Direct and inverse problems of heat transfer in soil Текст научной статьи по специальности «Физика»

ФИЗИКА ПОЧВ

UDK 631. 436

!Fariz D. Mikayilov, 2Ahmet Sami Erol DIRECT AND INVERSE PROBLEMS OF HEAT TRANSFER IN SOIL

1 University of Igdir, Agricultural Faculty, Department of Soil Science and Plant Nutrition, 76000 Igdir, Suveren Kampus, Turkey, e-mail: fariz.m@igdir.edu.tr, 2Selcuk University, Cumra High Educational College, Konya, Turkey, e-mail: aserol@selcuk.edu.tr Abstract. The procedures for determining soil thermal diffusivity coefficient based on the solution of inverse problems of heat transfer equation taking into account the boundary conditions on the surface which are described by two harmonics have been developed. These procedures enable estimating the thermal diffusivity in soil under natural conditions and that should increase the adequacy and expand the use of mathematical models of soil thermal regime. Keywords: soil, modeling, heat transfer, boundary conditions.

INTRODUCTION

To obtain comprehensive knowledge of soil thermal properties one should have the data which enables finding the values of the thermal characteristics for the specific composition and condition of the soil. The main thermal characteristics of soil are the coefficients of thermal conductivity, thermal diffusivity and thermal capacity. The knowledge of these soil characteristics may advance the solution of the topical problem of our time as forecasting soil thermal regime. In solving many problems related to soil thermal processes one should deal with soil thermal diffusivity coefficient (k). The determination of soil thermal diffusivity coefficient was discussed in many theoretical and experimental studies [1-4].

To determine soil thermal characteristics, two main groups of methods are used: computational and experimental methods. Some researchers consider the computational methods for determining thermal diffusivity and thermal conductivity coefficients to be the simplest and most convenient. Most often this is the temperature wave analysis method [2]. In most cases this method deals with the solution of heat transfer equations obtained without an initial condition and provided that T(œ,t) =To. However, when performing practical calculations, it is impossible to

set the soil temperature values at infinity as the initial values since they are unknown. Therefore, usually in such cases instead of T (ro, t) the temperature at a certain depth x = L should be set, starting with that, when x > L, the value of T (x, t) = const. Therefore, the calculation of the coefficient k by the equations obtained at the solution of a heat transfer model provided that T (L, t) = To at the lower boundary in the soil is of interest.

The goal of this work is to develop the procedure for determining soil thermal diffusivity coefficient (k) based on the solution of inverse problems of heat transfer equation with follow-on comparison of the existing methods. The calculation results were tested on some soil types of the Konya Province.

MATERIALS AND METHODS Setting the objective and the choice of heat transfer model in soil

To analyze the location of the temperature field in a soil profile, one may not apply the equation system of conductive, radiation and mass-exchange conductivity and use only the thermal conductivity equation with regard for the known heat transfer coefficients [4, 5]:

c,, I x

/ \дТ d

( x,t )-= —

v ' д t dx

чдТ X( x,t ) — v ' dx

It is possible to significantly simplify this equation if we take as constant the thermal capacity coefficient and the coefficients of thermal conductivity and thermal diffusivity into the soil depth. In this case, one-dimensional heat distribution in soil is described by the classical thermal conduc-

tivity equation which (in the absence of phase transitions of soil moisture and heat transfer with the moisture and with the assumption that the thermal gradients are associated with vertical heat transfer only, and in the absence of internal sources) is as follows [1, 3, 6, 8-11]:

dl d2 T — = k—t d t dx2

f

k = ■

Â

Â

PbCm Cv

v J

(1)

and its analytical solutions obtained with- boundary conditions on the surface are out initial condition and at periodic considered, i.e.:

m

T ( 0, t ) = ((t ) = T0 + 2 T J • cos ( ja t + Sj )

j=i

and provided that the soil temperature at the lower boundary (at infinity) is constant, i.e.:

lim T( x, t ) = T0

(2)

(3)

Here, T (x, t) is the soil temperature temperature; v = 2n/T0 is the circular dai-

at the point x at time point t; A is the ther- ly (or yearly) frequency; T0 is the period

mal conductivity coefficient; cv is the volu- (length) of wave expressed in days or

metric thermal capacity; pb is the soil den- years; s is the phase shift depending on

sity, k is the thermal diffusivity coefficient; zero-time reference.

T0 is the average daily (or yearly) the tem- The solution of the problem (1)-(3)

perature of soil active surface; Tj is the os- in dimensionless variables is as follows [1,

cillation amplitude of soil active surface 12-14]:

m

T(y,T) = To + £#j (y,bj)• cos

j=i

jo>T + si

(y h )]

(4)

where

y = x / L t = kt /

® j ( У, bj ) = !• e

L2 bj =yJja/ 2

a = aL / k

and

-Ibjy

Wj ( y, bj ) = bjy

(5)

However, when performing practical Therefore, in such cases instead of (3) the

calculations, it is impossible [7] to set the condition at the lower boundary is set in

soil temperature values at infinity as the the following form, which describes the

initial values since they are unknown. heat transfer process more realistically:

T( L, t ) = To

® y (b ) = T j

\

ch ( dJ )-cos ( dJ )

ch ( 2b} )- cos ( 2b} )

¥

( * b )

= arctan

Y2j(b, ) ' Yl( b, )

where

Y (b, )=ch (q, )cos (b,y)- ch (b,y)cos (q, ) , dj = 2b,(1 - y)

Y2j(*,b, ) = sh (q, ) sin (bj.y)-sh (b,,y) sin (q, ) , q, = b, (2 - y)

(8)

ch(z) = (ez + e-)/2, sh(z) = (ez -e-)/2-and are hyperbolic cosine and sine respectively

The study of soil average tempera- We define the average temperature in the ture is also important since, like other soil layer 0<y<1. To do this, we integrate the characteristics, the temperature values solution (4) from zero to one by the varia-vary with the depth to a lesser extent than ble y and obtain the mean integral solution the temperature values at a certain depth. of the equation (1) as follows:

T(r) = j T(y, z)dy = T0 + £ M, (b, ) • cos

0 J=1

Jor + s, -y

(b, )]

(9)

where Mj(bj) tpj(bj) and tpj(bj) under the boundary conditions (2) u (3) are determined by:

M, (b, )=T

Ï

ch (bj' ) cos (bj' ) ¥j (bj ) = arctan

Z.2 b

b2e

1 - e^ sin ( b )+ cos )]

1 + e-> [sin (bi )- cos fe )],

(10)

under the boundary conditions (2) and (6) are determined by:

M (b ) vVsh2fo)+sin2fo) ¥ (b x=arctan[shfc)-sinfc)

^ b [ ch (b ) + cos (b,)] ¥ b ' = arctan _ sh (b ) + sin (b )

(11)

1

The special cases of the solutions (4) and (9) with (7) and (10)-(11) are presented in the studies [3, 8, 12].

The solution of the inverse problem of heat transfer in soil

If the temperature of the soil surface within a day (a year) may be expressed by one harmonic, then the thermal diffusivity coefficient K may be found from the value of the daily temperature amplitude reduction with the depth or by the temperature

wave phase lag at different depths [1, 4, 11, 15]. This determination results in significant errors because the soil temperature does not always vary strictly sinusoi-dally. The introduction of the second harmonic (2) advances the soil active surface temperature variation to the actual state. By using the solution (4) and (7) for m=2, the equation to determine the thermal diffusivity coefficient K for arbitrary period To and dimensionless depth y may de-

rired. For this to be p°ssible, the tempera- time interval To should be known. Next, by

ture distribution in the soil layer [0, L] for using the solution (4) for m = 2: eight time points in the computational

T( y, r) = T0 + Oj (y, b ) • cos + + y, b2) • cos (2ar + a2)

(12)

first for an arbitrary dimensionless depth y and time ti=I • To/8 the following eight equations should written:

T (y, t,) = T0 + O (y, b ) • cos [—i + « j + O2 (y, b2) • cos J- i + «2 j , (i = 1,8)

since the following is the case

_ al2 k 2- r 2- r. _

jar = j• __•• —t, = j•ct, = j• — •7 • - = j• — •7 ccT = i-• i = - • i 2Cr = -• i k } ro 8 8 ' 84 ' 2

On some rearrangement of the equations (13) we obtain [10], (see. Appendix below):

4 ,

I [T( y, t )-T( y, t,+4)]2 = 8O2 ( y, b )

,=1 (14)

Taking into account both values (5) pressions which correspond to the bound-and (7)-(8) for the function bi) in the ary conditions (3) and (6): equation (14) we have the following ex-

X[T( * t )-T( * t+4)]2

8Tj2

. =

(15)

4 2

I[t(* )-T(* fi+4)] _ ch [2b (1 - *)] - cos [2b (1 - *)] 8T2 " ch ( 2b )- cos(2b)

(16)

Similarly, based on the solution (9) First we rearrange the solution (9)

for m=2 we may derive a mean integral for two harmonics, i.e. m=2 , and then sim-

equation to determine the thermal diffu- ilarly to the derivation of the equations (12)

sivity coefficient k for arbitrary period To. and (13) we obtain the Mowing equation:

4 ,,

I [T(t )-T(t,+4 )]2 = 8M2 (b)

i=1 (17)

The functions in the right-hand side There are several procedures to de-

of the equations (14) and (17), i.e. $i(y, termine the parameter k from the output

bi) and Mi(bi) depending on the bounda- curve of_dimensionless soil temperature

ry conditions are determined respectively T(y, t) or T(t) [1, 2, 4, 9, 12, 18] expressed

from (5) and (7) and (10)-(11). by the equations (4) or (9) of a soil profile.

1=1

In more detail these procedures are described by Mikayilov and Shein (2010) for the case when the soil surface temperature is expressed by one harmonic.

In this study we propose the determination of soil thermal diffusivity coefficient k based on the solution of inverse problems of heat transfer equation for the case when the soil surface temperature within a day (a year) may be expressed by two harmonics.

To determine the thermal diffusivity coefficient k (using the equations (15) and (16)), the following should be known: Ti -the oscillation amplitude of soil active sur-

The determination of k using the equation (16) is performed by computermediated fitting the values of bi* parameter provided that the values of the left-hand side coincide with the right-hand side calculated from the given data, i.e.

2[T(>,,t;)-T(*,c)] /8T2

i=i

By using the mean integral solution (9) it is also possible to find the thermal diffusivity coefficient k; its experimental basis is the data on the temperature in the soil layer [0, L] , that is T(t/), and Ti as well.

face temperature; To-the period (length) of a daily (yearly) wave expressed in days or years; T(y*t/*), (/=1,8) - the temperature values of the soil layer [0, L] at arbitrary depth y»=y=x»/L for eight time points: t* — I • To*/4 (I — Щ. For example, if To* = 24 hours, then t* — 3, 6, 9, ..., 24 hours. Having this data, first we calculate the differ-

Гт( j*, t* )-t( j*, С)] _

ences: for all / = 1,8.

Then from the equation (15) we obtain the value of thermal diffusivity coefficient at the depth x* — x by the equations:

(18)

^ , , b* = sjrnL2 /2к

From the relation we

find the value of the thermal diffusivity

coefficient k at the depth x* — x , and it

equals to

(19)

In this case, the fit of the value of bi parameter is performed using the equations which conform to the boundary conditions (3) and (6) respectively:

*

к =

n ( 2 x» )2

£ [т(х», 0-т(х„с )]2

ln2 -2-

8t2

since the following is the case:

- = ! b =fl =Ш '== ? bi = Lfê^2» =

» n к =-

^2

b*

T.

4 _ _ 2

Z[T(t)-T(t+4 )] ^ ch(b)-cos(b)

QT2 = A22

) T(t-+4 )] _ Sh2 (b )+sin2 (b ) 8Tf = 2b\ [ch(b ) + cos ( b )]2

As opposed to the previously developed procedures [9], here, to determine the thermal diffusivity coefficient k, one should know in advance the temporal distribution of temperature T(y»t,*J, in the soil layer [0, L] at an arbitrary dimension-less depth y» = x/L and T(t,) for eight time points; this distribution enables determining the parameter k by the equation (18)-(21) with higher accuracy

The parameters of soil surface temperature

To determine the soil surface parameters in (2), one and two harmonics were used. By using the measurement results and by using the least square method we determined the parameters of the surface temperature distribution of the soils under study. The preliminary calculation results and their comparison with the experimental data show that the introduction of the second harmonic enables more accurate determination of the temperature

distribution parameters on the soil surface. In future we plan a more detailed study of this procedure application in calculating soil temperature regime and determining thermophysical parameters and characteristics (soil thermal diffusivity coefficient and its dependence on moisture content).

CONCLUSIONS Based on the study of heat transfer model in soil taking into account the boundary conditions dynamics on the surface described by two harmonics, the following was obtained:

point and mean integral solutions; the theoretical foundations of the procedures for determining soil thermal diffusivity were proposed.

In the future we plan to experimentally test the adequacy of the proposed procedures and compare them with the existing procedures.

ACKNOWLEDGEMENT The presentation is supported by Selfuk University Scientific Research Projects Coordination Unit with Project number 15701650

APPENDIX A. FORMULAS

T(У, t ) = To +®i (y b )• cos I n + a | + $2 (y b2 )• cos I ni + «2 ], (* = I8)

i =l: T( y, ti ) = To+®i ( y, bi )•cos +a]+®2 (y, b2 )•cos |

n

•1+a

= T0 +°1 (У, b1 )• cos I n + a ]"°2 (У, b2 )• sin (a2 )

i = 2: T( y, t2 ) = To +Oi ( y, bi )• cos — 2 + a) + ®2 ( y, b2 )• cos + «21 = T0 - (У, bi ) • sin (a ) - (^ b2 ) • cos (a2 )

(13)

i = 3: T( y, t3 ) = To + 0, (y, b, )• cos \ f • 3 + a, J + 02 (y, \ )• cos if3 + a.

= T0(У, b, )• sin [f+ a, J + °2 (^ b )• sin (a2)

i = 4: t(y,t4) = t0 +0, (y,b,)• cos[f •4 + a, j + ®2 (y,¿2)• cos[f •4 + a. = t0 - o, (y, b, )• cos (a,) + o2 (y, ¿2 )• cos (a2)

i = 5: t( y, t5) = t+0( y, b )• cos [f • 5+ a j + ®2 (y, b )• cos [f •5+a = T0 -0, (y, b, )• cos [f+ a, j-02 ^ b2 )• sin (a2 )

i = 6: t(y,t6) = to + 0, (y,b,)• cos[f • 6 + a, j + ®2 (y,¿2)• cos[f • 6 + a2 = T0 +0, (У, b, )• sin (a, )-02 (У, b2 ) • cos (a2 )

i = 7: t(y, t7 ) = t +0, (y, b, )• cos [f • 7 + a, j + 0 2 (y, b2 )• cos [f • 7 + a.

= t0 + o, (y, b, )• sin [f + a, j + 02 (y, b2) • sin (a )

i = 8: t(y, t8 ) = t0 +0, (y, b, )• cos [f • 8 + a, j + ®2 (y, K )• cos [f • 8 + a2 = t0 +0, (y, b, )• cos (a,) + ®2 (y, b2 )• cos a )

q 1. T(y, ti )-T(У, t>+4 )f . i = ,,2,3,4

Subtracting for the , we obtain:

^ T,-T5 = 20, (y, b, )• cos (f/4 + 0, ) T2-T6 =-20! (y, b, )• sin (a,)

T3 -T7 = -201 (y, b, )• sin (f / 4 + a,) T4 -T8 = -201 (y, b,) • cos (a,) 3. 4. ,

and adding the value of their squares, we have:

£[T(y,^)-T(y,ti)]2 = 4[0, (y,b,)]2 • cos2 if + a, V 4[0, (y,b,)]2 • sin2 (a,)

+ 4 [0, (y, b,)]2 • sin21 f + a 0 (y, b )]2 • cos2 (a,)

= 4[0,(y,b,)] jcos2[f+aj+sin2(a)+sin2[f+a j+cos2(a)| = 4 [0, (y,, )]2 fl + ,} = 8 [0, (y, b )]2

i=,

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TYrnH

!фариз Микайылов, 2Ахмет Сами Ерол ТОПЫРАКТАГЫ ЖЫЛУ ЖЫЛЖУЫНЫН, Т1КЕЛЕЙ ЖЭНЕ КЕР1 М1НДЕТТЕР1 1Ыгдыр Университеттщ ауылшаруашылыц факультеттщ топырацтану жэне eciMdiK цopeKmeHyi кафедрасы, 76000 Ыгдыр, Туркия, e-mail: fariz. m @igdir. edu. tr 2СелчукyHueepcumemi жанындагы Чумра Жогары бШм беру колледжу 42500,

Конья, Туркия, e-mail: aserol@selcuk.edu.tr Бул жумыста жылу жылжудьщ Kepi мшдеттершщ тецеспрыушщ шешiмi негiзiнде, eKi Yйлeсiмдiлiгiмeн сипатталган бeткi к;абаттыц шекаралык; шарттарын есепке ала отырып топырак;тары жылу етгазпштж коэффициeнтiн аныщтау тэсiлдepi жасалды. Бул тэсiлдepмeн топырак;тары жылу режимшщ математикалык; модeльдepiн к;олдану шекараларын кeцeИтудi жэне сэикесппн кетepeтiн топырак;тары жылу етгазпштжт таботи жагдаида багалауга болады.

ТYйiндi свздер: топырак;, моделдеу, жылудыц жылжуы, шекаралык; шарттар.

РЕЗЮМЕ

!фариз Микаиылов , 2Ахмет Сами Ерол ПРЯМЫЕ Й ОБРАТНЫЕ ЗАДАЧИ ПЕРЕНОСА ТЕПЛА В ПОЧВЕ 1Кафедра почвоведение и питания растения сельскохозяйственного факультета Университета Ыгдыр, 76000 Ыгдыр, Турция, e-mail: fariz. m@igdir. edu. tr 2Высший образовательный колледж Чумра при университета Селчук, 42500,

Конья Турция, e-mail: aserol@selcuk.edu.tr В работе разработаны методики определения коэффициента температуропроводности почвы, основанные на решении обратных задач уравнения теплопереноса, при учете граничных условии на поверхности, описываемых двумя гармониками. Эти методы позволяют оценивать температуропроводность в почве в естественных условиях, что должно увеличить адекватность и расширить границы использования математических моделеи теплового режима почв.

Ключевые слова: почва, моделирование, перенос тепла, граничные условия.