# A Similarity Model of Specific Heat Loss Determined by Dimensional Analysis
## 6 Discussion and Conclusions  
The exponents in Equation (26) indicated that specific heat loss is most affected by the temperature criteria \(\pi_{2}\) and \(\pi_{1}\) and by the simplexes \(\pi_{5}\) and \(\pi_{6}\). A significantly weaker effect of the other criteria--\(\pi_{3}\), \(\pi_{4}\) and \(\pi_{7}\)--may be explained after a more detailed analysis of the process of heat transfer from hot water to the surrounding environment. A sensitivity analysis of the individual thermal resistances indicated a weak effect of the quantities that are present in those criteria--\(H\), \(C\) and \(\alpha_{\mathrm{e}}\)--on the heat loss value. Moreover, the effect of the quantity \(H\) was most reduced due to the fact that the measurements were carried out on two networks with identical pipe burial depths. The results obtained from the similarity model and the results obtained from the measurements exhibited a very good agreement. Figure 2 shows the correlation between the specific heat loss \(q_{l,\mathrm{meas}}\) calculated from the results of the measurements and \(q_{l,\mathrm{mod}}\) calculated using the criterial Equation (26). The correlation may be described by a regression line with the slope approaching 1, in particular 1.0004, with the coefficient of determination (square power of the correlation coefficient) \(R^{2}\) = 0.9502. The standard deviation of the difference was \(s_{\mathrm{A}}\) = 0.515 W\(\cdot\)m\({}^{-1}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Constant** & \multicolumn{6}{c}{**Exponent**} \\ \hline \(C\) & \(z_{1}\) & \(z_{2}\) & \(z_{3}\) & \(z_{4}\) & \(z_{5}\) & \(z_{6}\) & \(z_{7}\) \\ \hline
0.343 & 2.976 & \(-5.165\) & \(-2.016\cdot 10^{-4}\cdot-2.485\cdot 10^{-2}\) & \(-0.587\) & 0.155 & 2.641\(\cdot 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constant \(C\) and individual exponents. Figure 2: Correlation between the specific heat losses obtained by the measurements and from the model. The results of both methodologies were also tested by using a paired \(t\)-test at a significance level of \(p\) = 0.05. The value of the test criterion \(t\) was calculated using the following equation:  
\[t=\frac{|\overline{\Delta q_{l}}|\cdot\sqrt{m-1}}{s_{\Delta}} \tag{27}\]  
where \(m\) is the number of values. The quantity \(\overline{\Delta q_{l}}\) is an average value of the difference \(q_{l,\text{mea}}-q_{l,\text{mod}}\) calculated using the following equation:  
\[\overline{\Delta q_{l}}=\frac{\sum(q_{l,\text{mea},i}-q_{l,\text{mod},i})}{m} \tag{28}\]  
For the analysed datasets, \(m\) = 211 and \(\overline{\Delta q_{l}}\) = 0.0175 W\(\cdot\)m\({}^{-1}\), therefore, the value of the test criterion was \(t\) = 0.492. The critical value \(t_{\text{cr}}\) at a significance level of 0.05 was \(t_{0.05}\)(211-1) = 1.971. Since \(t<t_{\text{cr}}\), it may be stated that both of the methodologies led to identical results. The criterial Equation (26) represents a universal formula for expressing the specific heat loss for a twin pipe directly-burried heat network. The ranges of the quantities to which the derived model applies are listed in Table 3.  
An advantage of defining the specific heat loss using a criterial equation is that it facilitates the identification of specific heat loss, as well as the heat loss of the entire distribution system, without the necessity to perform repeated accurate measurements. An example of using the model, when the temperature of water in the feed pipe \(t_{1}\) = 81 \({}^{\circ}\)C, is shown in Figure 3, comparing the specific heat loss of the DN65 network and that of the DN100 network.