Keywords

1 Introduction

Safety of the roads in the permafrost region is largely determined by their thermal regime, especially when the road is built on soil with high ice content [1,2,3,4,5]. As the soil thaws, its strength deteriorates and melting of the inner ice layer may cause the road to collapse [6,7,8] as the strength of the foundation soil depends on the phase state of the water in the pores [9,10,11]. A way to manage the thawing depth is to add an insulation layer. The insulation layer can be composed of a single material, such as polystyrene, or a combination of thermal accumulation and thermal insulation materials [12,13,14]. One notable insulation material is foam glass ballast [15]. The properties of a thermal insulation layer are usually selected to fulfill the main requirements of preserving the foundation soil in a frozen state throughout the whole period of the road operation and allowing the soil to thaw up to a given depth while maintaining its load-bearing strength. The aim of this research is finding a function determine the thermal conductivity coefficient of insulation materials of the road that prevents the soil from thawing over the permissible depth.

2 Methodologies

An algorithm proposed in [16] is used to obtain the functions to determine the optimal properties of the insulation layer. The algorithm searches for a Biot number, yielded as a function of Fourier and Stefan numbers, guaranteeing that the thawing depth of the foundation soil will not exceed the depth allowed in the road design. The equations are presented in a dimensionless (criterion) form. The required thermal resistance of the insulation layer is determined using the obtained Biot number. The thawing depth with a thermal insulation layer applied can be determined using the formula [17, 18] resulting from solving a one-dimensional Stefan problem at boundary conditions of the third kind. The problem, in a dimensionless form, is:

$$ h = \sqrt {2Fo/St + 1/Bi^{2} } - 1/Bi $$
(1)

where \(Bi = h_{0 } /\left( {R \cdot \lambda_{\Pi } } \right)\), \(R = \delta_{i} /\lambda_{i}\), \(Fo = a\tau /h_{0}^{2}\), \(St = Lw/tC_{p}\), \(h = H/h_{0}\)

h is the thawing depth of the foundation soil, m. h0 is the typical dimension (size), m. R is the thermal resistance, m2K/W. \(\lambda_{\Pi }\) is the thermal conductivity coefficient of the thawed soil, W/mK. \(\delta\) is the thickness of the insulation layer, m. \(\lambda_{i}\) is the thermal conductivity coefficient of the insulation layer, W/mK. L is the latent heat of ice thawing, J/kg. w is the ice content in the soil, unitless. Cp is the total heat capacity of the soil, J/kgK. a is the thermal diffusivity of the soil, m2/s. t is the air temperature, °C. Bi is the Biot number. Fo is the Fourier number. St is the Stefan number.

Using the Eq. (1), a Biot number guaranteeing that the soil will not thaw beyond the permitted thawing depth over a given time period will be found:

$$ Bi = \left( {2h \cdot St} \right)/\left( {4Fo - h^{{2{ }}} St} \right) $$
(2)

In the Eq. (1) it is adopted that the temperature of the active layer of soil is equal to ice thawing temperature. As demonstrated in [19], in most typical cases this assumption is expedient for engineering calculations. The Eq. (1) can be further specified by including the concept of effective heat capacity of the rocks [19]:

$$ C = Lw + C_{p} \left| {T_{e} } \right| $$
(3)

where Te is the temperature of the frozen active layer of the soil, °C. In this case, the Eq. (1) is transformed into the form:

$$ h = \sqrt {2TFo/\left( {St + 1} \right) + 1/Bi^{2} } - 1/Bi $$
(4)

And the Biot number is found from the expression:

$$ Bi = 2h \cdot \left( {St + 1} \right)/\left( {4TFo - h^{{2{ }}} \left( {St + 1} \right)} \right) $$
(5)

The Stefan number notation changes. It will be equal to \(St = Lw/C_{p} \left| {T_{e} } \right|\). The dimensionless temperature simplex T, included in the Eqs. (4) and (5) that describes the relationship between the air and soil temperatures, will be equal to \(T = t/\left| {T_{e} } \right|\). The methods of determination of the Fourier and Stefan numbers when solving problems of heat exchange of atmospheric air with thawing or freezing rocks are considered in [20, 21].

Using the known Biot number, determined using the Eqs. (2) or (5), it is possible to quickly find the thermal resistance and choose a suitable material for the insulation layer of the road. The Biot number can be found using the equation:

$$ Bi = 2h\cdot\left( {St + 1} \right)/\left( {4TFo - h^{{2{ }}} \left( {St + 1} \right)} \right) $$
(6)

where \(R_{\Pi } = h_{0} /\lambda_{\Pi }\); \(R_{i} = \delta /\lambda_{i}\); \(Bi = {\upalpha }R_{\Pi }\). In a physical sense, \(R_{\Pi }\) is a unit of thermal resistance of a thawed layer of the road foundation in m2K/W. The heat transfer coefficient for a flat surface a is determined by the formula [22]:

(7)

where \(\alpha_{0}\) is a convective heat transfer coefficient that depends on the average air speed over a given time interval and can be determined using the Perlstein formula [23]. Should the thermal resistance of all layers of the road be considered, the thermal resistance of the insulation layer can be found from the expression:

$$ R_{i} = \frac{{R_{i} }}{Bi} - \frac{1}{{a_{{0{ }}} }} + \sum\nolimits_{i = 1}^{n} {Ri} { } $$
(8)

Here \({\text{R}}_{{\text{i}}}\) is the thermal resistance of an i-th individual layer of the road comprising n layers, excluding the thermal insulation layer, in m2/W°C.

The second member of the Eq. (8) is much smaller than the first one, Eq. (3) will be used for further analysis. This approach will provide some margin of safety as the excluded second member of the Eq. (8) decreases the thermal resistance of the insulation layer. Knowing the thermal resistance and thickness of the insulation layer, a material with the required thermal conductivity coefficient can be selected. A method to select an economically efficient material is proposed in [24]. The criterion of economic efficiency is equal to the product of the thermal conductivity coefficient of the material and the cost of one cubic meter of the material.

The physical and mechanical qualities of the foundation soil differ along the route and thus the thermal resistance of the insulation layer should be determined for individual sections of the road rather than for the entire road. Indicator m of the degree of change in the thermal resistance of the insulation layer m (\({\text{m}} = {\text{ R}}_{{{\text{i}}2}} /{\text{R}}_{{{\text{i}}1}}\)) and an indicator k (\({\text{k}} = {\text{h}}_{2} /{\text{h}}_{1}\)) of the change in permissible thawing depth of the foundation soil is introduced. The indicators on the baseline section of the road (index 1) and specific section of the road (index 2) are determined using the formula (6). It includes a Biot number dependent on the corresponding parameter h. After several transformations, the relationship between parameters m and h is found:

$$ m = k\left( {\frac{2Fo}{{St}} - \frac{{h_{1}^{2} }}{2}} \right)/\left( {\frac{2Fo}{{St}} - \frac{{k^{2} h_{1}^{2} }}{2}} \right) $$
(9)

Here \({\text{h}}_{1}\) is the permissible value of a dimensionless thawing depth of the foundation soil on the baseline section of the road.

3 Results and Discussion

An analysis of the equations shows that some of them reflect relationships relevant for practical applications. From the expression (6) a conclusion can be made that the thermal resistance of the insulation layer is directly proportional to the unit of thermal resistance of thawed foundation soil, soil and inversely proportional to the Biot number that restricts thawing to a maximum permitted depth. Calculations with varying data were done and their results are presented as charts in the Figs. 1, 2, 3 and 4. Figure 1 shows the dependence of the Biot number (Bi) on the dimensionless thawing depth of the foundation soil (h) at varying values of the complex 2Fo/St for A) small and B) large permissible dimensionless thawing depths. The shape of the charts shows that for small permissible depths (1-B), the relationship is close to linear. Independently of the value of the complex 2Fo/St, the degree of increase in the Biot number at change in the value of the Biot number is an almost constant value. For example, when the dimensionless thawing depth changes by 2.5x - from 0.2 to 0.5 - the degree of increase in Biot number for both a 2Fo/St complex equal to 4.0 and 2.5 remains a constant quantity equal to roughly 1.6. The shape of the curves describing the relationship of the Biot criterion on the dimensionless thawing depth for high thawing depth values (h > 0.5) is non-linear (Fig. 1A). For example, when the thawing depth increases twice, but within the interval 0.8 - 2.0, the degree of increase in the Biot number for the values of 2Fo/St complex of 4.0 is almost 2.0, and when the value of the complex is 2.5, the increase is 4.0. That is, almost twice as large, even though the degree of change in the dimensionless thawing depth remained the same and equal to 2.5, as in the first example.

Fig. 1.
figure 1

Biot number depending on the dimensionless thawing depth of the foundation soil at various values of the complex 2Fo/St. 1 – 2.5, 2 – 3.0, 3 – 3.5, 4 – 4.0. A – for large permissible dimensionless thawing depths, B – for small permissible dimensionless thawing depths.

Full size image

Figure 2 shows a 3D chart displaying the dependence of the Biot number on the Fourier and Stefan numbers at various values of the permissible thawing depths of the foundation soil.

Fig. 2.
figure 2

Biot numbers depending on the values of the Fourier and Stefan numbers at different permissible thawing depths of the foundation soil (h): 1 – 0.5, 2 – 0.75, 3 – 1.0.

Full size image

With the decrease in the dimensionless thawing depth, the dependence of the Biot number on the Fourier and Stefan criteria decreases. The greater the Stefan number and the smaller the Fourier number, the weaker the dependence. At small Stefan numbers and large Fourier numbers, the degree of change in the Biot number is insignificant, indicated by the planes merging in the figure. The greater the ice content in the foundation soil and the shorter the duration of the warm period during which the soil thaws, the smaller the thermal resistance of the insulation layer can be. When the layer is being designed, materials with higher thermal conductivity may be selected.

Figure 3 shows a dependence of the thermal resistance of the thermal insulation layer on the Biot number and the thermal resistance of thawed soil.

Fig. 3.
figure 3

The change in thermal resistance of the thermal insulation layer depending on the Biot number and the thermal resistance of a thawed soil layer.

Full size image

The chart indicates that when the Biot number increases, the influence of the thermal resistance of a thawed soil layer on the thermal resistance of a thermal insulation layer decreases. Compare the edges of the chart at Biot numbers of 0.2 and 1.0. The edge that is closer in the picture (Bi = 1.0) is almost parallel with the R axis. At small Biot numbers the curve rises sharply. In the first case, the range of thermal resistance of the thermal insulation layer varies from 2 to 4 (blue), in the second case from 2 to 10 (yellow).

Figure 4 demonstrates the change in thermal conductivity coefficient depending on the Biot number set in the design and the thermal conductivity coefficient of thawed soil layer at varying thickness of the thermal insulation layer.

Fig. 4.
figure 4

Thermal conductivity coefficient of a thermal insulation layer material (\(\lambda_{\eta }\), W/mK) depending on the Biot number set in the design and thermal conductivity coefficient of the thawed soil (\(\lambda_{\pi }\), W/mK) at varying thickness of the thermal insulation layer. 1 – 0.5 m, 2 – 0.2 m, 3 – 0.1 m.

Full size image

The selection of thermal insulation material depends on the restrictions on total thickness of the road. The greater the permissible thickness, the more likely regular materials, such as burnt rocks or dry sand, whose thermal conductivity coefficients are in the range of 0.4 – 1.0 W/mK could be sufficient as thermal insulation, depending on specific geocryological conditions. The Biot numbers must be in the ranges that do not permit soil thawing beyond the allowed depth. Figure 5 shows charts built on the basis of calculations done using the Eq. (9).

Fig. 5.
figure 5

The degree of change in thermal resistance \(m\) depending on the change in permissible thawing depth \(k\) at various values of the basic quantity \(h_{1}\). 1 – 0.5, 2 – 0.4, 3 – 0.3, 4 – 0.2, 5 – 0.5, 6 – 0.2 (1–4 when the 2Fo/St complex = 6, 5–6 when the 2Fo/St complex = 4).

Full size image

Figure 5 shows the dependence of the parameter \(m\) on \(k\) at various initial values of the base dimensionless quantity of the permissible thawing depth \(h_{1}\) for two values of the 2Fo/St complex. The charts show that in the considered range of data, the relationship between \(m\) and \(k\) is almost linear. The gradient Δm/Δk is close to one. This indicates that the thermal resistance of the insulation layer changes in a roughly equal proportion to the dimensionless thawing depth. If at some section of the road the permissible thawing depth can be increased twice, it means that the insulation layer can have a twice as large thermal conductivity coefficient as the material used for the rest of the road.

As the demonstrated by the chart, in the range of k (1, 0 ≤ k ≤ 2, 0) the two planes almost merge, even though the 2Fo/St complex increases twofold, and the thawing depth changes by 2.5x. An analysis of the charts confirms the possibility of applying the relationship between the thermal conductivity of materials used for the thermal insulation layer of a given thickness and the change in permissible thawing depth at various sections of the road. This relationship is described by the function \(\lambda_{2} = k\lambda_{1}\). Considering that the physical and mechanical properties of the soil are not constant along the route, in the design the required thermal resistance coefficient of the insulation layer should be calculated for parts of the route rather than the whole road. Correspondingly, materials applied at different sections can vary depending on the construction solutions adopted.

4 Conclusion

Engineering equations allowing to assess the main heat exchange factors and thermal physical properties of the foundation soils on the choice of construction materials for the roads in the permafrost area were devised. In particular, it was demonstrated that the thermal resistance of the thermal insulation layer of the road is directly proportional to a unit of thermal resistance of the thawed layer of the foundation soil and is inversely proportional to the Biot number that restricts soil thawing to a required depth. A possibility of applying the relationship between the thermal conductivity coefficient of the construction materials of the thermal insulation layer of a given depth and a change in the permissible soil thawing depth at varying sections of the road. Further research should be focused on surveying the influence of modification of design parameters of the roads and the thermal physical properties of the foundation soil on the road safety and reliability.