1. Introduction

2. Materials and Methods

2.1. Geometry of the Model

A two-dimensional model was developed to study the façade behaviour (Error! Reference source not found.). This choice was due to the need of keeping the model as simple as possible, and it was compatible with the geometry of the façade. In fact, since the air flows through many independent cavities, characterized by limited width (80 cm), horizontal air flows are negligible. For this reason, a 2D façade model, which neglects the third spatial dimension, was adequate for the study purpose.

Figure 3. Regions (solid and fluid) of the CFD model.

Figure 3. Regions (solid and fluid) of the CFD model.

2.5. Solver

The solver used (chtMultiRegionFoamIPPT, see Appendix A for details of the source code) allows to couple a transient fluid flow with heat transfer between regions, buoyancy effects, turbulence, and radiation modelling. The solver follows a segregated solution strategy, which means that the equations for each variable characterizing the system are solved sequentially and the solution of the preceding equations is inserted in the subsequent equation. The coupling between fluid and solid follows the same strategy: first the equations for the fluid are solved using the temperature of the solid of the preceding iteration to define the boundary conditions for the temperature in the fluid. After that, the equation for the solid is solved using the temperature of the fluid of the preceding iteration to define the boundary condition for the solid temperature. This iteration procedure is executed until convergence. For each fluid region the compressible Navier Stokes equation is solved, while for the solid regions only the energy equation has to be solved. The regions are coupled by a thermal boundary condition.

2.5.1. A New “Frozen-Unfrozen Flow” Solver

A novel algorithm, called the “frozen-unfrozen flow” solver, was developed to speed up the simulations, and consequently the overall model calibration process. The new solver features are explained in the following paragraph.

The purpose of developing this new solver was simply to speed up the simulation by switching the solution mode to “frozen” (i.e. no update of the velocity and pressure field, allowing large time steps) and “unfrozen” (i.e. solution of all transport equations, with normal time steps) sequentially. The normal time step in the “unfrozen” mode is determined by the Courant number and the solid diffusion numbers, while the time stepping in the “frozen” mode is set based on user input.

Error! Reference source not found. shows the schematic view of this algorithm. It starts with the initial time (${\tau }_{initial})$ and then several cycles with unfrozen (red zones) and frozen (blue zones) are repeated till the end of the simulation. For stability reasons, a transition mode (the grey zones) must be also considered when the flow mode is switched between “frozen” and “unfrozen”.

To do so, two periods were considered:

• “initial period”, when the fluid flow evolves according to time steps calculated based on Courant and solid diffusion numbers, to make the simulation stable at the beginning. It should be noted that the initial time not necessarily starts at time 0, since the algorithm is designed to work also in the case the simulation is re-started;
• “normal period”: in which the flow is sequentially set to frozen and unfrozen modes.

Algorithmic details related to these periods are summarized in Appendix B for the interested reader.

Figure 6. Illustration of the new solver algorithm by considering an “initial period”, and the “normal period” consisting of a “frozen” and “unfrozen” phase.

Figure 6. Illustration of the new solver algorithm by considering an “initial period”, and the “normal period” consisting of a “frozen” and “unfrozen” phase.

2.5.2. Solver Performance Evaluation

To test the new solver performance, new cases were created varying τ_frozen and τ_unfrozen in a systematic way to explore the effect of these numerical parameters on the accuracy of the results (i.e. the temperature values obtained in the model) and the time needed for the computation. The new cases were compared to a base case, t0001, identical to the new ones, but run with the old solver. Table 3 resumes the results obtained from the new cases.

All the simulations were run for 24 hours real time. ${\tau }_{initial}$ and ${\mathsf{\Delta }\mathrm{t}}_{Frozen}^{max}$ were set equal to 1s. The relative performance of the code, compared to the theoretical maximum performance that can be reached with the freeze/unfreeze setting used, was calculated as:

Relative performance = simulation speedup/(1+τ_frozen /τ_unfrozen)

(1)

As expected, the simulation speed increased by increasing the ratio τ_frozen /τ_unfrozen, while the accuracy did not seem to be inversely proportional to the speed. The relative performance of the code shows how close the speed increase was to the theoretical optimum. For example, for case t0011, the new algorithm reaches almost the maximum performance of the code (i.e., 90% of the theoretical optimum). Instead, for case t0016, the relative performance was only at 45%, indicating a potential to further increase the observed speedup of x45.

3. Results

3.2. Calibration Process

Starting from simulation t0104, the specific heat capacity (cp) of solid.2 (concrete) was systematically changed to find the optimum value that allowed to minimize the error between the experimental value and the CFD prediction of temperature TO. Simulation t0104 was chosen because it is one of the cases with lower average error and lower error in TO (see Table 5). Temperature TO was used as reference because, together with TR, it determines the heat flux entering the building through the timber-frame wall. As already mentioned, temperature TR is always well predicted by the CFD model in all the cases run during the sensitivity analysis. The results from the calibration are shown in Table 6 and Error! Reference source not found..

According to the model, the lowest error in the temperature TO is 6.5% and corresponds to case t4500, which considered a Cp for concrete equal to 4500 J/kgK, i.e. 4.5 times higher than the reference value prescribed by the Standard EN ISO 10456 [17]. This result is unrealistic and probably affected by a CFD model issue that leads to low temperatures prediction into the cavity (documented in paragraph 4.3).

Table 6. Steps and results of calibration process.

Table 6. Steps and results of calibration process.

Results of calibration process
Case	Description	Cp_concrete (J/kgK)	Error of TO prediction	Note
t1000 (= t0101)	1xCp_concrete_base	1000	11.59%	Cp value from EN ISO 10456:2007
t1500	1.5xCp_concrete_base	1500	9.97%
t1750	1.75xCp_concrete_base	1750	9.38%
t2000 (= t0104)	2xCp_concrete_base	2000	8.85%	starting case
t2250	2.25xCp_concrete_base	2250	8.40%
t2500	2.5xCp_concrete_base	2500	8.00%
t3000	3xCp_concrete_base	3000	7.36%
t3500	3.5xCp_concrete_base	3500	6.93%
t4000	4xCp_concrete_base	4000	6.66%
t4500	4.5xCp_concrete_base	4500	6.52%	best prediction
t5000	5xCp_concrete_base	5000	6.55%
t6000	6xCp_concrete_base	6000	7.02%

Figure 7. Graphical representation of the values listed in Table 6. According to the model, the minimum error is around 6.5% and corresponds to a Cp value of 4500 J/kgK for the concrete.

Figure 7. Graphical representation of the values listed in Table 6. According to the model, the minimum error is around 6.5% and corresponds to a Cp value of 4500 J/kgK for the concrete.

3.3. Validation Study

For the validation process, case t2000 (the best case from the sensitivity analysis) and t4500 (the calibrated model) were used, to highlight if the latter was more accurate than the former. The boundary conditions considered in this case refer to summer cloudy days, instead of summer sunny days. The models were run for 48 hours; the average errors of the temperature prediction reported in Table 7 are calculated considering only the last 24 hours, to exclude initialization errors. By looking at the results in Table 7, in general model t0321 predicts with lower error than the calibrated model t4500. However, the trend of model t4500 is more similar to the experimental one (Error! Reference source not found.): if one would consider an offset of approximately 4 °C for TO and TC, the experimental and simulated data would almost perfectly overlap. Similarly, an offset of approximately 1°C for TC_ext would greatly reduce the error.

Figure 8. Prediction of a) TO, b) TC, and c) TC_ext by model t2000 and t4500.

Figure 8. Prediction of a) TO, b) TC, and c) TC_ext by model t2000 and t4500.

3.4. Discussion of Persistent Underprediction of Surface Temperatures Inside the Cavity

The persistent underprediction of the surface temperatures inside the cavity by the CFD simulation is speculated to be mainly caused by an under resolved flow: specifically, we have assumed a laminar flow in the cavity, but used a relatively coarse computational mesh. Consequently, an instability that might develop in the flow, and increase flow resistance as well as the heat transfer rates were - most likely - not captured in our simulations. Thus, heat transfer from the concrete to the timber wall (specifically its surface) during heat up (noon and early afternoon) is strongly underpredicted. During the cool-off phase (late afternoon) this leads to significantly lower temperatures of the wall cavities, especially the one on the timber side.

3.4.1. Relative Importance of Radiative Heat Transfer Inside the Cavity

The transferred heat flux inside the cavity can be estimated via:

$$\dot{q}={\alpha }_{HEx}(T_{wall}-T_{fluid})$$

(2)

where

$${\alpha }_{HEx}=Nu\cdot \left(\frac{{\lambda }_f}{D_h}\right)D_h=4A_c/P=\frac{4ab}{2\left(a+b\right)}\approx 2a$$

(3)

$$\begin{gathered} Nu=7.54for{T}_{wall}=cte \\ Nu=8.23for\dot{q}=cte \end{gathered}$$

(4)

Equation (3) is taken from [19].

The cavity is modeled as a 2D domain with a depth of 3 cm. Hence, the heat transfer coefficient for the cavity is calculated as follows if a fixed temperature at the walls is considered:

$${\alpha }_{HEx}=Nu\cdot {\frac{{\lambda }_f}{D}}_h=7.54·0.02588/(2·0.03)=3.25W/m^{2}K$$

(5)

Now, we need to compute the average temperature of the cavity walls and air in the cavity. These values have been obtained from t0104 and t4500 cases (see Error! Reference source not found.). For an estimate of the patch-averaged heat fluxes due to heat transfer, we picked two time-snapshots on this plot (i.e. 8:00 am and 6:00 pm). We note in passing that the averaged values have been obtained by the “integrate variables” feature in ParaView, which performs an area-weighted calculation. As can be seen in Table 8, the patch-average radiative and transferred heat flux are of similar magnitude. This indicated that both the emissivity of the wall and the details of the flow in the cavity are important when predicting the heat fluxes, and consequently the surface temperatures in the cavity. It should be noted that a negative radiative heat flux (i.e. $q_r$) means that radiative heat leaves the patch.

Figure 9. Volume averaged temperature of fluid.1 as well as the patch averaged temperature of solid2.1 and solid.1.3 in contact with fluid.1 as a function of time case for case t0104 (left panel), and case t4500 (right panel).

Figure 9. Volume averaged temperature of fluid.1 as well as the patch averaged temperature of solid2.1 and solid.1.3 in contact with fluid.1 as a function of time case for case t0104 (left panel), and case t4500 (right panel).

4. Conclusions

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