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Previous air-infiltration models in silos have considered a depth gradient of oxygen concentration (*U/*x 0) and steady state (*U/*t = 0). The proposed model considers both a depth gradient and a time gradient of oxygen in the silo. Numerical solutions are obtained for oxygen concentration and respiration loss as a function of time and depth. Without any cover and after 10 days, losses are estimated as 21%, 13% and 6% of original dry matter (DM) in the top 0 to 50 mm, 50 to 100 mm and 100 to 200 mm, respectively. Without a cover and after 300 d, overall losses in a 3 m deep silo are estimated as 10 % of original DM. With typical polyethylene film (0.1 to 0.2 mm thickness), overall losses can be reduced to 2% DM or less, as estimated by the model. When considering the cost of polyethylene and the value of lost silage, no film is the least cost solution for storage up to two months. A 0.1 mm film is the least cost solution for storage during 125 d. A 0.2 mm thick film is the least cost solution for storage during 300 d. The model could be used to assess the value of substitute covers such as biofilms or organic products to replace polyethylene film.