# Multi-Physics Analysis of a Magnetorheological Valve Train with Experimental Validation
## 3 Magnetic and Thermal Studies  
The heart of this new valve is the magnetic plate block, and within this block, the required resistance force of the MR fluid depends on the current flowing through the magnetic coil. Owing to the electrical resistivity of the magnetic coil, some energy is dissipated as heat. This dissipated heat increases the device's temperature, which, in turn, affects the resistance of the coil and properties of the MR fluid. Therefore, an electromagnetic analysis and heat transfer analysis is needed to understand the performance of the proposed MR valve, and these two aspects of physics must be interlinked. For this reason, the magnetic plate block was analyzed and optimized in this study by performing a coupled magnetic and thermal multiphysics analysis. An early physics concept that relates electrical and thermal fields through a simple equation is the Joule-heating effect [20]:  
\[Q_{e}=I^{2}Rt. \tag{1}\]  
Figure 2: Working modes: (**a**) fully lifted, (**b**) fully closed, and (**c**) partially open. This equation could be used to determine the \(Q_{e}\), i.e., the amount of heat generated by a coil, where \(I\) is the electrical current, \(R\) is resistance in the coil, and \(t\) is time. Then, the energy balance equation could be used to determine the temperatures in the system (Equation (2)) [21]. \[\rho C_{p}\ \Delta T=\
abla\cdot(k\Delta T)+Q_{e}. \tag{2}\]  
where \(\rho\), \(C_{p}\), and \(k\) are the mass density, specific heat, and thermal conductivity of the medium respectively, and \(\Delta T\) is the temperature gradient. The concept of linear resistivity in metals could be used to obtain the relationship between the temperature and resistance of the coil. According to this concept, the resistivity of the conductor (\(\sigma\)) depends on the surrounding temperature as expressed in Equation (3) [21]. \[\sigma=\sigma_{0}(1+\alpha\left(T-T_{ref}\right), \tag{3}\]  
where \(T_{ref}\) is the reference temperature, \(\sigma_{0}\) is the reference resistivity, and \(\alpha\) is the resistivity temperature coefficient. By using the three presented equations, the aforementioned two-way coupled analysis could be performed. COMSOL Multiphysics GUI software was used to perform the two-way coupled magnetic field and thermal analysis. A three-dimensional model of the magnetic plate block was built, governing equations were incorporated, and simulations were conducted. The magnetic field simulation results were used to determine the shear stress in the MR fluid by using an empirical equation (Equation (4)). \[\tau_{y}=0.98(B)^{1.61}, \tag{4}\]  
where \(\tau_{y}\) is the yield stress of the MR fluid under the action of a magnetic field and \(B\) is the magnetic flux density. Finally, the resistance force of the MR block (\(F_{T}\)) was calculated using Equation (5). \[F_{T}=(n+1)(F_{a}+F_{d}), \tag{5}\]  
where \(n\) is the number of thin plates in MR block around which MR fluid flows, the MR fluid chaining force \(F_{a}\) is expressed as in Equation (6), and the viscous damping force (\(F_{d}\)) is expressed as in Equation (7). \[F_{a}=\frac{\dot{x}}{|\dot{x}|}\left(\frac{c_{1}\tau_{y}L}{h}\right)A_{p}, \tag{6}\]  
\[F_{d}=\left(1+\frac{wh}{2A_{p}}\right)\frac{12\mu LA_{p}{}^{2}}{wh^{3}}\dot{x }=k\frac{12\mu LA_{p}{}^{2}}{wh^{3}}\dot{x}, \tag{7}\]  
where the design factors \(L\) and \(w\) denote the height and width of the magnetic core, respectively; \(h\) is the thickness of the MR fluid layer; \(A_{p}\) is the effective area of the upper piston; \(\dot{x}\) is the liquid flow rate. The value of the parameter \(c_{1}\) ranged between 2.07 and 3.07, and it was determined using Equation (8). \[\left(c_{1}=2.07+\frac{12Q\mu}{12Q\mu+0.4 wh^{2}\tau_{y}}\right), \tag{8}\]  
In Equationits resistance force. These dimensions were optimized by performing simulations. Owing to design limitations, standard optimization techniques could not be employed. Therefore, the following local optimization procedure was used.