# A Similarity Model of Specific Heat Loss Determined by Dimensional Analysis
## 2 Analysis of the Problem  
To describe a process for which no exact physical equation exists, or if the application of a physical equation is assumed to be problematic, it is advisable to perform an experiment and to generalise its results while applying the laws of similarity. Similar processes that depend on several physical quantities are subject to certain correlations between the similarity constants, which cannot be arbitrarily changed [14; 15; 16; 17; 18; 19]. Such a condition must also apply to a description of the specific heat loss for directly-buried heat distribution systems. Following a detailed analysis, several physical quantities that affect the specific heat flow were selected for the twin pipe directly-buried heat network. The selected quantities relevant to the creation of a mathematical model using a dimensional analysis are listed in Table 1. In the process of creating a model law, the dimensions of physical quantities are always transformed to the seven SI units (kg; m; s; K; A; mol; cd). However, the quantities that are relevant to the creation of a model of specific heat loss are only four basic dimensions--kg, m, s and K.  
The table does not contain the coefficient of heat transfer from the flowing fluid to the pipe wall \(\alpha\), internal diameter of the pipe \(d_{1}\), pipe wall thickness \(b_{p}\), and the thermal \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Physical Quantity** & **Unit** \\ \hline \(\eta\) & Specific heat loss & W m\({}^{-1}\)\(\approx\) kg m s\({}^{-3}\) \\ \hline \(T_{1}\) & Average temperature of water in the feed pipe & K \\ \hline \(T_{2}\) & Average temperature of water in the return pipe & K \\ \hline \(T_{\mathrm{e}}\) & Ambient temperature & K \\ \hline \(d_{2}\) & External diameter of the pipe & m \\ \hline \(b\) & Insulation thickness & m \\ \hline \(\lambda_{\mathrm{m}}\) & Thermal conductivity coefficient for insulation & W m\({}^{-1}\)\(\approx\) kg m s\({}^{-3}\)-K\({}^{-1}\) \\ \hline \(H\) & Bural depth & m \\ \hline \(C\) & Pipe spacing & m \\ \hline \(\lambda_{\mathrm{s}}\) & Thermal conductivity coefficient for soil & W m\({}^{-1}\)\(\approx\) kg m s\({}^{-3}\)-K\({}^{-1}\) \\ \hline \(\alpha_{\mathrm{e}}\) & Coefficient of heat transfer to the surrounding environment & W m\({}^{-2}\)\(\cdot\)K\({}^{-1}\)\(\approx\) kg s\({}^{-3}\)\(\cdot\)K\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input quantities for the mathematical model. conductivity coefficient for the pipe wall \(\lambda_{\rm p}\). A detailed analysis, conducted for steel pipes through which hot water flowed at a rate of several metres per second, showed that the sum of the thermal resistance on the internal side of the pipe and the thermal resistance of the pipe wall was a value three orders lower than the value for the thermal resistance of the pipe insulation. Therefore, the quantities \(\alpha\), \(d_{1}\), \(b_{\rm p}\) and \(\lambda_{\rm p}\) can be neglected in the creation of the model. A similarity model for describing any phenomenon (a criterial equation) is created in the process of replacing selected dimensional quantities \(\varphi_{1}\) to \(\varphi_{n}\) with similarity criteria \(\pi_{1}\) to \(\pi_{l}\), and then by identifying functional correlations between the individual criteria either experimentally or by a numerical calculation. Such a criterial equation is then applicable to the entire group of similar phenomena. A general procedure for applying dimensional analysis, in order to describe a process for which no exact analytical solution is known, is described in detail in papers [20; 21; 22]. The authors of this article applied dimensional analysis when describing the formation of nitrogen oxides during a dendromass combustion process. The resulting criterial equation was verified in a physical experiment, conducted on the combustion equipment by [PERSON] with a power of 13 kW [23]. The results obtained from the model were in absolute agreement with the results obtained in situ using a HORYBA ENDA--680P analyser.