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I have a job on which it is difficult to clean up the fire. What should I do?

Check the fire to see whether it is off center (see discussionof off-center fires in next question).
Check the fan blades and if they are covered with lint and dirt, clean them.
If this is a conventional oil burner, check for a burned-off-end cone.
Check the oil pressure to be sure that it is at least 100 psi.
Check for a plugged line filter or pump strainer.
If the smoke is at the outside edges of the fire, it might be well to try a wider spray angle.
If the smoke occurs at the end of the fire, it might be well to try a wider spray angle.
The burner may have insufficient air capacity for the firing rate.
The burner may have too high air capacity for this firing rate. This would mean that the burner head couldn't handle a low firing rate.
It might just be a very inefficient burner, which cannot be made to burn clean.
The nozzle may be partially plugged, resulting i poor atomization.
The oil may have a high viscosity or it might be cold, resulting in collapsing spray.
Get sufficient air into the boiler room. If necessary, put in an air intake pipe with a screen and storm cover.

The fire in a burner is always off center and heavy on the right side. I can't find a nozzle that will straighten up the fire.

If the flame is always heavy on one side, the nozzle does not cause it. It would be impossible to make a nozzle spray that is heavy on one side and always have it come up on the same side of the burner.
Check the position of the nozzle in the burner head. The chances are it is off to one side or the other.
With a short burner tube, the air stream may be stronger on one side than the other. That will give the appearance of being heavy one side.
If the nozzle spray pattern is off center due to contamination or any other reason, it may mike a flame, which is heavy on one side, but it could be in any direction.

How do you cure pulsation?

A hollow cone nozzle generally gives greater freedom from pulsation than a solid cone nozzle.
A wider spray angle sometimes helps a pulsation condition. It may even be necessary to use a 90 degree nozzle. In this case watch for smoke at the outer edges of the fire.
Reduce the firing rate to the next smaller nozzle if it will carry the heating load.
The combustion chamber may be too large, allowing the fire to leave the burner.
Higher oil pressure sometimes helps because it gives a more stable spray pattern and smaller droplets, which burn closer to the burner.
Be sure there is air intake into the boiler room.
Sometimes a flame retention burner will cure or improve a pulsating or rumbling condition.

Soot particles around the boiler and other places in the basement. The fire is clean. What is the reason for this?

Make sure that you have adequate chimney draft so that there is not a back-pressure in the smoke pipe when the burner starts. This type of complaint occurs with a long smoke pipe between the boiler and the chimney. If the smoke pipe does not have enough pitch, there may be a back-pressure at the boiler connection even with sufficient chimney draft.
Check for an obstruction at the point where the smoke pipe enters the chimney.
Cement up all leaks in the boiler setting where soot might come out on the start.
Install a delayed opening solenoid valve or ProTek valve in the nozzle line to make smoother starts

If the burner sometimes fails to ignite smoothly and starts with a puff, what can be done to improve it?

Check the electrode points for proper spacing. The points should be 1/8" to 3/16" apart, 9/16" above the centerline of the nozzle and then spaced correctly forward from the face of the nozzle for each different spray angle.
Clean carbon and dirt from the points and from the insulators.
The transformer may be weak and not delivering full voltage or current. If you don't have another transformer immediately available, file the electrodes to a sharp point. This will give a better spark.
Check for cracked insulators. Sometimes an insulator may be cracked under the electrode bracket and it is difficult to find. This could cause a high voltage leak, thus reducing the voltage at the points.
A partially plugged nozzle causing off-center spray can cause delayed ignition.
Above about 2.50 gph a hollow cone spray sometimes may cause delayed ignition. Changing to a solid cone sometimes helps that situation.
The air setting on the burner may be wide open, thus tending to blow the fire out before it is established. Adjust the air correctly

I have tried firing fractional gallonage nozzles and they only work two or three weeks and plug up.

Fractional gallonage nozzles have become a very popular nozzle and are giving good satisfaction if properly used. First of all, do not remove a nozzle from it vial before you are ready to install it in the burner. That may contaminate it.
Be sure an adequate supply line filter is installed. This size nozzle should have a filter capable of removing particles over 50 microns. The problem is to know which filter will do that. Generally speaking, the paper filters are finer than the felt or woven filters that are generally available.
Install a Delavan line filter for extra nozzle protection. Make sure all trapped air is removed and follow installation instructions.
The Delavan Del-O-Flo nozzle may help. It was designed especially to minimize the usual plugging problems associated with low flow rates.
Flush or blow out the nozzle line and adapter before installing the nozzle.
If the nozzle runs exceptionally hot, find out why and remove the cause