# Quick Response Circulating Water Cooling of \(\pm\)3 mK Using Dynamic Thermal Filtering
## Dynamic Thermal Filtering to Achieve High Temperature Stability
### Modeling  
The temperature fluctuation attenuation ratio of the DTFM attenuator was modeled using a transfer function method, based on the equations proposed by [PERSON] et al. [8]. Transfer functions of the distributor, tube-tank, and accumulator, expressed as \(r_{\text{dis}}\), \(r_{\text{att}}\), and \(r_{\text{acc}}\), respectively, were defined as the temperature fluctuation ratios of the output CCW temperature to input CCW temperature of each part. In the distributor, convective heat transfer of CCW plays the key role in the temperature fluctuation attenuation. The temperature fluctuation attenuation of the tube-tank can be modeled through the double-pipe heat exchanger. The temperature fluctuation attenuation of accumulator resulted from CCW sub-streams mixing. Therefore, the transfer functions of the three parts of the DTFM attenuator can be derived as the following equations:  
\[r_{\text{dis}}(f)=\exp\left[\frac{u_{a}-\sqrt{u_{a}^{2}+4(\lambda/\rho c_{\text {w}})2\gamma rfC_{\text{a}}}}{2\lambda/\rho c_{\text{w}}}\right] \tag{1}\]  
\[r_{\text{att}}(f)=\exp\left[-\frac{1}{f_{\text{m}}c_{\text{w}}R}\left(\frac{2 j\pi fC_{a}}{2j\pi fC_{a}+1}\right)\right] \tag{2}\]  
\[r_{\text{acc}}(f)=\frac{1}{1+(\rho V_{\text{acc}}/F_{\text{m}})2j\pi f} \tag{3}\]  
where \(\rho\) is the density of water, \(V_{\text{acc}}\) is the volume of the accumulator, \(F_{\text{m}}\) is the mass flow rate of the CCW\({}_{f}f_{\text{m}}\) is the mass flow rate of each tube, \(u_{\text{a}}\) is the average water velocity in the accumulator, \(\lambda\) is the thermal conductivity of water, \(C_{\text{a}}\) is the thermal capacity of a single tube, \(L_{\text{a}}\) and \(L_{\text{d}}\) are height of  
Figure 3: Three-dimensional structure of the tube-tank heat exchanger. the accumulator and distributor respectively, \(c_{\text{w}}\) is the specific thermal capacity of water, and \(R\) is the thermal resistance of the attenuator. Then, the total temperature fluctuation attenuation ratio of the DTFM attenuator \(r\) is the product of transfer functions of the three parts. \[r=r_{\text{dis}}(f)\times r_{\text{acc}}(f)\times r_{\text{att}}(f) \tag{4}\]  
The main parameters of the designed DTFM attenuator are listed in Table 1.  
Before the DTFM attenuator is introduced into the CCW machine, the dynamic performance of the CCW temperature control is determined by the closed loop feedback temperature control of the CCW machine. After the DTFM attenuator is introduced, the response time of the CCW temperature control \(S_{\text{sys}}\) can be expressed as:  
\[S_{\text{sys}}=\max(S_{\text{mac}},S_{\text{med}}) \tag{5}\]  
where \(S_{\text{mac}}\) is the response time of temperature control before the DTFM attenuator is introduced and \(S_{\text{med}}\) is the updating time of the thermal capacity medium of the DTFM attenuator, expressed as:  
\[S_{\text{med}}=\frac{\left[b^{2}L-NL\pi(R_{\text{tube}}/2)^{2}\right]\rho}{F_ {\text{m}}} \tag{6}\]  
Therefore, \(S_{\text{mac}}>S_{\text{med}}\) should be satisfied to guarantee the dynamic performance of the CCW temperature control. In this research, the frequency band of the closed loop temperature feedback control of the CCW machine was \(\sim\)0.005 Hz, so \(S_{\text{mac}}\) = 200 s. According to parameters listed in Table 1, \(S_{\text{med}}\) = 82 s. Therefore, \(S_{\text{mac}}<S_{\text{med}}\) was well satisfied. The theoretical modeling results of the temperature fluctuation attenuation ratio of the designed DTFM attenuator are shown in Figure 4. It can be seen from the curve that the attenuation has the characteristic of low-pass filtering. For the CCW temperature fluctuation at low frequency, from 0 to 0.005 Hz, the temperature fluctuation is within the closed loop temperature feedback control bandwidth of the CCW machine, and can be suppressed through closed loop control of the machine. For the CCW temperature fluctuation above 0.005 Hz, it can be reduced by the DTFM attenuator to tens of dB.