With the potential application of Energy Recovery Linac (ERL), the superconducting (SC) cavities were developed to deliver much higher current than before. Nowadays, the current of the international SC accelerator designed has already exceeded 100 mA. This paper presents the design of a new 500 MHz 5-cell SC cavity (SINAP 5-cell cavity), in which the parameters r/Q= 515.5 Ω of the fundamental mode and the geometry factor G=275.8 are under an acceptable Radio Frequency (RF) field level (B_peak/E_acc=4.31 mT/MV/m and E_peak/E_acc=2.48). This design employs a larger beam pipe to propagate the Higher Order Modes (HOMs) out of the cavity and increases the damping efficiently for the dangerous HOMs. By simulation technique, it has been found that almost all the dangerous HOMs (including TE₁₁₁, TM₁₁₀, and TM₀₁₁) can be propagated into the beam pipe and are absorbed by ferrite absorbers, when the beam pile is enlarged. Finally, the loss factor for the new 5-cell cavity is also calculated.