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UFC 3-130-0616 January 2004

UNIFIED FACILITIES CRITERIA (UFC)

CALCULATION METHODS FOR

DETERMINATION OF DEPTH OFFREEZE AND THAW IN SOIL: ARTICAND SUBARCTIC CONSTRUCTION

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

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UFC 3-130-0616 January 2004

UNIFIED FACILITIES CRITERIA (UFC)

CALCULATION METHODS FOR DETERMINATION OF DEPTH OF FREEZE AND

THAW IN SOIL: ARTIC AND SUBARCTIC CONSTRUCTION

Any copyrighted material included in this UFC is identified at its point of use.Use of the copyrighted material apart from this UFC must have the permission of thecopyright holder.

U.S. ARMY CORPS OF ENGINEERS (Preparing Activity)

NAVAL FACILITIES ENGINEERING COMMAND

AIR FORCE CIVIL ENGINEER SUPPORT AGENCY

Record of Changes (changes are indicated by \1\ ... /1/)

Change No. Date Location

This UFC supersedes TM 5-852-6, dated 25 January 1988. The format of this UFC does notconform to UFC 1-300-01; however, the format will be adjusted to conform at the next revision.The body of this UFC is the previous TM 5-852-6, dated 25 January 1988.

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UFC 3-130-0616 January 2004

FOREWORD\1\The Unified Facilities Criteria (UFC) system is prescribed by MIL-STD 3007 and providesplanning, design, construction, sustainment, restoration, and modernization criteria, and appliesto the Military Departments, the Defense Agencies, and the DoD Field Activities in accordancewith USD(AT&L) Memorandum dated 29 May 2002. UFC will be used for all DoD projects and

work for other customers where appropriate. All construction outside of the United States isalso governed by Status of forces Agreements (SOFA), Host Nation Funded ConstructionAgreements (HNFA), and in some instances, Bilateral Infrastructure Agreements (BIA.)Therefore, the acquisition team must ensure compliance with the more stringent of the UFC, theSOFA, the HNFA, and the BIA, as applicable.

UFC are living documents and will be periodically reviewed, updated, and made available tousers as part of the Services responsibility for providing technical criteria for militaryconstruction. Headquarters, U.S. Army Corps of Engineers (HQUSACE), Naval FacilitiesEngineering Command (NAVFAC), and Air Force Civil Engineer Support Agency (AFCESA) areresponsible for administration of the UFC system. Defense agencies should contact thepreparing service for document interpretation and improvements. Technical content of UFC isthe responsibility of the cognizant DoD working group. Recommended changes with supportingrationale should be sent to the respective service proponent office by the following electronicform: Criteria Change Request (CCR). The form is also accessible from the Internet sites listedbelow.

UFC are effective upon issuance and are distributed only in electronic media from the followingsource:

Whole Building Design Guide web site http://dod.wbdg.org/.

Hard copies of UFC printed from electronic media should be checked against the currentelectronic version prior to use to ensure that they are current.

AUTHORIZED BY:

______________________________________DONALD L. BASHAM, P.E.Chief, Engineering and ConstructionU.S. Army Corps of Engineers

______________________________________DR. JAMES W WRIGHT, P.E.Chief EngineerNaval Facilities Engineering Command

______________________________________KATHLEEN I. FERGUSON, P.E.The Deputy Civil Engineer

DCS/Installations & LogisticsDepartment of the Air Force

______________________________________Dr. GET W. MOY, P.E.Director, Installations Requirements and

ManagementOffice of the Deputy Under Secretary of Defense

(Installations and Environment)

http://www.wbdg.org/pdfs/ufc_implementation.pdfhttp://www.wbdg.org/ccb/browse_cat.php?o=29&c=4http://dod.wbdg.org/http://dod.wbdg.org/http://www.wbdg.org/ccb/browse_cat.php?o=29&c=4http://www.wbdg.org/pdfs/ufc_implementation.pdf

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ARMY TM 5 - 8 5 2 - 6AI R FORCE AFR 8 8 - 1 9 , Vo l ume 6TECHNI CAL MANUAL

ARCT I C AND SUBARCT I CCONST RUCTI ONCAL CUL ATI ON MET HODSF OR DETERMI NATI ON OFDEPTHS OF FREEZE ANDTHAW I N SOI L S

DEPARTMENTS OF THE ARMY AND THE AI R FORCEJ ANUARY 1 9 8 8

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REPRODUCTI ON AUTHORI ZATI ON/ RESTRI CTI ONST h i s ma n u a l h a s b e e n p r e p a r e d b y o r f o r t h e Go v e r n me n t a n d i s p u b l i c p r o p e r t ya n d n o t s u b j e c t t o c o p y r i g h t .R e p r i n t s o r r e p u b l i c a t i o n s o f t h i s m a n u a l s h o u l d i n c l u d e a c r e d i t l i n e s u b s t a n t i a l l ya s f o l l o w s : " J o i n t D e p a r t me n t s o f t h e Ar my a n d A i r F o r c e USA, T e c h n i c a l Ma n u a lTM 5 - 8 5 2 - 6 / A FR 8 8 - 1 9 , Vo l u me 6 , C a l c u l a t i o n Me t h o d s f o r D e t e r mi n a t i o n o f De p t h so f F r e e z e a n d T h a w i n s o i l s , d a t e . "

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TECHNI CAL MANUAL HEADQUARTERSNO 5 - 8 5 2 - 6 DEPARTMENTSOF THE ARMYAI R FORCE REGULATI ON AND THE AI R FORCEAFR 8 8 - 1 9 , VOLUME 6 WASHI NGTON, DC , 2 5 J a n u a r y 1 9 8 8ARCTI C ANDSUBARCTI C CONSTRUCTI ON- CALCULATI ON METHODS FORDETERM NATI ON OF DEPTHS OF FREEZE ANDTHAW N SOI LS

" T h i s m a n u a l s u p e r c e d e s TM 5 - 8 5 2 - 6 / A F M 8 8 - 1 9 , C h a p 6 , d a t e d J a n u a r y 1 9 6 6

Pa r a g r a p h P a g eCHAPTER 1 . GENERALPu r p o s e a n d s c o p e . . . . . . . . . . . .. . . .. . . . .. . .. . .. . . 1 - 1 1 - 1Re f e r e n c es a n d s y mb o l s . . . . . . .. . . . . . . . . . . . . .. . .. . . .1 - 2 1 - 1Ba c k g r o u n d . . . . .. . . .. . . . . . . . .. . .. . . . .. . .. . .. . . . . 1 - 3 1 - 1

CHAPTER 2 . DEFI NI TI ONS AND THERMAL PROPERTI ESDef i n i t i on s 2 - 1 2 - 1T h e r ma l p r o pe r t i e s o f s o i l sa n d o t h er c on s t r u c t i o nma t e r i a l s . . . . .. . . . . . . .. . . . . . . . .. . . . . . . .. . . .. . . .. . . . - 2 2 - 2F u n d ame n t a l c on s i de r a t i o n s . . . . .. . . . .. . . .. . .. . . 2 - 3 2 - 3Fr e ez i n g a n d t h awi n gi n de x es . . . . .. . . . . . .. . . . . .. . . . . . . .. . . . . .. . . .. . . 2 - 4 2 - 5

CHAPTER 3 . ONE- DI MENSI ONAL LI NEARANDPERI OD I C MEAT FL OWT h e r ma l r e g i me . . . . . .. . . . .. . .. . . . . .. . .. . .. . .. . . 3 - 1 3 - 1Mod i f i e d Be r g g r e n e q ua t i o n . . . . .. . .. . . . .. . . . . . . . . 3 - 2 3 - 1Homog en eo us s o i l s . .. . . . . .. . .. . . . .. . . . .. . . . .. . . 3 - 3 3 - 3Mu l t i l a y er s o i l s . . . . . . .. . . . . .. . .. . .. . . . . .. . .. . . . 3 - 4 3 - 3E f f e c t o f s n ow a n dv ege t a t i v e c o v e r . . . . . .. . . . .. . . .. : .. . . . . .. . .. . . . 3 - 5 3 - 6Su r f a c e t emp er a t u r ev ar i a t i on s . .. . . . . . .. . . .. . . . .. . .. . . . . .. . . . . . . . 3 - 6 3 - 6Con v er t i n g i n de x es i n t oeq u i v a l e n t s i n e w a v e s o ft emp er a t u r e . . . . . . . . . . . .. . .. . . . . . .. . . .. . .. . . .. . . 3 - 7 3 - 8Pe n e t r a t i o n o f f r e e z e o rt h a w b en ea t h b u i l d i n g s . .. . . .. . . .. . . .. . . .. . .. . 3 - 8 3 - 1 1Us e o f t h e r ma l i n s u l a t i n gma t e r i a l s . . . . . .. . . . . . . . . . . .. . . .. . . . .. . .. . . .. . 3 - 9 3 - 1 8

CHAPTER 4 . TWO- DI MENSI ONAL RADI AL HEATFLOWGe n e r a l . . .. . . . . . . . . . .. . . . . . . . .. .. . .. . . . . . .. . . . .. . . . . .. .. . .. . . . 4 - 1 4 - 1P i l e i n s t a l l a t i o n i np e r ma f r o s t . . . . . . . . . .. . . . . . . . . . . .. . . .. .. . . .. . . .. .. . . . .. . . . . . .. 4 - 2 4 - 2U t i l i t y d i s t r i bu t i o n s y s t e msi n f r o z en g r o u nd . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . .. . . . .. 4 - 3 4 - 7Di s c us s i o n o f mu l t i -d i me n s i o na l h e a t f l o w . . . . . .. . . . . .. . . . . . . . . . . .. . . . . .. . . . .. 4 - 4 4 - 1 3

APPENDI X A . L I ST OF SYMBOLS A - 1APPENDI X B . THERMAL MODEL S FORCOMPUTI NGFREEZE ANDTHAWDEPTHS B - 1APPENDI X C . REFERENCES C - 1BI BL I OGRAPHY BI BL I O- 1

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F i g ur e 2 - 1 . Av e r a g e t h e r ma l c o n d uc t i v i t y f o r s a n d s a n d g r a v e l s , f r o z en . . . .. . . . . . .. . . . . . .. . . . .. . 2 - 32 - 2 . Av e r a g e t h e r ma l c o n d u c t i v i t y f o r s a n d s a n d g r a v e l s , u n f r o z e n .. . . .. . . .. . . . .. . . . .. . . 2 . 42 - 3 . Av e r a g e t h e r ma l c o n d u c t i v i t y f o r s i l t a n d c l a y s o i l s , f r o z en . . . . . .. . . . .. . . . . . . .. . . . .. . 2 . 42 - 4 . Av e r a g e t h e r ma l c o n d u c t i v i t y f o r s i l t a n d c l a y s o i l s , u n f r o z e n . . . . .. . . . . . . . .. . . . .. . 2 . 42 - 5 . Av e r a g e t h e r ma l c o n d u c t i v i t y f o r p e a t , f r o z e n .

. . . . . . . .. . . .. . . . . . .. . . . .. . .. . . .. . . .. . 2 . 52 - 6 . Av e r a g e t h e r ma l c o n d u c t i v i t y f o r p e a t , u n f r o z e n . .. . .. . . . .. . . . . .. . . . .. . . . . . .. . . . .. . 2 . 52 - 7 . Av e r a g e v o l ume t r i c h e a t c ap ac i t y f o r s o i l s . . . . . .. . . . . . .. . . . .. . . .. . . . .. . . . . . . .. . . . .. . . 2 . 82 - 8 . 7 do l ume t r i c l a t e n t h ea t f o r s o i l s . . .. . . . .. . . . . .. . . . .. . . .. . . . . .. . . . .. . . .. 2 . 92 - 9 . Av e r a ge mo n t h l y t e mp e r a t u r e s v e r s u s t i me a t Fa i r b an k s , Al a s k a- . .. . . . . . . .. . . . .. . . 2 - 1 02 - 1 0Re l a t i o n s h i p b e t we e n wi n d s p e e d a n d n - f a c t o r d u r i n g t h awi n g s e a s o n . . . . . .. . . .. . 2 - 1 12 - 1 1 . Re l a t i o n s h i p b e t we e n mean f r e ez i n g i n d e x a n d max i mum f r e ez i n g i n d e x f o r1 0 y e a r s o f r e c o r d , 1 9 5 3 - 1 9 6 2 ( a r c t i c a n d s ub ar c t i c r e g i o n s ) . . . . . . . .. . . . .. . . . . .. . . . .. . 2 - 1 22 - 1 2Re l a t i o n s h i p b e t we e n mean t h awi n g i n d e x a n d max i mum t h awi n g i n d e x f o r1 0 y e a r s o f r e c o r d , 1 9 5 3 - 1 9 6 2 ( a r c t i c a n d s ub ar c t i c r e g i o n s ) . . . . . . .. . . . .. . . . . . . .. . .. . 2 - 1 33 - 1 . . 1 o e f f i c i e n t i n mo d i f i e d Be r g g r e n f o r mu l a . 3 - 23 - 2 . Re l a t i o n s h i p b e t we e n ( x / 2 ; a t t -) a n d e r f ( x / 2 8 7 _ 1 ). . . . . .. . . .. . . . . . .. . . .. . . . .. . .. . . .. . . .. .. . . . . . .. . . . . .. . . . .. . . . .. . . . . . . .. . . . .3 - 73 - 3 . S i n us o i d a l t e mp e r a t u r e p a t t e r n . . . .. . .. . . . . .. . . . .. . . . . .. . . . .. . . . .. . . . .. . . . . . .. . . , . . . 3 . 93 - 4 . I n d e x e s a n d e qu i v al e n t s i n us o i d al t e mp e r a t u r e . . . . . .. . . . .. . . . .. . . . .. . . . . . .. . . .. . 3 - 1 03 - 5 . Av e r a g e mo n t h l y t e mp e r a t u r e s f o r 1 9 4 9 - 1 9 50 a n d e q u i v al e n t s i n e wave ,Fa i r b an k s , Al a s k a . . . . . . . . .. .. . . . . . . . .. . . .. . . . .. . . . .. . . .. . . . . .. . . .. . . . .. . . . . . . . .. . . . .. . . . 3 - 1 13 - 6 . L o n g - t e r m mean mo n t h l y t e mp e r a t u r e s a n d e q ui v a l e n t s i n e wave ,Fa i r b an k s , Al a s k a . . . . . . . . . . .. . .. . . . .. . . .. . . .. .. . . . . .. . . .. . . . . . .. . . .. . . . .. . . . . .. . . .. . 3 - 1 23 - 7 . Sc h e ma t i c o f d u c t e d f o un d a t i o n . . . .. . .. , . . . . . . . .. . . . .. . . . .. . . .. . . . .. . . . . ., . . . .. 3 - 1 43 - 8 . Pr o p e r t i e s o f d r y a i r a t a t mo s p he r i c p r e s s u r e . . . . .. . . . . .. . . . .. . . .. . . . .. . . . . .. . . . .. . 3 - 1 64 - 1 . I l l u s t r a t i o n f o r e x a mp l e i n p a r a g r a p h 4 - l a . . .. . . . .. . . . . . .. . . . .. . . .. . . . . .. . . . . . . .. . . . .. 4 - 24 - 2 . Te mp e r a t u r e a r o u n d a c y l i n de r h av i n g r e c e i v ed a s t e p c h a n g e i n t e mp e r a t u r e . . . .4 - 34 - 3 . Ge n e r a l s o l u t i o n o f s l u r r y f r e e z e- b a c k . .. . . . .. . . . . .. . . . . .. . .. . . . . . .. . . . .. . . . . . .. . . .. . 4 - 44 - 4 . Sp e c i f i c s o l u t i o n o f s l u r r y f r e e z e - b ac k . . . . . . . .. . . . . . .. . . . . .. . . . . .. . . .. . . .. . . . . . . . . .. . . . .4 - 54 - 5 . F r e e z e u p o f s t a t i o na r y wa t e r i n a n u n i n s u l a t e d p i p e . . .. . . . .. . . .. . . . .. . . . . . . .. . . . .4 - 94 - 6 . Te mp e r a t u r e d r o p o f f l o wi n g wa t e r i n a p i p e l i n e . . .. . . . .. . . . .. . . .. . . . . . .. . . . . . .. . . .4 - 1 2

Ta b l e 2 - 1 . Sp e c i f i c h e a t v al u e s o f v ar i o us ma t e r i a l s . . . . .. . . . .. . . . .. . . . . .. . . .. . . . .. . . . . . . .. . . . .2 . 22 - 2 . T h e r ma l p r o pe r t i e s o f c o n s t r u c t i o n ma t e r i a l s . . . . .. . . . .. . . . .. . . .. . . . .. . . . . . . .. . . . .2 . 62 - 3 . Ca l c u l a t i o n o f c u mu l a t i v e d e gr e e- d a y s . . . . . . .. . . .. . . . .. . . . . . . .. . . . . .. . .. . . . . . .. . . . .2 . 92 - 4 . n - f a c t o r s f o r f r e e z e a n d t h aw . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . .. . . . . .. .2 . 1 13 - 1 . Mu l t i l a y er s o l u t i o n o f mo d i f i e d Be r g g r e n e q ua t i o n . . . . .. . .. . . . . .. . . . .. . . . . . .. . . .. . 3 . 53 - 2 . Thaw p en et r a t i o n b e n ea t h a s l a b - o n - g r a de b u i l d i n g c o n s t r u c t e do n p e r ma f r o s t . . . . .. . . . . . .. . . . . . . . .. . . . .. . . . .. . . . .. . . . . .. . .. . . . . .. . . . .. . . . . . . . .. . . . . . .3 . 1 33 - 3 . I n s u l a t e d p a v eme n t d e s i g n , no f r o s t p en et r a t i o n . . . . . . .. . . . . .. . . .. . .. . . . . . . . . .. . .. 3 . 2 03 - 4 . I n s u l a t e d p a v eme n t d e s i g n , f r o s t p en et r a t i o n . . . .. . . . .. . . . .. . . .. . . .. . . . . . .. . . .. 3 . 2 0B - 1 . Th e r ma l , f l u i d a n d e l e c t r i c a n al o g s . . . . .. . . . . . .. . . . .. . . . . .. . . . . .. . . .. . . . .. . . . . . . . .. . . . .B- 2

L I S T OF FI GURES

L I S T OF TABLES

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1 - 1 . P u r p o s e a n d s c o p e .Th i s man ua l c o n t a i n s c r i t e r i a andme t h o d s f o r c al c u l a t i n g t h e de p t h s o ff r e e z e a n d t h a w i n s o i l s , wi t h c on s i de r a -t i o n o f t he e f f e c t s o f o t h e r a d j a c e n tma t e r i a l s , f o r t h e de s i g n o f m l i t a r yf a c i l i t i e s i n s e a s o n a l f r o s t , a r c t i c a n ds u b a r e t i c r e g i o n s . The c on t e n t s a r ea p p l i c ab l e t o b o t h Ar my a n d Ai r Fo r c ec on s t r u c t i o n i n a r c t i c , s u ba r e t i c a n ds e a s o n a l f r o s t a r e a s . The da t a p r e -s e nt e d i n t h i s man ua l r e l a t e t o a r c t i ca n d s u b a r e t i c f a c i l i t y de s i g n p r e s e n t e di n t h e o t h e r ma n u a l s o f t he Ar c t i c a n dSu b a r c t i c Co n s t r u c t i o n s e r i e s .1 - 2 . R e f e r e n c e s a n d s y m b o l s .Ap p end i x C l i s t s t he r e f e r e n c e s f o rt h i s man u a l ; a pp e n di x A c on t a i n s a l i s to f s y mb o l s .1 - 3 . B a c k g r o u n d .a . The de p t h s t o wh i c h s o i l s mayf r e e z e a n d t h a w i s v e r y i mp o r t a n t i nt h e de s i g n o f p a v e me n t s , s t r u c t u r e sand u t i l i t i e s i n a r e as o f s e a s o n a l f r o s tand p e r ma f r o s t . Me t h o d s o f c a l c u l a t i n gs u c h de p t hs , b a s e d on he at - t r a n s f e rp r i n c i p l e s , a r e p r e s e n t e d h e r e . F o rde r i v at i o n o f b a s i c e q u at i o n s , a n d t h e

CHAPTER 1GENERAL

* T M 5 - 8 5 2 - 6 / A F R 8 8 - 1 9 , V o l u m e 6

u n de r l y i n g t he or y , s e e a pp e n d i x C a n dt h e b i b l i o g r a p h y .b . H e a t t r a n s f e r i n s o i l s i n v o l v i n gp h a s e c h a n g e o f p o r e wa t e r i s an e x -t r e me l y c o mp l e x p r o c e s s a n d manyp r o b l e ms de f y r i g o r o u s ma t h e ma t i c a lt r e a t me n t . The me t h o d s p r e s e n t e d h e r ea r e s i mp l i f i e d p r o c e du r e s de v e l o p e df o r t h e s o l u t i o n o f e n g i n e e r i n g de s i g np r o b l e ms .c . Se v e r a l a s s u mp t i o ns h a v e beenmade i n d e v e l o p i n g p r ac t i c al me t h o d so f c a l c u l a t i n g de p t h s o f f r e e z e o r t h a wi n s o i l s . I t i s as s umed t ha t e a c h l a y e r o fma t e r i a l i s homoge neo us a n d i s o t r o p i c ,a n d t h a t t h e a v e r a g e t h e r ma l p r o p e r -t i e s o f f r o z e n and u n f r o z e n s o i l s a r ea p p l i c ab l e . Un l e s s s pe c i f i c da t a a r ea v a i l a b l e , i t i s a l s o as s umed t h a t a l l s o i lwa t e r i s c on v e r t e d t o i c e , o r a l l i c e i sc on v e r t e d t o wa t e r , a t a t e mp e r a t u r e o f3 2 F . Th i s l a t t e r a s s u mp t i o n i s s u b -s t a n t i a l l y c or r e c t f o r c oa r s e - g r a i n e ds o i l s b u t o n l y p a r t i a l l y t r u e f o r f i n e -g r a i n e d s o i l s .d . The s e r v i c e s o f t h e U. S . Ar my Co l dRe g i o n s Re s e a r c h and En g i n e e r i n gL ab o r a t o r y ( USACRREL ) , Ha no v e r , NewHa mp s h i r e , a r e a v a i l a b l e t o a s s i s t i nt h e d e v e l o p me n t o f s o l u t i o n s f o r he a t -f l o w p r o b l e ms i n s o i l s .

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2 - 1 . D e f i n i t i o n s .De f i n i t i o n s o f c e r t a i n s p e c i a l i z e d t e r ms_ a p p l i c a b l e t o a r c t i c a n d s u b a r c t i cr e g i o n s a r e c o n t a i n e d i n TM 5 - 8 _ 5 _ 2 /AFR_ S- 1 G, _ Vo l u me 1 . Fo l l o wi n g a r ea dd i t i o n a l t e r ms u s e d s p e c i f i c a l l y i nh e a t - t r a n s f e r c a l c u l a t i o n s .a . Th e r ma l c o n d u c t i v i t y , KTh eq u a n t i t y o f h e a t f l o w i n a u n i t t i met h r o u g h a u n i t a r e a o f a s u b s t a n c ec a u s e d b y a u n i t t h e r ma l g r a di e n t .b . S p e c i f i c h e a t , c . T h e q u a n t i t y o fh e a t a bs o r b e d ( o r g i v e n u p ) b y a u n i twe i g h t o f a s u b s t a n c e when i t s t e mp e r at u r e i s i n c r e as e d ( o r de c r e a s e d ) b y 1de g r e e Fa h r e n h e i t ( F ) di v i de d b y t h eq u a n t i t y o f h e a t a bs o r b e d ( o r g i v e n u p )b y a u n i t we i g h t o f wa t e r when i t st e mp e r a t u r e i s i n c r e as e d ( o r de c r e a s e d )b y 1 F .c . Vo l ume t r i c h e a t c a p a c i t y , C Th eq u a n t i t y o f h e a t r e q u i r e d t o c h a n g et h e t e mp e r a t u r e o f a u n i t v o l u me b y1 F .For u n f r o z e n s o i l s ,

L= 144 y d

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WCu = y d( c + 1 . 0 1 0 0For f r o z e n s o i l s ,WC f = y d ( c + 0 . 5 1 0 0 ) ( e q 2 - 2 )W

W1 0 0

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Av e r a g e v a l u e s f o r mo s t s o i l s ,c a v g = Yd ( c + 0 . 7 5 1 0 0 ) ( e q 2 - 3 )wh e r e c = s p e c i f i c h e a t o f t h e s o i l s o l i d s( 0 . 1 7 f o r m o s t s o i l s )

Yd = d r y u n i t we i g h t o f s o i lw = wa t e r c o n t e n t o f s o i l i n p e r -c e n t o f dr y we i g h t .d Vo l u me t r i c l a t e n t h e a t o f f u s i o n ,L Th e q u a n t i t y o f h e a t r e q u i r e d t ome l t t h e i c e ( o r f r e e z e t h e wa t e r ) i n au n i t v o l u me o f s o i l wi t h o u t a c h a n g e i nt e mp e r a t u r e - i n B r i t i s h t h e r ma l u n i t s( B t u ) p e r c ub i c f o o t ( f t 3 ) :

( e q 2 - 4 )

e . t h e r ma l r e s i s t a n c e , R T h e r e c i p r o -c a l o f t h e t i me r a t e o f h e a t f l o w t h r o u g ha u n i t a r e a o f a s o i l l a y e r o f g i v e n t h i c kn e s s d p e r u n i t t e mp e r a t u r e di f f e r e n c e :

f . Th e r ma l d i f f u s i v i t y , a . An i n d i c a -t o r o f h ow e a s i l y a ma t e r i a l w i l l u n de r g ot e mp e r a t u r e c h a n g e :cg . T h e r ma l r a t i o , a -

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( e q 2 - 7 )wh e r ev o a b s o l u t e v al u e o f t h e d i f f e r e n c eb e t we e n t h e mean a n n u a l t e m-p e r a t u r e b e l o w t h e g r ou nds u r f a c e a n d 3 2 F .v s o n e o f t wo p o s s i b l e me a n i n g s ,d ep e nd i n g on t h e p r o b l e mb e i n g s t u d i e d :( 1 ) v s = n F / t ( o r n l / t )wh e r en =c on v er s i on f a c t o r f r o m a i ri n de x t o s u r f a c e i n de xF= a i r f r e e z i n g i n de xI = a i r t h awi n g i n de xt = l e n g t h o f f r e e z i n g ( o r t h awi n g )s e a s o n .( 2 ) v s a b s o l u t e v a l u e o f t h e d i f -f e r e n c e b e t we e n t h e me a na n n u a l g r o u n d s u r f a c e t e mp -

e r a t u r e a n d 3 2 F .I n t h e f i r s t c a s e , v s i s u s e f u l f o rc o mp u t i n g t h e s e a s o n a l de p t ho f f r e e z e o r t h aw I n t h e s e c o n dc a s e , i t i s u s e f u l i n c o mp u t i n gmu l t i y e a r f r e e z e o r t h a wde p t h s t h a t ma y de v e l o p a s ar e s u l t o f s ome l o n g - t e r mc h a n g e i n t h e h e a t b a l a n c e a tt h e g r o u n d s u r f a c e .h . Fu s i o n pa r ame t e r , / j "

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* TM 5 - 8 5 2 - 6 / A F R 8 8 - 1 9 , V o l u me 6C

2 - 2

N - _ " s ( e q 2 - 8 )Lwh e r e v e ha s t h e t wo p o s s i b l e me a n i n g sn o t e d a b o v e .i . " L a mb d a " c o e f f i c i e n t , J I " A f a c t o ra l l o wi n g f o r h e a t c a p a c i t y a n d i n i t i a lt e mp e r a t u r e o f t he g r o u n d ( s e e f i g . 3 - 1 ) .A a d j ) . ( e q 2 - 9 )

j . T h e r ma l r e g i me . Th e t e mp e r a t u r ep a t t e r n e x i s t i n g i n a s o i l b o d y i n r e l a -t i o n t o s e a s o n a l v a r i a t i o n s .k . B r i t i s h t h e r ma l u n i t , B t u . Thequ a n t i t y o f he a t r e qu i r e d t o r a i s e t het e mp e r a t u r e o f 1 pound ( l b ) o f wa t e r 1 Fa t a b o u t 4 0 F .

T a b l e 2 - 1 . S p e c i f i c h e a t v a l u e s o f v a r i o u s m a t e r i a l s . *( U . S . A r my Co r p s o f E n g i n e e r s ) .

S p e c i f i c h e a t v a l u e s shown t o n e a r e s t 0 . 0 1 . A v e r a g e v a l u e s l i s t e d whe r e t e m p e r a t u r e i s n o t shown .

2 - 2 . T h e r ma l p r o p e r t i e s o f s o i l s a n d o t he rc o n s t r u c t i o n m a t e r i a l s .a Th e b a si c t h e r ma l p r o p e r t i e s o fs o i l s a n d o t he r c o ns t r u c t i o n ma t e r i a l su s e d t o c a l c u l a t e d e p t hs o f f r e e z e a n dt h a w a r e s p e c i f i c h e a t , t he r ma l c o n -du c t i v i t y a n d v o l u me t r i c l a t e n t h e a t o ff u s i o n . Ot h e r t e r ms u s e d i n he a t - f l o wc a l c u l a t i o n s a r e d e r i v e d f r o m t he s ed a t a a n d t he e l e me n t s o f we i gh t , l e n g t h ,t e mp e r a t u r e a n d t i me .b . t he s p e c i f i c h e a t o f mos t d r y s o i l sn e a r t h e f r e e z i ng p o i n t may be a s s u m e dt o b e c o n st a n t a t t h e v a l u e o f 0 . 1 7 Bt u / l b F . S p e c i f i c he a t s o f v a r i o us ma t e r i a l sa r e g i v e n i n t a b l e 2 - 1 ; s e e t h e ASFI RAEGu i d e a n d Da t a Book o f t he Ame r i c a n

M a t e r i a l T e m p e r a t u r e( O F ) S p e c i f i c h e a t( B t u l l b O F )Al u mi n u m - 2 7 . 4 0 . 2 0A s b e s t o s f i b e r s - - 0 . 2 5C o n c r e t e ( a v g . s t o n e ) - - 0 . 2 0C o n c r e t e ( d a ms ) - 0 . 2 2Cop p e r 4 4 . 6 0 . 2 0Co r k b o a r d 9 1 0 . 4 3

- 1 9 0 . 2 9C o r k , g r a n u l a t e d - - 0 . 4 2F i b e r g l a s b o a r d 1 1 1 0 . 2 4- 2 2 0 . 1 9F o a mg l a s - 2 0 0 . 1 6Gl a s s b l o c k , e x p a n d e d 1 1 2 0 . 1 8Gl a s s s h e e t s - - 0 . 2 0G l a s s w o o l - - 0 . 1 6I c e 3 2 0 . 4 8I r o n ( a l p h a ) 4 4 . 6 0 . 1 1Maso n r y _ _ 0 . 2 2Mi n e r a l w o o l _ _ 0 . 2 2P e r l i t e , e x p a n d e d _ _ 0 . 2 2P o l y s t y r e n e , c e l l u l a r f o am _ _ 0 . 2 7P o l y u r e t h a n e f o am - - 0 . 2 5Sawdus t - - 0 . 6 0S n o w - - 0 . 5 0S t e e l _ _ 0 . 1 2S t r a w - - 0 . 3 5Wa t e r - - 1 . 0 0Woods ( a v g . ) Y 6 8 0 . 3 3Woods f i b e r b o a r d 1 4 8 0 . 3 4

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So c i e t y o f He a t i n g a n d Ai r Co n d i t i o n-i n g En g i n e e r s f o r t h e s p e c i f i c h e a tv a l u e s o f c ommon ma t e r i a l s .c . Th e t h e r ma l c o n du c t i v i t y o f s o i l si s d e p e n d e n t u p o n a nu mb e r o f f a c t o r s :d e ns i t y ; mo i s t u r e c o n t e n t ; pa r t i c l es h a p e ; t e mp e r a t u r e ; s o l i d , l i q u i d a n dv a p o r c o n s t i t u e n t s ; a n d t h e s t a t e o f t h ep o r e wa t e r , wh e t h e r f r o z e n o r u n f r o z e n .Av e r a g e v a l u e s , e x p r e s s e d i n B t u / f th o u r F , f o r f r o z e n a n d u n f r o z e n g r a nu -l a r s o i l s , s i l t s a n d c l a y s s h o u l d b e r e a df r o m f i g u r e s 2 - 1 t h r o u g h 2 - 4 . Th ec h a r t s f o r s a n d s a n d g r a v e l s a r e a p p l i c -a b l e when t h e s i l t a n d c l a y c o n t e n tt o ge t h e r ma k e u p l e s s t h an 20 %o f t h es o i l s o l i d s . Th e c h a r t s f o r s i l t a n d c l a ya r e a pp l i c a b l e when t h a t f r a c t i o n i s a tl e a s t 5 0 % Fo r i n t e r me d i a t e s i l t - c l a yf r a c t i o n s , i t i s r e c o mme n d e d t h a t t h es i mp l e a v e r a ge o f t h e v a l u e s f o r t h et wo s e t s o f c h a r t s b e u s e d . I n a l l c a s e s ,t h e e r r o r i n t h e t h e r ma l c o n du c t i v i t ye s t i ma t e s ma y b e 2 5%, a n d e v e nh i g h e r when t h e p e r c e n t a ge o f q u a r t zg r a i n s i n t h e s o i l i s e x c e p t i o n a l l y h i g ho r l o w . Fi g u r e s 2 - 5 a n d 2 - 6 p r e s e n t e s t i -ma t e s o f t h e a v e r a g e t h e r ma l c o n d u c -t i v i t y o f f r o z e n a n d u n f r o z e n p e a t . Ane x c e l l e n t s o u r c e o f da t a f o r d r y c o n -s t r u c t i o n ma t e r i a l s i s t h e ASHRAHGu i d e a n d Da t a B o o k . Th e r ma l c o n -du c t i v i t y v a l u e s f o r a nu mb er o f c o m -mo n c o n s t r u c t i o n ma t e r i a l s a r e l i s t e di n t a b l e 2 - 2 .d . Th e l a t e n t h e a t o f f u s i o n i s t h ea mou nt o f h e a t r e q u i r e d t o c a u s e ap h a s e c h a n g e i n s o i l mo i s t u r e . Th i sa mou nt o f h e a t d o e s n o t c h a n g e t h et e mp e r a t u r e o f t h e s y s t e m whenf r e e z i n g o r t h awi n g t a k es p l a c e . Th eg r a v i me t r i c l a t e n t h e a t o f f u s i o n o fwa t e r i s a s s u m e d t o b e 1 4 4 Bt u / l b . Th ea mou nt o f h e a t e n e r g y r e q u i r e d t o c o n -v e r t 1 f t o f wa t e r t o i c e i s ( 1 4 4 X 6 2 . 4 = )9 0 0 0 B t u / f t 3 a n d t o c h a n g e 1 f t 3 o f i c e t owa t e r i s ( 1 4 4 X 0 . 9 1 7 X 6 2 . 4 = ) 8 2 4 0B t u / f t 3 . ( No t e : Th e de ns i t y o f wa t e r 6 2 . 4l b / f t 3 a n d t h a t o f p u r e i c e I s 5 7 . 2 l b . f t 3 ) .e . Fi g u r e s 2 - 7 a n d 2 - 8 ma y b e u s e dt o d e t e r mi n e t h e a v e r a ge v o l u me t r i ch e a t c a p a c i t y a n d v o l u me t r i c l a t e n th e a t o f f u s i o n , r e s p e c t i v e l y , o f mo i s ts o i l s .f . Th e f o l l o wi n g e x a mp l e i l l u s t r a t e st h e s i g n i f i c an c e o f l a t e n t h ea t o f f u s i o nr e l a t i v e t o t h e v o l u me t r i c h ea t c apa c i t yf o r a mo i s t s o i l . As s u me a s o i l h av i n g a

d r y u n i t we i g h t o f 1 2 0 1 b / f t 3 a n d a wa t e rc o n t e n t o f 1 5 p e r c e n t . I t s v o l u me t r i cl a t e n t h e a t o f f u s i o n , L , i s ( 1 4 4 X 1 2 0 X0 . 1 5 = 1 2 5 9 2 B t u / f t 3 a n d i t s a v e r a ge v o l u -me t r i c h e a t c a p a c i t y , C , i s ( 1 2 0 [ 0 . 1 7 +0 . 7 5 X 0 . 1 5 ] = ) 3 3 . 9 B t u / f t 3 OF . Th e q u a n -t i t y o f h e a t r e q u i r e d t o c h a n g e t h ep h a s e o f p o r e wa t e r i n 1 f t 3 o f t h i s s o i l a t3 2 F i s t h e s am e a s t h a t r e q u i r e d t oc a u s e a t e mp er a t u r e c h a n g e o f( 2 5 9 2 / 3 3 . 9 = ) 7 6 . 4 ' F when a p h a s ec h a n g e i s n o t i n v o l v e d .2 - 3 . F u n da me n t a l c o ns i de r a t i o ns .a . T h e o r e t i c a l b a s i s . Th e f r e e z i n go r t h a wi n g o f s o i l s i s t h e r e s u l t o f r e -mo v i n g o r a dd i n g h e a t t o a n e x i s t i n gs o i l m a s s . Th e mov e me nt o f h e a t i sa l wa y s i n t h e d i r e c t i o n o f l o we r t e mp e r -a t u r e . Th e t i me r a t e o f c h a n g e o f h e a tc o n t e n t d e p e n d s o n t h e t e mp e r a t u r ed i f f e r e n t i a l i n t h e d i r e c t i o n o f h e a t f l o wa n d o n t h e t h e r ma l p r o p e r t i e s o f t h es o i l .b . Ph y s i c a l f a c t o r s a n d da t a r e -q u i r e d . Ca l c u l a t i o n o f t h e d e p t h o ff r e e z e o r t h aw i s b a s e d o n k n o wl e d g eo f t h e p h y s i c a l a n d t h e r ma l p r o p e r t i e so f t h e s o i l i n t h e p r o f i l e , t h e e x i s t i n gt h e r ma l r e g i me , a n d t h e n a t u r e a n dd u r a t i o n o f b ou nd a r y c o n d i t i o n s

1 5 001 41 3

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T h e r ma l c o n d u c t i v i t y K i s e x p r e s s e d i n Bt u p e r h o u r p e r s q u a r e f o o t p e ru n i t t h e r m a l g r a d i e n t i n F p e r f o o t .Dash ed l i n e r e p r e s en t s e x t r a po l a t i o n .

( U . S . A r my Co r p s o f En g i n e e r s )Fi g u r e 2 - 1 . A v e r a g e t h e r ma l c on du c t i v i t y f o r s a n d s a n d

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1 0 2 0 30 4 0 5 0 6 0 7 0Mo i s t u r e C o n t e n t ( / , )w

Ther ma l c o n d u c t i v i t Y K i s e x p r e s s e d ' i q : $ t u p e r h o u r p e r s q u a r e f o o t p e ru n i t t h e r m a l g r a d i e n t i n F p e r f o o t .Da s he d l i n e r e p r e s en t s e x t r a po l a t i o n .( U . S . A r my Co r p s o f En g i n ee r s )F i g u r e 2 - 3 . Av e r a g e t h e r ma l c ond uc t i v i t y f o r s i l t and c l a y

s o i l s , f r o z en .

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( U . S . Ar my Co r p s o f En g i n e er s )

c a us i n g a c h a n g e i n t h e t h er ma l r e g i meDa t a pe r t i nen t t o t h e s o i l p r o f i l e i n -c l u d e g r a i n - s i z e d i s t r i b u t i o n , c l a s s i f i c a -t i o n , d ens i t y , wa t e r o r i c e c on t en t , a n dt e mp e r a t u r e o f e a c h s o i l s t r a t u mKnowl e d g e o f t h e t h er ma l p r ope r t i e so f a l l t h e ma t e r i a l s i n t h e h e a t f l o wpa t h i s a l s o r e q u i r e d . Me a s u r e d o r a s -s umed t e mpe r a t u r e s wi t h i n t h e s o i lmass d e t e r mi n e t h e i n i t i a l c ond i t i ons .I f s ur f a c e t emp er a t u r es c a n b e a s s u me dt o b e s p a t i a l l y u n i f o r ma nd t h e t h e r ma li n f l u en c e o f a n y b ur i e d s t r u c t u r e s c a nb e c o n s i d e r e d n e g l i g i b l e , o ne - d i me n -

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_ 02 0 0 400 6 0 0 8 0 0 1 0 0 0 1200 1 4 00 1600

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F i gu r e 2 - 5 . Av e r a g e t h e r ma l c ond uc t i v i t y f o r p e a t , f r o z e n .

6 0 0 8 0 0 1 0 0 0 1 20 0 1 4 0 0Wa t e r C o n t e n t ( %d r y w e i g h t )

F i g u r e 2 - 6 . Av e r a g e t h e r ma l c on du ct i v i t y f o r p e a t , u nf r o z en .

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s i o n a l h ea t f l owmay b e a s s u med , t h e r e -b y s i mp l i f y i n g t h e p r o b l e m a n d i t ss o l u t i on .2 - 4 . F r e e z i n g a n d t h awi ng i n d e x e s .a . Phy s i c a l c o n c e p t a n d q u a n t i t a -t i v e meas u r emen t . T h e pene t r a t i o n o ff r eez i ng o r t h a wi n g t emp er a t u r e s i n t os o i l pa r t l y d e p e n d s o n t h e ma gn i t u d ea n d d u r a t i o n o f t h e t empe r a t u r e d i f -f e r e n t i a l a t t h e a i r - g r o u n d i n t er f a c e .The ma gn i t u d e o f t h e t empe r a t u r e d i f -f e r e n t i a l i s e x p r e s s ed a s t h e nu mbe r o fd e gr e es t h a t t h e t empe r a t u r e i n t h e a i r

2 - 5

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T a b l e 2 - 2 . T h e r ma l p r o p e r t i e s o f c o n s t r u c t i o n( U . S . A r my C o r p s o f E n g i n e e r s ) . m a t e r i a l s

k Kc o n d u c t i v i t y c o n d u c t i v i t yT y p e o f U n i t w e i g h t ( B t u l f t 2 - h r " F ( B t u l f t "ma t e r i a l D e s c r i p t i o n ( I b l i t 3 ) p e r i n . ) h r - O F )A s p h a l t Mi x w i t h 6 % b yp a v i n g . we i g h t c u t -m i x t u r e b a c k a s p h a l t 1 3 8 1 0 . 3 0 . 8 6Co n c r e t e Wi t h s a n d a n d

g r a v e l o r s t o n ea g g r e g a t e ( o v e n -d r i e d ) 1 4 0 9 . 0 0 . 7 5Wi t h s a n d a n dg r a v e l o r s t o n ea g g r e g a t e ( n o td r i e d ) 1 4 0 1 2 . 0 1 . 0 0Wi t h l i g h t w e i g h ta g g r e g a t e s , i n - 1 2 0 5 . 2 0 . 4 3c l u d i n g e x p a n d e ds h a l e , c l a y o r 1 0 0 3 . 6 0 . 3 0s l a t e ; e x p a n d e d 8 0 2 . 5 0 . 2 1s t a g s ; c i n d e r s ; 6 0 1 . 7 0 . 1 4p um c e ; p e r l i t e ; 4 0 1 . 1 5 0 . 0 9 6v e r m i c u l i t e ; a l s o 3 0 0 . 9 0 0 . 0 7 5c e l l u l a r c o n - 2 0 0 . 7 0 0 . 0 5 8c r e t e s .

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B l a n k e t Mi n e r a l w o o l ,a n d b a t t f i b r o u s f o r m ,i n s u l a - p r o c e s s e d f r o m 1 . 5 - 4 . 0 0 . 2 7 0 . 0 2 2t i o n r o c k , s l a g , o rg l a s s

Wo o d f i b e r 3 . 2 - 3 . 6 0 . 2 5 0 . 0 2 1B oa r d C e l l u l a r g l a s s 9 . 0 1 ' 0 . 3 9 0 . 0 3 2a n d Co r k b o a r d ( w i t h -s l a b o u t a d d e d b i n d e r ) 6 . 5 - 8 . 0 1 0 . 2 7 0 . 0 2 2i n s u l a - G l a s s f i b e r 9 . 5 - 1 1 . 0 0 . 2 5 0 . 0 2 1t i o n Wo o d o r c a n e f i b e r -

i n t e r i o r f i n i s h 1 5 . 0 0 . 3 5 0 . 0 2 9( p l a n k , t i l e , l a t h )E x p a n d e d p o l y s t y r e n e 1 . 6 0 . 2 9 0 . 0 2 4E x p a n d e d u r e a f o r m a l -

d e h y d e 1 . 0 0 . 2 5 - -E x p a n d e d p e r l i t e 9 . 5 " 1 1 . 5 0 . 3 4 - -P o l y u r e t h a n e f o a m 1 . 5 - 3 . 0 0 . 1 7 - -Mi n e r a l w o o l w i t hr e s i n b i n d e r 1 5 . 0 0 2 8 0 . 0 2 3Mi n e r a l w o o l w i t h2 - 6 a s p h a l t b i n d e r 1 5 . 0 0 3 1 0 . 0 2 6

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Va l u e s f o r k a r e f o r d r y b ui l d i n g ma t e r i a l s a t a mean t e mp e r a t u r e o f 7 5 F e x c e p t a s n o t e d ; we t c on di t i o ns w i l l a dv e r s e l y a f f e c t v a l u e s o fmany o f t h es e ma t e r i a l s .

t Mea n t e mp er a t u r e o f 6 0 F .Mea n t e mp er a t u r e o f 3 2 0 F .

o r a t t h e g r o u n d s u r f a c e i s a b o v e ( p o s i -t i v e ) o r b e l o w ( n e g a t i v e ) 3 2 F , t h e a s -s umed f r e ez i n g p o i n t o f wa t e r . Thed u r a t i o n i s e x p r e s s e d i n d a y s .b . Ai r f r e e z i n g i n de x ( F ) a n d a i rt h awi n g i n de x ( 1 ) . The a i r f r e ez i n g a n da i r t h awi n g i n de x es , a s de f i n ed i n TM5 - 8 6 2 - 1 / AFR 8 8 - 1 9 , Vo l ume 1 , may b ed e t e r mi n e d b y t h e f o l l o wi n g me t h o d s .( 1 ) Summat i on o f de g r e e - d ay s o ff r e e z e a n d t h a w f r o m a v e r a g e d a i l yt e mp e r a t u r e s . I f T 1 i s t h e max i mumd a i l y a i r t e mp e r a t u r e a n d T 2 i s t h emi n i mum d a i l y a i r t e mp e r a t u r e . t h ea v e r a g e d a i l y a i r t e mp e r a t u r e T , mayb e t a k e n a s 1 / 2 ( T I + T 2 ) , a n d t h e nu mbero f d eg r e e - d ay s f o r t h e d a y i s ( T - _ 3 2 ) .Th e s u mma t i o n o f t h e d eg r e e - d ay s f o r "a f r e ez i n g o r t h awi n g s e a s o n g i v es t h ea i r f r e e z i n g o r a i r t h a wi n g I n d e x .Ta b l e 2 - 3 i l l u s t r a t e s t h e me t h o d u s e dt o o b t a i n t h e s u mma t i o n o f d eg r e e - d ay sf o r a 1 - we e k p e r i o d , a s s u mi n g t h a t - 4 56d eg r e e - d ay s h a d b e e n a c c u mu l a t e ds i n c e f r e ez i n g b e g a n . An a v e r a g e d a i l yt e mp e r a t u r e b a s e d on h o u r l y t e mp e r a -t u r e s wo u l d b e s l i g h t l y mor e a c c ur a t e ,b u t s u c h p r e c i s i o n i s n o t u s ua l l y wa r -r a n t e d . The n eg a t i v e s i g n , i n d i c a t i n g

f r e e z i n g d eg r e e - d a y s , i s u s ua l l yo mi t t e d .( 2 ) Ca l c u l a t i o n f r o ma v e r a g e mo n t h -l y t e mp e r a t u r e s . The f r e e z i n g o rt h awi n g i n de x may b e c a l c u l a t e d b yd e t e r mi n i n g t h e a r e a b e t we e n t h e 3 2 Fl i n e a n d t h e c u r v e o f a v e r a g e mo n t h l yt e mp e r a t u r e a n d t i me , t a k e n o v e r t h ea p p r o p r i a t e s e a s o n . The a r e a may b ed e t e r mi n e d b y p l a n i me t e r o r a s i mp l ea p p r o x i ma t i o n r u l e ( S i mp s o n ' s r u l e ,mi d o r d i n a t e r u l e , e t c . ) . Th e a r e as a r ee x p r e s s ed i n u n i t s o f d eg r e e - d ay s , r e -s u l t i n g i n a s u mma t i o n o f d eg r e e - d a y so r a f r e ez i n g o r t h awi n g i n de x . F o r a ne x a mp l e r e f e r t o f i g u r e 2 - 9 , a p l o t o ft h e mo n t h l y a v e r a g e t e mp e r a t u r e s a tF a i r b a n k s , A l a s k a, f r o m S e p t e mb e r1 9 4 9 t o Oc t o b e r 1 9 5 0 . De t e r mi n a t i o n o fa r e as b y p l a n i me t e r g a v e a f r e ez i n gi n d e x o f - 5 2 4 0 d e g r e e - d a y s and at h awi n g i n de x o f + 3 4 2 0 d eg r e e - d ay s .The u s e o f S i mp s o n ' s r u l e g a v e a f r e e z -i n g i n de x o f - 5 3 9 0 a n d a t h awi n g i n de xo f + 3 4 6 0 d e g r e e - d a y s . E i t h e r p a i r o fi n d i c es i s a de qu a t e f o r c o mp u t a t i o n s .c . Su r f a c e f r e e z i n g a n d s u r f a c e -t h a wi n g i n de x es . F o r d e t e r mi n i n g t h eh e a t f l o w wi t h i n t h e s o i l , i t i s n e c e s -

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s a r y t o d et e r mi n e o r es t i ma t e t h e t emp -e r a t u r e c on d i t i o n a t t h e g r o un d s u r -f a c e . Si n c e a i r t emp e r a t u r e s a r e gen -e r a l l y a v a i l a b l e a n d s u r f a c e t e mp e r a -t u r e s a r e n o t , a c or r e l a t i on b e t we e nt h es e t emp e r a t u r e s h el p s e s t a b l i s h t h et h e r ma l b o un d ar y c on d i t i o n a t t h eg r o un d s u r f a c e . Th e c o mb i n e d e f f e c t so f r a d i a t i v e , c o n v ec t i v e a n d c o n d u c t i v eh e a t e x c h a n g e a t t h e a i r - g r o u n d i n t e r -f a c e o f t e n mus t be c o n s i d er e d i n d e t e r -mi n i n g s u r f a c e t emp e r a t u r e .d Co r r e l a t i o n o f a i r a n d s ur f a c ei n d e xe s . No s i mp l e c or r e l a t i on e x i s t sb e t we e n a i r a n d s ur f a c e i n d e x es . Th ed i f f e r e n c e b e t we e n a i r a n d s u r f a c et e mp e r a t u r e s a t a n y s p e c i f i c t i me i si n f l u e n c e d b y l a t i t u d e , c l o u d c o v e r ,t i me o f y e a r , t i me o f d a y , a t mo s p h e r i ch umi d i t y a n d s t a b i l i t y , wi n d s p e e d ,s n o w c o v e r a n d g r o un d s ur f a c e c h a r -

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( 1 ) F r e e z i n g c o n d i t i o n s The n -f a c t o r i s v e r y s i g n i f i c a n t i n a n a l y t i c a lg r o u n d s t u d i e s . I t g e n e r a l l y i n c r e a s eswi t h wi n d s p e e d . Snow c o v e r r e f l e c t s al a r g e p a r t o f i n c o mi n g s o l a r r a d i a t i o nr e s u l t i n g i n h i g h e r f r e e z i n g i n d e x e sa t t h e s now s u r f a c e , b u t i t s i n s u l a t i n ge f f e c t c a n g r e a t l y r e d u c e t h e f r e e z i n gi n d e x a t t h e g r o u n d s u r f a c e . T h e e f f e c t so f t u r f o r a n o r g a n i c g r o u n d c o v e r o nt h e h e a t f l o w p r o c e s s e s a t t h e a i r -g r o u n d i n t e r f a c e a r e e x t r e me l y v a r i -a b l e a n d d i f f i c u l t t o e v a l u a t e . On t h e

b a s i s o f o b s e r v a t i o n s a n d s t u d i e s madet o d a t e , t h e n - f a c t o r s g i v e n f o r a v e r a g ec o n d i t i o n s i n t a b l e 2 - 4 s h o u l d b e u s e dt o c o n v e r t t h e a i r f r e e z i n g i n d e x t o t h es u r f a c e f r e e z i n g i n d e x i n t h e a b s e n c eo f s p e c i f i c me a s u r e me n t s a t t h e s i t e o fp l a n n e d c o n s t r u c t i o n .( 2 ) T h a wi n g c on d i t i o n s . The n -f a c t o r f o r t h a wi n g c o n d i t i o n s i s a f -f e c t e d b y t h e s a m e f a c t o r s a s t h o s e f o rf r e e z i n g c o n d i t i o n s . I t i s t h e r a t i o o fs u r f a c e d e g r e e - d a y s o f t h a w ( d e g r e esa b o v e 3 2 F ) t o a i r d e g r e e - d a y s o f t h a w .

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e . De s i g n i n d e x e s . F o r d e s i g n o f p e r -man en t p a v e me n t s , t h e d e s i g n f r e e z i n g( o r t h awi n g) i n d e x s ho u l d be t h e a v e ra g e a i r f r e e z i n g ( o r t h awi n g) i n d e x o ft h e t h r e e c o l d e s t wi n t e r s ( o r wa r me s ts u mme r s ) i n t h e l a t e s t 3 0 y e a r s o f

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n - f a c t o r s f o r f r e e z e a n d t h aw ( r a t i o o f s u r f a ce i n de x t o a i r i n d ex )( U . S . A r my Co r p s o f En g i n e e r s ) .

F o r f r e e z i n gc o n d i t i o n s F o r t h a wi n gc o n d i t i o n sS n ow s u r f a c e 1 . 0 _ -P o r t l a n d - c e me n t c o n c r e t e 0 . 7 5 1 . 5B i t u mi n ou s p a ve men t 0 . 7 1 . 6 t o 2 tB a r e s o i l 0 . 7 1 . 4 t o 2 tS h a d e d s u r f a c e 0 . 9 1 . 0T u r f 0 . 5 0 . 8T r e e - c o v e r e d 0 . 3 * * 0 . 4

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' TM 5 - 8 5 2 - 6 / AFR 8 8 - 1 9 , V o l ume 6r e c o r d I f 3 0 y e a r s o f r e c o r d a r e n o ta v a i l a b l e , t h e a i r f r e e z i n g ( o r t h a wi n g )i n d e x f o r t h e c o l d e s t wi n t e r ( o rwa r me s t s umme r ) i n t h e l a t e s t 1 0 - y e arp e r i o d may b e u s e d . F o r d e s i g n o ff o u n d a t i o n s f o r a v e r a g e p e r ma n e n ts t r u c t u r e s , t h e d e s i g n f r e e z i n g ( o rt h a wi n g ) i n d e x s h o u l d b e c o mp u t e df o r t h e c o l d es t ( o r wa r me s t ) wi n t e r i n3 0 y e a r s o f r e c o r d o r s h ou l d b e e s t i -mat ed t o c o r r e s p o n d wi t h t h i s f r e -q u e n c y i f t h e numbe r o f y e a r s o f r e c o r di s l i mi t e d . Pe r i o d s o f r e c o r d u s e ds h ou l d b e t h e l a t e s t a v a i l a b l e . To a v o i dt h e n ec es s i t y f o r a do p t i n g a n ew a n do n l y s l i g h t l y d i f f e r e n t f r e e z i n g ( o rt h awi n g ) i n d e x e a c h y e a r , t h e d es i g ni n d e x a t a s i t e wi t h c on t i n u i n g c o n -

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/ o ( KOr ZESUE )( F A I RBANKS!

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/ 6 ( F r YUKON/( BETTLES )

NOTE . r h u l r , b n e n l o n d d o l o f r o m / 9 52 t h r o u g h 1 9 6 1I I

8 0 0 0 9 0 0 000 5 0 0 0MEAN A I R- F REEZ I NG I NDEX , D e g r e e - d a y s

( U . S . A r my Co r p s o f En g i n e er s )

s t r u c t i on n e e d n o t b e c h a n g e d mor eo f t e n t h a n o n c e i n 5 y e a r s u n l e s s t h emo r e r e c en t t emp e r a t u r e r e c o r d s i n -d i c a t e a s i g n i f i c a n t c h a n g e . The d i s t r i -b u t i o n o f d es i g n f r e e z i n g a n d t h a wi n gi n d e x v a l u e s i n No r t h Ame r i c a i sp r e s e n t e d i n TM 5 - 8 6 2 - 1 / AF R 8 8 - 1 9 ,Vo l u me 1 . Th e r e l a t i v e l y l i n e a r r e l a t i o n -s h i p b e t we e n r e c o r d ed max i mum i n -d e x e s a n d mean f r e e z i n g a n d t h a wi n gi n d e x e s shown i n f i g u r es 2 - 1 1 a n d 2 - 1 2may b e u s e d i n c o n j u n c t i o n wi t h d i s -t r i b u t i on o f mean a n d f r e e z i n g a n dt h a w i n g i n d ex e s i n TM 5 - 8 6 2 - 1 / AFR8 8 - 1 9 , Vo l u me 1 t o d e t e r mi n e t h e d e s i g ni n d e x v a l u e s f o r a r c t i c a n d s ub a r c t i cr e g i o n s .

6000 7000F

F i g ur e 2 - 1 1 . Re l a t i on s h i p b e t we e n mean f r e ez i n g i n de x an d max i mum f r e e z i n g i n d ex f o r 1 0y e ar s o f r e c o r d , 1 9 5 3- 1 9 6 2 ( a r c t i c an d s ub ar c t i c r e g i o n s ) .

NWWW ( 6ULKANA) o6000 ( B / 6 DE L T A ! o ,Q

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I L 3500r

Xz 2500Z

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4000

3000

1 5 0 0

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500

( THUL E) 0- I BARROW0

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NOTE . ' Mu l e , a r e o n l e n d d ol o f r o m 1 9 5 2 / Amup h / o f / .Im- L j I 1 I 1 I I I t0 500 1 0 0 0 1 5 0 0 2000 2 5 0 0 3000 3500

MEAN AI R- THAW NG I NDEX, D" r wdo y sF( U . S . Ar my Co r p s o f En g i n e er s )

* TM 5 - 8 5 2- 6 / AFR 8 8 - 1 9 , Vo l ume 6

F i g u r e 2 - 1 2 . R e l a t i o n s h i p b e t w ee n mean t h a wi n g i n d e x a n d ma x i mum t h a wi n g i n d e x f o r 1 0 y e a r s o fr e c o r d , 1 9 5 3 - 1 9 6 2 ( a r c t i c a n d s u b a r c t i c r e g i o n s ) .

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CHAPTER 3ONE- DI MENSI ONAL L I NEAR AND PERI ODI C HEAT FL OW

3 - 1 . T h e r ma l r e g i me .The s e a s o n a l d ep t h s o f f r o s t a n d t h a wp en e t r a t i o n i n s o i l s d e p e n d s u p o n t het he r ma l p r o p e r t i es o f t h e s o i l mas s , t h es ur f a c e t emp er a t u r e ( u pp er b o un d a r yc o n d i t i o n ) a n d t he t h er ma l r e g i me o ft he s o i l a t t he s t a r t o f t he f r eez i n g o rt h a wi n g s e a s o n Many me t h o d s a r ea v a i l a b l e t o es t i ma t e f r o s t a n d t h a wp en e t r a t i o n d ep t h s a n d s ur f a c e t em-p er a t u r e s . So me o f t h es e a r e s um-ma r i z e d i n a p p e n d i x B . Th i s c ha p t e rc o n c en t r a t e s o n s o me t e c hn i q ues t h a tr e q u i r e o n l y r e l a t i v e l y s i mp l e h a n dc a l c ul a t i o n s . Fo r t he c o mp ut a t i o n a lme t h o d s d i s c u s s ed bel o w, t he i n i t i a lg r o u n d t emp er a t u r e i s a s s u m e d t o u n i -f o r ml y eq ua l t he mea n a n n u a l a i r t e m-p e r a t u r e o f t he p a r t i c u l a r s i t e u n d e rc o n s i d e r a t i o n . Th e u p p e r b o un d a r yc o n d i t i o n i s r ep r e s en t ed b y t h e s u r -f a c e f r eez i n g ( o r t ha wi n g ) i n d ex .3 - 2 . M o d i f i e d B e r g g r e n e q u a t i o n .

a Th e d ep t h t o wh i c h 3 2 F t em-p er a t u r es w i l l p en et r a t e i n t o t he s o i lma s s i s b a s e d u p o n t he " mo d i f i ed "Be r g g r e n eq ua t i o n , ex p r e s s e d a s :48Kn F

Lo r ( e q 3 - 1 )4 8 K n l

Lwh er eX= d ep t h o f f r eez e o r t h a w ( f t )K= t h e r ma l c on d uc t i v i t y o f s o i l( B t u / f t h r F )L = v o l ume t r i c l a t e n t hea t o f f us i o n( B t u / f t 3 )n=c onv er s i on f a c t o r f r o m a i ri n de x t o s ur f a c e i n dex ( d i men -s i o n l e s s )F= a i r f r eez i n g i n de x ( F - d a y s )I = a i r t h a wi n g i n d ex ( F - d a y s )

' TM 5 - 85 2 - 6 / AFR 8 8 - 1 9 , Vo l u me 6

= c o e f f i c i e n t t h a t c o n s i d e r s t hee f f e c t o f t emp er a t u r e c h a n g e si n t h e s o i l ma s s ( d i me n s i o n l e s s )The A c o e f f i c i e n t i s a f un c t i o n o f t hef r eez i n g ( o r t ha wi n g ) i n dex , t he meana n n u a l t emp er a t u r e o f t he s i t e , a n d t het he r ma l p r o p e r t i e s o f t he s o i l . Fr e ez ea n d t h a w o f l o w- mo i s t u r e - c o n t en t s o i l si n t he l o we r l a t i t u d e s i s g r e a t l y i n -f l uen c ed b y t h i s c o e f f i c i e n t I t i s d e t e r -mi n e d b y t wo f a c t o r s : t he t he r ma l r a t i oa a n d t he f u s i o n p a r a me t e r / j . T h e s eh a v e b e e n d ef i n ed i n p a r a g r a p h 2 - 1 .Fi g ur e 3 - 1 shows A a s a f u n c t i o n o f aa n d N .b A c o mp l e t e d e v e l o p me n t o f t h i seq ua t i o n a n d a d i s c us s i o n o f t h e n e c e s -s a r y a s s ump t i o n s a n d s i m p l i f i c a t i o n sma d e d u r i n g i t s d e v e l o p me n t a r e n o tp r e s e n t ed he r e . A f ew o f t h e mor e i m-p o r t a n t a s s ump t i o n s a n d s om e o f t hee q ua t i o n l i mi t a t i o n s a r e d i s c u s s e db e l o w . Th e a s s ump t i o n s a n d l i mi t a -t i o n s a pp l y r eg a r d l es s o f wh et h er t h eeq ua t i o n i s u s e d t o d et e r mi n e t h e d ep t ho f f r eez e o r t he d ep t h o f t h a w.( 1 ) As s ump t i o n s . Th e ma t hema t i c a lmo d e l a s s u me s o ne - d i men s i o n a l hea tf l o w wi t h t he en t i r e s o i l ma s s a t i t smea n a n n u a l t emp er a t u r e ( MAT) p r i o rt o t he s t a r t o f t he f r eez i n g s e a s o n . I ta s s um e s t h a t when t he f r eez i n g s e a s o ns t a r t s , t h e s u r f a c e t emp er a t u r ec h a n g e s s u d den l y ( a s a s t ep f un c t i o n )f r o m t he mean a n n u a l t emp er a t u r e t oa t emp er a t u r e v 3 d e g r e es b e l o w f r e e z -i n g a n d t h a t i t r ema i n s a t t h i s n ewt emp er a t u r e t h r o u g ho u t t he en t i r ef r eez i n g s e a s o n . L a t en t h e a t a f f e c t s t hemode l b y a c t i n g a s a hea t s i n k a t t hemo v i n g f r o s t l i n e , a n d t he mode l a s -s ume s t h a t t h e s o i l f r eez es a t a t emp er a -t u r e o f 3 2 F .( 2 ) L i mi t a t i o n s . Th e mo d i f i e dBe r g g r e n eq ua t i o n i s a b l e t o d et e r mi n ef r o s t p en e t r a t i o n i n a r ea s wh e r e t heg r o u n d b e l o w a d ep t h o f s ev e r a l f e e t

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* TM 5 - 8 5 2- 6 / AFR 8 8 - 1 9 , Vol ume 6

3 - 2

I I I MI dMI M%/I mp APEFAMUFAFI 19 ,aA P IMAOA

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l I /I%I% MEN. MMwM I 0; WAMIWI "I%L//w%

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F i g u r e 3 - 1 . A c o e f f i c i e n t i n t h e mo d i f i e d B e r g g r e n f o r m u l a .

NW4 - 10a0U

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r ema i n s p e r ma n e n t l y t h a we d , o r t od e t e r mi n e t h aw pene t r a t i o n i n a r e aswhe r e t h e g r o u nd b e l o w a d ep t h . o fs e v e r a l f ee t r e ma i n s p e r ma ne n t l yf r o z en . T h e s e t wo c o nd i t i o ns a r e s i m i -l a r i n t h at t h e t e mp er a t u r e g r a d i en t sa r e o f t h e same s h a p e , a l t h o ugh r e -v e r s ed wi t h r e s pe c t t o t h e 3 2 F l i n e . Nos i mp l e a n a l y t i c a l me t h o d e x i s t s t o d e t e r -mi n e t h e d ep t h o f t h aw i n s e a s o n a lf r o s t a r e as o r t h e d ep t h o f f r eez e i np er ma f r o s t a r e a s , a n d s uc h p r o b l emss h o ul d b e r e f e r r e d t o HQDA ( DAEN-ECE- G) o r HQAFESC . Nume r i c a l t e c h -n i q ue s a n d c o mp u t e r p r o g r ams a r eav ai l ab l e t o s o l v e mo r e c o mp l e x p r o b -l e ms . Ap p en d i x B d i s c us s es some t h e r -ma l c o mp u t e r mo d e l s f o r c o mp ut i n gf r ee z e a n d t h aw d ep t h s . The mo d i f i e dBe r g gr e n equa t i o n c a n n o t b e u s e d s u c -c es s f u l l y t o c a l c u l a t e pene t r a t i o n o v e rp a r t s o f t h e s e a s o n . Th e mo d i f i e dBe r g gr e n equa t i o n d o es n o t a c c o u n tf o r a n y mo i s t u r e mov emen t t h a t mayo c c u r wi t h i n t h e s o i l . Th i s l i m i t a t i o nwo u l d t e n d t o r e s u l t i n o v e r e s t i ma t e df r o s t pene t r a t i o n ( i f f r o s t h e a v e i s s i g -n i f i c an t ) o r u n d e r e s t i ma t e d t h awpene t r a t i o n( 3 ) App l i c ab i l i t y . Th e mo d i f i e dBe r g gr e n equa t i o n i s mo s t o f t e n a p p l i c -ab l e i n e i t h e r o f t wo w a y s : t o c al c ul a t et h e mul t i - y ea r d ep t h o f t h a w i n p e r ma -f r o s t a r e as o r t o c a l c u l a t e t h e d ep t h o fs e a s o n a l f r o s t pene t r a t i o n i n s e a s o n a lf r o s t a r e as . I t i s a l s o s ome t i me s u s e d t oc a l c ul a t e s e a s o n a l t h a w pene t r a t i o n( a c t i v e l a y er t h i c k n es s ) i n p er ma f r o s ta r e as .3 - 3 . Ho mo g en e o u s s o i l s .T h e d ep t h o f f r e ez e o r t h aw i n o n el a y er o f h o mo g e n e o u s s o i l may b e d e t e r -mi n e d b y mea n s o f t h e mo d i f i e dBe r g gr e n equa t i o n . A t h i n b i t umi n o usc o nc r e t e pa v emen t w i l l no t a f f e c t t h eh o mo g en e i t y o f t h i s l a y e r i n c a l c u l a -t i o n s , b u t a p o r t l a nd - c emen t - c o nc r e t ep a v eme n t gr ea t e r t h an 6 i n c h es t h i c ks h o u l d b e t r e a t e d a s a mul t i l a y er eds y s t em . I n t h i s e x amp l e f o r h o mo g e -n e o u s s o i l s , d e t e r mi n e t h e d ep t h o ff r o s t pene t r a t i o n i n t o a h o mo g e n e o u ss a n d y s i l t f o r t h e f o l l o wi ng c o n d i t i o ns :- Me an an n ua l t empe r a t u r e( MAT) = 3 7 . 2 F .

* TM 5 - 8 5 2 - 6 / AF R 8 8 - 1 9 , Vo l ume 6- Su r f a c e f r e e z i n g i n d ex ( n F ) _2 6 0 0 d e gr e e- d a ys .- L e ng t h o f f r e ez i n g s e a s o n ( t )= 1 6 0 d a y s .- So i l p r o pe r t i e s : y d = 1 0 0 1 b / f t 3 , w= 1 6 %.

The s o i l t h e r ma l p r o pe r t i e s a r e a sf o l l o ws :- Vo l ume t r i c l a t e n t h e a t o f f u s i o n ,L = 1 4 4 ( 1 0 0 ) ( 0 . 1 6 ) = 2 1 6 0 B t u / f t 3 . ( e q . 3 - 2 ) .- Av e r a ge v ol ume t r i c h e a tc a p a c i t y ,

Cavg = 1 00 [ 0 . 1 7 + + ( 0 . 7 5 X0 . l 5 ) ]= 2 8 . 3 B t u / f t 3 - F . ( e q 3 - 3 )- Av e r a ge t h e r ma l c o nduc t i v i t y ,K= 0 . 8 0 B t u / f t h r F ( f i g . 2 - 3 )K= 0 . 7 2 B t u / f t h r F ( f i g . 2 - 4 )Ka v = 1 / 2 ( Ku +K) - - ' 0 . 7 6 B t u / f th r f ' .

The , l c o e f f i c i e n t i s a s f o l l o ws :- Av e r a ge s ur f a c e t e mp er a t u r ed i f f e r e n t i a l ,v s = nF / t - 2 6 00 / 1 6 0- 1 6 . 6 F ( 1 6 . 6 F b el o w 3 2 F ) . ( e q 3 - 4 )- I n i t i a l t emp er a t u r e d i f f e r e n t i a l ,v o - MAT - 3 2 - 3 7 . 2 - 3 2 . 0 =6 . 2 F ( 6 . 2 a b o v e 3 2 F ) . ( e q 3 - 6 )- T h e r ma l r a t i o ,a = v o / v s = 6 . 2 / 1 6 . 6 - 0 . 3 3 . ( e q 3 - 6 )- F u s i o n p ar a me t e r ,p - v s ( C / L ) - 1 6 . 6 ( 2 8 . 2 / 2 1 6 0 ) - 0 . 2 0 . ( e q 3 - 7 )- L ambda c o e f f i c i e n t ,. 1 - 0 . 8 9 ( f i g . 3 - 1 ) . ( e q 3 - 8 )Es t i ma t e d d ep t h o f f r o s t pene t r a t i o n,

X4 8 Kn F = 0 . 8 9J 4 8 ( 0 . 7 6 ) ( 2 6 0 0 )= ~ 5 . 8 f tL 2 1 6 0 ( e q 3 - 8

3 - 4 . M u l t i l a y e r s o i l s .A mul t i l a y er s o l u t i o n t o t h e mo d i f i e dBe r g gr e n eq ua t i o n i s u s e d f o r n o n -homoge ne ous s o i l s b y d et e r mi n i n gt h at po r t i o n o f t h e s ur f a c e f r e e z i n g ( o rt h awi n g) i n d ex r e q u i r e d t o pene t r a t ee a c h l a y er . T h e s umo f t h e t h i c k n es s eso f a l l t h e f r o z e n ( o r t h a we d ) l a y er s i st h e d ep t h o f f r e e z e ( o r t h aw) . Th ep a r t i a l f r ee z i ng ( o r t h awi n g) i n d ex r e -q u i r e d t o pene t r a t e t h e t o p l ay er i sg i v en b y L i d , RF1 ( o r 1 1 ) - 2 ( 2 ) ( e q 3 - 1 0 )2 4 a 13 - 3

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TM 5 - 8 5 2 - 6 / AF R 8 8 - 1 9 , Vo l ume 6wh e r ed l = t h i c k n e s s o f f i r s t l a y er ( f t )R1 = d l / K 1 = t h er ma l r e s i s t a n c e o ff i r s t l a y er .The p a r t i a l f r e e z i n g ( o r t h a wi n g ) i n de xr e q u i r e d t o p en et r a t e t he s e c o n d l a y eri s

The p a r t i a l i n de x r e q u i r e d t o p en e t r a t et h e n t h l a y e r i s :

wh e r e I R i s t he t o t a l t he r ma l r e s i s t a nc ea b o v e t he n t h l a y er a n d eq ua l s

3 - 4

L 2 d 2 R2F 2 ( o r I 2 = . ( 1 1 1 + - ) .2 4 x 2 2

R nF n ( o r I n ) - ( MR + - )2 4 . 1 2 2L n d n( e q 3 - 1 1 )

( e q 3 - 1 2 )

R1 + R2 + R3 . . . R n . 1 " ( e q 3 - 1 3 )The s u mma t i o n o f t he p a r t i a l i n dex es ,

F 1 + F 2 + F 3 . .. +F n ( o r I 1 + I 2 + 1 3 . . .+ I n )I s e q u a l t o t he s ur f a c e f r e e z i n g i n de xt h a wi n g i n dex ) .a . I n t h i s e x a mp l e , d et e r mi n e t h ed e p t h o f t h a w p e n e t r a t i o n b en ea t h ab i t umi n ou s c o n c r e t e p a v eme n t f o r t h ef o l l owi n g c on d i t i o n s :- Mea n a n n u a l t emp e r a t u r e ( MAT)= 1 2 F .- A i r t h a wi n g i n de x ( I )= 7 8 0 d eg r e e- d a y s .- A v e r a g e wi n d s p e e d i n s umme r7 . 5 mi l e s p e r h o u r ( mp h ) .- L e n g t h o f t h a w s e a s o n ( t )= 1 05 d a y s .- So i l b o r i n g l o g - .

( e q 3 - 1 4 )

~ I n a c c o r d a n c e w i t h U n i f i e d S o i l C l a s s i f i c a t i o n Sy s t em .

S i n c e a wi n d s p e e d o f 7 - 1 / 2 mph r e s u l t si n a n n - f a c t o r o f 2 . 0 ( f i g . 2 - 1 0 ) , a s ur f a c et h a wi n g i n de x n I o f 1 5 6 0 d eg r e e - d a y si s u s e d i n t he c o mp u t a t i o n s . The v s , v oa n d a v a l u e s a r e d e t e r mi n ed i n t he s a mewa y a s t ho s e f o r t he h omo g e n e o u s c a s e :v s - 1 5 6 0 / 1 0 5 = 1 4 . 8 F ( e q 3 - 1 5 )v o = 1 2 . 0 - 3 2 . 0 = 2 0 . 0 F ( e q 3 - 1 6 )a s 2 0 . 0 / 1 4 . 8 - 1 . 3 5 . ( e q 3 - 1 7 )Th e t h er ma l p r o pe r t i e s C , K a n d L o ft he r e s p ec t i v e l a y er s a r e o b t a i n ed f r o mf i g u r e s 2 - 1 t h r o u g h 2 - 8 .b . Ta b l e 3 - 1 f a c i l i t a t e s s ol u t i o n o ft he mu l t i l a y er p r o b l e m, a n d i n t he f o l -l o wi n g d i s c us s i o n , l a y er 3 i s u s e d t oI l l u s t r a t e q ua n t i t a t i v e v a l ue s . Co l u mn s9 , 1 0 , 1 2 a n d 1 3 a r e s el f - e x p l a na t o r y .Co l umn 1 1 , T : , r e p r e s en t s t he a v e r a g ev a l u e o f L f o r a l a y er a n d i s e q u a l t oEL d / l d ( 2 5 8 1 / 5 . 0 = 5 1 7 ) . Co l umn 1 4 , Z s ,r e p r e s en t s t h e a v e r a g e v a l u e o f C a n di s o b t a i n ed f r o m XCd / l d ( 1 4 5 / 5 . 0 = 2 9 ) .Thus T : and - C r e p r e s e n t we i g h t e dv a l u e s t o a d ep t h o f t h a w p e n e t r a t i o ng i v en b y E d , wh i c h i s t he s um o f a l ll a y er t h i c k n e s s es t o t h a t d e p t h .Th e f u s i o n p a r a me t e r p f o r e a c h l a y eri s d e t e r mi n e d f r o m

v 8 ( C / L ) = 1 4 8 ( 2 9 / 5 1 7 ) = 0 . 8 3 . ( e q 3 - 1 8 )Th e A c o e f f i c i e n t i s e q u a l t o 0 . 5 0 8 f r o mf i g u r e 3 - 1 . Co l umn 1 8 , Rn , i s t he r a t i od / K a n d f o r l a y er 3 e q ua l s ( 3 . 0 / 2 . 0 ) o r1 . 5 . Co l umn 1 9 , F R , r e p r e s en t s t he s umo f t he Rn v a l u e s a b o v e t he l a y er u n d e rc on s i d e r a t i o n . Co l umn 2 0 , I R + ( R n / 2 ) ,e q ua l s t h e s um o f t he R n v a l u e s a b o v et he l a y er p l u s o n e- ha l f t h e Rn v a l u e o ft he l a y er b e i n g c o n s i d er e d . F o g , a y e r 3t h i s i s [ 1 . 3 2 + ( 1 . 5 0 / 2 ) ] = 2 . 0 7 . Co l umn 2 1 ,n I , r e p r e s en t s t he numbe r o f d eg r e e-d a y s r e q u i r e d t o t haw t he l a y er b e i n gc o n s i d er e d a n d i s d e t e r mi n e d f r o m

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* TM 5 - 8 5 2 - 6 / AF R 8 8 - 1 9 , Vo l ume 6

F o r l a y er 3 ,( 6 1 0 ) ( 3 . 0 )

The summa t i o n o f t h e numbe r o fd eg r e e - d a y s r e q ui r e d t o t h a w l a y e r s 1t h r o u g h 4 i s 1 2 9 7 , l e a v i n g ( 1 5 6 0 - 1 2 9 7 = )2 6 3 d eg r e e - d a y s t o t h a w a p o r t i o n o fl a y er 5 A t r i a l - a n d - e r r o r me t h o d i s u s e dt o d e t e r mi n e t h e t h i c k n es s o f t h et h a we d p a r t o f l a y er 5 . F i r s t , i t i s a s -s umed t h a t 1 . 0 f e e t o f l a y er 5 i s t h a we d( d e s i g n a t e d a s l a y er 5 a ) . Ca l c ul a t i o n si n d i c a t e 4 6 5 d eg r e e - d a y s a r e n e e d e dt o t haw 1 . 0 f o o t o f l a y er 5 o r ( 4 6 5 - 2 6 3 = )2 0 2 d eg r e e - d a y s mor e t h a n a v a i l a b l e .A new l a y e r , 5 b , i s t h e n s el ec t ed b y t h ef o l l owi n g p r o p or t i o nTh i s new t h i c k n es s r e s ul t s i n 2 6 0d eg r e e - d a y s r e q u i r e d t o t h a w l a y er 5 bo r 3 d eg r e e- d a y s l e s s t h a n a v a i l a b l e .F u r t h er t r i a l - a n d - e r r o r i s u n wa r r a n t e da n d t h e t o t a l es t i ma t e d t h a w p en et r a -t i o n wo u l d be 6 . 6 f e e t . A s i mi l a r t e c h -n i q u e i s u s e d t o e s t i ma t e f r o s t p en et r a -t i o n i n a mu l t i l a y er s o i l p r o f i l e .3 - 5 . E f f e c t o f s n o w a n d v e g e t a t i v e c o v e r .Th e r ma l p r o p er t i e s o f s n o w a n d v e g e t a -t i v e c o v e r s a r e e x t r eme l y v a r i a b l e i nbo t h t i me a n d s p a c e . Bo t h ma t e r i a l st e n d t o a c t a s i n s u l a t o r s a n d r e t a r dh e a t t r a n s f e r a t t h e a i r - g r o u n d i n t e r -f a c e . I n f r e e z e - t h awc omp ut a t i o n s , snowa n d v eg e t a t i v e s ur f a c e ma t e r i a l s a r et r e a t e d a s s ep a r a t e l a y e r s i n t h e mu l t i -l a y e r s ol u t i on o f t h e mod i f i e d Be r g g r e n

3 - 6

L dn 1 - ( 7 - R +2 4, 2

= 6 1 2 d eg r e e - d a y s

( e q 3 - 1 9 )

n 1 3 = ( 2 . 0 7 ) ( e q 3 - 2 0 )2 4 ( 0 . 6 0 8 ) 2

( 2 6 3 / 4 6 6 ) 1 . 0 = 0 . 5 7 f t ( t r y 0 . 6 f t ) . ( e q 3 - 2 1 )

eq ua t i on , wi t h snow c o v e r t h i c k n es se s t i ma t e d s ea s on a l l y . Th e t a bu l a t i o nb e l ow p r e s en t s a v e r a g e t h e r ma l p r o p e r -t i e s o f snow a p p l i c a bl e f o r c a l c u l a t i o ni n t h e n o t e d r e g i o n s i f a b e t t e r d a t ab a s e i s n o t a v a i l a b l e . I n t h e a b s e n c e o fs i t e - s p e c i f i c d a t a , f i g u r e s 2 - 5 a n d 2 - 6s h ou l d be u s e d t o e s t i ma t e t h e t h e r ma lc on d uc t i v i t i e s o f v eg e t a t i v e s ur f a c ec o v e r .3 - 6 . S u r f a c e t emp e r a t u r e v a r i a t i o n s .Th e t emp e r a t u r e s a t t h e a i r - g r o un di n t e r f a c e a r e s ub j ec t t o d a i l y a n d s e a -s o n a l f l u c t u a t i o n s . P r e c i p i t a t i o n , i n s o l a -t i o n , a i r t emp e r a t u r e a n d t u r bu l e n c ec on t r i bu t e t o t h e s e v a r i a t i o n s i n s u r -f a c e t emp e r a t u r e To f a c i l i t a t e ma t h e -ma t i c a l c a l c u l a t i o n s , t wo a s s u mp t i o n sa r e c ommonl y made r e g a r d i n g t h e t em-p er a t u r e s a t t h e u p p e r bo un d a r y : 1 ) as u d d e n s t ep c h a n g e o c c ur s i n s ur f a c et emp e r a t u r e o r 2 ) t h e s ur f a c e t e mp e r a -t u r e c h a n g e i s s i n us o i d a l . Th e s i n u -s o i d a l v a r i a t i o n o f t emp e r a t u r e o v e r ay e a r c l o s e l y a p p r o x i ma t e s a c t u a l c o n d i -t i o n s h o we v e r , i t i s a me n a b l e t o h a n dc a l c u l a t i on s o n l y i f l a t e n t h e a t e f f e c t sa r e n eg l i g i b l e . So l u t i o n s a n d e x a mp l e sf o r bo t h c on d i t i o n s a r e g i v e n be l o w .a S u d d e n s t e p c h a n g e Th i s i n v o l v e sa s u d d e n c h a n g e i n t h e s ur f a c e t em-p er a t u r e o f a mas s t h a t wa s i n i t i a l l y a ta c on s t a n t , u n i f o r m t emp e r a t u r e . Th es u d d e n s t ep c h a n g e wa s u s e d t o e s t a b -l i s h t h e b ou n d a r y c on d i t i o n s f o r h e a tf l o w i n t h e mod i f i e d Be r g g r e n e q ua t i o ng i v en i n p a r a g r a p h 3 - 2 I f t h e i n f l u e n c eo f l a t e n t h e a t i s n o t i n v ol v ed , o r i s a s -s umed n e g l i g i b l e , t h e f o l l owi n g e q u a -t i o n may be u s e d :

T( X t ) = T s + ( T o - T s ) e r f ( ) ( e q 3 - 2 2 )2 - f a t

Re g i o nU n i t

w e i g h t( I b l f t 3 )K

( B t u l f t )h r F ) C( B t u l f t 3 F ) L( B t u l f t 3 )I n t e r i o r A l a s k a 1 6 0 . 1 1 8 2 3 0 0Ca n a d i a n A r c h i p e l a g o ,N . A l a s k a n c o a s t ,a n d t e mp e r a t e r e g i o n s 2 0 0 . 1 8 1 0 2 8 8 0N o r t h e r n Gr e en l a n d 2 2 0 . 2 0 1 1 3 1 7 0

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wh e r eT ( X t ) = t e mp er a t u r e a t d e p t h x , a tt i me ( F )T S = s u d d e n l y a p p l i e d c o n s t a n ts u r f a c e t e mp er a t u r e ( F )T o = i n i t i a l u n i f o r m t e mp e r a -t u r e o f t h e m a s s ( F )e r f = ma t h ema t i c a l e x p r e s s i o n ,

- t e r me d t h e e r r o r f u n c t i o n ,wh i c h i s f r e q u e n t l y u s e di n h e a t f l o w c o mp u t a t i o n s( d i me n s i o n l e s s )x = d ep t h b e l o w s u r f a c e ( f t )

h14 'I

F i g u r e 3 - 2 . R e l a t i o n s h i p b e t w e e n ( x / 2 Y- a- t ) a n d e r f ( x / 2 a t ) .

* T M 5 - 8 5 2 - 6 / A F R 8 8 - 1 9 , Vo l u me 6a = t h er ma l d i f f u s i v i t y o f t h em a s s ( f t 2 / d a y = K / C)t = t i me a f t e r a p p l i c a t i o n o fs u d d e n c h a n g e i n s u r f a c et e mp e r a t u r e ( d a y s ) .F i g u r e 3 - 2 g i v e s t h e r e l a t i o n s h i p b e -t we e n ( x / 2 a t ) a n d e r f ( x / 2 1 r a t ) . Thee x p r e s s i o n f o r t h e e r r o r f u n c t i o n i s

shown i n a p p e n d i x A a n d f i g u r e 3 - 2 .I n t h i s e x a mp l e o f a s u d d e n s t e p c h a n g e ,a h i g h l y f r o s t - s u s c e p t i b l e s u b g r a d e i sc o v e r e d wi t h a 2 - f o o t t h i c k , n o n - f r o s t -s u s c e p t i b l e g r a v e l p a d . Bo t h s o i l s a r ea t a n i n i t i a l t e mp e r a t u r e o f 2 0 F I f t h e

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* TM 5 - 8 5 2 - 6 / AF R 8 8 - 1 9 , Vo l u me 6s u r f a c e o f t h e g r a v e l i s s u d d e n l y h e a t e dt o a n d ma i n t a i n e d a t 7 0 F f o r a nu mbe ro f d a y s , e s t i ma t e t h e t e mp e r a t u r e a tt h e g r a v e l - s u b g r a d e i n t e r f a c e a f t e r o n ed a y . The g r a v e l ma t e r i a l i s v e r y d r ya n d l a t e n t h e a t may b e i g n o r e d . T h et h e r ma l c on d u c t i v i t y o f t h e g r a v e l i s1 . 0 B t u / f t h r F a n d t h e v o l u me t r i c h e a tc a p a c i t y i s 25 B t u / f t 3 . T h e t h e r ma l d i f -f u s i v i t y o f t h e g r a v e l i s ( K / C = 1 . 0 / 2 5 )= = 0 . 0 4 f t 2 / h r . , o r 0 . 9 6 f t 2 / d a y , a n dx / 2 a t = ( 2 . 0 / 2 1 r 0. ~ X 1 ) = 1 . 0 2 . Fr omf i g u r e 3 - 2 , e r f ( x / 2 - a6 ) i s e q u a l t o 0 : 8 5 ,a n d t h e i n t e r f a c e t e mp e r a t u r e T i s[ 7 0 + ( 2 0 - 7 0 ) 0 . 8 5 1 = 2 7 . 5 F .b . S i n u s o i d a l c h a n g e . As u r f u t c e t e m-p e r a t u r e v a r i a t i o n t h a t i s n e a r l y s i n u -s o i d a l r e p e a t s i t s e l f p e r i o d i c a l l y f o r as u r f a c e e x p o s e d t o t h e a t mo s p h e r e . F o rmo s t p r o b l e ms i n t h i s ma n u a l , t h e s i n u -s o i d a l v a r i a t i o n o f c o n c e r n o c c u r s o v e ra n a n n u a l c y c l e . I f l a t e n t h e a t i s n o ti n v o l v e d o r i s a s s u me d n e g l i g i b l e , t h ef o l l o wi n g e q u a t i o n may b e u s e d :

wh e r eA = a mp l i t u d e o f t e mp e r a t u r e w a v ea t d e p t h x ( F ) .Ao a mp l i t u d e o f t h e s u r f a c e t e m-p e r a t u r e w a v e a b o v e o r b e l o wt h e a v e r a g e a n n u a l t e mp e r a -t u r e ( F )x = d e p t h b e l o w s u r f a c e ( f t )a = t h e r ma l d i f f u s i v i t y o f t h e m a s s( f t 2 / d a y )P= p e r i o d o f s i n e w a v e ( 3 6 5 d a y s )The s i n u s o i d a l t e mp e r a t u r e p a t t e r n i sa s s u m e d t o e x i s t a t a l l l e v e l s t o a d e p t hwh e r e t h e r e i s n o t e mp e r a t u r e c h a n g e .The t e mp e r a t u r e w a v e s l a g b e h i n d t h es u r f a c e w a v e , a n d t h e a mp l i t u d e o f t h es i n u s o i d a l w a v e s d e c r e a s e s wi t h d e p t h

b e l o w t h e s u r f a c e . T h e p h a s e l a g i sd e t e r mi n e d by t x = ( x / 2 ) ( 31 r6Ua) .T y p i c a l t e mp e r a t u r e - t i me c u r v e s f o r as u r f a c e a n d a t a d e p t h x a r e shown i nf i g u r e 3 - 3 .I n t h e f o l l o wi n g e x a mp l e o f a s i n u -s o i d a l t e mp e r a t u r e c h a n g e , t h e s u r f a c et e mp e r a t u r e o f a n 8 - f o o t - t h i c k c o n c r e t es l a b v a r i e s f r o m 6 0 t o - 4 0 F d u r i n g t h ey e a r . De t e r mi n e t h e max i mum t e m-p e r a t u r e a t t h e b a s e o f t h e s l a b a s -s u mi n g a d i f f u s i v i t y o f 1 . 0 f t 2 / d a y f o r3 - 8

A = A o e x p ( - x ) ( e q 3 - 2 3 )a P

t h e c o n c r e t e The a v e r a g e a n n u a l t e m-p e r a t u r e i s [ 6 0 + ( - 4 0 ) 1 / 2 = 1 0 F a n d t h es u r f a c e a mp l i t u de i s ( 6 0 - 1 0 ) = 5 0 F .T h e a mp l i t u de a t a n 8 - f o o t d e p t h e qu a l s5 0 e x p [ - 8 . V = 5 0 e ' 0 . 7 4 2( 1 . 0 ) ( 3 6 5 )

2 4 F . ( e q 3 - 2 4 )T h e max i mum t e mp e r a t u r e a t 8 f e e t i s( 1 0 + 2 4 ) = 3 4 F T h e t i me l a g t x b e t we e nt h e max i mum t e mp e r a t u r e a t t h e s u r -f a c e a n d 8 f e e t i s8 ~ 3 6 5t x = 4 3 d a y s2 n ( 1 . 0 ) ( a b ou t 6 w e e k s ) . ( e q 3 - 2 5 )( No t e : l a t e n t h e a t wo u l d i n c r e a s e t h et i me l a g a n d d e c r e a s e t h e a mp l i t u d e . )c . No n u n i f o r m l a y e r s . T h e me t h o do f e q u i v a l e n t t h i c k n e s s i s u s e d t o f i n dt h e t e mp e r a t u r e a t a p o i n t b e l o w a

nu mbe r o f l a y e r s o f d i f f e r e n t t h e r ma lp r o p e r t i e s . T h i s t e c h n i q u e a s s u m e s an e g l i g i b l e e f f e c t o f l a t e n t h e a t , a n d i n -v o l v e s r e d u c t i o n o f e a c h l a y e r t o a ne q u i v a l e n t ma t e r i a l t h i c k n e s s b y s e t -t i n g t h e r a t i o o f t h e t h i c k n e s s e s e q u a lt o t h e r a t i o o f s q u a r e r o o t s o f t h e t h e r -ma l d i f f u s i v i t i e s . F o r e x a mp l e , d e t e r -mi n e t h e e q u i v a l e n t g r a v e l t h i c k n e s sf o r t h e t h r e e l a y e r s s h o w n , a s s u mi n ga l l ma t e r i a l s a r e u n f r o z e n .The f o l l o wi n g t a b l e s h o w s t h a t 4 . 7 5f e e t o f t h e n o n u n i f o r m ma t e r i a l s c a nb e c o n s i d e r e d e q u i v a l e n t t o 5 . 4 f e e t o fg r a v e l f o r h e a t - f l o w p u r p o s e s . T h i se q u i v a l e n t t h i c k n e s s a n d t h e t h e r ma ld i f f u s i v i t y o f t h e g r a v e l a r e u s e d t oc a l c u l a t e t e mp e r a t u r e s a t t h e b a s e o ft h e g r a v e l l a y e r b y e i t h e r t h e s t e p -c h a n g e o r s i n u s o i d a l me t h o d .3 - 7 . Co nv e r t i n g i n de x e s i n t o e q u i v a l e n t s i n ewave o f t e mp e r a t u r e .Some p r o b l e ms may r e q u i r e t h e u s e o ft h e s i n u s o i d a l t e mp e r a t u r e v a r i a t i o nt e c h n i q u e , g i v e n o n l y t h e f r e e z i n g o rt hawi n g i n d e x e s a n d t h e a v e r a g ea n n u a l t e mp e r a t u r e . T h e s e i n d e x e smay b e c o n v e r t e d i n t o a s i n e c u r v e o ft e mp e r a t u r e t o g i v e t h e s a m e i n d e xv a l u e s a n d t h e s a m e m e a n t e mp e r a t u r e .F o r e x a mp l e , c o n v e r t t h e mo n t h l y a v e r -a g e t e mp e r a t u r e d a t a f o r F a i r b a n k s ,A l a s k a , shown i n f i g u r e 2 - 9 i n t o a ne q u i v a l e n t s i n e w a v e . T h e r e l a t i o n s h i pb e t we e n t h e s i n u s o i d a l a mp l i t u d e ,f r e e z i n g i n d e x , t h a wi n g i n d e x a n d a v e r -

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o f 5 2 4 0 d e g r e e - d a y s , t h e s i n u s o i d a la mp l i t u d e i s f o u n d t o b e 37 . 0 F . Th ee q u a t i o n o f t h e s i n e wa v e i sT = 2 7 . 0 - 3 7 . 0 s i n 2 v f t ( e q 3 - 2 6 )

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mi n i mu m )I f t he - t hawi n g i n dex o f 3 2 4 0 d eg r e e-d a y s h a d b e e n u s e d , t he s i n u s o i da l t em-p e r a t u r e amp l i t u de wo u l d b e 3 5 . 5 0 F .Th e ac t u al t emp er a t u r e c u r v e f o r F a i r -b a n k s , A l a s k a , a n d t he eq u i v al en t s i n ewave c o mp u t e d f r o m t h e f r e ez i n g i n dexa r e p l o t t ed i n f i g u r e 3 - 5 . Th i s i l l u s t r a -t i o n ma k e s u s e o f a i r i n dex e s b u t a s i n ew a v e c ou l d b e d e t e r mi n ed f o r s u r f a c ei n dex e s b y mu l t i p l y i n g t h e a i r i n de x e swi t h ap pr o p r i a t e n - f a c t o r s No t e t h a tt he me a n a n n u a l g r o u n d s u r f a c e t e m-p er a t u r e ma y b e s u b s t a n t i a l l y d i f f e r e n t( f r e q u en t l y h i g her ) f r o m t he me a n an -n u a l a i r t e mp er a t u r e b e c a u s e t hef r e ez i n g n - f a c t o r i s g en er a l l y n o t e q u a lt o t he t h awi n g n - f a c t o r . I f t he l o n g -t e r m me a n mo n t h l y t emp er a t u r e h a db e e n u s e d i n s t ead o f t he a v er a g emo n t h l y t emp er a t u r e s f o r t h