Kindergarten and First Grade start the year borrowing one book at a time. They are encouraged to return that books as soon as they are ready for a new one. Teachers provide time throughout the day for students to visit the library individually or in a small group to exchange books as needed.
For older students, I try to gauge check out amount by usage and need rather than by grade. We'll start the year with one or two books per visit and increase those numbers based on demonstrated responsibility and individual need. As students move up through the grades, they often require multiple resources; informational books for research and projects, as well as self-selected topics for personal reading.
The standard loan period for books in our library is one week. Books may be returned to the library and renewed for one additional week. This policy keeps our materials circulating and available for those who are next in line. Many times, if a child needs to keep a book longer than two weeks to finish reading it, that's a cue for me to work with the classroom teacher to help this student select a 'better fit' choice, (or to work with the student to find something so interesting it's worthy of flashlight reading in bed at night!) Inevitably, book pages suffer accidental damage (they're made of paper, after all!) I teach my students to bring damaged books back to the library and I will make the repair. Unlike the clear cellophane tape we have at home, libraries have special book tape that resists yellowing and is longer-lasting.