## 1 Introduction  
The Gurney model provided a great deal of improvement in the understanding of explosively-driven systems, delivering a fundamental insight of the coupling between the explosives and metallic flyer based on the conservation equations. It also introduced a well-known parameter of the mass of the flyer divided by mass of the explosive charge (M/C) ratio, and it became a common gate-way into related research. This approach begins with the following assumptions: (1) the constant output per unit mass of a charge, (2) one-dimensional, linear detonation, gas velocity profile, (3) volume detonation of the given charge in the system, (4) no shock effect in the projection of the flyer, etc. [1]. This approach was further developed by many researchers and expands the application into areas including various system configurations of cylinders, symmetric or non-symmetric sandwiches, spheres, etc. [2]. Despite providing great potential to related research fields, the Gurney model reveals minor limitations because of the assumptions addressed above. For example, the Gurney model only predicts the terminal velocity of the flyer based on the initial system configuration, meaning that the acceleration profile of the explosively-driven flyer is not readily available. If the explosives system functions in an early stage of detonation, then the Gurney model is not suitable because the system is still in an acceleration stage, and it may reach the terminal velocity later. Another limitation of the Gurney model is that the model originated from a one-dimensional point of view, and prohibits a detailed understanding of the side rarefaction (or edge effect) upon detonation of the charge, which eventually will affect the terminal velocity. This is an important subject for applications where the speed of projection is considered as the measure of performance, i.e., shaped charges, fragmentation of casing, slapper detonators, etc. It is obvious that the design of engineered explosive devices cannot be a one-dimensional configuration, and this fact may affect the use of the conventional Gurney equations, without the consideration of multi-dimensional configuration, or no consideration of the side rarefaction, delivering inaccuracy. In terms of the side rarefaction in a typical flat flyer projection, there is a study by [PERSON], to understand the effect from the side rarefaction (or side loss) by applying a new term of \"effective charge mass\" (\(C_{e}\)), and this study is based on a semi-empirical observation delivering more advanced applications in this field [3; 4]. In this study, \(C_{e}\) is calculated based on a cone-shaped charge volume with 60\({}^{\circ}\) base angle (dotted line in Figure 1). This approach can be applied to the Gurney model to optimize the charge amount as follows  
\[\frac{v}{\sqrt{2E}}=\left\{\frac{\left(1+2\frac{M}{C_{e}}\right)^{3}+1}{6 \left(1+\frac{M}{C_{e}}\right)}+\frac{M}{C_{e}}\right\}^{-\frac{1}{2}} \tag{1}\]  
where \(M\) is the mass of the flyer, \(\sqrt{2E}\) is the Gurney velocity constant, and \(v\) is the speed of the flyer. This approach addresses the effect from the side rarefaction (or side loss, Figure 1) in order to calculate the flyer velocity, and the side rarefaction is empirically calculated based on the 60\({}^{\circ}\) base angle (or 30\({}^{\circ}\) cone angle). The 60\({}^{\circ}\) base angle is closely related to the speed of rarefaction coming from the surface of the charge, or a half-speed of detonation velocity in the charge, and this angle is deduced upon consideration of the rarefaction speed. This is a somewhat simplified approach, but it greatly emphasizes the importance of side rarefaction that affects the flyer projection. Later, [PERSON] proposed a 45\({}^{\circ}\) base angle based on a series of numerical simulations under the same concept [5].