They're pretty much equivalent.
That said, omitting the article has a slight feeling of playing with a group or orchestra, wherein the instrument is a synecdoche referring to the position the person occupied within the group.
I used to play the flute.
I used to play flute in the Civic Orchestra.
Omitting the article also can carry the feeling of playing an instrument in the general sense.
I play woodwinds.
In this case it would sound strange to use the article because you are speaking of a class of instruments.
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).
Best Answer
When you say "The laboratory", you are referring to a specific object, which fulfils the role expressed by laboratory. Contrast this with "a laboratory", which can refer to any object that fulfils the role conveyed by the word laboratory.
Biochemistry refers to a general topic or idea—not to a specific object, nor to any object.
Thus,
is correct usage.
On the other hand, since this is also the name of a thing,
would be correct as well.