**4. Thermal Stresses**

The temperature distribution throughout the pipeline wall, obtained by solving the IHCP, is used to determine the thermal stresses in the whole surface and considered on boundary surfaces.

The radial, longitudinal and circumferential thermal stresses are determined, assuming that the pipeline ends can move freely. The thermal stress equations are given by [33].

$$\sigma\_r = \frac{E\beta\_T}{2(1-\nu)} \left( 1 - \frac{r\_{in}^2}{r^2} \right) \left[ T\_w(t) - T\_w(r,t) \right] \tag{20}$$

$$\sigma\_{\varphi} = \frac{E\beta\_T}{2(1-\nu)} \left[ \left( 1 - \frac{r\_{in}^2}{r^2} \right) \overline{T}\_w(t) + \left( 1 - \frac{r\_{in}^2}{r^2} \right) \overline{T}\_w(r,t) - 2T\_w(r,t) \right] \tag{21}$$

$$
\sigma\_z = \frac{E\beta\_T}{1-\nu} \left[ T\_w(t) - T\_w(r,t) \right] \tag{22}
$$

Equations (20)–(22) giving the thermal stress components consider the radial temperature distribution in the pipeline, as the temperature drop in the pipeline wall in the direction of steam flow is minor.

In Equations (20)–(22), the symbols *Tw*(*t*) and *Tw*(*r*, *t*) denote the mean wall temperature that is given by the following formulas.

$$\overline{T}\_w(t) = \frac{2}{r\_{out}^2 - r\_{in}^2} \int\_{r\_{in}}^{r\_{out}} rT\_w dr \approx \frac{2\Delta r}{r\_{out}^2 - r\_{in}^2} \left[ r\_2 \frac{T\_{w\_1} + T\_{w\_2}}{2} + r\_{n+1} \frac{T\_{w\_n} + T\_{w\_{n+1}}}{2} + \sum\_{i=2}^{n-1} \frac{r\_i + r\_{i+1}}{2} T\_{w\_i} \right] \tag{23}$$

$$\overline{T}\_w(r,t) = \overline{T}\_w(r\_i, t) = \frac{2}{r^2 - r\_{in}^2} \int\_{r\_{in}}^r rT\_w dr \approx \frac{2\Delta r}{r\_i^2 - r\_{in}^2} \left[ r\_2 \frac{T\_{w\_1} + T\_{w\_2}}{2} + \sum\_{j=2}^i \frac{r\_j + r\_{j+1}}{2} T\_{w\_j} \right] \tag{24}$$

where the symbols *r*2, *ri*, *i* = 3 ... *n* + 1, and *rj*, *j* = 1 ... *n* + 2 denote the radiuses (Figure 2).

The radial stresses *σ<sup>r</sup>* is equal to zero on the inner and outer surface of the tube ( *σ*|*r*=*rin* = *σ*|*r*=*rout* = <sup>0</sup>). The circumferential σ<sup>ϕ</sup> and the axial σ<sup>z</sup> thermal stresses on these surfaces are equal *σϕ <sup>r</sup>*=*rin* <sup>=</sup> *<sup>σ</sup>z*|*r*=*rin* , *σϕ <sup>r</sup>*=*rout* = *<sup>σ</sup>z*|*r*=*rout*).
