Bhastrika is sometimes known as the breath of fire, whereas kapalabhati is also known as shining skull breath, or breath of glow. I have also seen a fast, easily paced, kapalabhati, being called kapalabhati, with the slower, more forceful kapalabhati being called bhastrika.
Bhastrika is rapid breathing in & out of the nose. Kapalabhati is also nostril breathing, but with a passive inhale, with the emphasis being on the more active exhale, like a sneeze. Usually 1 sneeze per inhale. Dr Swami Gitananda from Ananda Ashram, Pondicherry, advised 1-2 of either of these breaths per minute, using a steady rhythm. Each done up to 120 breaths. As a yoga teacher, I teach both of them in rounds of 5 breaths. Starting with maybe 3 rounds, then as the student can do more, I do 10 breaths, then 15, then 20. Often this is sufficient in a beginners or general class of about an hours duration. Obviously, for more advanced students, a higher number is fine. When I am teaching like this, the pranayama is at the end of a class, before meditation, &/or relaxation.
We have looked at the use of bhastrika (september & august 2016 posts), in sadhana & classes. When the warmer weather comes, I like to use kapalabhati, instead. It is more cooling. Bhastrika is heating & kapalabhati is more cooling. However, if you like to use both, do bhastrika 1st.
If you are teaching bhastrika on it's own in summer, follow each round with one deep cooling breath, such as sitali or sheetkari. This helps to reduce the heat which is so important for those who are adversely affected by heat. These people can also do this in winter, but for those who are sensitive to cold & wet, this use of a cooling breath should be avoided in winter.
There are other differences between these two breaths. Bhastrika can be used as an energy building pranayama, as a lung purifier, or both, as mentioned here.
In this case, you would do the bhastrika pumping the navel, then move into pumping the chest area, doing each an equal number of times. Frequently, I will be crosslegged, & will twist to the right to do 20 bhastrikas, then repeat to the left side. Now face forwards, pump the navel 20x, then do chest bhastrika 20x. Finish with a deep breath in & out & be aware of the effects of this before doing another pranayama.
This gives 80 bhastrikas. Sufficient on it's own, for most people, without kapalabhati. For advanced students, you could just add about 30-60 kapalabhatis, after.
In summer, I do the twist to right, then left, then facing forwards, 20 kapalabhatis each, making 60 all up. If you make them smooth & rhythmic, rather than loud & slow, you will get a better cooling effect. And in the same post mentioned above, there are directions for adding bhastrika to some poses. In summer, do kapalabhati instead. And then just do kapalabhati at the end for pranayama, rather than adding bhastrika. Consistency is key with teaching, & it's also important not to try to fit everything in, as this does not create a nice effect.
Actually, bhastrika is a pranayama, but kapalabhati is one of the 6 main purification techniques (shat karmas) of hatha yoga. What does kapalabhati purify? It's the frontal lobes, the area of discrimination, of good judgements & choices. And with the frontal lobes we have the forehead. There is a square portal in the forehead which is the gateway to the inner realm of Ajna chakra. Ajna is also known as the 3rd eye, & the eye of the soul (whatumanawa). Ajna is a cosmic realm.
And, the inside of the forehead is where the manifestations of mind, energy, chakras, tattwas, koshas...everything! manifests. Kapalabhati is an exemplary aid for this. It's easy to see why it is also called shining skull breath, & breath of glow.