## 5 Simulations with DeltaEC  
To model the thermoacoustic engine and get information on the acoustic field neat the test HXs the computer code DeltaEC has been used. The code integrates the linear theory of thermoacoustics, firstly formulated by [PERSON] [18] and subsequently refined by [PERSON] [1; 2], which has demonstrated accurate precision for pressure amplitudes up to 10% of the mean pressure. The code performs 1D numerical integration of the momentum, continuity and energy equations through each segment of the acoustic network. The solutions found for adjoining segments are then matched by imposing the continuity of pressure and volume flow rate at their junction. For further details the reader is addressed to ref. [16].  
In Figure 6 the axial distribution of the acoustic field (acoustic pressure, \(p\), and volumetric velocity, \(U\)) is shown. The \(p\) and \(U\) profiles resemble those characterizing a half-wavelength resonator. The graph shows how the HX location falls between a velocity node and antinode. The acoustic particle velocity near the HX can be modulated by varying the heat input to the system. This also influences the operative temperature of the heater, \(T_{H}\), the acoustic pressure amplitude, \(p\), and the amplitude of the particle oscillation displacement  
\[x_{1}=\frac{U}{A_{HX}\omega} \tag{7}\]  
\(A_{HX}\) being the cross section area of the HX open to gas flow. The last is an important parameter in the design of HXs since, being the flow oscillatory, an excessively long HX (along the longitudinal direction of the particle acoustic oscillation) could only lead to additional thermoviscous losses without incrementing the surface area available for heat transfer.  
Conversely, a much too short HX could be ineffective in transferring the required thermal load. It is generally assumed, as a rule of thumb, that the optimal length of the HX (\(L_{HX}\)) should be of the order of the peak-to-peak acoustic displacement amplitude [1], although this problem remains to date an open issue.  
In Figure 7 the variables \(T_{H}\), \(DR\) and \(2x_{1}/L_{HX}\) are reported as a function of the heat input for the two typologies of HX. In both cases the drive ratio is lower than 10% even at the highest power delivered by the heater so the growth of non linear acoustic effects should not considerably affect the engine operation. But, more interesting, is the circumstance that at a given \(\hat{Q}_{H}\) value the  
Figure 6: The axial distribution of the acoustic pressure amplitude and volumetric velocity amplitude.  
peak-to-peak particle displacement amplitude, \(2x_{1}\), surpasses the length of the HX. This \"transition\" should allow to investigate on the optimal length along the longitudinal direction which maximizes the HX performance.  
As a concluding remark it has to be underlined that DeltaEC relies, as for the calculation of the gas side heat transfer coefficient, on a simple boundary layer model expressed by the equation  
\[h=\frac{K}{y_{eff}}, \tag{8}\]  
where  
\[y_{eff}=\min\{\delta_{x},R_{h}\}, \tag{9}\]  
It is retained that this expression may be quite inaccurate and be affected by an error of about a factor of two.