Definite article before scientific terms
The is appropriate in each of the places you have a question about:
In this experiment, the heat transfer coefficient was calculated, allowing to estimate the frost thickness.
Here, allowing is incorrectly used, since it needs an object (*allowing someone to estimate...*). It would be better to use something else:
In this experiment, the heat transfer coefficient was calculated in order to estimate the frost thickness.
...
Firstly, the air properties are defined.
...
As the crystals grow in radius, the amount of air in the ice is reduced, due to the ice becoming more packed.
As you said, a would not work because in each case, you're talking about a specific pre-introduced or pre-defined entity. There are times when you would want no article, but in these examples, you would want the. See the Wikipedia article:
The definite article in English is the, denoting person(s) or thing(s) already mentioned, under discussion, implied, or familiar. It is often used as the very first part of a noun phrase.
The article "the" is used with singular count nouns (the car) and with singular uncountable nouns (the coffee) and plural nouns (the cars) when both the speaker and hearer would know the identity of the thing or idea already.
However, in English, unlike in some other languages such as French, the definite article is absent before familiar but intangible concepts such as "happiness": Happiness is contagious is correct, whereas The happiness is contagious is not, unless a very specific example of happiness is referred to. The is also omitted when the noun refers to a generic mass object (Coffee grows in Colombia) or to a generic collection of countable objects (Cars have accelerators).
Daniel is right about the article use. However.
Each of the examples has another problem.
In the first example, the gerund clause ", allowing to estimate the frost thickness" is ungrammatical as it stands. Allow does not take a subjectless infinitive complement with to.
- *Bill allowed to leave.
It should be ", allowing [somebody] to estimate the frost thickness". The identity and description of [somebody] is up to the author, but there does have to be an agent NP before the infinitive.
In the second example, notice that I didn't say "Secondly". Don't say Firstly, either.
In the third example, the final gerund clause ", due to the ice becoming more packed", with Acc subject the ice (instead of Poss subject the ice's), is an ill fit. Due to prefers a nounier object, especially a factive one.
This is a good place for a the fact that S complement, like ", due to the fact that the ice becomes more packed". Or ", due to the increasing compaction of the ice".