That big black cylindrical object sticking out of pot belly stove is referred to as a stove pipe. The stove pipe helps direct harmful fumes out of the area of the burning stove. The differences between a stove pipe and a chimney pipe are surprisingly subtle but exist none the less. Here are a few facts about each.
Some stove pipes are a straight vertical pipe while others may form an elbow shape, but most of the pipe should be straight. If an elbow is necessary it should be a short turn. A stove pipe is designed to function from the stove or fireplace to an exterior chimney outlet. A stove pipe never penetrates a wall or ceiling. It cannot handle the intense heat. It connects to the chimney pipe.
A chimney pipe, which is designed to sustain ultra high temperatures, runs through a chimney, directing the caustic gasses outside of the building. A chimney may have several chimney pipes or none at all. The chimney pipe protrudes out of the chimney to connect to a stove pipe and approximately two feet out the top of the chimney for ventilation.