# Design, Modeling, and Analysis of Piezoelectric-Actuated Device for Blood Sampling
## 2 Physical Modeling of the Device
### Working Principle  
The micropump is actuated by supplying sinusoidal voltage to FQPB. This will move the piezoelectric bimorph up and down, moving the silicon diaphragm the same way in the pump chamber, creating suction and pressure. During the suction cycle, the diaphragm moves upward, creating back pressure inside the pump chamber, resulting in the suction of blood in the chamber through the microneedle. During the pressure cycle, the diaphragm moves downward, creating pressure inside the pump chamber, causing a part of the extracted volume to move towards the biosensor location. One micropump with the integrated microneedle was used for blood glucose measurement only.  
The volume flow rate was calculated based on the diaphragm expansion and contraction in the chamber by using the analytical formulation [29] and computational simulations [30]. A more detailed analytical formulation could be very complex for the micropump flow behavior in non-Newtonian fluid. Thus, in this work, micropump flow was analyzed numerically by using finite element modeling. In the first step, an FE model of the four quarters' piezoelectric bimorph actuator (FQPB) was created. The parametric study was carried out with various load conditions. In the second step, the fluid-structure behavior was considered in the pump chamber. The constitutive equations relating the electric and mechanical fields in piezoelectric media are given by [PERSON] et al. [30]:  
\[\{T\}=[C^{E}]\{S\}-[e]\{E\} \tag{1}\]  
\[\{D\}=[e]^{T}\;\{S\}+\left[\varepsilon^{S}\right]\{E\} \tag{2}\]  
where \(T\) is the stress vector, \(C^{E}\) is the elasticity matrix at the constant electric field, \(S\) is the elastic strain vector, \(e\) is the piezoelectric stress matrix, \(E\) is the electric field intensity vector, and \(\varepsilon^{S}\) is the dielectric matrix at constant mechanical strain. Thus, in the piezoelectric actuator analysis, these coupled piezoelectric equations are solved numerically through the following finite element formulation [31]:  
\[M_{u}\ddot{U}+C_{u}\dot{U}+K_{u}U+K_{u\varnothing}\Phi=F \tag{3}\]  
\[K_{u\varnothing}^{T}U+K_{\varnothing\varnothing}\Phi=Q \tag{4}\]  
where \(C_{u},M_{u}\) and \(K_{u}\) are the damping, the mass matrices, and stiffness matrices, respectively; \(Q\) and \(F\) are the electrical and mechanical loads, respectively; and \(\Phi\) and \(U\) are the electrical potentials and the displacement vectors, respectively. The dielectric conductivity and the piezoelectric coupling, are given by \(K_{\varnothing\varnothing}\), \(K_{u\varnothing}\), respectively. The fluid-flow studyis based on the momentum and continuity equations. It is assumed that blood flow is governed by Navier-Stokes equations [32; 33]:  
\[\rho\left[\frac{\partial u}{\partial t}+(u.\
abla)u\right]=-\
abla P+\tau_{e} \
abla^{2}u \tag{5}\]  
\[\frac{\partial\rho}{\partial t}+\
abla.(\rho u)=0 \tag{6}\]  
In the above equations, \(\rho\) is the density, \(u\) is the velocity vector, \(\tau_{e}\) is the viscosity, \(t\) is the time, and \(P\) is the pressure. The software ANSYS(r) was used for solving the governing equations numerically using the finite volume method, as implemented in the software.