Study on the performance analysis and optimization of funnel concept in wave-energy conversion

Abstract

Wave-energy converters (WECs) usually comprise a wave-energy absorber and a reaction supporter on which the wave forces can work. The reaction supporter can promote the accumulation and conversion of wave energy, making WECs more feasible. In this article, we propose a novel WEC structure consisting of a submerged disk and a coaxially moon-pool paddock. In this structure, the disk functioned as the energy absorber and the paddock functioned as the corbelled and wave-energy accumulation device. The power take-off system comprises a linear damper and a spring, which can be activated by the heaving motion of the bodies. The heaving motion equation of the corresponding system in frequency domain is given according to the literature. The optimal reactive spring stiffness and resistive damping coefficients are determined by their vibration characteristics. Furthermore, the corresponding oscillation amplitudes and the wave-energy absorption capture width ratios are also determined approximately. Based upon eigenfunction expansion matching method in the linear potential theory, the analytical expressions for the diffraction and radiation are obtained. In addition, this research proposed the theoretical expressions for added mass, damping coefficient, and exciting force. Finally, the effects of the geometrical parameters and submerged depth of the moon-pool paddock on the wave-energy conversion are illustrated and discussed. It is found that the moon-pool paddock can assist accumulate wave energy and increase the energy conversion. The results also demonstrate the importance of tuning the geometries of the paddock to maximize total energy capture.

Appendix A

The expression for known function Z ₀(·), Z _n (·), and ε _λ is given by

$${Z_0}(z)=\cosh {k_0}h\cosh {k_0}(z+h)/(2{k_0}h+\sinh 2{k_0}h)$$

$${Z_n}(z)=\cos {k_n}h\cos {k_n}(z+h)/(2{k_n}h+\sin 2{k_n}h)$$

$${\varepsilon _\ell }=\left\{ \begin{gathered} 1/{Z_0}(z)\;\;\;(\ell =0) \hfill \\ 2{i^\ell }/{Z_0}(z)\;\;\;(\ell \geqslant 1). \hfill \\ \end{gathered} \right.$$

The known functions Q(·) in diffraction and radiation potentials in sub-domain III _(1,2) are given by

$${Q^{\ell ,0}}(r)=\left\{ \begin{gathered} \ln (r/{R_1})/\ln ({R_2}/{R_1})\;\;\;(\ell =0) \hfill \\ [{(r/{R_1})^\ell } - {({R_1}/r)^\ell }]/[{({R_2}/{R_1})^\ell } - {({R_1}/{R_2})^\ell }]\;\;\;(\ell \geqslant 1) \hfill \\ \end{gathered} \right.$$

$${\tilde {Q}^{\ell ,0}}(r)=\left\{ \begin{gathered} \ln ({R_2}/r)/\ln ({R_2}/{R_1})\;\;\;(\ell =0) \hfill \\ [{({R_2}/r)^\ell } - {(r/{R_2})^\ell }]/[{({R_2}/{R_1})^\ell } - {({R_1}/{R_2})^\ell }]\;\;\;(\ell \geqslant 1) \hfill \\ \end{gathered} \right.$$

$${Q^{\ell ,n}}(r)=K_{1}^{{n\ell }}{I_\ell }({\lambda _n}r) - I_{1}^{{n\ell }}{K_\ell }({\lambda _n}r)$$

$${\tilde {Q}^{\ell ,n}}(r)=I_{2}^{{n\ell }}{K_\ell }({\lambda _n}r) - K_{2}^{{n\ell }}{I_\ell }({\lambda _n}r)$$

$$\left\{ {K_{1}^{{n\ell }},I_{1}^{{n\ell }},K_{2}^{{n\ell }},I_{2}^{{n\ell }}} \right\}=\frac{{{K_\ell }({\lambda _n}{R_1}),{I_\ell }({\lambda _n}{R_1}),{K_\ell }({\lambda _n}{R_2}),{I_\ell }({\lambda _n}{R_2})}}{{{K_\ell }({\lambda _n}{R_1}){I_\ell }({\lambda _n}{R_2}) - {K_\ell }({\lambda _n}{R_2}){I_\ell }({\lambda _n}{R_1})}}$$

$${\lambda _n}={{n\pi } \mathord{\left/ {\vphantom {{n\pi } {(h - d)}}} \right. \kern-0pt} {(h - d)}}.$$

The known functions Ω(·) in diffraction and radiation potentials in each sub-domain are given by

$$\Omega _{7}^{{\ell ,{\rm I}}}(r,z)={\varphi _{0,\ell }}\quad {\text{and}}\quad \Omega _{7}^{{\ell ,I{I_{(1,2)}},II{I_{(1,2)}},IV,V,VI}}(r,z)=0$$

$$\Omega _{{3(p)}}^{{\ell ,{\rm I},IV}}(r,z)=0\quad (p=1,2)$$

$$\Omega _{{3(1)}}^{{\ell ,I{I_{(1,2)}}}}(r,z)=0{\text{,}}\quad \Omega _{{3(2)}}^{{\ell ,I{I_{(1,2)}}}}(r,z)={{(z{\omega ^2}+g)} \mathord{\left/ {\vphantom {{(z{\omega ^2}+g)} {{\omega ^2}}}} \right. \kern-0pt} {{\omega ^2}}}$$

$$\Omega _{{3(1)}}^{{\ell ,V}}(r,z)={{(z{\omega ^2}+g)} \mathord{\left/ {\vphantom {{(z{\omega ^2}+g)} {{\omega ^2}}}} \right. \kern-0pt} {{\omega ^2}}},\quad \Omega _{{3(2)}}^{{\ell ,V}}(r,z)=0$$

$$\Omega _{{3(1)}}^{{\ell ,II{I_{(1,2)}}}}(r,z)=0,\quad \Omega _{{3(2)}}^{{\ell ,II{I_{(1,2)}}}}(r,z)={{[2{{(z+h)}^2} - {r^2}]} \mathord{\left/ {\vphantom {{[2{{(z+h)}^2} - {r^2}]} {4(h - d)}}} \right. \kern-0pt} {4(h - d)}}$$

$$\Omega _{{3(1)}}^{{\ell ,VI}}(r,z)={{[2{{(z+h)}^2} - {r^2}]} \mathord{\left/ {\vphantom {{[2{{(z+h)}^2} - {r^2}]} {4(h - d)}}} \right. \kern-0pt} {4(h - d)}},\quad \Omega _{{3(1)}}^{{\ell ,VI}}(r,z)=0.$$

The above generalized d denotes the vertical distance between seabed and bottom of the hull in each sub-domain.