US Army Corps of Engineers

ι

REPORT 88-2 Cold Regions Research & Engineering Laboratory

Freezing of soil with an unfrozen water content and variable thermal properties

Cooler Temperature warmer

Soil Surface

Ice and Water (no phase change)

Minimum Freeze Temρerafure

Ice and Water (phase change)

Maximum Freeze Temperature

Thawed

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 Materi-als, 1916 Race St., Philadelphia, Pa. 19103.

Cover: Geometry of a semi-infinite soil mass,

initially at a temperature above freez- ing, that freezes due to a constant

surface temperature below freezing.

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

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17. COSATI CODES

Frozen soils Phase change

18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)

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 ac-counted for by a proper choice of thermal properties used in the constant-thermal-property solution.

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11. TITLE (include Security Classification)

Freezing of Soil with an Unfrozen Water Content and Variable Thermal Properties

NOMENCLATURE

A 2 λ ο B

B _ δ-X 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 + σο - βτι) 1 ^0

F2 1321

3αοβΡι

k thermal conductivity kij ki/ki kο any specified constant-conductivity mushy zone value of k; otherwise k0 ku K Ρ-A

N 1 +213,P

Π 2

f latent heat of fusion of water m mass

mw, 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

Tf, Tm highest and lowest temperatures for phase change

Τ0 , TS initial and surface temperatures

x Cartesian coordinate xf volumetric water fraction

ii

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

phase change parameter defined by eq 23 2 α3t

Υd dry unit density of soil solids (mass of soil solids per unit volume)

77 2'/J phase change parameter

Tf — T 9

Tf — Tm

, dimensionless temperature

λ ψ0 k3 2

λ0 ψ0 k30

^ ratio of unfrozen water mass to soil solid mass

Εο, f' Εs values of at Tf, Tm, Ts

Ο C32 Φ ST

C30ψ σ0 T

Í'f— Ts =

Τf— Τm

TO — Tf Ο0

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

PREFACE

This report was prepared by Dr. Virgil J. Lunardini, Mechanical Engineer, Applied Re-search 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 techni- cal reviews of this report.

The contents of this report are not to be used for advertising or promotional purposes. Ci-tation of brand names does not constitute an official endorsement or approval of the use of

such commercial products.

iv

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 10 5. 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. 10 3. Effect of phase change temperature 14 4. Comparison of exact and heat balance integral solutions with linear and

constant k and C 14

v

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 mini- mum 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

1 2 3

Ice + Water Ice + Water Thawed

To Νο Phase Change Phase Change

- - - - - -

F—a- Thawed

e0 r-- —

1 m

TS ' • 1 f X

x 6 Tm T f

Figure 1. Geometry for solidification with a phase change zone.

Figure 2. Unfrozen water vs temperature.

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 fully frozen and fully thawed thermal conductivities and fully frozen and fully thawed specific heats. Cf, Cu

Obviously these properties are functions of the particular form of the unfrozen water func- tion (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, how-ever. 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

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

Cat' (7) (k óx^— Q ηΡd yt

The most general case will be a problem with three regions as shown in Figure 1. The equa-tions for region 1 are

82 7', axe αι at

1 a Τ, (8)

7', (0, t) Ί s (8a)

T,(X,, t) Tm (8b)

, a Τ,(Χ,) k^X^)

a Τ2(Χ, ) (8c) αχ ax

For region 2 (the mushy zone)

α— (k z2 ) C + Pld d _ r at

T2 (9)

T2(X,, t) Tm (9a)

T2(X, t) Tf (9b)

k(X) aT2(X) k,

aT,(X) (9c) αχ αχ

The thawed zone is governed by

a2 Τ3 axe «, at

1 aτ, (10)

lim Τ3 (χ, t) To (lOa) X-• 00

Ρ,(χ, ο) Ρo (lOb)

Τ, (X, t) Tf. (10c)

3

These equations can be partially nondimensionalized by using the following relations:

Tf — Ti Tf — TS TO — Tf θi Tf —Tm 1,2,3 Tf Tm 00 Tf Tm

σ C3 2 φ σ0

C3 0 0 λ ψ0 k32 λ ψ0 k30

SηΡ- SτΡ

kfŪ «0 ko ^.T

C3( Tf - Ts) β(^

1 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) (ha)

C o C0 (1 +13202). (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:

e,

k^ S k(Θ21 dΘ2. 1

(12) 0

For the linear Ε case the function V can be evaluated explicitly:

ψ o 02 + 2

612 . 1

(i2a)

Then

k ^ k0 '/1+ 20,,1 (13)

^/1 +- 1 2β,ψ ^z (14)

The transformed equations are as follows:

a28, 1 αο, ax2

(15) α, at

0, (0, t) Q ψ (15a)

8,(X,, t) — 1 (15b)

4

k, ί ' (X,, t) k(Χ,) (X' ' t).

(15c) x

The equations for region 2 are written only for the linear ξ assumption (the details are given in Appendix A):

'/1 +2,^,^ αx 2 (16)

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:

8203 3χ2 «3 āt

1 11 03 (18)

him 83(χ, t) - ψ0 (18a) χ- οο

83 (χ, 0) - ψο (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 two-zone 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 ex-amined 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 ade-quate for the general problem.

Linear unfrozen water function

Variable thermal properties Equations 16-18 are valid for this case except that the boundary condition in eq 16b be-

comes

5

ψ(0, t) ο + P. (18d) 2

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

‚/1 + 2β2ψ ' 2 dx 2

F(x, t)dx - F2X), (19) dt l 0 0

(X t) dt Ir

f α

φο δl -

α3 óx x 83 dx + (20)

where

FZ (32 ,

3 α0 J3 ,

For the heat balance integral treatment, eq 18a and b become

83 (δ, t) — φο (18e)

0. (180 cm

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)

^ο [` . a — x) ) 2 ii (22)

where

b 2λ0 δ -X

cX 2 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

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:

1 (25) 7 B + 1)

ΚQ, Β (B + 1) _ (F, (2 + + FF (Q2 — 1)1«3 (26) /

(1 _ Α2Κβ,

Q ` + A (,/Ν —1 ) +

In Q3 4Κ 2ν2 ,Κ

Α '(Ν Α (Ν —1) 3 (1 _2Κ ιn Q3

Q2 __ 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+σο 1 — ί32n). «ο

Constant thermal properties

The constant-thermal-properties solution follows from the preceding case if 0 2 , —0 and β, -- 0. It can be shown that, for 02 , = 0,

lim Q, 0 , — Ο

1 (27a)

lim Q 2 β1 --0

1 (27b)

lim 1 0,— Ο

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«3322 0. (28)

The parameter y is again found from eq 25.

7

For this case an exact solution is possible by using a similarity transformation, as was shown by Lunardini (1985). The solution is

02 erfx/2 i +

om) 1

(29) erf -1/α32 (1 + σ)

erfc 1 8 3 2 t ) ^3 (30)

erfc y

λ erf γ'α32 (1 + = 'iα32(1 + σ) erfc 7 ē Υ 2 Γα32(Ι + "] — 1] σ) (31)

where

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 approxima-tion will be used for constant thermal properties.

The equations for region 2, using eq 2, are

α2 aX2

2

t ci [(1 + 2002 — σ8i] (32)

82(X, t) 0 (32a)

02(X,, t) 1 (32b)

k2 2 (Χ, t)

k3 (Χ, t). ó^

(32c)

The heat balance integral form of eq 32 is

2(0 ,

x

al Η (Χ, t) áx t) l dt 1 [(1 + 2σ)82 — σ82] dx.

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

02 b,(Χ—x) + c,(Χ — χ) 2 (34)

where

b,X 2 ̂ B

8

c,Χ 2 φ B

The equation for Β is now

(00 - 2λ)(Β + 3) α32[(D ο) 1 + + 2σ - 5) 2

(35) 5 ^ Z

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)

0 0.20 gram water gram solids

Εf 0.0782

d 1.68 gram solids/cm 3

P 80 cal/gram water

ST 0.1539.

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% varia-tions 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

1 0.63 0.0058 0.431 -0.1429 0.3988 Base case with variable k, C. Constant specific heat, C Constant specific heat, C Constant k, C.*

2 0.63 0.0058 0.431 0 0.3996 0.2 0.63. 3 0.54 0.0058 0.431 0 0.4016 0.7 0.54. 4 0.54 0.0083 0 0 0.4575 14.7 5 0.585 0.0071 0 0 0.4126 3.5 Constant k, C. t

6 0.54 0.0083 0 0 0.4277 7.2 Constant k, C,* exact solution.

* k2 kf = 0.0083 C2 Cf 0.54 k3 ku C3 Cu

kf+ ku Cf+Cu t k2 = 2 0.0071 C2 = 2 0.585.

9

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, 132 7 (%) Comment

1 0.63 0.0058 1 -0.50 0.4626 Extreme properties, linear variable k, C. Constant specific heat, C = 0.63, linear Ε.

Constant specific heat, C = 0.315, linear Ε Constant k, C,* linear Ε.

Ε , 2 0.63 0.0058 1 0 0.4607 -0.4 3 0.315 0.0058 1 0 0.4696 1.5 4 0.315 0.0116 0 0 0.5671 22.6 5 0.473 0.0087 0 0 0.4726 2.2 Constant k, C, t linear Ε. 6 0.315 0.0116 0 0 0.5304 14.7 Constant k, C, * linear Ε, exact solution. 7 0.315 0.0116 0 0 0.5226 13.0 Constant k, C*, quadratic Ε.

* k2 kf = 0.0116 C2 Cf 0.315.

kf +k„ C„+Cf t k2 = 2 0.0087 C2 = 2 0.473.

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 ac- ceptable 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.

1.4 1.4 I Γ

1.2 0 33 0.1 1.2 033=1.0

0.3 - 0.6 -

Η LO

Χ= ο 0.8 0.8

i x tσ \^ ̂ 0.5

0.5 N 0.6 0. 6 Η

0.4 0.4 2 2 _ _

0.2 5 0.2 5 i

IO - - 10

° ° ^o p oo I ^o k οο ^_ C 32 i

l0J sT

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

0.6 0.6^Lj

06 \ ̂ =ο.6

^ α 32 = 0 . 3

α32 = 1 . 0 Χ= 0.5 \ 0.6-- \

\λ=0.5 \ 0.4 \ 0.4 Ν

y

2 \ 2

0.2 - 0.2

5 _ 5

10 1^

0 ο 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, 0.1.

I 0.15 0.3 λ = ο α32= 1. o

0.6 — — ; 32 =0.1

\ 0.3- -

0.1 0.2

Υ I.0

Υ

- --- 0.5 - -

0.05 2

` 2 5 10

10 ^

Ι 00 I t o Ι 00 ι

0-

l ^

0-

e. ψ 0.1, α32 0.1, 0.3. f. φ 0.1, 32 1.0, 0.6.

Figure 4 (cont'd).

THREE-ZONE PROBLEMS

Since the variable-property case can be adequately handled by an appropriate constant-property 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

αΡ2 a e

a ο2 (1 + σ)

ót a82

(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, t̂

(37) erf

erf γ,/α32(1 + σ) —

(38) erf-ν/α32(1 +σ) —erf η '/α,2(1 + σ)

03 = erfc — 1 Ψo (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 equa- tions:

,%«32(1 + σ) (erfcγ)ē Υ2Γα,=(1 +σ)-1] λ[erfγ,iα32(1 + σ) — erfη,/α12(1 + σ)] (42)

(φ — 1)e-7/2[1— α120 + Οi[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 calcula-tion.

Heat balance integral solution The heat balance integral equations for the three-zone problem are as follows:

α (0 t) j x,

0 (44) αΡ,

l α Ι (Χ1, t) — S (8 — l)dx

8 1 (0,t) — Φ (44a)

12

Θ, (Χ, , t) — 1 (44b)

αθ, (Χ, , t) ^ αθ2 (Χ, , t) (44c) αχ

k2, αχ

α, 1322 (Χ, t) — α Θ 2 (χ, , t)1 ^ d Θ Ζdχ + Xl (45)

Θ2(Χ, t) 0 (45α)

Θ2(Χ,, t) ° 1 (45b)

082 (Χ, t) ^ 083 ίΧ, t) (45c)

αχ k32 αχ

383 (Χ' t)

δ (46)

- ί 3 dt χ θ3dχ +

θ3(δ, t) Q φο (46α)

Θ3(Χ, t) 0 (46b)

(δ, t) Ο. (46c) αχ

Quadratic temperature profiles for the three regions lead to

^ φo ! δ_ X (47) θ 3 —

Θ 1 + b 2(Χ, — x) + c2(Χ, — χ)2 (48)

Θ2 ο e2(Χ —x) + f2 (Χ — χ) 2 (49)

where

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

Table 3. Effect of phase change temperature. (After Lunardini 1985.)

Tm Χ• Χ? Case (°C) η y (cm) (cm) ∆Χ = Χ - Χ,

1 -4 0.0617 0.3029 33.33 8.13 25.2 2 -2 0.1135 0.2576 28.34 14.95 13.39 3 φ 0.1376 0.2272 25.0 18.12 6.88 4 -0.5 0.1492 0.2106 23.17 19.65 3.52 5 -0.1 0.1571 0.1946 21.41 20.69 0.72

Neumann 0 0.1606 - 21.15 21.15 0

• For t 24 hours.

Τ, = 4°C 7 -6 Τf 0 °C k, = 0.00828 cal/s-cm- °C

k2 = 0.00703 k3 0.00578 C, === C2 3 C 0.165 cαl/cm'-°C

ST 0.0605.

The solutions to eq 44-46 are straightforward; the details will be omitted. The results are

given below:

^ (50) 7

B 1B

+

(1 - R) 2 72 α3, (51)

(Φ - 1)(3 - 77 2) - (6 + 77 2 ) (1 - R)k2 , - B 0 (52)

1 2B 1 3 ^) R 0132 7 2 3 2R + 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 solu-tions with linear Ε and constant k and C.

Heat balance

Tm Exact solution integral solution Variation (%) (°C) Case η γ η 7 η γ

1 -4 0.0617 0.3029 0.0622 0.2938 0.8 -3.1 2 -2 0.1135 0.2576 0.1150 0.2411 1.3 -6.4 3 φ 0.1376 0.2272 0.1390 0.2168 1.0 -4.6 4 -0.5 0.1492 0.2106 0.1505 0.2050 0.9 -2.6 5 -0.1 0.1571 0.1946 0.1595 0.1958 1.5 0.6

Τ0 4°C 7 -6 Τf = 0 °C k, = 0.00828 cal/s-cm- °C

k2 0.00703 k3 0.00578 C, = == 2 3 CC 0.165 cal/cm 3 - °C

ST 0.0605.

14

Quadratic unfrozen water function The equations for the quadratic unfrozen water relation are identical to eq 44-46 except

x

that the energy equation for the mush region (eq 45) is

Iráχ2 2 t) J1

x, ^2 (X, t) - χ̂ (,, f [(1 + 2σ)Θ2 — σθZ]dx + (1 + σ)Χ, (54)

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

1 2z — R 2 α32 72 + 1

(2z2 7z + 18) ] 0 (55) — 1 3 - 5 -

where

z R λ Β

Case 1 of Table 4 was evaluated for the quadratic Ε, and it was found that η 0.0572 and -γ 0.2730. 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 k2 ,. Graphical so-lutions 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 con-verged exactly to the Neumann solution as Tf - Tm approached zero. Thus, the mathe- matical 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 assump-tion 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 con-stant-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 com-pensate for the actual variable thermal conductivity.

15

1.4

Ι.2 λ = 0.5

φ=, χρ .,ο Ι.2

Ι.ο λ= ο.5 y

φ = 5 , χρ =0.1 , ξ =.

H η -

H

0.8 2 0.8 2

0 . 6 0.5 ^ ^^05

2 I υ . \ ^ ^ 1.0

^ _

0.4 2 ^ ̂ ^ 0.4 5 \\ - - Ι 0

-- ^ ŌΙΟ ^i^ - _

ΙΟ --^ 0.2 ΙΟ -- - - ^_ - ̂ -

0 0 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.

1.4

1.50 λ=0.5 φ= ΙΟ,χ^ =0.5, f/ξō 0.2 Ι . 2 λ = 05

Ι 0 , xp^0. Ι, f /ξ0 =0.2

H Ι .Ο η - η - -

Ι.ο 2 2

\ I Ο

1.00 0.8 2 \^ _ ̂

5

^^ 0.6^-_ 5

0.50 0.4 V 5 - -^^ ^ 10 ^

10 _--^ ^ ^- 0.2

0 Ι ΙΟ 100 0

Ι l ο ΙΟο

σ σ

c. φ 10, x g 0.5, ξf /ξο 0.2. d. φ 10, χρ 0.1, ξf /^ ο 0.2.

Η

2.4 150 I λ =ο 5 φ =30, χ ^ =0.1, f/ξ= 0 . 2 ο

^ =30, χ^= 0.5,f/ξ =0.2 2.0 1.0 λ =0.5

2 - ^.\ 1.6 Ι .οο

5 -̂ \\

\

η

= \ \ 10 - 1.2 - Ι Ο -^^ ^

^ \ ^ - -

0.50 0.8 5 - -^ --- _

ΙΟ 0.4 ^^^ -~

0 0 L Ι

σ I 0 1 00 Ι

σ Ι Ο Ι 00

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

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 ther-mal properties.

LITERATURE CITED

Anderson, D.M. and A. Tice (1973) The unfrozen interfacial phase in frozen soil water sys- tems. In Analysis and Synthesis. Ecological Studies, vol. 4 (A. Nodos et al., Ed.). New York: Springer-Verlag, pp. 107-124. Berggren, W.P. (1943) Prediction of temperature distribution in frozen soils. Transactions, American Geophysical Union, 24(3): 71-77. 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): 214-217. 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. 354-373. 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): 598-600. Lunardini, V.J. (1982) Freezing of soil with surface convection. In Proceedings of Third In- ternational Symposium on Ground Freezing, Hanover, N.H., pp. 205-212. 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 Sym- posium, II: 38-46. American Society of Mechanical Engineers. Lunardini, V.J. (1988) Heat conduction with freezing or thawing. USA Cold Regions Re- search and Engineering Laboratory, CRREL Monograph 88-1. Nakano, Y. and J. Brown (1971) Effect of a freezing zone of finite width on the thermal re- gime of soils. Journal of Water Resources Research, 7(5): 1226-1233. Neumann, F. (ca. 1860) Lectures given in 1860s. (See Riemann-Weber. Die partiellen Differ- ential-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: 331-334. 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. 11-14. 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: 230-243.

17

APPENDIX A: DERIVATION OF THE MUSHY ZONE EQUATION

The energy equation in the mushy zone (region 2) is

x a a (k ΞΞ

ax ) C + PΥd d r

2 at A1)

The dimensionless temperature in region 2 is defined as

Tf — TZ 82

Τf - Τm

(A2)

Equation 1 transforms immediately to

x a (k

dx cae2

at (Tf — Tm) doe at Qηd d^ a82 (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 con-stant thermal properties different than those of the thawed state. Thus, we will define k0 and C0 to be any constant values of the thermal conductivity and specific heat desired. Then

k k0 (1 + 13182) (A5a)

C C0(1 + '3282). (Α5b)

Clearly, for the mushy zone with variable thermal properties, k 0 = ku and C0 = Cu . How- ever, 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 k0 and C0 appropriately.

If the thermal conductivity varies with temperature, the Kirchoff transformation can be used to define a new temperature variable:

82

ψ o 0 kf k(y) dy (A6)

where y is a dummy variable. From eq A6

aΘ2 ko aψ at k at

(A7)

k aθ2 ko aψ (A8) ax ax

19

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(1 — 021 + 021 ' l1 + 20,x). (All)

Equation Α3 then becomes

kk

οo

,/1 + 20,,E óx2 a 2

' (- Co(T^ — Πυd ∆ ξ

+ 132,'/1 + 213, ψ ót ^ ιΡ^ (Α12) 1321+

T) ^t

Equation Α12 can be put into a more convenient form by noting that

',I1 + ót

2ι4 t ó

[(1 + 2β,ψ) 3/2]. (Α13) 313

Finally the energy equation for the mushy zone with a linear unfrozen water content function

is

2 αο '/ + 2β ι ψ óx

[ (1 - 1321 + σο)ψ + 3 βΡi (1 + 2Ι 4) (Α14)

20

APPENDIX B: SOLUTION OF THE TWO-ZONE PROBLEM WITH A LINEAR Ε AND VARIABLE THERMAL PROPERTIES

The equations for zones 2 and 3 are

2Ο,,, - ,/1 + ax a 2

' ' at

(Β1)

ψ(0, t) 22

P (B 1 a)

ψ(X, t) Ο (Bib)

k0 ax (X'

t) k3 x (X't) ^ (Bic)

a203

axe = α3 at 1 αθ3

( B2 )

61 3 (ό , t) — Ψο (Β2α)

(δ t) Ο (B2b) ax

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

a2 ^ ^ adx aF dx. o f ^/1 + 2^,,

o at (B4)

Using Leibniz's rule this is

dΧ (Β5) 2β, dx ú-./1 + axe f F(χ, t)dx — F(X, t) _ dt

Now

F(X,t) F2 (Β6)

21

and the HBA equation for region 2 is

^ 2

0 ,i + 2β, 02 dx

l 0

F(χ, t)dx — F2Χj .

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:

d (δ + 2Χ). (Β13) δ — Χ t

The solution to eq Β13 follows easily with the aid of eq 11 and 12:

1 (Β 14) B B

+1

Equation Β7 can be solved to yield an algebraic equation for the unknown quantity B. From eq Β9 we note

a ψ 2c

2( b) (Β15) ^x

xZ

where c is only a function of time. Then eq Β7 is

Χ 2có ,/ 1 + 2^, ,^dx F(x, t)dx — F2Χ . (Β 16)

dt Ó

22

The solution of eq Β16 is straightforward but rather tedious. The results are x

f ,/1 +2,3, ψ dχ Q, Χ (Β17) 0

0 f Fdx F,

( +

3 1 X + F2 f (1 + 2β,ι^) /2 dx

0 (Β18)

x f (1 + 2β,ψ) 3/'2 dχ Q2Χ (Β19) 0

where

Q. ^N + (^ 1 )+ -

3 1 — 2i

Κ `42 β1

4 + 1 — 2K + g (N — 1) + In Q3 Q2

Q3 - ,/2 j3, Κ + ί , (2Κ + A )

'!2J3,Κ + β,A

A _ 2ψ0 k3o

B

Κ P - Α.

The equation for B is then

ΚQ, Β 3 + + 3^

+ F2 (Q 2 — 1) α 3 . (Β20)

23

A facsimile catalog card in Library of Congress MARC format is repro-duced below.

Lunardini, Virgil J. Freezing of soil with an unfrozen water content and variable thermal pro-

perties/ by Virgil J. Lunardini. Hanover, N.H.: U.S. Army Cold Regions Research and Engineering Laboratory; Springfield, Va.: available from Na- tional Technical Information Service, 1987.

v, 30 p., illus.; 28 cm. (CRREL Report 88-2.) Bibliography: p. 17. 1. 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 88-2.

* U. S. GOVERNMENT PRINTING OFFICE: 1988--500-050--62079