Comparison with measured data

In the first part of the work, the models were used to reproduce the propagation of a tempera-ture front in a 470 m long pre-insulated pipe, as logged during an experiment that was carried out by Ciuprinskas and Narbutis [83] on the district heating system of Vilnius (Lithuania) during May 1997. This dataset was already used by other researchers for comparing and vali-dating thermal networks models ([84], [85]). A schematic view of the experimental site is depicted in Figure 3.7 [83]. The inlet temperature and the mass flow rate of both supply (case 1) and return (case 2) pipes are set as a boundary conditions and the resulting outlet tempera-tures are compared with the measured values. Due to the very low water velocity in the con-sidered period (about 0.036 m/s), it takes hours for the temperature front to reach the pipe outlet.

Table 3.4 - Parameters of the pre-insulated pipes (valid for both cases).

Variable Value

Pipe length, L (m) 470

Internal diameter of the pipe, 𝑑_𝑖 (mm) 312.7 External diameter of the pipe, 𝑑_𝑒 (mm) 323.9

Diameter of casing, D (mm) 450.0

Thermal conductivity of insulation, λ (W/(m K)) 0.033 Thermal conductivity of the ground, λ_𝑔(W/(m K)) 2.0

Depth, z (m) 1.0

Temperature of the undisturbed ground, 𝑇_𝑔(°C) 8.0 Table 3.5 - Boundary conditions.

Case 1 2

Initial temperature at inlet node, 𝑇_{𝑠𝑡,𝑖𝑛} (°C) 67.7 64.1 Initial temperature at outlet node, 𝑇_{𝑠𝑡,𝑜𝑢𝑡} (°C) 66.3 62.6

Mass flow rate (constant), 2.646 2.646

Figure 3.7 - Schematics of the experimental site (taken from [83]).

This unusual condition affects the accuracy of the models as discussed in [84]. The parame-ters and the boundary conditions used in the simulations are reported in Tables 3.4 and 3.5, respectively. The time-step of simulation is 5 minutes and the sample-time of the dataset is 10 minutes. Values at times between two consecutive loggings were interpolated linearly. In or-der to simulate the propagation of the temperature front in a single pipe, the pipe was discre-tized into sub-pipes in the pre-processing phase. The Courant number -Eq. (3.21)- is the ratio between the velocity of the flow and the velocity at which the system is “observed”. The latter is the ratio between the length of the pipe elements and the time step of the solver and thereby depends on how the system is discretized in time and space.

𝐶𝑜 = 𝑣 𝛥𝑡 𝛥𝑥

(3.21)

where 𝑣 is the flow velocity, 𝛥𝑡 is the simulation time step and 𝛥𝑥 is the discretization inter-val of the pipe branches, i.e. the spatial resolution of the grid. The Courant–Friedrichs–Lewy (CFL) condition is a necessary condition for the convergence of partial differential equations solved by explicit finite difference schemes; this condition for one dimensional models reduc-es to Co < 1. This means that the velocity of the observer must be greater or equal than the velocity of the flow. If this condition is not met, finite difference schemes based on explicit integration are unstable. The solution is unstable when the errors made at one time step of the calculation are magnified as the computations are continued. The model presented here uses an implicit method, that is not sensitive to stability issues. However, although the CFL condi-tion must not be necessarily met, the accuracy of the numerical solucondi-tion depends on the reso-lution of the grid. An accurate discretization of the system help reducing numerical diffusion

at the expense of a higher computation time, as discussed above. The temperature wave ob-tained with the model and with the measurements have been adimensionalized with Eq. (22), as suggested in [84]

𝑇_𝑑1= (𝑇𝑡𝑟− 𝑇𝑠𝑡)𝑜𝑢𝑡

(𝑇_{𝑡𝑟,𝑚𝑎𝑥}− 𝑇_𝑠𝑡)_𝑖𝑛

(3.22)

where 𝑇_𝑡𝑟 is the temperature during the transient period, 𝑇_𝑠𝑡 during steady state. Since the measurements where likely affected by thermal stratification due to very low velocities [85]

according to the criteria given in [86], the peak temperature of the wave at pipe outlet was not considered as a reliable indicator. The accuracy was assessed by measuring the time of wave start and the wave width. The time of wave start was considered as the time when the adimen-sionalized wave reaches 0.05, and the wave width was measured at 𝑇_𝑑1 = 0.20. Ten simula-tions were carried out: five with 𝛥𝑡 = 30 s and five with 𝛥𝑡 = 300 s. In every simulation the pipe was modelled using different number of nodes. The Courant number was varied from 0.04 to 2. Figure 3.8 shows the temperature values at the outlet of the supply pipe with the two considered time steps. It can be seen that by increasing the number of nodes from 3 to 21 there is a significant improvement both in terms of phase and of wave amplitude. Due to the aforementioned effect of thermal stratification, it is difficult to understand whether the case with 46 nodes brings a further improvement. According to the criteria given in [86], the cross-sectional temperature difference could reach 50% of the wave amplitude under the considered conditions. The modelled temperature front arrives at pipe outlet long before the real front if the pipe is not discretized with a sufficient number of nodes. The discrepancy from measured values is also evident in terms of wave width –see Table 3.6. However, the computational time increases with the number of nodes, as shown in Table 3.6 and Figure 3.9. In Table 6 the computational time is also split between the hydraulic calculation (the SIMPLE loop) and the thermal calculation (that includes both the calculation of the stiffness matrix K and the numer-ical solution of the ODE equation).

Figure 3.8 -Measured data at pipe outlet versus models with (a) 𝛥𝑡 = 300 s and (b) 𝛥𝑡 = 30 s.

For a given Courant number, the CPU time increases linearly with the number of time steps.

The CPU time is spent mostly in the thermal balance when the number of nodes is low. When the latter grows, the hydraulic calculations become predominant and most of the time is spent to reach the convergence of the SIMPLE algorithm. Figure 3.9 also shows that it is counter-productive to refine the mesh above a certain Courant number, because it leads to unneces-sarily high CPU time without improving the accuracy of the results. The solution seems to get even worst for Courant number higher than 1 (with 𝛥𝑡 = 300 s) or higher than 0.1 (with 𝛥𝑡 = 30 s).

Table 3.6 - Resume of accuracy and computational time.

Time

Figure 3.9 -Error of the models and computational time with (a) 𝛥𝑡 = 300 s and (b) 𝛥𝑡 = 30 s.

The simulations were run on a computer equipped with an Intel Core i7-4510U with 2.0 GHz (with Turbo Boost up to 3.1 GHz) and a 8 GB DDR memory.