# A Similarity Model of Specific Heat Loss Determined by Dimensional Analysis
## 3 Similarity Model Based on the Criterial Equation  
Applying a criterial equation to any process always brings a smaller number of criteria \(\pi\) than the number of relevant quantities \(n\) on which the process depends. The basic equation describing a correlation between \(n\) relevant quantities \(\varphi_{1}\), \(\varphi_{2}\)... \(\varphi_{i}\)... \(\varphi_{n}\) of various dimensions is as follows:  
\[f(\varphi_{1},\ \varphi_{2}\ \ldots\varphi_{i}\ \ldots\varphi_{n})=0 \tag{1}\]  
For each quantity \(\varphi\), it is possible to write a dimensional formula based on a defining equation. It is the product of the symbols of the base units with the respective exponents. In the case of four base units of selected physical quantities (kg, m, s, K), a defining equation is as follows:  
\[[\varphi]=\ {\rm m}^{x_{1}}\cdot{\rm kg}^{x_{2}}\cdot{\rm s}^{x_{3}}\cdot{\rm K }^{x_{4}} \tag{2}\]  
In Equation (2), the dimensional exponents \(x_{1}\) to \(x_{4}\) are rational numbers, which must follow from the solution--see the procedure below. Equation (1) is dimensionally homogeneous; this means that the variables \(\varphi_{i}\) cannot be individually present in the equation, but only in the form of products:  
\[\pi=\prod_{i=1}^{n}\varphi_{i}^{x_{i}} \tag{3}\]  
where in \(\pi\) is the dimensionless variable quantity (similarity criterion); \(x_{i}\) is the exponent (a rational number); and \(\varphi_{i}\) is physical quantities with the respective dimensions. According to Equation (1), and while taking the physical quantities listed in Table 1 into account, the following correlation must apply:  
\[(q_{l},T_{1},T_{2},\ T_{\rm e},\ d_{2},\ b,\lambda_{\rm in},\ H,\ C,\lambda_{ \rm s},\alpha_{\rm e})=0 \tag{4}\]  
In general, for a process described by \(n\) relevant quantities, a total of \(l\) similarity criteria may be created. The number of searched criteria \(\pi\) is identified using the following equation:  
\[l=n-h \tag{5}\]  
where in \(h\) is the rank of a dimensional matrix. According to Equation (3), the following may be written:  
\[\pi=q_{l}^{x_{1}}\cdot T_{1}^{x_{2}}\cdot T_{2}^{x_{3}}\cdot T_{\rm e}^{x_{4}} \cdot d_{2}^{x_{3}}\cdot b^{x_{6}}\cdot\lambda_{\rm in}^{x_{27}}\cdot H^{x_{8} }\cdot C^{x_{9}}\cdot\lambda_{\rm s}^{x_{10}}\cdot\alpha_{\rm e}^{x_{11}} \tag{6}\]  
The dimensional formula of Equation (6) is as follows:  
\[1=(\rm kg\cdot m\cdot s^{-3})^{x_{1}}\cdot(K)^{x_{2}}\cdot(K)^{x_{3}}\cdot(K) ^{x_{4}}\cdot(m)^{x_{9}}\cdot(m)^{x_{6}}\cdot(kg\cdot m\cdot s^{-3}\cdot K^{-1 })^{x_{7}}\cdot(m)^{x_{8}}\cdot(m)^{x_{9}} \tag{7}\]  
Since the left side of Equation (7) equals one, the sum of the dimensional exponents for each base quantity must equal zero. Therefore, for individual dimensions of the physical quantities (m, kg, s, K) that affect a value of the specific heat dissipation rate in the analysed heat network, the set of Equations (8)-(11) applies:  
\[\rm For\ a\ unit:\ ``metre''\qquad x_{1}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10 }=0 \tag{8}\]  
\[``kilogram\"\qquad x_{1}+x_{7}+x_{10}+x_{11}=0 \tag{9}\]  
\[``second\"\quad-3x_{1}-3x_{7}-3x_{10}-3x_{11}=0 \tag{10}\]  
\[[\varphi]=\ {\rm m}^{x_{1}}\cdot{\rm kg}^{x_{2}}\cdot{\rm s}^{x_{3}}\cdot{\rm K }^{x_{4}} \tag{2}\]0  
Equations (9) and (10) are in a linear correlation. Therefore, in a further solution, one of them must be omitted, and the rank of the dimensional matrix then equals \(h=3\). According to Equation (5), out of the eleven relevant quantities it is then possible to create eight similarity criteria. In order to obtain the eight independent criteria \(\pi\), it is necessary to make eight independent solutions of the set of Equations (8)-(11), and in each case to select eight unknowns \(x_{i}\). When selecting the unknowns, the usual procedure is that one unknown is determined as equal to one, while the others are equal to zero. Such a selection has only one constraint--the selected unknowns cannot be mutually correlated.