Experimental validation of a heat exchanger model for thermoacoustic applications

Thermoacoustics is a promising technology for energy conversion purposes. Among the bottlenecks limiting a large diffusion of thermoacoustic devices, there are heat exchangers, whose behaviour in oscillatory flows is rather different than those working in stationary flows. Furthermore, the classical linear acoustic theory in the frequency domain cannot predict with high-fidelity the thermo-fluid dynamics of such heat exchangers. In this article, a CFD model based on a standing wave device including a parallel plate heat exchanger is proposed. The setup is inspired by a similar prototype of a thermoacoustic engine in which the performance of the ambient heat exchanger was tested. The results of the CFD model are therefore compared, in terms of the temperature difference between fluid and solid wall in the heat exchanger, average heat flux and Nusselt number, with experimental data showing a satisfying agreement.


Introduction
Thermoacoustic engines are energy devices converting heat into electricity through acoustic waves [1].They are based on thermoacoustic effect: if a temperature difference across a particular porous media called stack overcomes a threshold value (the onset temperature), pressure and velocity instabilities arise in the system [2].This stack, the core where the energy conversion between heat and acoustic power takes place, is generally sandwiched between the hot Heat eXchanger (HX), the source at high temperature, and the cold one representing the thermal sink.The working principle and therefore the optimal design of heat exchangers in oscillatory flows is distinct from those one functioning in steadystate conditions.In oscillatory flows, the geometrical characteristics (the length and the pore dimension) of the heat exchangers are respectively bounded by the peak-to-peak-particle displacement, and the thermal penetration depth, strictly depending on the working frequency [3].For this reason, both standard heat transfer correlations, which rely on frequency, and simple one-dimensional models, hindered by sudden changes in section area that impede fluid flow and the development of thermal fields, fail to accurately represent the observed physical phenomena [4]- [6].
In this article, a thermoacoustic heat engine model based on Computational Fluid Dynamics is presented and validated against experimental data of a finned tube heat exchanger (ambient one).A comprehensive review of CFD modelling in thermoacoustics is provided in ref. [7].While the ambient heat exchanger is modelled microscopically taking into account the fluid flow across the fins, the hot heat exchanger and the stack are modelled by means of an already validated porous media macroscopic approach for oscillatory flows.This has been done in order to reduce the computational costs of the simulations [8].

Physical system and Experimental setup
In this section the experimental setup, described in a more detail in ref. [9], is briefly described.The physical system is made of a standing wave thermoacoustic engine.The working fluid is air at atmospheric.Its resonance frequency stands at about  0 = 145  and by an onset temperature difference of approximately 300 •C.A sketch of the standing wave engine is shown in Figure 1.The resonator is made of stainless-steel tube.The main components are the hot duct (between the right wall the hot HX) also known as thermal buffer tube (diameter of 10.5 cm and 10 cm long), the stack (diameter of 10.5 cm and length of 8 cm).The thermoacoustic core is a ceramic (cordierite) honeycomb stack 7.9 cm long in the oscillating flow direction.The pores have a squared shape with Rh = 0.28 mm while the cell density is near 400 Cell Per Square Inch.Even though it is previously shown that suck kind of stacks are not among the most efficient ones [10], they are commercial being used as filter in the automotive sector.An electric heater is adopted as input energy source into the system in place of a classical heat exchanger.The ambient HX is a finned-tube type 2 cm long in the oscillatory flow direction.The fins are made of strips of laminated copper 0.45 mm thick and 2 cm wide.These strips have been welded to the external walls of the pipe with a fin spacing of 1.1 mm (resulting in Rh = 0.55 mm), as shown in Figure 2. The porosity of this HX (ϕ) is calculated to be around 54%.A photograph of the presented finned-tube HX is pictured in Figure 2 .Figure 2 The finned tube Ambient Heat Exchanger (on the left) and position of the thermocouples

Governing equations and numerical model
Thermoacoustic phenomena are generally described the Navier-Stokes equations system.At the macroscopic level, in the porous media such as the stack and the HXs, specific source term and virtual mass/ thermal inertia corrections are respectively considered (  ,   ) as shown in the following equations (1-3) [8].
In order to generalize the model for porous media and free-fluid zones, a specific application solver has been developed in OpenFOAM environment.The variables density, Darcian velocity, pressure, and temperature are indicated as ,   , ,  respectively.The parameter  is equal to 1 in the macroscopic porous zone and zero otherwise.The virtual mass coefficient and thermal inertia correction factor are: The viscous and thermal penetration depths depend on frequency and fluid properties, while the complex Darcy () and Nusselt number () depend on also on the microstructure of the porous media through the thermos-viscous functions (  ,   ).For a square pore stack, such functions have an analytical expression [11].
In order to calculate the thermal power exchanged and the temperature difference between the average fluid temperature and solid wall one, it is necessary to model the ambient Heat Exchanger at the microscopic level.In theory, a 3D unitary cell at least should be taken into consideration due to the presence of the water tubes.However, to reduce the computational cost of the simulations, three 2D unitary cells representing the fins of the HX are simulated, neglecting the three-dimensional effects of the water tubes.It was shown that three fluid channels represent a good compromise between accuracy and simulation cost [12].On the other hand, the hot HX and the stack are modelled with the porous media macroscopic approach based on the thermos-viscous functions developed in a previous article (  ,  ) [8] and briefly summarized above.It must be noted that non-linear effects in the stack region are not considered as made in ref. [13] because Reynolds number values are below the threshold value for considering also Forchheimer effects The heat generation term in the equation ( 3) is proportional to the real part of Nusselt number and the temperature difference between fluid and solid phase: To effectively reproduce the experimental conditions, the boundary conditions of the numerical model have been obtained from the experimental measurements.More specifically, for each simulation, the pressure amplitude of the microphone "P1" has been used on the right boundary to set a sinusoidal input boundary condition.The other side of the resonator has been set a solid wall.In terms of energy equation, a linear assumption has been assumed between the measured hot temperature  ℎ (hot temperature) in the heater and the wall temperature   (cold temperature) for the stack.
where   = 0.09 ,  0 = 0.11  and the temperature data corresponding to each pressure are shown in Table 1.Similarly, the wall temperature of the ambient HX is prescribed to the experimental value   .Such experimental data, used as boundary conditions in the numerical model, are summarized in Table 1.

Results and discussion
In the experimental setup, the heat exchanged between the thermoacoustic working fluid and cold water has been measured directly from inlet and outlet of water temperatures and its mass flow rate, as explained in a more detail in ref. [9].On the other hand, from a numerical perspective, it is possible to calculate the average heat flux by integrating the normal temperature gradient along the solid wall of the heat exchanger and dividing over the total exchange area considered in the numerical model.While on the water side, after a transient phase, the steady-state condition is reached, on the air side, the thermoacoustic working fluid is oscillating.For this reason, the flux data have to be also averaged over an acoustic period of time (1/ 0 ).
In formulas: This quantity can be compared to the experimental values once the thermal power available from the experimental data provided is divided over the total heat exchange area of the physical prototype.It is estimated to be around 0.39 m 2 .The comparison between experimental and numerical heat flux is illustrated in Figure 3.The uncertainty bar included the experimental data ranges from 14% to 7%, related to the lowest and highest Reynolds number investigated.In most cases, the numerical data fall into the bar uncertainty of the experimental data.Both the curves show a quadratic dependence of the heat flux on the velocity inside the ambient HX, expressed in terms of Reynolds number.As a consequence of the previous results, the Nusselt number can be calculated from the ratio between the above heat fluxes and temperature difference using the hydraulic diameter  ℎ and the fluid thermal conductivity   as scaling factor.It has to be specified that this Nusselt number should be not confused with the one shown in ref. [4], in which a logarithmic temperature difference involving also the water temperature was considered.For this reason, a different absolute uncertainty model of the Nusselt number has to be considered in the present Nusselt calculation.
where   ,   ,  Δ are the absolute uncertainties of Nusselt, heat flux and temperature difference respectively.From the trend pictured in Figure 5 it is possible to understand that, for Reynolds number larger than about 1000, the temperature difference and heat flux data fit almost the same quadratic dependence.Even if the trend for the Nusselt number (in which the logarithmic mean temperature difference method was applied) is opposite, it is visible that, for the same threshold Reynolds number, a saturation value is reached.According to the authors, this behaviour is justified by the fact that, for that Reynolds number, the particle displacement is approximately equal to the HX length [9].

Figure 5
Comparison between numerical and experimental Nusselt number versus Reynolds number

Conclusions
In this article, a numerical model of a standing wave thermoacoustic engine is proposed.The main conclusions can be summarized by the following bullet points.
• The generalized model for macroscopic porous media, adapted for oscillatory flows and already validated in a previous article against experimental literature data, has been extended for a complex computational domain involving not only porous media but also free fluid regions.• The experimental setup for the standing wave engines has permitted to perform a systematic comparison between numerical and experimental data, using pressure and temperature data as boundary conditions in the numerical model.• The results show that numerical data almost always fall in the experimental uncertainty range both in terms of temperature and heat flux, as well as Nusselt number.

Figure 1 A
Figure 1 A sketch of the thermoacoustic engine

Figure 3
Figure 3 Comparison between numerical and experimental heat flux versus Reynolds number

Figure 4
Figure 4 Comparison between numerical and experimental fluid-solid temperature difference versus Reynolds number

Table 1 .
[9]erimental data used in the numerical model as boundary conditions, data from ref.[9]and private conversation