## 4 The Measurement Methodology  
To get information of the heat transfer rates sustained by the HXs and on the related gas-side heat transfer coefficient experimental and calculation procedures based on standard energy balance measurement are developed. As for the first parameter, the methodology involves the determination  
Figure 4: (**a**) The ambient heat exchanger with rectangular pore geometry; (**b**) The ambient heat exchanger with circular pore geometry. Figure 5: The standing wave test engine; (**a**) without thermal insulation (rock wool) and (**b**) with thermal insulation. of the temperature at the inlet section of the water flow (\(T_{wi}\)), the one at the outlet section (\(T_{w0}\)) and the mass flow rate of water (\(\dot{m}_{w}\)). This allows to calculate the thermal power captured by the HX as  
\[\dot{Q}_{A}=\dot{m}_{w}c_{w}(T_{w0}-T_{wi}), \tag{2}\]  
where \(c_{w}\) is the water specific heat. Note that the standing wave test rig discussed so far is not connected to any acoustic load. In other words, there is no component (loudspeakers, linear alternator, resistance/compliance series assembly, etc.) which could extract a fraction of the acoustic power produced by the stack from the engine. More exactly, the acoustic load is constituted by the internal surfaces of the resonator and by the HXs solid walls themselves over which the acoustic power is entirely consumed by viscous dissipation and thermal relaxation processes. Therefore, in the absence of thermal and/or acoustic power leakages to/from the outside environment, the heat extracted by the ambient HX, and represented by Equation (2), should precisely coincide with the heat delivered by the heater to the system (\(\dot{Q}_{H}\)). To reduce uncertainty in measurements the system (from the \"hot\" duct to the AHX) is thermally insulated from the external ambient by rock wool. To estimate the gas-side heat transfer coefficient a number of thermocouples need to be placed/embedded in the HXs with a radial distribution to measure the temporally and spatially averaged temperatures of the solid walls and of the gas locally oscillating. A mean heat transfer coefficient can be then calculated from the ratio of the heat transfer rate per unit exchange area and the temperature difference between the gas and the solid walls of the HX. For AHX1 a different procedure, involving the measurement of the temporally and spatially averaged temperature of the gas in the HX (\(T_{g}\)) near the ambient side of the stack can be applied [11]. For a cross-flow heat exchanger, in fact, the heat transfer rate can be expressed as  
\[\dot{Q}_{A}=U^{\prime}A\theta_{ml}, \tag{3}\]  
where \(U^{\prime}A\) is the overall thermal conductance of the HX and \(\theta_{ml}\), the log-mean temperature difference, is defined as  
\[\theta_{ml}=F\frac{T_{w0}-T_{wi}}{\ln\left(\frac{T_{g}-T_{wi}}{T_{g}-T_{w0}} \right)}, \tag{4}\]  
\(F\) being a correction factor for cross flow HXs and having taken into account that, since the flow is oscillatory, the inlet and outlet temperatures of the primary fluid (the gas) are the same (\(T_{g}\)). Once calculated the values of \(\dot{Q}_{A}\) and \(\theta_{ml}\) from the experimental measurements it is possible to evaluate the overall thermal conductance of the HX from Equation (3). To estimate the gas-side heat transfer coefficient the overall conductance \(U^{\prime}A\) must now be expressed in an explicit form. A simplified heat transfer model provides [17].