# Temperature Fluctuation Attenuation of Circulating Cooling Water Using Dynamic Thermal Filtering
## 3 Modeling  
The DTFM attenuator is designed using a tube-tank heat exchanger for its high heat exchanging efficiency with compact structure. The structure of the tube-tank heat exchanger is shown in Figure 2. It consists of a square tank and an array of parallel tubes evenly arranged in the tank. The volume of the tank is divided into two chambers. One chamber is between the tubes and the wall of the tank, serving as the thermal capacity medium chamber. The other chamber is given by the inner volume of all tubes, serving as the CCW chamber. So CCW flows through the tubes, and the tubes are evenly immersed in the thermal capacity medium. A distributor is set in the upper part of the heat exchanger to guide CCW to flow into the tube array. A collector is set in the lower part of the heat exchanger to converge CCW from the tube array. The effect of temperature fluctuation attenuation is mainly contributed by the tube array, and partly contributed by the upper distributer and lower collector. The designed tube-tank heat exchanger is composed of 36 tubes and a stainless-steel tank. The length of each tube \(L_{\rm tube}\) is 0.7 m, the diameter of each tube \(R_{\rm tube}\) is 8 mm, and the center spacing between two neighboring tubes \(d\) is 32 mm. The height of the tank \(L\) is 0.7 m, and the width \(b\) is 0.23 m. The total flow rate of CCW \(F_{\rm m}\) is 16 L/min. CCW flows into the upper distributor from the inlet header, and then flows through all tubes with same flow rate. Since the tubes are evenly immersed in the thermal capacity medium, spatial structural symmetry results in the same thermal characteristics of all tubes. The flow rate is kept small enough to make the laminar flow dominant in the tubes. Therefore, only convective heat transfer is considered to calculate the temperature fluctuation attenuation of the tube array, while the effect of mixing in the tubes is ignored. On the other hand, in the lower collector, sub-streams from the tubes act as jets and converge together, thus water mixing rather than heat transfer is the main effect contributing to CCW temperature averaging. The tank is covered with thermal isolation material to avoid heat exchange with ambient air. Figure 2: Axial and radial sectional views of tube-tank heat exchanger. The temperature fluctuation attenuation of the DTFM attenuator is mathematically modeled by using transfer function method with reference to [14]. Transfer functions of the distributor, the tube-tank part, and the collector are defined as the fluctuation ratios of output temperature to input temperature of CCW. The total temperature fluctuation attenuation ratio of the DTFM attenuator is the product of transfer functions of all the three parts. The transfer functions of the distributor, tube-tank part and collector are expressed as follows:  
\[r_{\text{dis}}(f)=\text{exp}\Bigg{(}\frac{u_{\text{a}}-\sqrt{u_{\text{a}}^{2} +4(\lambda/\rho c_{\text{w}})2\eta fC_{\text{a}}}}{2\lambda/\rho c_{\text{w}} }L_{\text{d}}\Bigg{)} \tag{4}\]  
\[r_{\text{cel}}(f)=\frac{1}{1+(\rho V_{\text{acc}}/F_{\text{m}})2\eta f} \tag{5}\]  
\[r_{\text{att}}(f)=\text{exp}\Bigg{(}-\frac{1}{f_{\text{m}}c_{\text{w}}R}\text {Re}\Bigg{(}\frac{2\jmath\pi fRC_{\text{a}}}{2\jmath\pi fRC_{\text{a}}+1}\Bigg{)} \Bigg{)} \tag{6}\]  
where \(\rho\) is the density of water, \(V_{\text{dis}}\) is the volume of distributor, \(F_{\text{m}}\) is the total volume flow rate of CCW, \(f_{\text{m}}\) is the volume flow rate of each tube, \(u_{\text{a}}\) is the average water velocity in the collector, \(\lambda\) is thermal conductivity of water, \(C_{\text{a}}\) is the thermal capacity of a single tube, \(L_{\text{a}}\) and \(L_{\text{d}}\) are the height of collector and distributor respectively, \(c_{\text{w}}\) is the specific thermal capacity of water, and \(R\) is the thermal resistance of the heat exchanger. The CCW temperature fluctuation is defined as the maxima of peak-to-peak value of temperature oscillations of CCW temperature.