When the house exists and you want to build a construction chimney, it will have to be located under a smoke outlet, shunt, that allows the smoke and carbon monoxide to go outside and that the outside oxygen enters to help combustion and that the indoor air is not contaminated, in the case that that smoke outlet does not exist, it will have to be located with the proper licenses and permits.
Resolved the location of the shunt, the draft of the chimney should have the appropriate section, proportional and calculated to the volume where combustion is going to be carried out and the length to be traversed until the exit of the gases and in addition the shunt should have a cusp that allows the air to come in clean and smoke out smoothly, so that it does not allow the dirty air to come back in through the draft and thus have a gas inhalation problem.
The fireplace can be installed free of wall or attached to it. The material with which the chimney forms must be refractory, that is to say that it supports the high temperatures that are formed and to which all the materials in contact are going to be submitted.
The materials usually have an RF card, usually come as RF-index, it will be a number that indicates the average fire resistance of the material or training in concrete, in minutes, for example RF-120. The hearth will be installed and over it the cladding and the hood, the rock wool will help to isolate the walls of the hood and then attach to the outside plasterboard plates, the joints will be treated and the whole chimney will form a perfect and unique composition. All the areas to which the chimney is attached should be perfectly insulated and conditioned for heat and fire.
The chimneys can be of open or closed home, the closed home is less romantic and aesthetic, but it is certainly safer and with greater calorific and distribution power.