# Effect of Magnetic Nanofluids on the Performance of a Fin-Tube Heat Exchanger
## 3 Methodology  
### Governing Equations  
In this study, numerical analysis was performed using COMSOL Multiphysics, a commercial program for CFD (Computational Fluid Dynamics). Numerical analysis was conducted in a Reynolds-averaged Navier-Stokes (RANS) model, 2D, steady state. Continuity, momentum, and energy equations as the governing equations are as follows [14]:  
Continuity:  
\[\
abla\cdot\left(\overset{\rightarrow}{\rho V}\right)=0 \tag{1}\]  
Momentum:  
\[\
abla\cdot\left(\overset{\rightarrow}{\rho VV}\right)=-\
abla p+\
abla\cdot \left(\mu\left(\
abla\overset{\rightarrow}{V}+\left(\
abla\overset{ \rightarrow}{V}\right)^{T}\right)-\frac{2}{3}\mu\left(\
abla\cdot\overset{ \rightarrow}{V}\right)I+F \tag{2}\]  
Energy:  
\[\
abla\cdot\left(\overset{\rightarrow}{V}(\rho E+p)\right)=\
abla\cdot \left(k\
abla T-\sum h_{j}\overset{\rightarrow}{J}_{j}+\left(\overset{ \rightarrow}{\overline{\tau}}\cdot\overset{\rightarrow}{V}\right)\right)+S_{h} \tag{3}\]  
Since the \(F\) term of the momentum equation means the body force due to buoyancy and magnetic force, it has \(\rho\beta\left(T-T_{ref}\right)\), which is the body force due to buoyancy, and \(\left(\overset{\rightarrow}{M}\cdot\
abla\right)\overset{\rightarrow}{B}\)  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Properties** & **Value** \\ \hline Thermal conductivity & 0.19 W/(m\(\cdot\)K) \\ Thermal expansion coefficient & 0.00086 1/K \\ Relative permeability & 2.552 \\ Heat capacity at static pressure & 1840 J/(kg\(\cdot\)K) \\ Density (T = 298.15 K) & 1221 kg/m\({}^{3}\) \\ Dynamic viscosity & 0.00727 Pa\(\cdot\)s \\ \hline \hline \end{tabular}
\end{table}
Table 1: The properties of EFH-1 [13].  
Figure 1: Geometry and figures of the reference model applied in this study. (**a**) 3D configuration of a fin-tube heat exchanger model; (**b**) 2D configuration of a fin-tube heat exchanger model.  
which is the body force due to magnetic force. Therefore, by substituting two body forces into the momentum Equation (2), the following equation can be finally obtained.  
Momentum:  
\[\
abla.\left(\overrightarrow{\rho VV}\right)=-\
abla p+\
abla.\left(\mu\left( \
abla\overrightarrow{V}+\left(\
abla\overrightarrow{V}\right)^{T}\right)- \frac{\delta}{\mu}\left(\
abla.\overrightarrow{V}\right)I+\left( \overrightarrow{M}.\
abla\right)\overset{\rightharpoonup}{B}+\rho\beta \left(T-T_{ref}\right)\right) \tag{4}\]  
Magnetic flux density, \(B\), and magnetic field strength, \(H\), related to [PERSON]'s equation, are defined as below [15]:  
\[\overrightarrow{\
abla}.\overset{\rightharpoonup}{B}=0 \tag{5}\]  
\[\overrightarrow{\
abla}\times\overset{\rightharpoonup}{H}=0 \tag{6}\]  
Additionally, B can be used to express [PERSON]'s law as follows:  
\[\overset{\rightharpoonup}{B}=\mu_{0}\left(\overset{\rightharpoonup}{M}+ \overset{\rightharpoonup}{H}\right) \tag{7}\]  
Here, \(\mu_{0}\left(=4\pi\times 10^{-7}\right)\) is a constant value, permeability in a vacuum, and M is magnetization with a unit of A/m.  
\[\text{M}=\chi_{m}\overset{\rightharpoonup}{H} \tag{8}\]  
Here, \(\chi_{m}\) is magnetic susceptibility, and is defined as follows [16]:  
\[\chi_{m}=\frac{\chi_{0}}{1+\beta(\text{T}-\text{T}_{0})} \tag{9}\]  
Here, \(\beta\) and \(\chi_{m}\) are the thermal expansion coefficient and magnetic susceptibility at a reference temperature, respectively. From Equation (9), it is shown that \(\chi_{m}\) and T are inversely proportional. The convection phenomenon using this property is thermomagnetic convection.