REPORT Freezing of soil with an unfrozen water content and variable thermal properties. Temperature. Soil Surface

Transcription

1 ι REPORT 88-2 US Army Corps of Engineers Cold Regions Research & Engineering Laboratory Freezing of soil with an unfrozen water content and variable thermal properties Soil Surface Cooler Temperature warmer Ice and Water (no phase change) Minimum Freeze Temρerafure Ice and Water (phase change) Maximum Freeze Temperature Thawed

2 For conversion of S/ metric units to U.S./British customary units of measurement consult ASTM Standard Ε380, Metric Practice Guide, published by the American Society for Testing and Materials, 1916 Race St., Philadelphia, Pa Cover: Geometry of a semi-infinite soil mass, initially at a temperature above freezing, that freezes due to a constant surface temperature below freezing.

3 CRR Ε L R 88-2 March 1988 Freezing of soil with an unfrozen wa ter con ten t and variable therm αlp ro p er nes Virgil J. Lunardini Prepared for OFFICE OF THE CHIEF OF ENGINEERS Approved for public release; distribution is unlimited.

4 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE la. REPORT SECURITY CLASSIFICATION Unclassified REPORT DOCUMENTATION PAGE lb. RESTRICTIVE MARKINGS 2a. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAILABILITY OF REPORT 2b. DECLASSIFICATION/DOWNGRADING SCHEDULE 4. PERFORMING ORGANIZATION REPORT NUMBER(S) CRREL Report 88-2 Approved for public release; distribution is unlimited. 5. MONITORING ORGANIZATION REPORT NUMBER(S) Form Approved 0MB Νο Exp. Date: Jun 30, a. NAME OF PERFORMING ORGANIZATION U.S. Army Cold Regions Research and Engineering Laboratory 6c.. ADDRESS (City, State, and ZIP Code) Hanover, New Hampshire b. OFFICE SYMBOL (if applicable) CECRL 7a. NAME OF MONITORING ORGANIZATION 7b. ADDRESS (City, State, and ZIP Code) 8a. NAME OF FUNDING/SPONSORING ORGANIZATION U.S. Cold Research and Engineering Laebrgooanstory 8c. ADDRESS (City, State, and ZIP Code) Hanover, New Hampshire b. OFFICE SYMBOL (if applicable) CECRL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ILIR 6 ΧΧ SOURCE OF FUNDING NUMBERS PROGRAM ELEMENT NO. PROJECT NO. TASK NO. WORK UNIT ACCESSION NO. 11. TITLE (include Security Classification) Freezing of Soil with an Unfrozen Water Content and Variable Thermal Properties 13a. TYPE OF REPORT 13b. TIME COVERED FROM 16. SUPPLEMENTARY NOTATION TO 14. DATE OF REPORT (Year, Month, Day) 15. PAGE COUNT March COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number) Frozen soils Phase change Soils 19. ABSTRACT (Continue on reverse if necessary and identify by block number) While many materials undergo phase change at a fixed temperature, soil systems exhibit a definite zone of phase change. The variation of unfrozen water with temperature causes a soil system to freeze or thaw over a finite temperature range. Exact and approximate solutions are given for conduction phase change of plane layers of soil with unfrozen water contents that vary linearly and quadratically with temperature. The temperature and phase change depths were found to vary significantly from those predicted for the constant-temperature or Neumann problem. The thermal conductivity and specific heat of the soil within the mushy zone varied as a function of unfrozen water content. It was found that the effect of specific heat is negligible, while the effect of variable thermal conductivity can be accounted for by a proper choice of thermal properties used in the constant-thermal-property solution. 20. DISTRIBUTION /AVAILABILITY OF ABSTRACT 13 UNCLASSIFIED/UNLIMITED SAME AS RPT. DTIC USERS 22a. NAME OF RESPONSIBLE INDIVIDUAL Virgil J. Lunardini DD FORM 1473, 84 MAR 83 APR edition may be used until exhausted. All other editions are obsolete. 21. ABSTRACT SECURITY CLASSIFICATION Unclassified 22b. TELEPHONE (include Area Code) c. OFFICE SYMBOL CECRL-EA SECURITY CLASSIFICATION OF THIS PAGE UNCLASSIFIED

5 NOMENCLATURE 2 λ ο A B _ δ-x B X C specific heat Cu Ci/Ci C0 arbitrary value of specific heat for constant-property mushy zone, otherwise Cο= Cu F defined by eq 17 F, 1 (1 + σο - βτι) ^0 F αοβΡι k thermal conductivity kij ki/ki k ο any specified constant-conductivity mushy zone value of k; otherwise k0 ku K Ρ-A N ,P Π 2 f latent heat of fusion of water m mass m w, mass of water q heat flux qg latent heat flux during solidification R 1- &,3 1 - C3( Τf - Τ) ST, Stefan number Υd f t time Τ temperature highest and lowest temperatures for phase change Τ0, TS initial and surface temperatures x Cartesian coordinate xf volumetric water fraction Tf, Tm ii

6 Χ, Χ, phase change interface positions for Tf, Tm z Rλ/B α thermal diffusivity k/c αi/αj αο k0 /CO β, kfu 1 02 Cfu 1 5 temperature penetration depth Υd 2 α3t phase change parameter defined by eq 23 dry unit density of soil solids (mass of soil solids per unit volume) 77 2'/J 9 Tf T λ ψ0 k3 2 λ0 phase change parameter Tf Tm, dimensionless temperature ψ0 k30 ^ ratio of unfrozen water mass to soil solid mass Εο, f' Εs values of at Tf, Tm, Ts Ο σ0 Ο0 = C32 Φ ST C30ψ T Í'f Ts Τf Τm TO Tf Tf Tm ψ dimensionless temperature defined by eq 12 Subscripts 123,, regions of soil f,, s u frozen value, surface value, and thawed value iii

7 PREFACE This report was prepared by Dr. Virgil J. Lunardini, Mechanical Engineer, Applied Research Branch, Experimental Engineering Division, U.S. Army Cold Regions Research and Engineering Laboratory. This study was conducted under ILIR 6 ΧΧ 71462, Heat Transfer with Freezing or Thawing. The author thanks Dr. Yin-Chao Yen and F. Donald Haynes of CRREL for their technical reviews of this report. The contents of this report are not to be used for advertising or promotional purposes. Citation of brand names does not constitute an official endorsement or approval of the use of such commercial products. iv

8 CONTENTS Page Abstract i Nomenclature ii Preface iv Introduction 1 Basic equations 2 Two-zone problems 5 Linear unfrozen water function 5 Quadratic unfrozen water function 8 Three-zone problems 11 Linear unfrozen water function 11 Quadratic unfrozen water function 15 Conclusions 15 Literature cited 17 Appendix A: Derivation of the mushy zone equation 19 Appendix B: Solution of the two-zone problem with a linear Ε and variable thermal properties 21 ILLUSTRATIONS Figure 1. Geometry for solidification with a phase change zone 1 2. Unfrozen water vs temperature 1 3. Heat flow in the mushy zone 3 4. Quadratic solution for the two-zone problem Quadratic solution for the three-zone problem 16 TABLES Table 1. Effect of thermal properties on freeze of soil with average properties and linear Ε 9 2. Effect of thermal properties on freeze of soil with extreme property variations Effect of phase change temperature Comparison of exact and heat balance integral solutions with linear and constant k and C 14 v

9 Freezing of Soil with an Unfrozen Water Content and Variable Thermal Properties VIRGIL J. LUNARDINI INTRODUCTION The mathematical theory of conductive heat transfer with solidification has been largely confined to materials that change phase at a single temperature. The best-known problem of this type is that of Neumann, and its solution has been widely used for the freezing of soils (Neumann 1860, Berggren 1943, Carslaw and Jaeger 1959). However, for media such as soils, the phase change can occur over a range of temperatures (Anderson and Tice 1973, Tice et al. 1978, Lunardini 1981a). In other words, at any temperature below the normal freezing point, there will be an equilibrium state of unfrozen water, ice and soil solids. Figure 1 shows the geometry for a semi-infinite soil mass, initially at a temperature above freezing, that freezes due to a constant surface temperature held below the freezing point. The phase change is assumed to occur within the temperature limits of Tm and Tf, representing minimum and maximum phase change temperatures. Figure 2 shows the unfrozen water as a function of temperature for a typical soil. At Tf all of the water is in the liquid form, while at Tm the free water is all frozen. There may be a residual amount of bound water, denoted by, that will remain unfrozen even at very low temperatures. It will be assumed that for Τ < Tm, unfrozen water may exist but no phase change will occur. The region Tm s Τ < Tf is called the zone of phase change, or the mushy zone. In this region, water will solidify to ice, and unfrozen water and ice will coexist. As (Tf Tm) 0, the phase change will approach the typical Neumann-type problem, which is To Ice + Water Ice + Water Thawed Νο Phase Change Phase Change e0 r-- F a- Thawed 1 m TS ' 1 f X x 6 T m T f Figure 1. Geometry for solidification with a phase change zone. Figure 2. Unfrozen water vs temperature.

10 particularly valid for coarse materials such as sands and gravels. The form of the ξ function for soils can be expressed by different functional relations. The simplest relation is linear: ^ Εο + (T Tf) (1) where Ε Εο Εf and Tm Tf- Tm. Another functional relation, which can closely model the data and is easy to manipulate analytically, is a quadratic form: ^ Εο + 2 (Τ Τf)+ (Τ Τf) z. (2) Tm If ξο, Εf and Tm are the same for these functions, then the mean unfrozen water slope dε/dτ will be identical. The thermal conductivity and the specific heat within the mushy zone are functions of the unfrozen water and may be represented by k ku (kf ku) (^ Εο) (3) C Cu (Cf Cu) ( ) (4) where kf, ku Cf, Cu fully frozen and fully thawed thermal conductivities and fully frozen and fully thawed specific heats. Obviously these properties are functions of the particular form of the unfrozen water function (Frivick 1980). Within the fully frozen region (zone 1) it is assumed that the thermal properties are constant and equal to the frozen values, while for the thawed region (zone 3) the properties are constant and equal to the thawed soil values. Tien and Geiger (1967) and Ozisik and Uzzell (1979) used an unfrozen liquid content that varied with position within the two-phase zone. They did not deal with soil systems, however. Cho and Sunderland (1974) found a solution for the freezing of a material with the thermal conductivity varying linearly with temperature; however, the material changed phase at a single temperature. BASIC EQUATIONS Consider a small volume of material within the mushy zone. Energy will be conducted in and out of the volume, and latent heat will be released during solidification (Fig. 3). Thus the problem is one of conduction with a distributed energy source. The governing equations were derived by Lunardini (1985, 1988). The net conduction is qχ qχ + Αx a x (k a t Χ. (5) The latent energy released due to solidification of a mass of water mw is as follows qg P mw P'd Ε χ. (6) 2

11 Latent Heat Χ q x+ Χ Χ Χt X Figure 3. Heat flow in the mushy zone. The energy equation in the mushy zone is (k óx^ Q ηρd yt Cat' (7) The most general case will be a problem with three regions as shown in Figure 1. The equations for region 1 are 82 7', 1 a Τ, axe αι at (8) 7', (0, t) Ίs (8a) T,(X,, t) Tm (8b), aτ,(χ,) αχ k^x^) a Τ2(Χ, ) ax (8c) For region 2 (the mushy zone) α (k z2 ) C + Pld _ T2 d r at (9) T2(X,, t) Tm (9a) T2(X, t) Tf (9b) k(x) at2(x) αχ k, at,(x) (9c) αχ The thawed zone is governed by a2 Τ3 axe 1 aτ, «, at (10) lim Τ3 (χ, t) To (loa) X- 00 Ρ,(χ, ο) Ρo (lob) Τ,(X, t) Tf. (10c) 3

12 These equations can be partially nondimensionalized by using the following relations: θi Tf Ti Tf TS TO Tf T f T m 1,2,3 Tf Tm 00 Tf Tm σ C3 2 φ SηΡ- σ0 C3 0 0 SτΡ λ ψ0 k32 λ ψ0 k30 β(^ 1 kfū «0 ko ^.T C3( Tf - Ts) CO β2 Cf. With these relations the thermal conductivity and specific heat in region 2, for the linear ^ case, are k o k0(1 + β, 02) C o C0( ). (ha) (jib) For the mushy zone with variable thermal properties, eq 11 a and b will use k 0 ku and Co Cu. However, the use of the general notation k 0 and Cu allows the properties in region 2 to be specified independently of the frozen or thawed values if the thermal properties are constant (Appendix A). In the mushy zone the following transformation will be used: 1 e, k^ S k(θ21 dθ2. 0 (12) For the linear Ε case the function V can be evaluated explicitly: ψ o (i2a) Then k ^ k0 '/1+ 20,,1 (13) ^z ^/1 2β,ψ +- 1 (14) The transformed equations are as follows: a28, 1 αο, ax2 α, at (15) 0, (0, t) Q ψ (15a) 8,(X,, t) 1 (15b) 4

13 k, ί' (X,, t) k(χ,) (X' (15c) x ' t). The equations for region 2 are written only for the linear ξ assumption (the details are given in Appendix A): '/1 +2,^,^ 2 (16) αx t ψψ(χ, t) Ο (16α) ψ(x,, t) 1 + ι, /2 (16b) k0 αx (X ' t) k3 3x (X ' t) (16c) αοf (1 + σο - 020)0 + ^^ (1 + 2βι ψ) (17) The thawed region has the following relations: χ (18) «3 āt him 83(χ, t) - ψ 0 (18a) χ- οο 8 3(χ, 0) - ψ ο (18b) 03(Χ, t) Ο. (18c) Exact solutions for the three-zone problem with variable thermal properties even with a linear ξ function have not been found. It is possible to obtain approximate solutions using the heat balance integral; however, before doing this it is useful to examine the simpler twozone problem. If the surface temperature 1 S is greater than or equal to the minimum phase change temperature, then a completely frozen zone will not exist. Thus we need only examine regions 2 and 3. TWO-ZONE PROBLEMS The two-zone problem is simpler than the three-zone case and will lead to results that can simplify the need for the full three-zone problem. The linear unfrozen water case will be examined for both variable and constant thermal properties, while the quadratic water content case will be evaluated only for the constant property problem. This will be shown to be adequate for the general problem. Linear unfrozen water function Variable thermal properties Equations are valid for this case except that the boundary condition in eq 16b becomes 5

14 ψ(0, t) ο + 2 P. (18d) An approximation of the solution may be obtained with the heat balance integral method. The heat balance integral method has been adapted to problems of freezing in soils systems by Lunsrdini (1981b, 1982, 1983) and Lunardini and Varotta (1981). The equations for the heat balance integral method are well known and will not be derived here; the interested reader can consult Lunsrdini (1981a) for details. Referring to Figure 1 and Appendix B, eq 16 and 18 become where 0 2 /1 + 2β2ψ x ' 2 d r α (X t) dt I f - α3 óx x dt F(x, t)dx - F2X), (19) l 0 83 dx + φο δl (20) FZ (32, 3 α0 J3, For the heat balance integral treatment, eq 18a and b become 8 3 (δ, t) φο (18e) cm 0. (180 Quadratic temperature profiles are assumed for ψ and 03 since experience has shown that they yield good results for the heat balance integral method: ψ b(χ x) + c(χ χ) 2 (21) where ^ο [`.a x) ) 2 ii (22) b cx 2 2λ0 δ -X P - bx. To simplify the algebra the following parameters are defined: Χ 2γ ^ 3 t (23) δ - Χ ΒΧ. (24) The use of eq 23 and 24 is not a constraint on the final solution. 6

15 The solution of eq 19 and 20 is straightforward but tedious. The algebraic manipulations are given in Appendix B. The unknown parameters y and B can be found from the following equations: 7 1 B + 1) (25) Κ Q, Β (B + 1) _ (F, (2 + + F(Q 2 1)1«3 (26) / Q ` + _ Α2Κβ, A (,/Ν (1 4Κ 1 ) + 2ν2,Κ In Q3 Q 2 Α '(Ν Α (Ν 1) 3 (1 _2Κ ιn Q3 4 ^N + (1 _ 2Κ ) + 8 Κ + 8'1K 213, Q3 -,/23,Κ + 0, (2Κ + A ),/2ο.Κ + 13,A Ν 1 + 2β,Ρ A 2 λ0 B Κ Ρ Α Π 2 F, 1 (1+σο «ο ί32n). Constant thermal properties The constant-thermal-properties solution follows from the preceding case if 0 2, 0 and β, It can be shown that, for 0 2, = 0, lim Q, 0, Ο lim Q 2 β1 --0 lim 1 0, Ο 1 (27a) 1 (27b) 82. (27c) For the constant-property case, «0 becomes α2 and σο σ, λο λ. Then the solution is (Β )(B + 1) (1 + )(1 ++ σσ))α « (28) The parameter y is again found from eq 25. 7

16 For this case an exact solution is possible by using a similarity transformation, as was shown by Lunardini (1985). The solution is 02 1 erfx/2 i + om) erf -1/α 32(1 + σ) (29) where 8 3 erfc 2 ^3 t ) 1 erfc y λ erf γ'α 32(1 + = 'iα32(1 + σ) erfc 7 ē Υ 2 Γα32(Ι + "] 1] σ) (31) (30) erfx fe 2 dz 0 is the familiar error function. Quadratic unfrozen water function When a quadratic unfrozen water function is used (for example, eq 2), it is not possible to find a closed-form solution for variable thermal properties. Thus a heat balance approximation will be used for constant thermal properties. The equations for region 2, using eq 2, are α2 2 ax2 ci t [( σ8i] (32) 82(X, t) 0 (32a) 02(X,, t) 1 (32b) k2 2 (Χ, t) k3 ó^ (Χ, t). (32c) The heat balance integral form of eq 32 is x al Η (Χ, t) áx dt 1 [(1 + 2σ)82 σ82] dx. 2(0, t)l 0 (33) Equation 20 is still valid for region 3. The temperature profile for region 3 is again eq 22. The temperature in region 2 is assumed to be where 02 b,(χ x) + c,(χ χ) 2 (34) b,x 2 ^ B 8

17 c,χ2 φ B The equation for Β is now (00-2λ)(Β + 3) α32[(d + ο) 1 + 2σ - 5) 2 5 ^ Z (35) The parameter -γ is again found from eq 25. The two-zone solutions can be compared by considering some specific cases. For example, consider a typical soil with the following properties suggested by Nakano and Brown (1971): Τo 4 C Ίs -4 Tf 0 Tm - 4 (thus φ Ψο 1) Εf gram water gram solids d 1.68 gram solids/cm 3 P 80 cal/gram water ST The soil thermal properties and the results of several cases are summarized in Tables 1 and 2. Cases 1-3 show that the effect of specific heat variation is not important and can safely be neglected. However, case 4 indicates that the thermal conductivity can cause 15-25% variations in the rate of growth of the freezing zone. Case 5 uses average values of k and C within Table 1. Effect of thermal properties on freeze of soil with average properties and linear. Difference Co k from case 1 Case (cal/cm 3- C) (cal/s-cm- C) (32 (%) Comment Base case with variable k, C Constant specific heat, C Constant specific heat, C Constant k, C.* Constant k, C. t Constant k, C,* exact solution. * k2 kf = C2 Cf 0.54 k3 ku C3 Cu kf+ ku Cf+Cu t k2 = C2 =

18 Table 2. Effect of thermal properties on freeze of soil with extreme property variations. Difference Co ko from case 1 Case (cal/cm'- C) (cal/s-cm- C) 1, (%) Comment Extreme properties, linear Ε, variable k, C C = 0.63, linear Ε. Constant specific heat, Constant specific heat, C = 0.315, linear Ε k, C,* linear Ε. Constant Constant k, C, t linear Ε Constant k, C, * linear Ε, exact solution Constant k, C*, quadratic Ε. * k 2 kf = C2 Cf kf +k t k2 = 2 C +Cf C2 = the mushy zone, and the effect of variable properties can be adequately taken into account by using the constant-property solution with the average of the fully frozen and fully thawed thermal properties. Cases 4 and 6 compare the heat balance approximation to the exact solution (eq 31). The approximate solution is within about 7% of the exact solution. This tends to verify the acceptable accuracy of the heat balance integral method. The effect of the different unfrozen water content functions can be deduced from cases 4 and 7 of Table 2. The growth rate for the quadratic water function lags behind that of the linear water function by about 9%. This was also noted by Lunardini (1985). Since the quadratic unfrozen water function will be a more accurate representation of an actual soil, the quadratic solution is presented in Figure 4 for the two-zone problem I Γ = Η i x tσ \^ ^ 0.5 N Η LO 0.5 Χ= ο IO i ^o p oo I ^o k οο ^_ C 32 i st l 0 J α.φ 1, α32 0.1, 0.3. b. ψ 1, α32 1.0, 0.6. Figure 4. Quadratic solution for the two-zone problem. 10

19 ^ Lj =ο.6 06 \ ^ ^ α 32 = 0. 3 α32 = 1. 0 Χ= 0.5 \\ \ λ=0.5 \ 0.4 \ 0.4 Ν y \ ο 0 5 _ ^ I I 0 ι οο I 10 ι οο 0- ιτ C. ψ 0.6, α32 1.0, 0.6. d. ψ 0.6, α32 0.3, I 0.3 ; 32 = \ α32= 1. o 0.6 λ = ο Υ I.0 Υ ^ ` I t o Ι 00 ι 0-0- l ^ Ι 00 e. ψ 0.1, α32 0.1, 0.3. f. φ 0.1, , 0.6. Figure 4 (cont'd). THREE-ZONE PROBLEMS Since the variable-property case can be adequately handled by an appropriate constantproperty solution, only the constant-property problem will be examined. Linear unfrozen water function Exact solution Equations 15 and 18 are the governing equations for regions 1 and 3, while the equations for region 2 are 11

20 αρ2 a ο2 a e (1 + σ) a82 ót (36) 8^(X, t) 0 (36a) 82(X,, t) 1 (36b) k2 (X, t) k2 áx' (X, t). (36c) The similarity method was used by Lunardini (1985) to obtain an exact solution to this problem. The solution is as follows: 8, ( 1 0) 1 + 2\ία, t^ erf (37) erf γ,/α32(1 + σ) erf-ν/α32(1 +σ) erf η'/α,2(1 + σ) (38) 03 = erfc 1 Ψ o (39) X, 0 2η'/α, t (40) Χ o 2γ,%α 3 t. (41) The parameters η and η are found from the simultaneous solution of the two following equations:,%«32(1 + σ) (erfcγ)ē Υ 2Γα,=(1 +σ) -1] λ[erfγ,iα32(1 + σ) erfη,/α12(1 + σ)] (42) (φ 1)e-7 /2[1 α120 + Ο i[erf^ (1 + σ) erfη'/α,2(1 + 0)] Q k2,,/α,2(1 + 0)erfη. (43) Lunardini (1985) showed that this solution approached the Neumann solution as the phase change zone decreased (Table 3). It is clear from Table 3 that the solution does converge to the Neumann solution as (Tf Tm) 0. The thaw/freeze interface greatly exceeds the value for the Neumann solution if phase change occurs over a 4 C zone as in the example calculation. Heat balance integral solution The heat balance integral equations for the three-zone problem are as follows: αρ, l α Ι (Χ1, t) α (0 t) j x, S (8 l)dx 0 (44) 8 1 (0,t) Φ (44a) 12

21 Θ, (Χ,, t) 1 (44b) αθ, αχ (Χ,, t) ^ k2, αθ2 αχ (Χ,, t) (44c) α, 1322 (Χ, t) α Θ 2 Θ2(Χ, t) 0 Θ2(Χ,, t) 1 (χ,, t)1 ^ d Θ Ζdχ + Xl (45) (45α) (45b) 082 (Χ, t) ^ k32 αχ - ί (Χ' t) θ3(δ, t) Q φο 083 (45c) ίχ, αχ t) δ θ3dχ + (46) dt χ (46α) Θ3(Χ, t) 0 (46b) αχ (δ, t) Ο. (46c) Quadratic temperature profiles for the three regions lead to ^ φo! δ_ X (47) θ 3 where Θ 1 + b 2(Χ, x) + c2(χ, χ)2 (48) Θ2 ο e2(χ x) + f2(χ χ) 2 (49) b2χ ο 2k2, R Β c2χ 2(1 R) 2 φ 1 2k2,(1 R) f2χ2r 2 Q 2λR R ο 1,/α, 3. 13

22 Table 3. Effect of phase change temperature. (After Lunardini 1985.) Tm Χ Χ? Case ( C) η y (cm) (cm) Χ = Χ - Χ, φ Neumann For t 24 hours. Τ, = 4 C 7-6 Τf 0 C k, = cal/s-cm- C k2 = k C, === C2 C 3 ST cαl/cm'- C The solutions to eq are straightforward; the details will be omitted. The results are given below: 7 ^ (50) B 1 B + (1 - R) α3, (51) (Φ - 1)(3-77 2) - ( ) (1 - R)k2, - B 0 (52) 1 2B 1 3 ^) - - R R + 0. (53) B These four equations can be easily solved for η and y. Table 4 shows that the approximate solution is in error by less than 6% when compared to the exact solution. Table 4. Comparison of exact and heat balance integral solutions with linear Ε and constant k and C. Heat balance Tm Exact solution integral solution Variation (%) Case ( C) η γ η 7 η γ φ Τ0 4 C 7-6 Τf = 0 C k, = cal/s-cm- C k k C, = 2 CC == cal/cm 3- C ST

23 Quadratic unfrozen water function The equations for the quadratic unfrozen water relation are identical to eq except that the energy equation for the mush region (eq 45) is x ^2 I r áχ2 (X, t) - ^χ 2 t) J 1 (,, f [(1 + 2σ)Θ2 σθz]dx + (1 + σ)χ, (54) x, The quadratic temperature approximations (eq 47-49) are used again. The solution is again given by eq 50-52, with the mushy zone equation given by where 1 2z R 2 α (2z 2-7z + 18) ] 0 (55) z R λ Β Case 1 of Table 4 was evaluated for the quadratic Ε, and it was found that η and -γ These values differ from the linear Ε approximation by about 8%. The quadratic Ε, three-zone problem is a function of λ, σ, φ, α31, α32 and k 2,. Graphical solutions are shown in Figure 5 for typical soil parameters. CONCLUSIONS The mathematical model used assumed the latent heat to be a source of energy distribution throughout the volume of a soil with phase-change temperature limits of Tf and Tm. This contrast with the treatment of the latent heat is totally released at the upper phase-change temperature Tf. A comparison of the exact solution for the former case showed that it converged exactly to the Neumann solution as Tf - Tm approached zero. Thus, the mathematical model is based on sound physical principles. The existence of a mushy zone can have a significant effect on the mechanical strength of freezing or thawing soils. For many predictions of the strength of a freezing soil, the assumption is made that the soil temperature is that calculated from the Neumann solution. A soil with a mushy zone would have a completely frozen layer of soil that is thinner than that for the Neumann case and therefore would have less bearing capacity. In addition a thick zone of frozen soil would exist that has a variable unfrozen water content. Again the presence of this liquid water will decrease the mechanical strength of the soil. The thawing case would also tend to yield a bearing strength of soil that is less than that predicted with the Neumann solution temperatures. The effect of variable specific heat on the rate of phase change is negligible for the cases examined; thus, it is acceptable to use an average specific heat value in the mushy zone. The variation of the thermal conductivity with water content is much more significant and can cause a 15% change in the rate of freezing of the soil. However, it was shown that the constant-property solution, with average values of the thermal properties in the mushy zone, gives a solution that is quite close to that for the variable-property solution. It is acceptable to use the much simpler, constant-property solution with average thermal properties to compensate for the actual variable thermal conductivity. 15

24 1.4 φ=, χρ., Ι.2 λ = 0.5 ο Ι.2 Ι.ο λ= ο.5 y H ^ ^^05 ^ ^ 2 _ I υ. ^ \ ^ ^ ^ \\ η - H - ^ ŌΙΟ ^i^ -- Ι 0 φ = 5, χρ =0.1, ξ =. - _ -- ΙΟ --^ ^_ - ^ ΙΟ I ΙΟ Ι 00 ^ ΙΟ 100 σ σ a. φ 5, χ 0.5, ξf/ξο 0.2. b. φ 5, X 0.1, ξf /ξο 0.2. φ= ΙΟ,χ^ =0.5, f/ξō λ=0.5 Ι.Ο η - H η Ι. 2 λ = 05 Ι.ο 2 \ I Ο _ ^ 5 2 \^ Ι 0, xp^0. Ι, f /ξ0 = ^-_ 5 ^^ ^^ ^ V ^ _--^ ^ ^ Ι ΙΟ 100 σ 0 Ι l ο ΙΟο c. φ 10, x g 0.5, ξf /ξο 0.2. d. φ 10, χ ρ 0.1, ξf /^ ο 0.2. σ λ =ο 5 φ =30, χ ^ =0.1, f/ξ= 0. 2 I ο ^ =30, χ^= 0.5,f/ξ = λ = ^.\ 1.6 Ι.οο ^^ - 10 Ι Ο ^ - Η \\ η = \\ \ ^ -- \ 0.8 _ 5--^ ^ --- ΙΟ 0.4 ^^^ -~ L Ι I Ι Ι Ο Ι 00 σ σ e. φ 30, χ 0.5, ξf,/ξο 0.2. f. φ 30, χ 0.1, ξf,/ ξο 0.2. Figure 5. Quadratic solution for the three-zone problem. 16

25 A series of graphs are presented of the constant-property, three-zone problem for typical ranges of soil parameters. These graphs allow rapid predictions to be made for the freezing of soils with an unfrozen water content that is a function of temperature and variable thermal properties. LITERATURE CITED Anderson, D.M. and A. Tice (1973) The unfrozen interfacial phase in frozen soil water systems. In Analysis and Synthesis. Ecological Studies, vol. 4 (A. Nodos et al., Ed.). New York: Springer-Verlag, pp Berggren, W.P. (1943) Prediction of temperature distribution in frozen soils. Transactions, American Geophysical Union, 24(3): Carslaw, H.S. and J.C. Jaeger (1959) Conduction of Heat in Solids. Oxford: Clarendon Press. Cho, S.H. and J.E. Sunderland (1974) Phase change problems with temperature-dependent thermal conductivity. Journal of Heat Transfer, 96(2): Frivik, P.E. (1980) State-of-the-art report, Ground freezing: Thermal properties, modeling of processes and thermal design. In Proceedings, International Symposium on Ground Freezing, pp Lunardini, V.J. (1981a) Heat Transfer in Cold Climates. New York: Van Nostrand Reinhold Company. Lunardini, V.J. (1981b) Phase change around a circular cylinder. Journal of Heat Transfer, 103(3): Lunardini, V.J. (1982) Freezing of soil with surface convection. In Proceedings of Third International Symposium on Ground Freezing, Hanover, N.H., pp Lunardini, V.J. (1985) Freezing of soil with phase change occurring over a finite temperature zone. In Proceedings, 4th International Offshore Mechanics and Arctic Engineering Symposium, II: American Society of Mechanical Engineers. Lunardini, V.J. (1988) Heat conduction with freezing or thawing. USA Cold Regions Research and Engineering Laboratory, CRREL Monograph Nakano, Y. and J. Brown (1971) Effect of a freezing zone of finite width on the thermal regime of soils. Journal of Water Resources Research, 7(5): Neumann, F. (ca. 1860) Lectures given in 1860s. (See Riemann-Weber. Die partiellen Differential-gleichungen. Physik (5th ed., 1912), 2: 121.) Ozisik, M.N. and J.C. Uzzell (1979) Exact solution for freezing in cylindrical symmetry with extended freezing temperature range. Journal of Heat Transfer, 101: Tice, A.R., C.M. Burrows and D.M. Anderson (1978) Phase composition measurements on soils at very high water content by the pulsed nuclear magnetic resonance technique. In Moisture and Frost-Related Soil Properties. Washington, D.C.: Transportation Research Board, National Academy of Sciences, pp Tien, R.H. and G.I. Geiger (1967) A heat transfer analysis of the solidification of a binary eutectic system. Journal of Heat Transfer, ASME, Ser. C, 89:

26 APPENDIX A: DERIVATION OF THE MUSHY ZONE EQUATION The energy equation in the mushy zone (region 2) is a a x (k ΞΞ ax ) C + P Υd d r 2 at A1) The dimensionless temperature in region 2 is defined as 82 Tf TZ Τf- Τm (A2) Equation 1 transforms immediately to a x (k dx cae2 at Qηd d^ a82 (Tf Tm) doe at (A3) With the parameters defined in the text, it is also clear that, for the linear case, k ku(1 + 0, 82) (Α4α) C Cu(1 + (3282). (Α4b) Equations Α4α and b are written explicitly for the definitions of k and C discussed earlier. However, it will be advantageous to allow for the case where the mushy zone will have constant thermal properties different than those of the thawed state. Thus, we will define k 0 and C0 to be any constant values of the thermal conductivity and specific heat desired. Then k k0( ) (A5a) C C0(1 + '3282). (Α5b) Clearly, for the mushy zone with variable thermal properties, k 0 = ku and C0 = Cu. However, if we wish to examine the constant thermal property case, then we can simply set k 0, Co to any convenient values. For example we could let the properties of the mushy zone be the mean values of the frozen and thawed states and define k 0 and C0 appropriately. If the thermal conductivity varies with temperature, the Kirchoff transformation can be used to define a new temperature variable: 82 ψ ko f k(y) dy 0 (A6) where y is a dummy variable. From eq A6 a Θ2 at ko a ψ k at (A7) k aθ2 ax ko aψ ax (A8) 19

27 The derivation will proceed for the case of a linear function. The quadratic assumption can also be used, of course, but the equations will be more complicated. From eq Α4 and Α6 an explicit relation between ψ and 02 can be found: V 02 + (Α9) It follows that k k0'/1 + 2 β, t/i (Α10) C Cο( ' l1 + 20,x). (All) Equation Α3 then becomes k k ο o,/1 + 20,,E a 2 óx2 ' ( Πυd ξ Co(T^ T) ^t + 132,'/ , ψ ^ ιρ^ ót (Α12) Equation Α12 can be put into a more convenient form by noting that ',I1 + 2ι4 ót 313 ó t [(1 + 2β,ψ) 3/2]. (Α13) Finally the energy equation for the mushy zone with a linear unfrozen water content function is αο'/ + 2β ι ψ 2 óx [ ( σο)ψ + 3 βρi (1 + 2Ι 4) (Α14) 20

28 APPENDIX B: SOLUTION OF THE TWO-ZONE PROBLEM WITH A LINEAR Ε AND VARIABLE THERMAL PROPERTIES The equations for zones 2 and 3 are a 2,/1 + 2Ο,,, - ax ' ' at (Β1) ψ(0, t) 22 P (B 1 a) ψ(x, t) Ο (Bib) k0 t) k3 ax (X' ^ x (X't) (Bic) a203 1 αθ3 axe = α3 at ( B2 ) 61 3 (ό, t) Ψο (Β2α) ax (δ t) Ο (B2b) where F F1 ψ+ F2(1 +2β, tig) 12 (Β3α) Fi (1 + σο 02,) (B3b) F2 ί 2ι (Β3c) 3αοβι The heat balance integral method ( ΗBΙ) integrates the energy equation over the volume of interest. For region 2 f ^/1 + 2^,, ^ a adx 2^ o o af dx. at (B4) Using Leibniz's rule this is ú-./1 + 2β, dx axe dχ f F( χ, t)dx F(X, t) _ dt (Β5) Now F(X,t) F2 (Β6) 21

29 and the HBA equation for region 2 is ^ 2 0,i + 2β, 02 dx F(χ, t)dx F2Χj. l 0 The ΗΒΙ equation for region 3 is derived in the same manner and is á (X, t) α α3 83 dx + 0 l (Β8) t x Quadratic temperature profiles for regions 2 and 3 that satisfy the boundary conditions are ψ b(x x) + c(χ χ) 2 (Β9) 83 _ Ψ δ X - 1 I (Β 10) where b = 2φ0 k3 δ Χ cx 2 P bx. The moving interfaces Χ and δ are assumed to have the forms Χ 2η α3 t (Β11) δ (B + 1)Χ (Β 12) where B is a constant. Using eq Β10 with eq Β8 leads to the following equation: δ Χ d t (δ + 2Χ). (Β13) The solution to eq Β13 follows easily with the aid of eq 11 and 12: 1 B B +1 (Β 14) Equation Β7 can be solved to yield an algebraic equation for the unknown quantity B. From eq Β9 we note a ψ ^x 2c 2( xz b) (Β15) where c is only a function of time. Then eq Β7 is Χ 2có,/ 1 + 2^,,^dx F(x, t)dx F2Χ. dt Ó (Β 16) 22

30 The solution of eq Β16 is straightforward but rather tedious. The results are x f,/1 +2,3, ψ dχ Q, Χ (Β17) 0 where f 0 Fdx F, ( X + F2 f (1 + 2β,ι^) /2 dx 0 (Β18) x f (1 + 2β,ψ) 3/'2dχ Q2Χ (Β19) 0 Q. Q2 ^N + (^ - 1) K + g `42 β1 (N 1) + 2i Κ In Q3 3 1 A _ 2ψ0 k3o B Q3 -,/2 j3, Κ + ί, (2Κ + A ) '!2J3,Κ + β,a Κ P - Α. The equation for B is then ΚQ, Β ^ + F2(Q 2 1) α 3. (Β20) 23

31 A facsimile catalog card in Library of Congress MARC format is reproduced below. Lunardini, Virgil J. Freezing of soil with an unfrozen water content and variable thermal properties/ by Virgil J. Lunardini. Hanover, N.H.: U.S. Army Cold Regions Research and Engineering Laboratory; Springfield, Va.: available from National Technical Information Service, v, 30 p., illus.; 28 cm. (CRREL Report 88-2.) Bibliography: p Frozen soils 2. Phase change 3. Soils I. United States Army. Corps of Engineers. II. Cold Regions Research and Engineering Laboratory. III. Series: CRREL Report * U. S. GOVERNMENT PRINTING OFFICE: