**2. Transient Flow Equations in Viscoelastic Pipelines**

The standard system describing the one-dimensional transient flow of a compressible liquid in an elastic pipe consists of the continuity Equation (1) and momentum Equation (2) [13,14]:

$$
\frac{
\partial h
}{
\partial t
} + v \frac{
\partial h
}{
\partial \mathbf{x}
} + \frac{c^2}{g} \frac{
\partial v
}{
\partial \mathbf{x}
} = 0
\tag{1}
$$

$$\frac{\partial v}{\partial t} + v \frac{\partial v}{\partial \mathbf{x}} + g \frac{\partial h}{\partial \mathbf{x}} + \frac{f}{2D} v|v| = 0 \tag{2}$$

where *f* is the friction factor, *c* is the pressure wave velocity, *h* is the piezometric head, *v* is the average flow velocity, *g* is the gravity acceleration, *x* is the space coordinate, *t* is time and *D* is the internal pipe diameter.

The system of Equations (1) and (2) may be applied for steel pipelines. However, this mathematical description is no longer suitable when transient flow in polymeric pipes is considered. Polymers exhibit both viscous and elastic characteristics. In practice, to include the viscoelastic behaviour of the pipe wall during transient flow, the approach of "mechanical analogues" is usually used. In this approach, the one-dimensional mechanical response of an elastic solid is represented by the mechanical analogue of a spring, and the linear viscous response is represented by a viscous dashpot [15] (Figure 1).

**Figure 1.** Generalised Kelvin–Voigt model.

Figure 1 shows the so-called generalised Kelvin–Voigt model, which consists of *N* elements. The total strain can be described as a sum of instantaneous (elastic) strain and retarded strain [4,5]:

$$
\varepsilon = \varepsilon\_0 + \varepsilon\_r \tag{3}
$$

where *ε*<sup>0</sup> is the instantaneous (elastic) strain and *ε<sup>r</sup>* is the retarded strain. The total retarded strain results from the behaviour of each of the *N* elements:

$$
\varepsilon\_r = \sum\_{k=1}^{N} \varepsilon\_k \tag{4}
$$

The total strain generated by the constant stress is as follows:

$$\varepsilon(t) = \frac{\sigma}{E\_0} + \sum\_{k=1}^{N} \frac{\sigma}{E\_k} \left[ 1 - \mathbf{e}^{\frac{t}{\tau\_k}} \right] \tag{5}$$

where *σ* is the constant applied stress, *E*<sup>0</sup> is the elastic bulk modulus of the spring, *Ek* is the elastic bulk modulus of the *k*-th Kelvin–Voigt element, *τ<sup>k</sup>* is the retardation time of the *k*-th Kelvin–Voigt element defined by *τ<sup>k</sup>* = *μk*/*Ek* and *ƒ*⊂<sup>k</sup> is the viscosity of the *k*-th dashpot.

The creep compliance function corresponding to the generalised Kelvin–Voigt model is equal to the following:

$$J(t) = J\_0 + \sum\_{k=1}^{N} J\_k \left[1 - \mathbf{e}^{\frac{-t}{T\_k}}\right] \tag{6}$$

where *J*<sup>0</sup> is the instantaneous creep compliance of the first spring, defined by *J*<sup>0</sup> = 1/*E*0, and *Jk* is the creep compliance of the *k*-th Kelvin–Voigt element, defined by *Jk* = 1/*Ek*.

According to the Boltzmann superposition principle, the total strain is as follows:

$$\varepsilon(t) = J\_0 \sigma(t) + \int\_{\varrho}^{t} \sigma(t - \xi) \frac{\partial J(\xi)}{\partial \xi} d\xi \tag{7}$$

where *ξ* is a dummy parameter required for integration.

The hoop stress is related to pipe wall thickness and internal diameter:

$$
\sigma = \frac{dp \, D \, \mathfrak{a}}{2s} = \frac{(p - p\_0) \, D \, \mathfrak{a}}{2s} \tag{8}
$$

where *α* is a dimensionless parameter (dependent on pipe diameter and constraints), *s* is the pipe wall thickness, *p* is the current pressure and *p*<sup>0</sup> is the steady-state pressure.

After calculating the creep function time derivative in Equation (7) and including Equation (8) in it, one obtains the following:

$$\varepsilon(t) = \frac{[p(t) - p\_0]D(t)\,\mathrm{a}}{2\,\mathrm{s}\,\mathrm{E}\_0} + \sum\_{k=1}^{N} \frac{I\_k}{\tau\_k} \int\_{\boldsymbol{\varrho}}^{\boldsymbol{t}} \frac{[p(t - \boldsymbol{\xi}) - p\_0]D(t - \boldsymbol{\xi})\,\mathrm{a}}{2\,\mathrm{s}} \,\mathrm{e}^{-\frac{\boldsymbol{\xi}}{\tau\_k}} \,d\boldsymbol{\xi} \tag{9}$$

This strain influences the cross-sectional area of the pipe, and therefore, it must be included in the continuity equation. In order to take into account the viscoelastic properties of the polymer pipe, the continuity Equation (1) has to be derived from the Reynolds transport theorem [16]. The final form of the continuity equation is as follows:

$$\frac{\partial h}{\partial t} + v \frac{\partial h}{\partial x} + \frac{c^2}{g} \frac{\partial v}{\partial x} + \frac{2c^2}{g} \frac{\partial \varepsilon\_r}{\partial t} = 0 \tag{10}$$

Equations (2) and (10) constitute the mathematical description of transient flow in a polymeric pipe. A separate problem is how friction is represented in the momentum Equation (2). The way of representing friction in Equation (2) has a significant impact on the results of the numerical calculations of the water hammer equations [17]. The most common approach is to include one of the many unsteady friction models in the continuity equation [18,19]. However, the problem of unsteady friction falls outside the scope of this investigation, as energy dissipation was taken into account by applying a viscoelastic model, and the friction factor was calculated using the Darcy–Weisbach equation:

$$h = f\_s \frac{v|v|}{2D} \tag{11}$$

where *fs* is the Darcy friction factor.
