An Introductory Review of Numerical Methods for Ground Thermal Regime Calculations

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An Introductory Review of Numerical Methods for Ground Thermal

Regime Calculations

JBR Paper No. 1061 Iivision of Building

Res

ISSN

0381

P

National Research

Conseil national

i

GO.

la61

Council Canada

de recherches Canada

1

c . 2 . BLSG

u'

AN

INTRODUCTORY REVIEW

OF

NUMERICAL METHODS FOR

GROUN

THERMAL

REGIME CALCULATIONS

ly L.E. (

NATIONAL RESEARCH COUNCIL OF CANADA DIVISION OF BUILDING RESEARCH

AN

INTRODUCTORY REVIEW OF NUMERICAL METHODS FOR GROUND THERMAL REGIME CALCULATIONS

L.E. Goodrich

DBR

Paper No. 1061 of the

Division of Building Research

SUMMARY

This paper reviews the literature on numerical methods for ground temperature calculations. In addition, an introductory discussion of both finite difference and finite element methods for conduction heat flow is given. The mathematical description has deliberately been kept readable and yet sufficient detail has been given to make comprehensible the main notions of these methods as well as their practical realization.

Methods of treating freezing and thawing are discussed. The

description is confined to the relatively simple case of "in situ" phase change.

The paper closes with a presentation of formulae for estimating the components of the ground surface heat balance and discusses some of the difficulties that arise in their application as a boundary condition for practical numerical model computations.

Ce rapport fait la synthsse dlune Ctude bibliographique sur les mCthodes numCriques pour le calcul des tempdratures dans le sol. Afin d'initier ltCtudiant, une partie importante du rapport est consacrCe aux notions de base des mCthodes par diff'erences finies et par Clhents finis dans le cas du transfert de chaleur par conduction. Le raisonnement

mathO&natique a 'et'e consciemment r'eduit au strict minimum mais suffisamreent de dCtails sont donngs pour que le lecteur comprenne les notions

principales de ces techniques ainsi que leur application pratique.

Dans beaucoup de probl2mes rgels dlanalyse thermique des sols, l'on slint'eresse surtout au gel. Ce rapport comprend une section explicant diffdrents aspects de la modClisation numgrique de ce ph6nodne. Par souci de simplicit'e la pr'esentation se limite au cas de changement de phase "in situ".

Dans la section finale du rapport on considsre la question de la mod'elisation numCrique des conditions aux limites.

On

pr'esente des

formules qui peuvent dtre utilisges pour estimer les diverses composantes du bilan de chaleur en surface. On expose aussi les difficult'es que soul2ve leur application aux problhes pratiques de ggnie.

CONTENTS

Page No.

Introduction Numerical Methods

F i n i t e Difference Methods for Conductive Heat Transfer

F i n i t e Element Methods for Conductive Heat Transfer

Comparison of F i n i t e Element and F i n i t e Difference Methods

The Numerical Treatment of Freezing and Thawing Surface Boundary Conditions

Radiation

Sensible Heat Transfer Evaporative Heat Transfer References

AN

INTRODUCTORY REVIEW OF NUMERICAL METHODS FOR GROUND THERMAL REGIME CALCULATIONS INTRODUCTION

In soils the dominant mechanism of heat transfer is that of thermal conduction. Frequently, even when other mechanisms such as moisture migration in the liquid or the vapour phase are important, these can be accounted for by defining apparent thermal properties and "pure" conduc- tion models can still provide useful results.

The release or absorption of latent heat during freezing and thawing strongly affects the pattern of seasonal ground temperature variations. In geotechnical problems, determining the location of the freezing or thawing front(s) is often the principal reason for carrying out thermal analysis. Although in many cases freezing and thawing will be accompanied by moisture redistribution along with geometrical changes, as a starting point it is usually sufficient to consider the case of pure heat

conduction including latent heat of freezing or thawing of in situ water. This review is limited to these cases. A discussion of surface heat balance boundary conditions is also given.

NUMERICAL METHODS

Although analytical methods provide useful solutions in simple cases, frequently it is necessary to consider problems involving a great deal more physical detail than can be readily treated by these methods. The usual range of geotechnical problems involves layered systems with latent heat, temperature-dependent thermal properties and time-dependent boundary conditions. In many cases it is essential to consider two- and even

three-dimensional heat flow. Numerical methods are capable of handling most of these complexities. In addition, these methods are very flexible and, once established, the same program can be used to solve a range of problems without the need to devise a new solution for each case.

The most widely used numerical methods are the finite difference and the finite element methods. Douglas (1961) presents a thorough survey of finite difference methods for parabolic differential equations. The book of Richtmyer and Morton (1967) is a relatively modern work on finite difference methods for initial value problems while the classic book of Dusinberre (1961) is devoted specifically to heat transfer calculations. Books by Desai and Abel (1971), Zienkiewicz (1971), Oden (1972), and

Norrie and DeVries (1973) describe the finite element method with emphasis on applications in structural and continuum mechanics. Zienkiewicz (1977) is an expanded and revised version of the 1971 book which presents many developments relevant to heat flow computations not contained in the original version. Hinton and Owen (1980) give a very readable account of the finite element method and provide a wealth of useful information for those who write their own programmes. Myers (1971) presents an

techniques as they apply to conductive heat transfer problems and gives much practical information as well as extensive references.

FINITE DIFFERENCE METHODS FOR CONDUCTIVE HEAT TRANSFER

The fundamental differential equation of conductive heat flow is the Fourier equation:

where T = temperature at time t and position x C = heat capacity per unit volume, j

'

kj = components of thermal conductivity tensor along the principal axes,

and

I

= rate of internal heat generation per unit volume. aT

The term -k

-

represents the heat flux in the

x

direction. In

-i ax, j

Equation

1 ;he dpace coordinates are assumed to coincide with the

principal axes of the material.

To illustrate the basic concepts of the finite difference method, consider the simple case of one-dimensional heat flow with no internal sources:

Space may be discretized into elements of width Axi =

-

xi, where xi are the locations of nodes i at which the temperature will be evaluated. To allow treatment of layered media, distinct thermal properties C. and ki are associated with each element. Integrating Equation 2 once witk

respect to x gives the nodal heat balance equation:

where xi+ = xi-x _{i- 1}

-

and xi++

-

.

Replacing derivatives by differences:

and ignoring the spatial dependence of aT/at, the finite difference equation for node i becomes:

Equation

4

represents the heat balance at a node in terms of temper- atures at neighbouring nodes.

A

similar equation is required for each node in the mesh with suitable modification for boundary nodes. If

temperatures are known at time level "m", solving the system of equations gives the new temperatures at time level "m

+

1". By repetition of the process the complete temperature-time-history can be calculated. A

procedure similar to this is common to all finite difference and finite element techniques even though they may differ considerably as regards both the basic equations developed and the techniques used for their solution.

Equation

4

contains a time weighting parameter whose value lies between zero and one.

If

is set equal to zero then the right hand side contains only terms that are functions of temperature at time level "m". Since these are known already, the new temperature at node i, T T 1 , can be calculated directly. The calculation is rapid and requires no extra

storage for intermediate results. For these reasons the "explicit", or "forward difference" time-stepping procedure is appealing but,

unfortunately, the results may not be correct unless the time-step chosen is small enough; the solution will diverge from the true solution and the method is said to be "unstable". The most accurate results are obtained if the time weighting parameter is given the value 4 =

f.

With this choice Equation

4

becomes the Crank-Nicholson central difference equation. For each time-step it is necessary to solve simultaneously' the complete system of linear equations since

T T ~

is only known implicitly. The usual methods of solution are by Gaussian elimination or by iterative

techniques, and Myers (1971) gives basic details of these methods as they apply to the heat flow equation. Complete information on solution

techniques may be found in Jennings (1977). The number of computations per time-step with implicit methods is considerably greater than with explicit methods, but since there is no theoretical limit on the size of time-step permitted, overall calculation times may be greatly reduced and the Crank-Nicholson scheme is generally preferred for practical

computations. In practice, an upper limit is set to the usable time-step length by the occurrence of oscillations which, although stable, lead to degradation of solution accuracy. Oscillation can be avoided altogether by choosing = 1, but at the cost of seriously increasing the truncation error.

A

compromise value = 213 which results from a finite element procedure has also sometimes been recommended.

Equation 4 can be extended to two and three dimensions by adding terms to account for heat flows in the y and z directions. In the

simplest formulation, a rectangular mesh is used and solution of the matrix equations can be carried out rapidly and efficiently using the Alternating Direction Implicit (ADI) method. In this method, the relatively high cost of solving the complete two-dimensional system of equations is avoided by the expedient of dividing the time-step into two parts. During the first part the components involving differences in one of the coordinate directions are replaced by their known values at the previous time-step. The resulting tri-diagonal equations are then solved with the same efficient algorithm as that used for one-dimensional

problems. After updating, the equations are solved in the second coordinate direction. The method, although not exact, is rapid and

efficient and has enjoyed wide popularity. The same technique can be used with cylindrical and spherical meshes.

Two-

and three-dimensional AD1 methods are discussed at length by Douglas (1961) and reviewed by Spanier

(1968) and Smith (1970).

A

different type of alternating direction method is the Alternating Direction Explicit Method of Saul' yev (1 960) (also Larkin, 1964) which uses both a forward and a reverse sweep for each coordinate direction. At each sweep all but one of the spatial difference terms is evaluated at the known time level, so that the method is an unconditionally stable explicit time-stepping scheme. The accuracy of the approximate solution is,

however, poor and the Saul'yev method is not widely used.

A

one-dimensional programme intended for ground thermal regime

applications is given in Goodrich (1974a,b). Two-dimensional AD1 computer programmes are given by Clark and Peterson (1969), and Peterson (1969a) while three-dimensional programmes are described by Clark et a1 (1969) and Peterson (1969b). A two-dimensional programme using an iterative solution technique and capable of treating radiation, free and forced convection as well as conduction is described by Pierce and Stumpf (1969).

For practical applications it is often necessary to treat problems where the geometry is not simple; boundary or internal surfaces may be inclined to the coordinate axes or may have an irregular shape. Special programming is required to treat these surfaces if suitable accuracy is to be maintained. If the mesh spacing is shortened in one coordinate

direction to permit treating a region where the geometry is irregular or the thermal properties change rapidly with position, many additional unnecessary nodes must be introduced in norrcritical regions of the

problem merely to maintain the regular grid. This is clearly inefficient and can be very costly even for problems of moderate complexity.

These difficulties are mitigated with methods which use irregular mesh shapes. In the past the principal disadvantage of irregular mesh

techniques has been their requirement for large amounts of computer memory and long computation times as compared to AD1 methods. With the power of present day computers, these considerations have become less restrictive, and irregular mesh formulations are now widely used.

The s i m p l e s t i r r e g u l a r mesh f o r m u l a t i o n s a r e based on elements whose s h a p e s a r e g e n e r a l t r i a n g l e s o r q u a d r i l a t e r a l s . Computing t h e h e a t

balance f o r each node r e q u i r e s summing t h e c o n t r i b u t i o n s from a l l a d j a c e n t neighbouring nodes. In g e n e r a l t h e number of neighbouring nodes w i l l be d i f f e r e n t from one r e g i o n of t h e problem s p a c e t o a n o t h e r . The t y p i c a l nodal e q u a t i o n i s g i v e n by:

where CC r e p r e s e n t s t h e h e a t c a p a c i t y a s s o c i a t e d w i t h node i weighted

j i j

f o r t h e volumes a s s o c i a t e d w i t h neighbouring nodes j , and t h e K a r e

i j

conductances between nodes i and j. The complete s e t of e q u a t i o n s i s asssembled i n t o a g l o b a l m a t r i x e q u a t i o n which h a s t h e form:

where

[ c ]

and [ K ] a r e g l o b a l h e a t c a p a c i t y and conductance m a t r i c e s , {TIrn i s a v e c t o r of n o d a l t e m p e r a t u r e s a t time l e v e l m , and @ i s t h e time

weighting parameter.

I n c o n t r a s t t o r e g u l a r g r i d methods t h e nodal e q u a t i o n s a r e n o t w r i t t e n down e x p l i c i t l y b u t , i n s t e a d , a r e a u t o m a t i c a l l y g e n e r a t e d d u r i n g

t h e computation. T h i s g r e a t l y enhances programme f l e x i b i l i t y b u t r e s u l t s i n some l o s s of programme e f f i c i e n c y . I f t h e t h e r m a l p r o p e r t i e s and problem geometry do n o t vary w i t h time, t h e n assembly of t h e g l o b a l c o e f f i c i e n t m a t r i c e s i s only r e q u i r e d a t t h e o n s e t of t h e computation. G e n e r a l l y , however, t h e problem w i l l n o t be s o simple and, s i n c e t h e assembly p r o c e s s i n v o l v e s a c o n s i d e r a b l e number of m a n i p u l a t i o n s , t h e c o s t s r i s e r a p i d l y . A complete i r r e g u l a r mesh f i n i t e d i f f e r e n c e programme c a p a b l e of h a n d l i n g a n i s o t r o p i c m a t e r i a l s , temperature-dependent thermal p r o p e r t i e s , i n t e r n a l h e a t g e n e r a t i o n , and convection and r a d i a t i o n h e a t t r a n s f e r f o r multi-dimensional problems i s d e s c r i b e d by Thomas and MacRoberts (1965). The programme u s e s an i t e r a t i v e scheme t o s o l v e t h e backward d i f f e r e n c e equation. A s i m i l a r l y comprehensive i r r e g u l a r mesh programme i n c l u d i n g f a c i l i t i e s f o r t r e a t i n g simultaneous h e a t and mass t r a n s f e r a s w e l l a s o t h e r r e f i n e m e n t s i s d e s c r i b e d i n Edwards (1969). This programme u s e s an e x p l i c i t forward d i f f e r e n c e scheme. R e s t r i c t i o n s o n time-step l e n g t h a r e a l l e v i a t e d , however, by t r e a t i n g boundary nodes w i t h an i m p l i c i t time-stepping procedure.

FINITE ELEMENT METHODS FOR CONDUCTIVE HEAT TRANSFER

I n r e c e n t y e a r s h e a t flow problems have g e n e r a l l y been t r e a t e d by t h e f i n i t e element method. Although t h i s t e c h n i q u e l e a d s u l t i m a t e l y t o a system of m a t r i x e q u a t i o n s which i s analogous t o t h a t o b t a i n e d w i t h

i r r e g u l a r g r i d f i n i t e d i f f e r e n c e methods, t h e manner i n which t h e e q u a t i o n s a r e d e r i v e d i s fundamentally d i f f e r e n t . Rather t h a n w r i t i n g d i r e c t l y t h e d i f f e r e n c e e q u a t i o n f o r t h e h e a t b a l a n c e a t a node, t h e whole

problem, i n c l u d i n g boundary c o n d i t i o n s , i s r e c a s t a s a problem of f i n d i n g t h e t e m p e r a t u r e d i s t r i b u t i o n which minimizes a n a p p r o p r i a t e f u n c t i o n a l . I n t h e e a r l i e s t works t h i s was achieved u s i n g t h e v a r i a t i o n a l c a l c u l u s ( V i s s e r , 1965; Wilson & N i c k e l l , 1966). A more powerful a l t e r n a t i v e t o e s t a b l i s h i n g t h e a p p r o p r i a t e f u n c t i o n a l i s by u s e of a weighted r e s i d u a l process.

I n t h e method of weighted r e s i d u a l s , f o r a g e n e r a l l i n e a r o p e r a t o r

( L ) , t h e r e s i d u a l ( E ) i s d e f i n e d by:

where Y = t r u e s o l u t i o n and O = t h e approximate s o l u t i o n . The approximate s o l u t i o n w i l l , i n g e n e r a l , be expressed a s a s e r i e s i n v o l v i n g x, t and a number of undetermined c o e f f i c i e n t s . The r e s i d u a l i s minimized i n a d i s t r i b u t e d s e n s e by r e q u i r i n g :

where W i s a s e t of a r b i t r a r y weight f u n c t i o n s and R r e p r e s e n t s t h e problem space-time domain. T h i s y i e l d s a s e r i e s of e q u a t i o n s from which t h e unknown c o e f f i c i e n t s can be c a l c u l a t e d .

For a p p l i c a t i o n t o t h e h e a t e q u a t i o n i t i s most convenient t o c o n s i d e r t h e e q u a t i o n a t a s i n g l e i n s t a n t of time and c a r r y o u t t h e weighting w i t h r e s p e c t t o t h e s p a c e c o o r d i n a t e s only. The approximate s o l u t i o n i s d e f i n e d a s :

where T i ( t ) a r e t i m e dependent n o d a l t e m p e r a t u r e s and Ni(%), are piecewise continuous f u n c t i o n s of t h e s p a c e c o o r d i n a t e s 2 , c a l l e d shape f u n c t i o n s . The Galerkin weighted r e s i d u a l p r o c e s s c o n s i s t s of choosing f o r t h e a r b i t r a r y w e i g h t i n g f u n c t i o n s t h e shape f u n c t i o n s themselves. For t h e i s o t r o p i c h e a t e q u a t i o n w i t h i n t e r n a l s o u r c e I per u n i t time p e r u n i t volume, t h i s i m p l i e s :

(i = 1, 2,

...

n ) . The boundary c o n d i t i o n t o be s a t i s f i e d by T i s :

where i s t h e outward normal component of t h e t e m p e r a t u r e g r a d i e n t a t an

the boundary surface, h is the surface heat transfer coefficient, To is the environmental temperature and q is a prescribed outward normal surface heat flux. Additionally, the solution may be constrained either at

internal points or at points on the boundary. Introducing Greens identity:

and making use of Equations

8

and 10, then Equation

9

becomes:

(i = 1, 2,

. . . .

.

n).

In Equation 12 the surface integrals apply only to appropriate sections of the external boundary surface. The volume integrals are evaluated by considering space divided into elements within which material properties are taken as constant. Nodes are located at the element vertices and, in higher order accurate formulations, along the element sides as well as internally.

The system of equations represented by (12) can be written in matrix form as:

where the subscript B implies that these terms occur only for certain

elements on the boundary. In Equation 13 the matrices

[K]

,

[ c ]

and

[H]

and vectors 1 { Q } ~ and {FIB have components which are evaluated by

adding contributions from each element. The components of the individual element matrices are:

where

V

and

A

are element volume and surface area respectively and the thermal properties and transfer coefficient are assmed to be constant within the particular element. The components of the internal heat source vector are:

where

I

is the heating rate per unit volume of the element.

The elemental contribution to the component of the vector of prescribed boundary heat flux is:

where q is the prescribed heat flux in the outward normal direction for the element e.

Finally, the elemental contribution to the environmental temperature vector has the ith component:

In Equations 14(a-f) the indices i and j range over values

corresponding to all the nodes associated with the element e. In practice these global node numbers are paralleled by a system of local node numbers defined for each element.

As

well as being notationally more convenient, this makes for more efficient processing of the element coefficient

matrices.

Once a specific element spatial discretization has been chosen and the relationship between element numbers and global and local node numbers established, the global coefficient matrices of Equation 13 can be

evaluated. This is done by evaluating Equations 14(a-f) for each element in turn, adding the elemental contribution to the matrix coefficients at all global locations ij, and to the vector terms at all locations i associated with the element.

In Equation

13

the finite element method has been applied only to the spatial domain. The most commonly used procedure for treating the time domain is to replace the time derivative by a centred-difference:

Defining

where the bar denotes the time average value for the time-step, then Equation 13 can be put in the form:

where

and

I;}

₌

₊

_+{T}~

.

_(16d)

-

Solving Equation 16a for

{T)

,

Equation 16d can be solved trivially to t+A t

yield the unknown temperatures

{T}

.

Other time-stepping schemes which have been considered include finite elements in time (Gray et.al, 1973; Bruch and Zyvoloski, 1974; KShler and Pittr, 1974; Warzge, 1974) and least squares procedures (Zienkiewicz and Lewis, 1973; Lewis and Bruch, 1974). These procedures lead to schemes which may require considerably more computational effort as well as increased computer storage when compared with the centred-difference method. Higher order multistep methods have also been proposed (Douglas,

1961; Zienkiewicz, 1971). Unfortunately, the higher order methods are more susceptible to oscillation than single step methods (Zla'mal, 1977; Wood, 1978; Goodrich, 1980). It is also found that, to ensure stability, a relatively poor truncation error has to be accepted. For example, it can be shown that it is not possible to have an unconditionally stable three-time-level method with truncation error as small as that of the two- level centred-difference method (Lambert, 1973). For all these reasons, and because of its simplicity, the centred-difference scheme is almost always chosen in practice.

In certain problems it is possible to avoid time-stepping altogether. For example, in the case where boundary conditions are periodic and

thermal properties are independent of temperature, if only the periodic steady solution is of interest, the time domain can be handled by

application of Fourier series while the space domain is treated by the finite element method. Details of one such method and its application to calculation of heat loads for insulated foundations are described by Mikhailov et a1 (1976).

A

reduction in computing time by a factor of 10 compared with a typical time-stepping procedure, was found for this case.

Equations 16(a-c) with Equations 14(a-f) constitute the essential steps of the Galerkin finite-element method. In order to clarify the

details of their application it is useful to consider a specific spatial discretization scheme. For two-dimensional problems the simplest choice is that of linear triangular elements. For this element, unknown temper- ature nodes are located at the vertices of an arbitrary triangle.

Temperature is assumed to vary linearly with position within the element.

A

local node numbering system is introduced, as defined in Fig.

1.

The temperature within an element is given by:

where TI, T2, T3 are the nodal temperatures and N1, N2, N3, are the

element shape functions. Using the fact that T(x, y) is a linear function and imposing T(xi, yi) = Ti at the vertices leads to:

where ai = xjyk

-

xkyj bi = yj

-

Yk Ci = _Xk

_-

_Xj

=area of the element

and the indices i,j,k represent the integers 1,2,3 and their cyclic permutations.

Consistent with the choice of linear triangular elements for internal nodes, external boundaries are treated using linear line elements. The temperature on a boundary segment associated with nodes i, j is given by:

where Li = 1

-

x' /R

R = separation of i,j measured along the boundary

Using Equations 18 and 20 and simplifying the equations to correspond to two-dimensional flow, Equations 14a and 14b become:

and Equation 14c becomes:

where 6 is the Kroneker delta: iJ

For a point source located at node r the internal heat source vector has components:

or, if the source is of uniform strength throughout the element, then:

If q is the average heat flux out through the boundary segment (e), then the contribution to the ith component of the prescribed boundary heat flux vector is:

Similarly, if the environmental temperature is assumed uniform for an element, then:

Assembly of t h e g l o b a l m a t r i c e s [A] and

[c]

can be i l l u s t r a t e d w i t h r e f e r e n c e t o Fig. 2. Beginning w i t h element number p t h e l o c a l m a t r i c e s [ K ~ ] and [Ce] a r e evaluated. For t h i s element, t h e l o c a l nodes 1, 2, 3

correspond t o t h e g l o b a l nodes c n, a , r e s p e c t i v e l y . The c o n t r i b u t i o n s of t h i s element t o t h e g l o b a l [ A m a t r i x are:

'31 + k c 3 1 + Anc

~ 3 ~t 2 + ~ ~+ fAnn 2

E v a l u a t i o n of t h e g l o b a l

[c]

m a t r i x f o l l o w s an i d e n t i c a l p a t t e r n . Since t h e element m a t r i c e s a r e symmetric i t f o l l o w s t h a t [ A ] and

[c]

a r e a l s o symmetric and o n l y elements on and t o one s i d e of t h e main d i a g o n a l need be c a l c u l a t e d and s t o r e d . I f element p c o n t a i n s a uniformly

-

Since the element has no external boundaries there are no contributions to

{i}

and

{i}.

In similar fashion, all the other element contributions are evaluated

and added into the appropriate locations in the global

[A]

and

[ c ]

matrices.

For an element such as s, (Fig. 2) external boundary conditions must be accounted for. If the boundary condition is that of linear heat

transfer along the segment of length

R

joining nodes a and b, then the

A

matrix is modified by adding:

hR to matrix elements Aaa and Abb

-

3

and

-

he to matrix elements Aab and Aba

.

6

-

_hT

_R

-

At the same time the vector

{P}

is modified by the addition of

2

to Pa

-

2

and Pb.

The linear triangular element formulation is the simplest possible in two dimensions. More elaborate formulations exist and one of the most generally recommended for field problems is that based on isoparametric

quadrilateral elements (Zienkiewicz and Parekh, 1970). In its most

general form, this second order element can have curved sides and, therefore, can closely follow arbitrary boundary shapes. Higher order element formulations are well established in structural mechanics applications but may be less useful for dynamic field problems. The additional computational effort involved detracts from their practicality and, in the typical ground temperature problem where thermal properties are not constant, the cost may be worse than for a greater number of linear elements. These methods also do not generally perform well when material properties change abruptly or when treating discontinuities such as arise in freezinglthawing problems. Through the use of transition elements, elements having different orders of accuracy may be used in the same problem, and this may make higher order accurate elements more

practical.

Methods which combine analytic solutions with finite elements have also been devised, and these "finite strip" and "singularity programming" techniques can be very helpful in improving accuracy while reducing

computation costs, although a penalty must be paid in loss of generality of the solution. Zienkiewicz (1 977) contains useful background material for this as well as for many other specialized finite element topics. Two examples for heat flow problems, chosen because they illustrate widely different cases, are the paper by Emery (1973) on the treatment of discontinuous boundary conditions or point heat sources and that of Padovan (1974) which discusses a method for axisymmetric heat flow in anisotropic materials.

In addition to being symmetric, the fully assembled global matrices

[A]

and

[c]

are banded, (non-zero elements exist only along the main diagonal and along a limited number of additional diagonals). In a practical calculation, storage and computation associated with the assembly process is minimized by taking advantage of the symmetry and bandedness. Solution of the equations will usually be by one of the direct methods, or for larger systems or nonlinear problems, by iterative methods. These topics are well covered in Jennings (1977), and program- ming techniques are described in Desai and Abel (1971), Zienkiewicz (1971,

1977), and Hinton and Owen (1980).

COMPARISON OF FINITE ELEMENT

AND

FINITE DIFFERENCE METHODS

The finite element and finite difference methods both lead to a

matrix equation to be solved for the unknown vector of nodal temperatures. The two methods are fundamentally related insofar as both are derivable as weighted residual processes.

An abundant literature exists regarding the relative merits of the two procedures. See for example, Lemmon and Heaton (1969), Emery and Carson (1971), Yalamanchili and Chu (1970, 1971, 1973), Gray and Schnurr (1975) and Schroeder (1975). Unfortunately, the significance of these and other comparison studies is sometimes reduced by lack of a clear

definition of what constitutes a finite difference as opposed to a finite element method. Certain authors imply, for example, that irregular grids are unique to the finite element procedure. In one case, the definition of finite difference method restricts this class to first order difference operators but makes the comparison with higher order accurate finite

element formulations. In other cases, comparisons are made using different matrix solution techniques, thus further clouding the issue.

With the definition of both methods as specific weighted residual processes, it can be stated that the essential difference in the

formulations shows up in the coefficient matrices. For the particular case of linear triangular elements, the

[K]

matrix is identical to what can be found directly by finite difference procedures. The

[c]

matrix however contains of f-diagonal elements whereas in the usual finite difference formulations the neighbouring nodes are assumed not to

contribute to the time derivative of the temperature and the

[c]

matrix is diagonal. The implications of this are that while the finite element formulation will possess a somewhat smaller truncation error, since the 6T/6t terms are being more accurately represented, at the same time the

f i n i t e element f o r m u l a t i o n i s more s u s c e p t i b l e t o o s c i l l a t i o n

(Yalamanchili and Chu, 1973). E v i d e n t l y , o s c i l l a t i o n , even i f s t a b l e , can l e a d t o d e g r a d a t i o n of o v e r a l l s o l u t i o n accuracy. This s u g g e s t s t h a t i t

w i l l o f t e n b e d e s i r a b l e t o e l i m i n a t e t h e o f f - d i a g o n a l terms i n t h e

[ c ]

m a t r i x by lumping them on t o t h e main d i a g o n a l , and Zienkiewicz (1977) b r i e f l y d i s c u s s e s a p p r o p r i a t e p r o c e d u r e s i n t h e c o n t e x t of s t r u c t u r a l dynamics problems.

Lumping f o r h e a t flow problems h a s been considered by Lemmon (1973) and Mahata and McNary (1974). It t u r n s o u t f o r t h e c a s e of l i n e a r

t r i a n g u l a r elements ( t h e most widely used c h o i c e ) w i t h lumping, t h e f i n i t e element e q u a t i o n s become i d e n t i c a l t o t h e f i n i t e d i f f e r e n c e equations. A t t h i s p o i n t , t h e arguments r e g a r d i n g t h e r e l a t i v e merits of one o r t h e o t h e r method b e g i n t o a p p e a r academic. The formalism of t h e f i n i t e element approach i s , however, by f a r t h e most powerful and, f o r t h i s r e a s o n a l o n e , t h a t method should be p r e f e r r e d .

THE NUMERICAL TREATMENT OF FREEZING AND THAWING

I n almost a l l c a s e s t h e thermal behaviour of t h e ground i s s t r o n g l y a f f e c t e d by l a t e n t h e a t a s s o c i a t e d w i t h f r e e z i n g o r thawing of t h e s o i l water. Often t h e p r i n c i p a l o b j e c t i v e of ground thermal a n a l y s i s i s t o d e t e r m i n e t h e amount of f r e e z i n g o r thawing t h a t w i l l occur o v e r a g i v e n p e r i o d of t i m e . Various methods have been d e v i s e d f o r t h e numerical t r e a t m e n t of l a t e n t h e a t . The problem i s a d i f f i c u l t one, however, and t h e advantages of any p a r t i c u l a r numerical f o r m u l a t i o n a r e i n v a r i a b l y accompanied by more o r less s e r i o u s d i s a d v a n t a g e s .

The s i m p l e s t methods t o programme a r e t h o s e which c o n s i d e r t h e l a t e n t h e a t a s a d i s t r i b u t e d h e a t s o u r c e ( s i n k ) . E i t h e r t h e moving s o u r c e ( I l a t ) can be t r e a t e d e x p l i c i t l y a s an a d d i t i o n a l term i n t h e h e a t b a l a n c e

equation:

u s i n g

where u i s unfrozen water c o n t e n t and Lat i s t h e l a t e n t h e a t p e r u n i t mass of f r e e z i n g l t h a w i n g f o r w a t e r , o r , s i n c e t h e u n f r o z e n w a t e r c o n t e n t i s p r i m a r i l y determined by t h e t e m p e r a t u r e of t h e p a r t i a l l y f r o z e n s o i l :

Usually, no a t t e m p t i s made t o d i s t i n g u i s h an i n t e r f a c e s e p a r a t i n g f r o z e n and thawed p h a s e s w i t h i n a s i n g l e element and t h e h e a t s o u r c e i s

considered t o be d i s t r i b u t e d e v e n l y over a l l nodes of t h e element. With t h e s e methods t h e computed l o c a t i o n of t h e f r e e z i n g o r thawing i n t e r f a c e o s c i l l a t e s around t h e t r u e p o s i t i o n and t h e temperature f i e l d i n t h e immediate neighbourhood i s d i s t o r t e d . The o s c i l l a t i o n and d i s t o r t i o n a r e most s e v e r e i f t h e f r e e z i n g range i s narrow. These d i f f i c u l t i e s a r e d i s c u s s e d by P r i c e and Slack (1954). I n o r d e r t o m a i n t a i n e r r o r s w i t h i n a c c e p t a b l e bounds, i t i s n e c e s s a r y t o use s h o r t time-steps and small g r i d d i v i s i o n s . The i n t r o d u c t i o n of l a t e n t h e a t makes t h e problem n o n l i n e a r which, i n t u r n , g e n e r a l l y r e q u i r e s t h e u s e of i t e r a t i v e s o l u t i o n

t e c h n i q u e s , and t h e r e f o r e c o s t s r i s e r a p i d l y , p a r t i c u l a r l y f o r two- and three-dimensional problems.

Numerous v a r i a n t s and r e f i n e m e n t s of t h e a p p a r e n t h e a t c a p a c i t y

approach a r e p o s s i b l e . The f u n c t i o n a l r e l a t i o n s h i p between u n f r o z e n w a t e r c o n t e n t and temperature i s o f t e n n o t known and, s i n c e t h e f r o s t o r thaw d e p t h i s c o n t r o l l e d mainly by t h e t o t a l amount of w a t e r which changes phase, a simple s t e p - f u n c t i o n i s sometimes used t o r e p r e s e n t t h e a p p a r e n t h e a t c a p a c i t y . The s t e p - f u n c t i o n t a k e s on a l a r g e v a l u e f o r t e m p e r a t u r e s w i t h i n some a r b i t r a r y range T 1

<

T

<

Tf and a p p r o p r i a t e thawed and f r o z e n v a l u e s f o r t e m p e r a t u r e s above and below t h i s i n t e r v a l . Dempsey and

Thompson (1969) and Ho e t a 1 (1970) d e s c r i b e one-dimensional models based on t h i s approach. A s i m i l a r approach f o r two-dimensional problems i s used by Couch e t a 1 (1970) and Fleming (1971). Nakano and Brown (1971)

d e s c r i b e a one-dimensional c e n t r a L d i f f e r e n c e method u s i n g t h e a p p a r e n t h e a t c a p a c i t y approach i n which a l o c a l g r i d r e f i n e m e n t i s used t o improve accuracy and reduce s o l u t i o n o s c i l l a t i o n . I t e r a t i o n normally r e q u i r e d when u s i n g t h e a p p a r e n t h e a t c a p a c i t y approach c a n b e avoided by u s i n g t h r e e time l e v e l s . A three-time-level f i n i t e d i f f e r e n c e scheme f o r two- dimensional n o n l i n e a r problems i s d e s c r i b e d by Bonacina and Comini (1973)

.

Bonacina e t a 1 (1973) p r e s e n t a three-time-level scheme f o r one-

dimensional problems w i t h phase change u s i n g t h e f r e e z i n g range approach. A t h r e e - l e v e l scheme which i s more a c c u r a t e and much l e s s s u b j e c t t o o s c i l l a t i o n i s d e s c r i b e d i n Goodrich (1980).

An a l t e r n a t i v e t o t h e a p p a r e n t h e a t c a p a c i t y method i s t o r e p l a c e t h e h e a t c a p a c i t y t e r m i n Equation 24 w i t h t h e time d e r i v a t i v e of t h e

e n t h a l p y :

The e n t h a l p y i s a continuous f u n c t i o n of t e m p e r a t u r e , even w i t h i n t h e f r e e z i n g range, and i t s u s e i s expected t o l e a d t o more smoothly v a r y i n g temperature d i s t r i b u t i o n s . Atthey (1975) p r e s e n t s a method i n which b o t h H and T a r e c a l c u l a t e d simultaneously. I n Comini e t a 1 (1974)

Equation 28 i s used simply t o e s t i m a t e t h e a p p a r e n t h e a t c a p a c i t y .

Apparent h e a t c a p a c i t y f o r m u l a t i o n s have t h e advantage of programming s i m p l i c i t y . They have, u n f o r t u n a t e l y , a number of drawbacks; s e e , f o r example, P r i c e and Slack (1954). The phase i n t e r f a c e i s p r e d i c t e d t o

advance i n a more o r l e s s smoothed step-wise f a s h i o n , t h e width of t h e s t e p s correspon'ding t o t h e element dimensions. The p r e d i c t e d t e m p e r a t u r e f i e l d i n t h e v i c i n i t y of elements undergoing phase change may be s e r i o u s l y d i s t o r t e d , and i n o r d e r t o c o n t r o l t h e e r r o r , very s m a l l element s i z e s may be r e q u i r e d . But, i f t h e element s i z e i s t o o s m a l l , t h e r e i s a danger of

" ~ n i ~ u L n g " t h e l a t e n t h e a t c o n t r i b u t i o n e n t i r e l y u n l e s s t h e time-step

Fa

correspondingly l i m i t e d . The l a t e n t h e a t c o n t r i b u t i o n may a l s o be missed i n problems where two phase change i n t e r f a c e s c o e x i s t i n c l o s e proximity. The u s u a l a p p a r e n t h e a t c a p a c i t y f o r m u l a t i o n i s i n c a p a b l e of c o r r e c t l y a c c o u n t i n g f o r t h e o c c u r r e n c e of more t h a n one phase-boundary w i t h i n a s i n g l e element. This c a n l e a d t o s e r i o u s e r r o r s i n c e r t a i n s i t u a t i o n s such a s d u r i n g a n n u a l f r e e z e b a c k of t h e a c t i v e l a y e r i n permafrost

problems and t h i s i s n o t e a s i l y c o r r e c t e d by merely r e f i n i n g t h e element s i z e s .

The t r e a t m e n t of phase change i n f i n i t e element methods i s v e r y s i m i l a r t o t h a t used w i t h f i n i t e d i f f e r e n c e methods. Wheeler (1973) and Guymon and Hromadka (1977) u s e a b a s i c f r e e z i n g r a n g e approach w h i l e Comini e t a 1 (1974) d e s c r i b e a n a p p a r e n t h e a t c a p a c i t y e n t h a l p y scheme u s i n g t h r e e time l e v e l s . D e l Guidice e t a 1 (1978) and Morgan e t a 1 (1978) s u g g e s t f u r t h e r improvements t o t h i s method. h a n g e t a 1 (1972) and Svec and Charlwood (1973) p r e s e n t methods i n which t h e l a t e n t h e a t i s t r e a t e d a s a moving h e a t source. I n t h e method of h a n g e t a l , f r e e z i n g i s assumed t o occur a t a f i x e d temperature w h i l e i n t h a t of Svec and Charlwood, t h e unfrozen-water-content t e m p e r a t u r e r e l a t i o n i s used. Zienkiewicz e t a 1 (1973) u s e an approach s i m i l a r t o t h a t d e s c r i b e d i n Doherty (1970). T h i s method, which i n v o l v e s computing a pseudo energy balance a t nodes undergoing phase change w h i l e a t t h e same time r e s e t t i n g t h e i r t e m p e r a t u r e t o t h e f r e e z i n g p o i n t may, however, n o t y i e l d p h y s i c a l l y c o n s i s t e n t r e s u l t s and must be used w i t h c a u t i o n . Kliewer (1973)

d e s c r i b e s a m o d i f i c a t i o n which d o e s a s s u r e p h y s i c a l c o n s i s t e n c y .

It i s a l s o p o s s i b l e t o t r e a t l a t e n t h e a t a s a moving boundary

problem. I n t h i s approach t h e p o s i t i o n of t h e phase change i n t e r f a c e i s determined e x p l i c i t l y a s i t moves c o n t i n u o u s l y through t h e network of node p o i n t s . In most f o r m u l a t i o n s t h e l a t e n t h e a t i s assumed t o be e n t i r e l y absorbed o r g e n e r a t e d on t h e moving i n t e r f a c e . With t h i s approach, i t i s n e c e s s a r y t o s o l v e s i m u l t a n e o u s l y t h e h e a t e q u a t i o n and t h e n o n l i n e a r phase change i n t e r f a c e e q u a t i o n :

+

-h

where F1 and F2 a r e normal h e a t f l u x e s on e i t h e r s i d e of t h e i n t e r f a c e ,

4 L i s t h e l a t e n t h e a t p e r u n i t volume l i b e r a t e d o r absorbed a t t h e

-b

i n t e r f a c e and

!:

i s t h e i n t e r f a c e v e l o c i t y . The programming e f f o r t

r e q u i r e d i s c o n s i d e r a b l y g r e a t e r t h a n t h a t involved w i t h t h e a p p a r e n t h e a t , c a p a c i t y approach. A v a r i e t y of schemes have been proposed t o t r e a t t h e

Hanley, 1970) which i g n o r e t h e n o n l i n e a r i t y implied i n Equation 28 a s w e l l

a s v a r i a b l e time-step methods and schemes i n which t h e node s p a c i n g i s

made t o vary w i t h t i m e . D e t a i l s of v a r i a b l e time-step o r space-grid

methods may be found i n Douglas and G a l l i e (1955a, b ) , Kazemi and P e r k i n s (1971) and Murry and Landis (1969).

Formulations which permit t h e phase plane t o l i e between n o d a l boundaries of t h e f i x e d s p a c e g r i d a r e a l s o p o s s i b l e . The scheme of E h r l i c h (1958) i s of t h i s t y p e , a s i s t h a t of Meyer e t a 1 (1972). E h r l i c h ' s scheme i s , however, l i m i t e d t o homogeneous systems and, i n a d d i t i o n , r e q u i r e s i t e r a t i o n o v e r a l l s p a c e nodes. That of Meyer e t a l , which i s n o t a f i n i t e d i f f e r e n c e scheme, i n v o l v e s a c o n s i d e r a b l e amount of

computing e f f o r t a t each time-step. The method i s l i m i t e d t o problems

i n v o l v i n g a s i n g l e phase plane. A method which a v o i d s some of t h e s e d i f f i c u l t i e s , i s d e s c r i b e d i n Goodrich (1974a, 1978). I n t h i s scheme t h e c e n t r e d - d i f f e r e n c e form of Equation 28 i s coupled w i t h a Crank-Nicholson f o r m u l a t i o n a t o r d i n a r y nodes. The method produces r e s u l t s of h i g h numerical a c c u r a c y , and i f only a s i n g l e phase p l a n e i s involved, t h e s o l u t i o n can be arranged i n such a way t h a t s t a n d a r d Gaussian e l i m i n a t i o n can be used a t a l l nodes w h i l e a s p e c i a l non-linear scheme i s used f o r t h e phase i n t e r f a c e . The method c a n , t h e r e f o r e , be v e r y e f f i c i e n t f o r one- dimensional problems.

Moving boundary methods f o r two- and three-dimensional problems a r e d e s c r i b e d i n L a z a r i d i s (1970) and i n F i s h e r and Medland (1974). The f i n i t e d i f f e r e n c e scheme of L a z a r i d i s u s e s e x p l i c i t time-stepping and a r e g u l a r r e c t a n g u l a r g r i d and t h i s w i l l l i m i t i t s p r a c t i c a l a p p l i c a t i o n . The f i n i t e element f o r m u l a t i o n i s used i n t h e paper by F i s h e r and Medland, which d e s c r i b e s s e v e r a l methods of t r e a t i n g l a t e n t h e a t , i n c l u d i n g b o t h moving boundary and moving s o u r c e f o r m u l a t i o n s . U n f o r t u n a t e l y , a l t h o u g h

t h e paper g i v e s a w e a l t h of i n f o r m a t i o n f o r t h e one-dimensional c a s e , no d e t a i l s a r e g i v e n r e g a r d i n g e x t e n s i o n t o multidimensional problems.

Apparent h e a t c a p a c i t y methods a r e s i m p l e and t h e i r implementation i s

q u i t e s t r a i g h t f o r w a r d and need n o t b e d i s c u s s e d f u r t h e r . Moving i n t e r f a c e methods a r e much more d i f f i c u l t t o implement and r e q u i r e f o r b i d d i n g l y

cumbersome (and hence c o s t l y ) coding f o r a l l b u t t h e s i m p l e r one-

dimensional c a s e s . The moving s o u r c e method of Hwang e t a 1 (1972) i s

i n t e r m e d i a t e between t h e two above methods a s r e g a r d s coding complexity and a t t h e same time produces r e s u l t s of a c c e p t a b l e a c c u r a c y a t r e a s o n a b l e c o s t i n computing time. For t h e s e r e a s o n s , only t h i s method i s d i s c u s s e d i n d e t a i l . I n Hwang's method t h e l a t e n t h e a t s o u r c e v e c t o r a s s o c i a t e d w i t h a p a r t i c u l a r element i s c a l c u l a t e d a t each time-step u s i n g

temperatures determined a t t h e p r e v i o u s time-step. Assuming a l i n e a r

temperature d i s t r i b u t i o n o v e r t h e element, and t h a t a l l t h e w a t e r f r e e z e s a t a f i x e d t e m p e r a t u r e , t h e f r a c t i o n of t h e e l e m e n t a l a r e a i n t h e f r o z e n phase a s w e l l a s t h e r a t e of change of t h i s f r a c t i o n c a n b e estimated. The r a t e of change of f r o z e n f r a c t i o n a l a r e a i s used t o compute t h e

i n t e r n a l h e a t g e n e r a t i o n term f o r t h e element. Following Hwang, a l i n e a r t r i a n g u l a r element undergoing phase change c o n t r i b u t e s a h e a t s o u r c e term, t o be added t o t h e r i g h t hand s i d e of Equation 16:

Bk

= (Ti

-

Of )/(Ti

-

Tk)

ej

= (Ti

-

Bf)/(Ti

-

Tj)

L = latent heat per unit volume of soil and d = elemental area (see Equation 18e).

The direct use of Equation 29 leads, however, to a non-linear system of matrix equations requiring iterative solution techniques. But since:

where [L?(t)] is a time-dependent

3

x

3

matrix, the non-linearity can be removed by evaluating [~f(t)] at the "olds* time level m using the tinite- difference form:

Hwang further replaces the unsymmetric matrix

[L;]~

by a diagonal form:

Bj Bk

₀

₀

-

Tk

Bj (1

-

Bk)

+

Bk(l

-

B j

r l

T I T k Ti-;. BkBj

I*)

Ti

-

Tj

This is physically equivalent to lumping the latent heat contribution at the element nodes and probably does not strongly affect accuracy while permitting efficiencies in the solution technique. Linearization, while eliminating the need for iteration, has the disadvantage that convergence to the true frost or thaw depth may not be completely assured. For

s i t u a t i o n s where temperatures approach q u a s i s t e a d y - s t a t e c o n d i t i o n s , such a s w i t h very wet m a t e r i a l s o r w i t h c o n s t a n t boundary t e m p e r a t u r e s , t h e e r r o r should be n e g l i g i b l e , a t l e a s t a f t e r a s u f f i c i e n t number of time- s t e p s . With r a p i d l y changing s u r f a c e temperatures t h e e r r o r s could be more s e r i o u s , p a r t i c u l a r l y when d e a l i n g w i t h r e l a t i v e l y d r y m a t e r i a l s . The method a l s o produces some e r r o r when moving t h e phase boundary a c r o s s l a y e r s of d i f f e r e n t m a t e r i a l s ; t h e e r r o r becomes worse when moving between d r y and wet l a y e r s . It should a l s o be remembered t h a t t h e method only roughly accounts f o r t h e phase change i n t e r f a c e p o s i t i o n w i t h i n an element and i s n o t c a p a b l e of a l l o w i n g f o r more t h a n one i n t e r f a c e . A l l t h e s e e r r o r s can be diminished by reducing t h e s i z e of t h e t i m e s t e p and r e f i n i n g t h e mesh a t t h e expense of i n c r e a s e d computer c o s t s .

SURFACE BOUNDARY CONDITIONS

I n n a t u r e t h e ground s u r f a c e i s d r i v e n by t h e n e t f l u x of h e a t a r i s i n g from absorbed s o l a r and thermal r a d i a t i o n and from s e n s i b l e and l a t e n t h e a t t r a n s f e r between t h e ground and t h e o v e r l y i n g a i r . This i s expressed by t h e s u r f a c e h e a t b a l a n c e equation:

where Q = t h e ground h e a t f l u x a t t h e s u r f a c e , g

= t h e normal g r a d i e n t of ground temperature e v a l u a t e d a t t h e s u r f ace.

RS = n e t f l u x of s o l a r r a d i a t i o n (dependent on s u r f a c e r e f l e c t i v i t y

and o r i e n t a t i o n a s w e l l a s on a v a i l a b l e incoming s o l a r r a d i a t i o n ) ,

RL = n e t thermal ( l o n g wave) r a d i a t i o n (dependent on s u r f a c e and a i r

t e m p e r a t u r e s a s w e l l a s cloud c o n d i t i o n s , and a i r temperature and humidity g r a d i e n t s ) ,

QH = s e n s i b l e h e a t f l u x a s s o c i a t e d w i t h a i r flow over t h e s u r f a c e (dependent on a i r and s u r f a c e temperature a s w e l l a s wind speed and n a t u r e of t h e flow regime),

QE = l a t e n t h e a t f l u x a s s o c i a t e d w i t h e v a p o r a t i o n of m o i s t u r e from t h e s u r f a c e (dependent on t h e a v a i l a b i l i t y of m o i s t u r e i n a d d i t i o n t o t h e f a c t o r s which c o n t r o l t h e s e n s i b l e h e a t t r a n s f e r ) .

While RS i s n e c e s s a r i l y p o s i t i v e o r z e r o , RL, QH, and QE may be of e i t h e r s i g n . The d a i l y and s e a s o n a l v a r i a t i o n s of t h e s u r f a c e h e a t f l u x terms l e a d t o a s i m i l a r v a r i a t i o n of t h e ground s u r f a c e temperature. I n

Equation 33 t h e i n d i v i d u a l h e a t f l u x components a r e v e r y much l a r g e r t h a n t h e n e t ground h e a t f l u x (Qg) and f u r t h e r m o r e e r r o r s i n e s t i m a t i o n of t h e components a r e cumulative and may e a s i l y be of t h e same magnitude o r g r e a t e r than Q i t s e l f . Three of t h e f o u r components e x h i b i t a non-linear r e l a t i o n s h i p

t8

s u r f a c e temperature. S i n c e t h e u s e of t h e s u r f a c e h e a t

balance e q u a t i o n r e q u i r e s i n c r e a s e d computational e f f o r t a t c o n s i d e r a b l e c o s t , and s i n c e , i n a d d i t i o n , t h e n e c e s s a r y i n f o r m a t i o n i s g e n e r a l l y n o t a v a i l a b l e , where p o s s i b l e , c a l c u l a t i o n s a r e based on an assumed p r e s c r i b e d s u r f a c e temperature. A l i n e a r i z e d form of Equation 33:

Qg = Rs

+

h (T ambient

-

T s u r f a c e ) (34 i s a l s o sometimes used. S i t u a t i o n s do, however, a r i s e where i t i s n e c e s s a r y t o c o n s i d e r t h e complete s u r f a c e h e a t balance.

The purpose of t h i s s e c t i o n i s t o p r e s e n t formulae f o r c a l c u l a t i n g t h e components of t h e h e a t b a l a n c e which a r e s u i t a b l e f o r u s e i n numerical models and t o p o i n t o u t some of t h e d i f f i c u l t i e s which a r i s e . Much

a d d i t i o n a l i n f o r m a t i o n , i n c l u d i n g d i s c u s s i o n of most a s p e c t s of t h e broad s u b j e c t of micrometeorology, i s g i v e n by S u t t o n (1953), Geiger (1965), Munn (1966), S e l l e r s (1965), S t r i n g e r (1972) and Pavlov (1976).

R a d i a t i o n

S o l a r r a d i a t i o n r e a c h i n g t h e e a r t h ' s s u r f a c e l i e s p r i m a r i l y i n t h e wave l e n g t h r a n g e 0.3 pm t o 3 pm. The v i s i b l e r e g i o n , 0.3 pm t o 0.7 pm, comprises about h a l f t h e t o t a l f l u x . The remainder i s p r i m a r i l y near

-

i n f r a r e d r a d i a t i o n . Under c l e a r s k y c o n d i t i o n s t h e s o l a r r a d i a t i o n

r e c e i v e d a t t h e e a r t h ' s s u r f a c e i s made up of a d i r e c t beam and a d i f f u s e component r e s u l t i n g from s c a t t e r i n g due t o atmospheric w a t e r vapour and d u s t p a r t i c l e s . When t h e sky i s covered by c l o u d , o n l y d i f f u s e r a d i a t i o n a r r i v e s a t t h e s u r f a c e . Although t h e d i r e c t beam r a d i a t i o n f o r c l e a r s k i e s can be c a l c u l a t e d t h e o r e t i c a l l y , provided t h e atmospheric

t r a n s m i s s i v i t y f a c t o r i s known, t h e d i f f u s e component c a n o n l y b e roughly e s t i m a t e d . S u i t a b l e formulae a r e g i v e n by S t r i n g e r (1972) and d i s c u s s i o n of c a l c u l a t i o n s f o r d i f f e r e n t s u r f a c e o r i e n t a t i o n s may b e found i n S c o t t

(1964). D e p l e t i o n of s o l a r r a d i a t i o n by clouds adds c o n s i d e r a b l e

u n c e r t a i n t y t o t h e c a l c u l a t i o n and e r r o r s of 30% o r more c a n b e expected ( S e l l e r s , 1965). Actual measurements should be used whenever p o s s i b l e f o r model c a l c u l a t i o n s . Data f o r Canadian s t a t i o n s a r e p u b l i s h e d i n

Environment Canada Monthly R a d i a t i o n Summary.

Only p a r t of t h e t o t a l incoming s o l a r r a d i a t i o n (Rsi), i s a c t u a l l y absorbed by t h e ground, t h e remainder i s r e f l e c t e d skyward a s outgoing s h o r t wave r a d i a t i o n (Rso). For s u r f a c e s such a s b a r e r o c k o r m i n e r a l s o i l s , s o l a r r a d i a t i o n i s absorbed e s s e n t i a l l y a t t h e s u r f a c e . On t h e o t h e r hand, f o r n a t u r a l covers s u c h a s v e g e t a t i o n , w a t e r , snow o r i c e , no w e l l d e f i n e d s u r f a c e e x i s t s , and t h e r a d i a t i o n i s absorbed i n s t e a d

throughout a s h a l l o w zone. I n d e n s e l y t r e e d a r e a s , most of t h e s o l a r r a d i a t i o n i s absorbed w i t h i n t h e f o r e s t canopy; i n open t u n d r a , w i t h i n t h e f i r s t few c e n t i m e t r e s of t h e o r g a n i c mat. For i c e and snow c o v e r s , t h e n e a r - i n f r a r e d f r a c t i o n i s absorbed w i t h i n a few c e n t i m e t r e s of t h e s u r f a c e while t h e v i s i b l e f r a c t i o n may p e n e t r a t e t o t h e u n d e r l y i n g water o r s o i l .

The r e f l e c t i v i t y o r albedo ( A ) , of a s u r f a c e f o r s h o r t wave r a d i a t i o n i s d e f i n e d a s t h e p o r t i o n of t h e incoming t o t a l s o l a r r a d i a t i o n t h a t i s r e f l e c t e d and i s expressed a s a p e r c e n t a g e :

Albedo depends on the nature of the surface and the angle of incidence of the radiation, and can vary from more than 90% for a new snow surface to less than 10% for a dark, wet surface.

Every body emits radiation over a range of wavelength that depends on its temperature. For the temperatures associated with the earth's surface and atmosphere, this thermal or long wave radiation is in the wavelength range of

4

pm to 120 pm. The outgoing long wave radiation (RLo) emitted by the earth's surface follows closely the Stefan-Boltzmann equation for a black body :

where T is the absolute surface temperature and o is the Stefan-Boltzmann constant. Within the atmosphere, thermal radiation is emitted in all directions. The net downward flux (RU) available at the earth's surface is determined by the distribution of temperatures and water vapour, carbon dioxide and dust particle concentrations throughout the entire atmospheric column. In contrast to incoming solar radiation, the incoming long wave radiaton is increased when clouds are present. It also increases when temperatures increase skyward (inversion conditions). The emitted flux is usually dominated by radiation produced near the earth's surface where air temperatures are warmest and water vapour concentration is highest. For climatological computations, the incoming long wave radiation can be estimated from empirical formulae using temperature and humidity measured at screen height. These and other factors relating to atmospheric fong wave radiation are reviewed by Sellers (1965) and are discussed in detail by Kondrat'yev (1965).

A

widely used empirical relation is the Brunt formula:

-

₄

R~

(clear sky) = (a

+

b Je) UT, (37

where

T

and e are air temperature and water vapour pressure measured at screen height

,

and a and b are empirical constants. This -equation is usually corrected by multiplying by a "cloud cover" factor:

R ~ i

-

= l + a l * c 2 R~i(clear sky)

where c is the fractional cloud cover and a' is an empirical coefficient which varies with cloud type and height. Formulae such as Equations 37 and

38

evidently have at best only a statistical validity. They may have an accuracy of 15-20% for time periods of weeks or months (Sellers, 1965). Larger deviations are probable for shorter periods or if the formulae are used to calculate values for a specific site.

The net radiation (R), of all wavelengths available for heating the ground is :

Incoming and outgoing s o l a r r a d i a t i o n can be measured d i r e c t l y u s i n g pyranometers, and a p r e c i s i o n approaching 5% can be achieved i f t h e measurements a r e made c a r e f u l l y . The n e t r a d i a t i o n n e a r t h e ground

s u r f a c e c a n b e measured d i r e c t l y u s i n g n e t radiometers. The accuracy of n e t r a d i a t i o n measurements i s r e l a t i v e l y poor.

S e n s i b l e Heat T r a n s f e r

The s e n s i b l e h e a t f l u x i s dependent on t h e s t a t e of t h e atmosphere and t h e n a t u r e of t h e s u r f a c e . During p e r i o d s when t h e s u r f a c e i s c o o l i n g and t h e r e i s l i t t l e wind, t h e t e m p e r a t u r e of t h e a i r n e a r i t w i l l i n c r e a s e w i t h h e i g h t and i t s d e n s i t y w i l l decrease, r e s u l t i n g i n a s t a b l e

c o n d i t i o n . S e n s i b l e h e a t flow f o r t h i s c o n d i t i o n i s p r i m a r i l y by molecular conduction, and i s r e l a t i v e l y small.

I f t h e s u r f a c e of t h e e a r t h i s warmer t h a n t h e a i r above, due, f o r example, t o t h e a b s o r p t i o n of s o l a r r a d i a t i o n , t h e a i r i n c o n t a c t w i t h i t

w i l l be warmed. Its d e n s i t y w i l l be reduced and i t w i l l have a tendency t o rise, i n d u c i n g f r e e convection. S e n s i b l e h e a t t r a n s f e r w i l l now b e p r i n c i p a l l y by c o n v e c t i o n , t h e conductive c o n t r i b u t i o n b e i n g s e v e r a l o r d e r s of magnitude smaller.

I f t h e a i r i s moving h o r i z o n t a l l y , t h e s e n s i b l e h e a t t r a n s f e r w i l l be i n c r e a s e d by mechanical t u r b u l e n c e . The c o n t r i b u t i o n of t u r b u l e n c e

depends on b o t h roughness of t h e s u r f a c e and t h e s t a b i l i t y of t h e a i r . S t a b l e c o n d i t i o n s a s s o c i a t e d w i t h t e m p e r a t u r e i n v e r s i o n s have a damping e f f e c t ; buoyancy-induced convection t e n d s t o i n c r e a s e t u r b u l e n c e . R a d i a t i v e p r o c e s s e s i n t h e atmosphere c a n a l s o have a n i n f l u e n c e , a c t i n g a s l o c a l s o u r c e s o r s i n k s of s e n s i b l e h e a t . By analogy w i t h h e a t t r a n s f e r i n s o l i d s , i t i s o f t e n assumed t h a t t h e s e n s i b l e h e a t f l u x (QH), i s g i v e n by: where p i s t h e d e n s i t y of a i r , T i s t h e a i r t e m p e r a t u r e , c i s t h e h e a t c a p a c i t y of a i r a t c o n s t a n t p r e s s u r e ,

r

i s t h e d r y a d i a b a t f c l a p s e r a t e and KH i s a d i f f u s i o n c o e f f i c i e n t t h a t depends on a t m o s p h e r i c s t a b i l i t y , roughness of t h e s u r f a c e , wind speed and h e i g h t above t h e ground f o r which QH i s c a l c u l a t e d . For use w i t h h e a t b a l a n c e boundary c o n d i t i o n s

Equation 40 i s u s u a l l y approximated by a n e q u a t i o n of t h e form:

where Ta i s t h e a i r temperature, Ts i s t h e s u r f a c e t e m p e r a t u r e , and CH i s t h e s e n s i b l e h e a t t r a n s f e r c o e f f i c i e n t . CH depends on t h e same f a c t o r s a s

Assuming that the mechanism of heat transfer is similar to that of

momentum transf er, equations can be derived based on aerodynamic boundary

layer theory which relate KH or CH to windspeed, stability of the

turbulent airflow and aerodynamic roughness of the surface (Z,). For

neutral stability, it can be shown theoretically that the wind speed near the ground surface should vary logarithmically with height. This leads to the following expression for the sensible heat flux:

where

k

is von Karman's constant (= 0.4), U2 is the wind speed at height

Z2, T1 is the air temperature at height Z l and:

Scott (1957) used an empirical power law wind profile formula

proposed by Deacon (1949) to extend the evaluation to stable and unstable conditions. Using Deacon's wind profile law leads to:

where 8 is an empirical coefficient whose value reflects the stability.

For stable conditions B G 1, and for unstable conditions B 2 1, while

the limit 8 + 1 applies to neutral conditions. Stable conditions

generally prevail at night or during the winter while unstable conditions are associated with the daytime and summer season. Very large changes in the sensible heat transfer coefficient occur as a result.

More advanced theoretical expressions for sensible heat transfer

coefficients are described by Dyer (1974). The subject is difficult,

however, and there is still considerable controversy over some of the most fundamental aspects. Equation 42 describes only mechanical turbulence generated by the wind but, even when there is no wind, buoyant convection can occur. This is reflected in empirical formulae for the sensible heat transfer coefficient which usually contain an additional term independent of wind speed.

Evaporative Heat Transfer

The mechanism of mass transfer of water vapour leaving a surface is the same as that which controls the movement of dry air. The equations for the transfer of latent heat of evaporation are similar, therefore, to those for the transfer of sensible heat. The latent heat flux can be described by the following, which is analogous to Equation 42: