## 4 Active Systems
### Hot Water Production and Swimming Pools Heating
#### 4.2.1 Existing Thermal Energy Demand  
Firstly, the adopted approach for the calculation of the annual thermal energy required for the production of hot water will be presented. It was based on the data provided by the Stadium's Management, such as:  
* on a daily basis, roughly 1400 users make use of the Stadium's available sports facilities; around 500 of them are swimming pools users and 900 of them make use of the rest sports halls;
* it is estimated that each user of the swimming pools consumes V = 30 L of hot water per visit, while each user of the rest sports halls consumes V = 20 L per visit;
* it is additionally estimated that 50% of the sports halls users, namely 450 users per day, do not make use of the changing rooms, hence they do not contribute to the daily hot water consumption;
* the hot water temperature, density and specific heat capacity are adopted equal to T\({}_{\text{h}}\) = 45 \({}^{\circ}\)C [53], \(\varrho\) = 990 kg/m\({}^{3}\) and c\({}_{\text{p}}\) = 4180 J/kgK, respectively;
* following the above assumptions and facts, the hot water consumption daily profile was developed, presented in Figure 12a. Eventually, the final thermal energy consumption Q is calculated on a daily basis with the following essential relationship:  
\[\text{Q = m\text{-c}_{\text{p}}\text{-(T}_{\text{h}}-\text{T}_{\text{c}})}, \tag{5}\]  
where m is the hot water mass consumption, calculated as the product of the consumed hot water volume V with its density \(\varrho\) and T\({}_{\text{c}}\) is the water temperature from the supply network. The annual fluctuation of the average monthly values of the temperature T\({}_{\text{c}}\) is given in the Greek Directive of Buildings' Energy Performance [53] for the climatic zone of Crete. By replicating the thermal power demand daily profile day by day for the whole annual period, the corresponding annual time series is developed, presented in Figure 12b. By integrating this curve, the annual final thermal energy demand for hot water production is calculated at 261,015 kWh. Figure 12: (**a**) Daily profile of final thermal power demand and (**b**) annual time series of final thermal power demand for hot water production. The calculation of the heat losses from the swimming pools' free surfaces to the indoor ambient environment leads to the calculation of the thermal power required for the maintenance of the water temperature in the swimming pools at the desirable level. To this end, firstly, the calculation of the indoor space heating loads to the outdoor ambient is required, from which the annual fluctuation of the indoor space temperature \(\mathrm{T_{in}}\) will be calculated. Secondly, the swimming pools heating loads \(\mathrm{\dot{Q}_{sp}}\) are calculated with the following general relationship:  
\[\mathrm{\dot{Q}_{sp}}=\mathrm{\ A\cdot U\cdot(T_{w}-T_{in})}, \tag{6}\]  
where:  
1. the swimming pools upper surface area, from which heat is transferred to the ambient environment (in m\({}^{2}\)). 2. the U-factor determining the heat transfer rate from the swimming pools free surface to the ambient environment (in W/m\({}^{2}\)K). 3. the desirable water temperature in the swimming pools, adopted equal to 26 \({}^{\circ}\)C.
4. the indoor space temperature (in \({}^{\circ}\)C). In the existing operation condition, heat is transferred directly from the water to the ambient with convection and radiation, hence the U-factor is given by the relationship:  
\[\mathrm{U_{w}}=\mathrm{\ h_{rw}}+\mathrm{h_{c}}, \tag{7}\]  
where \(\mathrm{h_{c}}=2.5\) W/(m\({}^{2}\cdot\)K) the convective heat transfer factor for indoor space air flows above horizontal surfaces [52; 54] and \(\mathrm{h_{rw}}\) the heat transfer factor with radiation, given by the following relationship:  
\[\mathrm{h_{rw}}=\mathrm{\ 4\cdot\varepsilon_{w}\cdot\sigma\left(\frac{T_{w}+T_{a }}{2}\right)^{3}}, \tag{8}\]  
where:  
1. \(\varepsilon_{w}=0.957\) is the emissivity coefficient of water;
2. \(\sigma\) is the Stefan-Boltzman constant, equal to \(5.67\times 10^{-8}\) W/(m\({}^{2}\cdot\)K\({}^{4}\));
3.