Question: In *[The Path of the Just](http://rads.stackoverflow.com/amzn/click/1583306773)* (Mesilat Yesharim) by [Rabbi Moshe Chaim Luzzato](https://en.wikipedia.org/wiki/Moshe_Chaim_Luzzatto) (ch.19), I noticed he sometimes uses the term "nachat ruach" and other times "korat ruach." 

What is the meaning of both expressions and their distinction in this context?

Answer: 
In kesubos 40b the gemara explains that a rapist has to pay the raped woman damages (פגם) based on the her loss in value, which is estimated by how much a person would pay more for a virgin slave woman to marry her off to a slave that gives him korat ruach. Rashi says that because the slave gave him korat ruach the master wants to give him nachas ruach by marrying him off to virgin.

It appears from here that korat ruach is needed to cool down a person who is angry or worried. Nachas ruach seems to be relevant when a person has a desire for something, his ruach isn't quiet until he gets it. This last point is spoken out by the GRA on Mishlei 1:23. The concept of ruach is really a kabalistic concept (נר"נ - נשמה רוח ונפש) and is spoken about extensively by the GRA in Mishlei, and many other books.
Korat Ruach is only cited in two places in Mesilat Yesharim

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> בנפש, שישתדל לעשות לחבירו כל **קורת רוח** שיש בידו, בין בעניני הכבוד, בין בכל שאר הענינים, כל מה שהוא יודע שאם יעשהו לחבירו הוא מקבל **נחת רוח** ממנו, מצות חסידות הוא לעשותו, כל שכן שלא יצערנו בשום מיני צער כלל, יהיה באיזה אופן שיהיה.‏
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2. > 
> ובכלל הענין הזה שלא לצער לשום בריה אפילו בעלי חיים ולרחם ולחוס עליהם, וכן הוא אומר (משלי יב, י) יודע צדיק נפש בהמתו, וכבר יש שסוברים (שבת קכח, ב; בבא מציעא לב, ב) צער בעלי חיים דאורייתא, ועל כל פנים דרבנן. כללו, של דבר הרחמנות וההטבה צריך שתהיה תקוע בלב החסיד לעולם, ותהיה מגמתו תמיד לעשות **קורת רוח** לבריות, ולא לגרום להם שום צער וכו'. ‏
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In the first occurence, almost immediately the author uses as synonym nachat ruach (in the above quotes, nachat ruach and korat ruach are in bold). This two occurences of korat ruach regard Nachat Ruach for humans. As mentioned in the answer of @mevakesh, the word korat refers to cooling, calming. For persons, the happyness is linked to recover from anxiety, doubt. 

So, may be that the author uses this expression regarding persons, or animals, *to enlight that there is a particular feature: to make good for other is almost alway to calm his distress, physic or mental*. Humans and animals are weak and vulnerable.