# 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  
#### 2.1.1 Introduction to the Electrical Analogy  
Although many mechanisms have been proposed, it is widely agreed that the fundamental aspect of evaporation-induced electricity lies in ion transport within microstructures, driven by pressure differences or concentration gradients [5; 17; 18]. In this context, the electricity generation process inevitably involves the interaction between liquid and solid surfaces, forming an electrical double layer that contributes to charge buildup and transport [19], thereby influencing the overall device performance. As reviewed in our previously published work, several mechanisms exist beyond streaming potential theory, including ion concentration gradient, microbial electricity, ionovoltaic, pseudo-streaming, and evaporation potential [13]. While each of these mechanisms provide valuable insights into the underlying physical processes, we believe it is crucial to study the electrical output characteristics using a circuit model based on the capacitor-current framework, which [PERSON] et al. established as a solid foundation [14].  
The benefits of capacitor-current model can be highlighted from three main perspectives. First, it directly captures the characteristics of electrokinetic processes, building on fundamental phenomena such as streaming currents and electrical double layers, which are easy to understand. Second, the capacitor-current model offers relative simplicity and physical clarity. By having an equivalent electrical circuit model, we can create a straightforward analogy to understand the essential behaviors of ion transport and charge storage. Third, this model aligns well with experimental data, as we will demonstrate later. In our wood-based nanogenerator, we observed an electricity output resembling the response of a combination of a capacitor and a current source, characterized by transient charge storage and gradual discharge over time. These results confirm the suitability of this model in explaining evaporation-induced electricity.  
Based on the framework developed by [PERSON] et al. [14], the core elements of the electrical analogy consist of an internal capacitor (\(C_{\text{int}}\)), a streaming current source (\(I_{\text{str}}\)), and the device's internal resistance (\(R_{\text{int}}\)), which are arranged in parallel. \(C_{\text{int}}\) represents charge accumulation within the electrical double layer of the microchannels. \(I_{\text{str}}\) is generated by ion transport, driven by pressure difference caused by evaporation. \(R_{\text{int}}\) accounts for ion transport resistance along the channels. The complete electrical analogy, including the external load resistance (\(R_{\text{ext}}\)) and external current (\(I_{\text{ext}}\)), is illustrated in Figure 1. This simplified electrical analogy effectively highlights the characteristics of the electricity output, capturing the practical aspects of power generation within the device.  
As shown in Figure 1, we present the simplified capacitor-current model used to describe the electrical behavior of the system. This model simplifies the complex electrokinetic processes by representing them as a combination of resistive, capacitive, and current-generating elements. Such a representation allows for straightforward predictions of steady-state current (\(I_{\text{steady}}\)) and time-dependent behaviors. For example, the interaction between \(R_{\text{int}}\), \(C_{\text{int}}\), and \(R_{\text{ext}}\) directly influences the transient response of the system and its steady-state performance. In subsequent sections, we use this model to analyze the experimental data and extract parameters critical to device optimization.