A method for purification of crude terephthalic acid comprising: a) obtaining a digester feed slurry of particles of crude terephthalic acid, comprising terephthalic acid, 4-carboxybenzaldehyde and p-toluic acid in a solvent liquid comprising aqueous acetic acid and a catalyst system comprising at least one heavy metal compound from a primary oxidation reactor; b) feeding the crude terephthalic acid slurry to a first digestion zone of a bubble column system in downstream reaction flow from the primary oxidation reactor; c) heating the crude terephthalic acid slurry to a temperature of from 150°C to 280°C either before entry to the first digestion zone or when within the first digestion zone; d) supplying a gas comprising oxygen to the first digestion zone, wherein the superficial velocity of the gas rising near the top of the first digestion zone is in a range of from 0.1 cm/s to 8 cm/s; e) at least partially dissolving particles of crude terephthalic acid in the acetic acid thereby releasing at least some 4-carboxybenzaldehyde and p-toluic acid from the particles and exposing the dissolved 4-carboxybenzaldehyde and p-toluic acid to the oxygen to effect oxidation to terephthalic acid, and to obtain a first stage digester slurry; f) passing the first stage digester slurry to a second digestion zone, which is optionally located vertically beneath the first digestion zone; g) supplying a gas comprising oxygen to a lower portion of the second digestion zone; wherein a supply rate of the gas to the second digestion zone is less than the rate of supply to the first digestion zone; h) dissolving and releasing additional 4-carboxybenzaldehyde and p-toluic acid from the particles and exposing the dissolved 4-carboxybenzaldehyde and p-toluic acid to the oxygen to effect additional oxidation to terephthalic acid, and to obtain a second stage digester slurry; i) optionally, moving the second stage digester slurry through one or more further digestion zones structured similar to the second digestion zone and optionally vertically beneath the second digestion zone; j) removing the resulting terephthalic acid crystal slurry from the last digestion zone; and k) isolating the obtained terephthalic acid crystal particles; wherein the first digestion zone, the second and optional one or more further digestion zone(s) are free of mechanical agitation. a) obtaining a digester feed slurry of particles of crude terephthalic acid, comprising terephthalic acid, 4-carboxybenzaldehyde and p-toluic acid in a solvent liquid comprising aqueous acetic acid and a catalyst system comprising at least one heavy metal compound from a primary oxidation reactor; b) feeding the crude terephthalic acid slurry to a first digestion zone of a bubble column system in downstream reaction flow from the primary oxidation reactor; c) heating the crude terephthalic acid slurry to a temperature of from 150°C to 280°C either before entry to the first digestion zone or when within the first digestion zone; d) supplying a gas comprising oxygen to the first digestion zone, wherein the superficial velocity of the gas rising near the top of the first digestion zone is in a range of from 0.1 cm/s to 8 cm/s; e) at least partially dissolving particles of crude terephthalic acid in the acetic acid thereby releasing at least some 4-carboxybenzaldehyde and p-toluic acid from the particles and exposing the dissolved 4-carboxybenzaldehyde and p-toluic acid to the oxygen to effect oxidation to terephthalic acid, and to obtain a first stage digester slurry; f) passing the first stage digester slurry to a second digestion zone, which is optionally located vertically beneath the first digestion zone; g) supplying a gas comprising oxygen to a lower portion of the second digestion zone; wherein a supply rate of the gas to the second digestion zone is less than the rate of supply to the first digestion zone; h) dissolving and releasing additional 4-carboxybenzaldehyde and p-toluic acid from the particles and exposing the dissolved 4-carboxybenzaldehyde and p-toluic acid to the oxygen to effect additional oxidation to terephthalic acid, and to obtain a second stage digester slurry; i) optionally, moving the second stage digester slurry through one or more further digestion zones structured similar to the second digestion zone and optionally vertically beneath the second digestion zone; j) removing the resulting terephthalic acid crystal slurry from the last digestion zone; and k) isolating the obtained terephthalic acid crystal particles;