# Development and Validation of a Capacitor-Current Circuit Model for Evaporation-Induced Electricity
## 2 Development of the Simplified Capacitor-Current Circuit Model
### Overview of the Mansouri Model
#### 2.1.2 Exploring Streaming Current: From the Basics to Complex Models  
The core of this electrical analogy is the streaming current, which arises from the pressure-driven fluid flow transporting ions through a channel. In the context of evaporation-induced electricity, the fluid of interest is water. As water flows along the channel, the surface material undergoes hydrolysis, releasing charged ions. Then, the water flow carries these released charges, generating an electric current that can be harvested  
Figure 1: Electrical analogy for evaporation-induced electricity. The arrows indicate the flow direction.  
through an external load. For a single channel, the physics governing this process can be summarized by using the following equation [14]:  
\[I_{\text{str}}=-\frac{\varepsilon\zeta}{\mu}\Delta P\frac{A}{L}f(\kappa a) \tag{1}\]  
where \(I_{\text{str}}\) is the streaming current, \(\varepsilon\) is the fluid permittivity reflecting its ability for charge storage, \(\zeta\) is the zeta potential of the channel surface material representing surface charge density, \(\mu\) is the dynamic viscosity of the fluid indicating the resistance to flow, \(\Delta P\) is the pressure difference applied across the channel, and \(A\) and \(L\) are the cross-section area and length of the channel. Specifically, \(f(\kappa a)\) is a correction factor based on the Debye length \(\kappa^{-1}\) and the channel radius \(a\). When the channel radius is much larger than the Debye length, i.e., \(\kappa a\gg 1\), this correction factor simplifies to 1 and then we have the Smoluchowski equation [19].  
The above equation demonstrates how key properties, like surface charge, fluid viscosity, and pressure, affect the streaming current. Specifically, high pressure difference or high surface zeta potential leads to a high current. And low fluid viscosity is favored for generating a high streaming current. This single-channel formulation offers a starting point to understand the electrokinetic phenomena central to evaporation-induced electricity. In real-world applications, a working device always involves complex multi-channel or microstructure systems, where fluid-surface interactions and electrode effects further complicate the physical picture.  
Therefore, [PERSON] et al. developed a more comprehensive model incorporating transient dynamics and multiple channels [14]. To maintain simplicity, we present a simplified approximation of the Mansouri model. This full model is described in the Laplace domain as:  
\[I_{\text{ext}}(s)=I_{\text{conv}}(s)\times Y(s) \tag{2}\]  
where \(s\) is the complex frequency variable used in Laplace transforms, \(I_{\text{ext}}(s)\) is the external current, \(I_{\text{conv}}(s)\) the convection current, and \(Y(s)\) the system's admittance:  
\[Y(s)=\frac{1}{R_{\text{int}}+R_{\text{E}}+\frac{1}{S_{\text{C-E}}}+sC_{\text{ int}}} \tag{3}\]  
in which \(R_{\text{E}}\) accounts for electrode resistance and \(C_{\text{E}}\) the electrode double layer capacitance at the fluid-electrode interface.  
This approximation reflects how resistive and capacitive elements interact to determine the transient and steady-state behavior of the circuit. Mathematically, this model shows that when fluid flow starts, the circuit would experience an initial current spike, followed by a gradual decay to a steady-state value, i.e., a plateau. This characteristic is determined by the interplay between resistive and capacitive components, with time constants dependent on specific material properties. Though accurate, in practice, a simplified capacitor-current circuit model can also capture the essential electrokinetic behaviors and avoid the computational complexities of the full model by [PERSON] et al. [14].