a n t s r e q u ir e p e r io d ic d i u r n a l tem p e r a t u r e f l u c t u a t i o n f o r v ig o ro u s grow th [ 1 0 ] 13^1. S in c e a s o i l warm ing sy stem can have an e f f e c t on a v e ra g e s o i l p r o p e r t i e s i t sh o u ld be d e s ig n e d f o r s t e a d y - s t a t e c o n d i t i o n s c o rre s p o n d in g to s u i t a b l e a v e ra g e v a l u e s. T hus, th e t r a n s i e n t c h a r a c t e r o f th e sy stem n eed n o t l i m i t th e u s e f u l n e s s o f a s t e a d y - s t a t e m odel. l.ty C lo s u re. In t h i s in t r o d u c t o r y c h a p te r, th e c o n c e p t o f s o i l w arm ing u s in g pow er p l a n t w aste h e a t h a s b een d is c u s s e d. Two im p o rta n t c r i t e r i a f o r a s s e s s in g th e f e a s i b i l i t y o f s o i l w arm ing have been i d e n t i f i e d. Of p rim a ry c o n c e rn i s w h e th e r th e a g r i c u l t u r a l o b je c tiv e s o f h ig h and u n ifo rm s o i l te m p e ra tu re a lo n g w ith s u f f i c i e n t m o is tu re l e v e l s to enhance p l a n t g ro w th c a n be a c h ie v e d. I f s o, th e c a p a b i l i t y o f m e e tin g th e pow er p l a n t o b je c tiv e o f w aste h e a t

27 11 d is p o s a l must be a sse sse d as a second c o n s tr a in t. The rem ainder of th e d i s s e r t a t i o n w ill now he review ed. A c o n s ta n t p ro p e rty h e a t co n duction model o f a s o i l warming system i s p re se n te d in C hapter 2. The model i s u s e fu l f o r d eterm in in g s o i l tem p eratu re p r o f i l e s and h e a t d is s ip a tio n c a p a b ility. However, sin c e th e im p o rta n t f a c to r of s o i l m o istu re d is t r i b u t i o n i s d isre g a rd e d in th e m odel, a v a r ia b le p ro p e rty tr a n s p o r t model i s p re se n te d in C hapters 3» ^t and 5 th a t, f o r th e f i r s t tim e, en ab les the problem of m o istu re m ig ra tio n to be s tu d ie d. The r e s u l t s of th e c o n s ta n t and v a r ia b le p r o p e r tie s models a re used in C hapter 6 to a s s e s s th e c a p a b ility of a s o i l warming system to meet th e s ta te d o b je c tiv e s. F in a lly, th e m ajor c o n c lu sio n s a re summarized and some recom m endations f o r f u r t h e r stu d y a re made in C hapter 7.

28 CHAPTER 2 CONSTANT PROPERTY MODEL FOR SOIL TEMPERATURE 2.1 In tro d u c tio n The s o i l warming system analyzed here i s i l l u s t r a t e d in F ig u re (2.1 ); the system c o n s is ts of a s e r ie s of e q u a lly spaced p a r a l l e l p ip e s b u rie d a t a uniform d epth in a le v e l f i e l d. Warm w ater i s su p p lie d to th e p ip e s from a nearby power p la n t. To m ain tain more uniform s o i l tem p eratu re, w ater flow s in o p p o site d ir e c tio n s through a d ja c e n t p ip e s. The purpose of the a n a ly s is in t h i s c h a p te r i s to model the h e a t tr a n s f e r in th e s o i l around th e p ip e s and to e v a lu a te the h e a t d is s ip a tio n r a t e, under the assum ptions of pure h e a t conduction and c o n s ta n t p h y s ic a l p r o p e r tie s of the s o i l medium. S p e c if ic a lly, in S e c tio n 2.2 a c o n s ta n t p ro p e rty h e a t co nduction model, based upon [133* i s p re sen ted in d im en sio n less form and g ra p h ic a l r e s u l t s are d isc u sse d. In S ec tio n 2.3 the u t i l i t y of t h i s sim ple model i s co n sid ered and some lim ita tio n s a re i d e n t i f i e d. A lso, the need f o r a more d e ta ile d a n a ly s is of the sim ultaneous h e a t and mass tr a n s f e r in the s o i l i s in d ic a te d. 12

29 13 buried pipe -'" N soil surface flow direction SOI F ig u re (2.1 ) Layout o f a s o i l warming system.

30 The C onstant P ro p e rty Model A c o n s ta n t therm al c o n d u c tiv ity, s te a d y - s ta te h e a t co n d u ctio n model f o r s o i l h eated by a system o f su b -su rfa c e p ip e s, as shown in F igure (2.1 ), can be developed by e x p lo itin g the l i n e a r i t y of the tw o-dim ensional L aplace E quation governing the h e a t tr a n s f e r. The tech n iq u e, in v o lv in g the method o f im ages, i s o u tlin e d by Jakob E ll] and r e c e n tly was a p p lie d to the case under c o n s id e ra tio n by K endrick and Havens [13 ]. Here, f o r th e f i r s t tim e, th i s problem i s analyzed in term s of non-dim ensional e q u a tio n s, a form t h a t f a c i l i t a t e s the drawing of some g e n e ra l conc lu s io n s ab o u t the o p e ra tio n of th e system. A p o rtio n o f the s o i l warming system i s shown in F igure (2.2 ). Water e n te rs a t a tem p eratu re o f Tj, coming in to a d ja c e n t p ip e s a t o p p o site ends o f the f i e l d. Since the s o i l su rfa c e tem p eratu re Tg i s le s s than T^ when the system i s in u se, the w ater co o ls as i t flow s through the p ip e s. The tem perature a t p o in ts in the s o i l, T (x,y,z ), depends upon th e lo c a l w ater tem p eratu res Tw^ (z ) and Tw2 ( z ) as w ell a s th e s o i l su rfa c e tem p eratu re. The r a te of h e a t t r a n s f e r i s determ ined by th e lo c a l tem perature p o te n tia ls (Twi ( z )" Ts ) and (Tw 2 ^ Ts^ 311(1 th e fchermal r e s is ta n c e of th e s o i l. This h e a t flow in tu r n p la y s a ro le in d eterm in in g the w ater tem p eratu res. Other

31 soil surface at Ts 15 water in at T water in at plane of symmetry F ig u re (2.2 ) S o il warming model

32 16 p aram eters in v o lv ed in th e h e a t tr a n s f e r problem a re the mass flow r a t e in th e p ip e s, m, and the la y o u t p a ra m e te rs: depth d, sp acin g s, p ipe ra d iu s R, and the le n g th o f each p ipe L. K endrick and Havens [133 p re s e n t an e q u a tio n f o r th e s o i l tem p eratu re ab o u t 2N+1 (N i s an even number) p ip e s. The d im en sio n less form of t h a t eq u atio n, in term s o f s u i t ab le d im en sio n less v a r ia b le s and p aram eters, i s 4> fx2+ (l-y)2 *1/ x2+(i+y)2 N/2 r E i J n=l > (l-y )2+ (2ns-x)2 -\2 (l+y)2+(2ns-x) (l-y)2+(2n +x)2 (l+y)2+(2ns+x)2 /AV BTi\ \ a2- b2 / inv / ( l - y ) 2+ ( ( 2 n - l ) i - 5 ) 2 (l+y)2+((2n-l)s-x) - - n2 N/2 I E x ny i (1-y)2+((2n-1)s+x)2 -,-\2 n=l (l+y)2+((2n-l)s+x) ( )

33 17 where * - T ~Ts d im en sio n less s o i l Xi~Tg..... tem p eratu re T 11 Tw r Ts 0?T -T s 1 d im en sio n less w ater m m tem p eratu res * 1I i S m _ 2 w2~ s T -T z - v /x ~ / a - /a d im en sio n less p o s itio n ' ' y -y /d, z = z / d... c o o rd in a te s r;_r/ d d im en sio n less p ip e '..... s iz e p aram eter ( r c l ) d im en sio n less p ip e s= s/d... sp acin g p aram eter (S>2r) and A ( s,r ) = l n ( 2 / r - l ) + N/2 _ 2 _- 2' 2n V ' f(2 -r) +(2ns) I 2-JIn r-p -l (2.2.2) n=l L r + (2ns) J N/2 E H (2 -r) + (2 n -l) s 1 In ~o'~o (2.2.3 ) n=l L r + ( 2 n - l) s J From E quation (2.2.1 ) i t can be seen t h a t $ i s a fu n c tio n of p o s itio n, lo c a l w ater tem p eratu res and T2, and th e la y o u t p aram eters s and r. S y m bolically, $ = $ ( x,y,z ;T 1,T2, i l r ) (2.2.^ )

34 18 The d im en sio n less e q u a tio n d e s c rib in g w ater tem p eratu re t h a t fo llo w s from K endrick and Havens* developm ent i s Tj = cosh(nz) + B-A cosh(n)- / A -B sinh(n ) A -B co sh (n ) + A sinh(n) sin h (B z) (2.2.5 ) where n =2nX L/tfiC / A2-B2 w ith A. b ein g th e th erm al conduc- P t i v i t y of th e s o i l medium and C th e s p e c if ic h e a t o f w ate r. P A p h y s ic a l i n t e r p r e t a ti o n of th e d im en sio n less p aram eter n i s g iv en i n S e c tio n 2,2.1 a f t e r F ig u re (2.4-) i s p re s e n te d. The r a d ic a l /A 2-B2, ap p earin g in E quation (2.2.5 ) and th e d e f in it io n o f n, i s a fu n c tio n o f s and? as d efin e d in term s of E quations (2.2.2 ) and (2«2o3)» This dependence i s shown g ra p h ic a lly i n F ig u re (2.3 ) From E quation (2.2.5 ) i t can be seen t h a t T^ depends upon a x ia l p o s itio n z and th e la y o u t p aram eters s, r, and n. S y m b o lically, T^ = T^zjs,?,n ) (2.2.6)

35 CM CD ^ < S F ig u re (2.3 ) V alues of th e r a d ic a l -B^ as fu n c tio n of and?.

36 20 Only one e q u a tio n need be p re se n te d because the symmetry of the system re q u ire s t h a t f o r p ip e s w ith lo c a l w ater tem p eratu res Tw l ^ and Tw2 ^ ' Tl ( i ) = T2 ( 1 - i) T h erefo re, u sin g E quations (2.2.1 ) and (2.2.5 ) along w ith th e symmetry r e la tio n s h ip, th e system tem p eratu res a re co m p letely d e s c rib e d. A co n v en ien t s e t o f cu rv es w ill now be o b tain ed from th e non-dim ensional e q u a tio n s P re s e n ta tio n of G raphical R e s u lts. E quations (2.2.1 ) and (2.2.5 ) in d ic a te t h a t th e system tem p eratu res depend n o t only upon th e system la y o u t p aram eters s, r, and h, b u t a ls o upon p o s itio n w ith in the systems x, y, z. A conveni e n t s e t o f graphs can be developed upon e lim in a tio n of the p o s itio n dependence. The f i r s t s te p in t h i s d ir e c tio n w ill be i d e n t i f i c a t i o n of c e r t a in c r i t i c a l lo c a tio n s w ith in th e system. The v alu e o f T^ a t z=l i s a measure o f th e amount of w ater tem p eratu re drop achieved by th e system, where T^(1)=0 in d ic a te s maximum c o o lin g and T ^ (l)= l co rresp o n d s to the case of no c o o lin g. When E quation (2,2.5 ) i s e v a lu ated a t z= l, th e r e s u l t i n g e x p ressio n f o r w ater tem p eratu re drop

37 21 through th e system depends only upon the la y o u t p aram eters s, r, and n» TjCl) = T^i.r.n) (2.2.7) This r e la tio n s h ip i s p re se n te d g ra p h ic a lly i n F ig u re ( 2.4 ). A p h y s ic a l in t e r p r e t a ti o n of the p aram eter n can he o b tain ed by stu d y in g lin e s of fix e d S and r on F ig u re (2.4 ). For sm all n, co rresp o n d in g to low therm al c o n d u c tiv ity, s h o r t p ip e s, an d /o r a high mass flow r a t e, th e re i s n e a rly iso th e rm a l flow in th e p ip e s t h a t i s, T ^ (l)~ l. On the o th e r hand, when n i s la rg e, co rresp o n d in g to h ig h er!th erm al c o n d u c tiv ity, lo n g ru n s of p ip e, an d /o r a low mass flow rate* th e re i s in c re a se d c o o lin g of th e w ate r. Thus, n can be i n te r p r e te d as a gauge of th e system *s a b i l i t y to co o l th e w a te r. Next, c r i t i c a l lo c a tio n s f o r s o i l tem p eratu re w ill be i d e n t i f i e d. For ty p ic a l f i e l d cro p s, th e ro o ts can extend s e v e ra l m eters in to th e ground see F ig u re (2.5 ) And, as shown in F igure ( l. l ), t h e tem perature a t which th e r o o ts a re m ain tain ed can in flu e n c e a g r ic u ltu r a l y ie ld. The y ie ld d a ta in F ig u re ( l. l ) a re somewhat id e a liz e d, however, because th ey r e p re s e n t th e r e s u l t s of t e s t s ru n w ith the p la n ts in c o n s ta n t tem perature b a th s, in s te a d of i n s o i l.

38 2 ^ F igure (2.4-) D im ensionless w ater tem perature drop as a fu n c tio n of the la y o u t p aram eters.

39 ROOT SYSTEMS OF CROPS IN DEEPLY IRRIGATED SOIL 0 -sugar beet potato wheat 1m 3 L alfalfa F igure (2.5 ) Root system s of se v e ra l cro p s, (from J. Janick, H o rtic u ltu ra l S cie n ce, 2nd E d., W. H. Freeman Co. 1972) corn

40 2b So, th e y ie ld v a lu e s a re s t r i c t l y v a lid only when th e re i s e s s e n t i a l l y uniform tem p eratu re throughout th e r o o t zone. U n fo rtu n a tely, t h i s c o n d itio n can n o t he achieved w ith "buried p ip e s. T his i s c l e a r l y re v e a le d by F ig u re (2.6 ) which g iv e s ex p erim en tal d a ta f o r th e v a r ia tio n o f s o i l tem pera tu re w ith d epth in a p l o t h ea te d w ith iso th e rm a l c a b le s F ig u re (2.6 ) shows th a t, depending upon where the p ip e s a re lo c a te d in th e s o i l - r o o t system, th e r o o ts can ex p erien ce tem p eratu res anywhere from Tg a t th e s o i l s u rfa c e to Tj a t the p ip e s. For th e purpose of t h i s d is c u s s io n i t would be d e s i r ab le to have a s in g le measure f o r th e tem p eratu re of th e r o o ts. Then, u sin g d a ta such as t h a t p re se n te d i n F ig u re (1.1 ) as a g u id e, th is measure could be invoked to g iv e some assu ran ce t h a t th e tem p eratu res th ro u g h o u t th e r o o t re g io n a re in a b e n e f ic ia l ra n g e. A s a t is f a c t o r y m easure of t h i s k in d i s d i f f i c u l t to d e fin e, however, because th e r o o ts g e n e ra lly extend over a la rg e re g io n of s o i l (F igure ( 2.5 )) i n which the r o o t tem p eratu res v a ry over a c o n s id e ra b le range as w e ll. The n e a rly l in e a r n a tu re o f the tem p eratu re p r o f i l e in F ig u re (2.6 ) su g g e sts t h a t the v alu es o f $ a t y=0.5» i.e» h a l f the d is ta n c e from the s o i l su rfa c e to the p ip e s, g iv e a measure of th e average s o i l tem p eratu re in t h a t re g io n. For s o i l warming a p p lic a tio n s, the p ip e s a re

41 25 T - T s 0 Tf T S = y/d O heated V unheated F ig u re (2.6 ) E xperim ental v e r t i c a l tem pera tu re p r o f i l e about a "buried h e a t so u rce.

42 26 l i k e l y to "be p laced so th a t the s o i l "between th e su rfa c e and th e p ip e s encompasses the "bulk of the r o o t zone. Thus, the tem perat'ure in t h a t re g io n, c h a ra c te riz e d "by the v alu e a t y=0,5, w ill a r b i t r a r i l y be c a lle d the r o o t zone tem pera t u r e. There i s a p e rio d ic f lu c tu a tio n of r o o t zone tem pera tu re w ith x, due to th e p resen ce of warm p ip e s a t r e g u la r i n t e r v a l s. I t i s rea so n a b le to e lim in a te th e x dependence by a v era g in g a c ro ss the f i e l d. The average r o o t zone temp e r a tu re, th en, depends on ly upon z, when the la y o u t p aram eters s, r, and n a re fix e d. F igure (2.7 ) i s a s e t of cu rv es d e p ic tin g the v a r ia tio n of the average r o o t zone tem p eratu re w ith z f o r th e p aram etric v alu es in d ic a te d on the f ig u r e. I t can be seen t h a t th e v alu es a re lo w est a t m id fie ld. For the purpose of the subsequent d isc u ssio n, z=0.5 w ill be chosen as th e c r i t i c a l lo c a tio n fo r s o i l te m p e ra tu re. The average m id fie ld r o o t zone tem p eratu re, denoted as <$>, has no p o s itio n dependence and i s on ly a fu n c tio n of i, r, and r\. This dependence i s p re se n te d g ra p h ic a lly in F ig u re s (2.8 ), (2.9)» and (2.1 0 ) f o r v a rio u s v alu es o f th e p ara m ete rs.

43 I T AVERAGE ROOT ZONE TEMPERATURE vs. AXIAL POSITION (s = 2.1/r=25) ^ ".4 + y\ / K. I I f H ^ f Z F ig u re (2.7 ) A x ial v a r ia tio n of dimens io n le s s ro o t zone tem p eratu re.

44 28 a v e r a g e r o o t z o n e t e m p e r a t u r e AT MIDFIELD (s =.5),1 /r= 1 0 =25 1/r=50 = i = 27Tkl_ 1 rh Cn \/ A2-B2 F igu re (2.8) D im en sio n less s o i l tem perature a s a fu n c tio n o f th e la y o u t p ara m eters..

45 a v e r a g e r o o t z o n e t e m p e r a t u r e AT MIDFIELD ^ /r=10 Vr=25! 1/r=50 j 1/r= f kl 1 m C p / A2 -B 2 F igu re (2.9) D im en sio n less s o i l tem perature a s a fu n c tio n o f th e la y o u t p aram eters.

46 a v e r a g e r o o t z o n e t e m p e r a t u r e ; AT MIDFIELD /r=10 1/r=25 1/r=50 1/7= yj = 2/TkL 1 m C p / A2 -B 2 F igu re ( ) D im en sio n less s o i l tem perature a s a fu n c tio n o f th e la y o u t p aram eters.

47 Q u a lita tiv e O b serv atio n s. The graphs p re se n te d in the p re v io u s s e c tio n, whose v a l i d i t y i s s u b je c t to the assum ptions in volved in th e c o n s ta n t p ro p e rty model, w ill now be examined in o rd er to determ ine the ran g es of p a ra m e tric v a lu e s t h a t a re most d e s ira b le f o r m eeting a g r i c u l t u r a l and power p la n t c o o lin g needs. F i r s t, re g a rd in g the a g r ic u ltu r a l o b je c tiv e s of hig h and uniform s o i l tem p eratu re, F ig u re s (2.8 ), (2.9 ), and (2.1 0 ) show t h a t the h ig h e s t average r o o t zone tem p eratu res a re o b tain ed by d esig n s in which s, l / r, and n a re sm all, achieved f o r example when th e re i s c lo se sp acin g, la rg e p ip e s, and a s h o rt f i e l d (an d /o r la rg e flow r a t e ). Close sp acin g i s a ls o d e s ira b le f o r tem perature u n ifo rm ity. That i s, f o r c o n fig u ra tio n s in which th e p ip e s a re w idely sp aced, th e re i s l i k e l y to be f lu c tu a tio n of tem perature from p o in t to p o in t in the r o o t zone (y=0.5) due to th e in te r m itte n t lo c a tio n s of th e warm p ip e s. F urtherm ore, f o r o v e r a ll u n i fo rm ity sm all v a lu e s o f n» co rresponding to n e a rly i s o t h e r mal flow, would be b e s t; a ls o, th e re would be g r e a te r u n ifo rm ity w ith la rg e p ip e s ( l / r sm a ll). These observatio n s on the b e s t p aram etric ran g es fo r s o i l tem perature a re summarized in Table 1. Turning to th e power p la n t o b je c tiv e, the amount o f w ater c o o lin g o b tain ed by th e system i s im p o rta n t. From F ig u re (2 A ) i t can be seen t h a t la rg e v a lu e s o f n e f f e c t

48 32 th e most c o o lin g, fo r any given s and r. A lso, h ig h e r v a lu e s of i and l / r le a d to g r e a te r tem perature d ro p s, so w idely spaced, sm all p ip e s are th e most fa v o ra b le f o r maximum c o o lin g. P h y s ic a lly, th i s i s a consequence o f th e f a c t t h a t fo r such a c o n fig u ra tio n, the tem p eratu re g ra d ie n ts away from the p ip e s a re th e s te e p e s t. These r e s u l t s f o r w ater tem p eratu re drop are summarized in Table 1. In s p e c tio n o f Table 1 re v e a ls th a t d i f f e r e n t p a ra m e tric ran g es a re suggested to meet the in d ic a te d o b je c tiv e s. That i s, th e o b je c tiv e s of h ig h and uniform s o i l tem p eratu re f o r a g r ic u ltu r e a re a t odds w ith the d e s ire to t r a n s f e r la rg e amounts o f energy from condenser w ater in su b -su rfa c e p ip e s. This o b se rv a tio n i s a d i r e c t r e s u l t of th e f a c t t h a t h e a t co n d u ctio n depends d i r e c t ly upon th e tem p eratu re g r a d ie n t. Thus, as j u s t n o ted, sm all, w idely spaced p ip e s would y ie ld the l a r g e s t g ra d ie n ts, and th e re fo re would co o l most re a d ily. However, th e a g r ic u ltu r a l o b je c tiv e s are b e s t met by la r g e r, c lo s e ly spaced p ip e s, sin c e then the tem p eratu re g r a d ie n t n e a r the p ip e s i s low er and the h e a tin g e f f e c t p ro p ag a tes as f a r as p o s s ib le tow ards the s o i l s u rfa c e. I t i s c le a r, th en, t h a t th e se o b je c tiv e s a re a t odds, and to d esig n sim u lta n eo u sly fo r b o th re q u ire s some compromise.

49 33 TABLE 1 P aram eter B est Range f o r M eeting: A g ric u ltu ra l O bjective (high and uniform s o i l tem p eratu re) Power P la n t O b jectiv e (la rg e w ater tem pera tu re drop) s sm all la rg e l / r sm all la rg e n sm all la rg e 2.3 E v a lu a tio n of th e Model In t h i s s e c tio n, the a b i l i t y o f the c o n s ta n t p ro p e rty model to p r e d ic t a c c u ra te ly tem perature p r o f i l e s and h e a t d is s ip a tio n i s d isc u sse d. A lso, an im p o rtan t lim ita tio n o f th e model i s id e n tif ie d, namely the i n a b i l i t y to p r e d i c t m o istu re v a r ia tio n in the s o i l, in d ic a tin g th e need f o r a more d e ta ile d a n a ly s is. F i r s t, th e v a l i d i t y of the tem perature p re d ic tio n s i s co n sid ered. The c o n s ta n t p ro p e rty h e a t conduction model p r e d ic ts tem p eratu re p r o f i l e s t h a t a re i n s a ti s f a c t o r y agreem ent w ith a c tu a l d a ta, as evidenced by F ig u re (2.1 1 ). This i s n o t unexpected, sin c e the experim ents in which th e d a ta were o b tain ed in clu d ed su b -su rfa c e i r r i g a t i o n n e a r th e p ip e s in

50 3 ^ T Ts T - T O.2.4 S ~G >,.6!>* & o data from C2^l D data from [263 constant property model F igure (2.1 1 ) Comparison of c o n s ta n t p ro p e r ty model w ith experim ental d a ta.