The CDC published a study on this recently, but it's not doubling the same mask.
Double mask refers to a three-ply medical procedure mask covered by a three-ply cloth cotton mask. A knotted and tucked medical procedure mask is created by bringing together the corners and ear loops on each side, knotting the ears loops together where they attach to the mask, and then tucking in and flattening the resulting extra mask material to minimize the side gaps.
Basically, the whole study was more geared to improving mask fit. Wearing a different type mask on top is basically one way to achieve that.
During January 2021, CDC conducted experimental simulations using pliable elastomeric source and receiver headforms to assess the extent to which two modifications to medical procedure masks, 1) wearing a cloth mask over a medical procedure mask (double masking) and 2) knotting the ear loops of a medical procedure mask where they attach to the mask’s edges and then tucking in and flattening the extra material close to the face (knotted and tucked masks), could improve the fit of these masks and reduce the receiver’s exposure to an aerosol of simulated respiratory droplet particles of the size considered most important for transmitting SARS-CoV-2. The receiver’s exposure was maximally reduced (>95%) when the source and receiver were fitted with modified medical procedure masks. These laboratory-based experiments highlight the importance of good fit to optimize mask performance. Until vaccine-induced population immunity is achieved, universal masking is a highly effective means to slow the spread of SARS-CoV-2** when combined with other protective measures, such as physical distancing, avoiding crowds and poorly ventilated indoor spaces, and good hand hygiene. Innovative efforts to improve the fit of cloth and medical procedure masks to enhance their performance merit attention.
At least two recent studies examined use of mask fitters to improve the fit of cloth and medical procedure masks. Fitters can be solid (2) or elastic (3) and are worn over the mask, secured with head ties or ear loops. The results indicated that when fitters are secured over a medical procedure mask, they can potentially increase the wearer’s protection by ≥90% for aerosols in the size range considered to be the most important for transmitting SARS-CoV-2 (generally <10 μm). Other studies found that knotting and tucking a medical procedure mask or placing a sleeve made of sheer nylon hosiery material around the neck and pulling it up over either a cloth or medical procedure mask (3,4) also significantly improved the wearer’s protection by fitting the mask more tightly to the wearer’s face and reducing edge gaps. A recent expert commentary (5) proposed double masking as another means to improve the fit of medical procedure masks and maximize the filtration properties of the materials from which they are typically constructed, such as spun-bond and melt-blown polypropylene. Based on experiments that measured the filtration efficiencies of various cloth masks and a medical procedure mask (6), it was estimated that the better fit achieved by combining these two mask types, specifically a cloth mask over a medical procedure mask, could reduce a wearer’s exposure by >90%.
[...]
Results from the first experiment demonstrated that the unknotted medical procedure mask alone blocked 42.0% of the particles from a simulated cough (standard deviation [SD] = 6.70), and the cloth mask alone blocked 44.3% (SD = 14.0). The combination of the cloth mask covering the medical procedure mask (double mask) blocked 92.5% of the cough particles (SD = 1.9).
In the second experiment, adding a cloth mask over the source headform’s medical procedure mask or knotting and tucking the medical procedure mask reduced the cumulative exposure of the unmasked receiver by 82.2% (SD = 0.16) and 62.9% (SD = 0.08), respectively. When the source was unmasked and the receiver was fitted with the double mask or the knotted and tucked medical procedure mask, the receiver’s cumulative exposure was reduced by 83.0% (SD = 0.15) and 64.5% (SD = 0.03), respectively. When the source and receiver were both fitted with double masks or knotted and tucked masks, the cumulative exposure of the receiver was reduced 96.4% (SD = 0.02) and 95.9% (SD = 0.02), respectively.
[...] these experiments did not include any other combinations of masks, such as cloth over cloth, medical procedure mask over medical procedure mask, or medical procedure mask over cloth.
Don't ask me why they haven't tested the other combos.
Most of the media headlines reporting on this study are already (perhaps by necessity) already glossing over the details and reporting that double masking gives >90% protection or something like that. :/
If you're curious, one of those previous studies cited has these more elaborate details on improving mask efficiency, but not [really] by double masking (unless you want to consider the nylon hosiery on top as such):
As expected based on data from our previous work, a National Institute for Occupational Safety and Health–approved 3M 9210 N95 respirator used as a reference control provided very high mean FFE (98.4% [0.5%]; n = 1) (Table). The medical procedure masks with elastic ear loops tested in this study had a mean (SD) FFE of 38.5% (11.2%), which was lower than that of medical surgical masks with tie strings (71.5% [5.5%]; n = 4). Tying the ear loops and tucking in the corners of the procedure mask to minimize gaps in the sides of the mask increased the mean (SD) FFE to 60.3% (11.1%). The “fix-the-mask” 3–rubber band modification and the nylon hosiery sleeve modifications, which were also intended to reduce gaps between the mask and the wearer’s face, improved mean (SD) FFE to 78.2% (3.3%) and 80.2% (3.1%), respectively.
I've cropped part of the table to the most relevant mods (which are illustrated in the subsequent figure):
I'm not sure how much one can learn from this bit alone, but that study
also contrasted/tested a commercial mask with (and without) inserts:
For example, the 2-layer nylon mask with ear
loops was tested under various conditions, including with and
without an aluminum nose bridge, with and without a commercially
available insert, and after 1 wash cycle in a standard
household washing machine (air-dried on a drying rack). The
unwashed nylon mask without a nose bridge or insert had an
FFE of 44.7%. The addition of a nose bridge reduced visible
gaps around the nose and increased FFE to 56.3%. Adding a
filter insert to the mask with the nose bridge in place resulted
in a further increase in FFE to 74.4%. Interestingly,
the FFE of the nylon mask (with the nose bridge but without
the filter insert) improved slightly to 79.0% after washing.
It is unclear why washing alone improved the FFE from
56.3%to 79.0%. It may be that the washing/drying process unraveled
some of the fibers to increase the overall filtration surface,
and thus filtration efficiency, of the medium, or perhaps
it modified the mask shape or size in away that improved fit
or both. The washing/drying test was not repeated with additional
nylon masks. Further investigation to assess the association
of single and multiple washing with mask integrity and
material disposition would be necessary to validate any improvement
in FFE.
The mask discussed in that last para looks like this, i.e. has fairly good side fit:
although that image doesn't really show all the [inner] details (nose bridge, inserts).
As the paper noted, the odd bit is that while inserts surely helped in the new state, washing it achieved the same effect (without inserts)...
